Geophysics of Soil Mapping Using Airborne Gamma-Ray Spectrometry(Billings,PhD Thesis, 1998)

GEOPHYSICAL ASPECTS OF SOIL MAPPING USING AIRBORNE GAMMA-RAY SPECTROMETRY by

Stephen David Billings

A thesis submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy

Department of Crop Sciences Faculty of Agriculture The University of Sydney NSW, 2006, Australia

October 1998

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Statement

This thesis is an account of my own research undertaken during the period March 1995 to October 1998, while I was a full-time student in the Department of Crop Sciences at the University of Sydney.

Except as otherwise indicated in the acknowledgements and in the text, the work described is my own.

This thesis has never been submitted to another university or similar institution.

Stephen David Billings Canberra October 1998

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ABSTRACT Airborne gamma-ray spectrometry is a remote sensing technique that is increasingly being used to supplement or replace existing techniques for soil mapping. Gamma-rays from radioactive decay of potassium, uranium and thorium in the soil are counted with a large crystal detector mounted in a low-flying aircraft. However, even with recent developments in multichannel processing techniques, there are difficulties in creating accurate images from these noisy, blurred and unevenly sampled data. Overcoming these difficulties is the scope of this thesis.

First, a mono-energetic model of gamma-ray spectrometry is developed that accounts for detector shape and aircraft movement during data collection.

The detection probability depends on

gamma-ray energy, and the area and thickness that the detector projects in the direction of a source. The resulting geometrical model is approximately radially symmetric and predicts a reduced measurement footprint compared to a model that ignores detector shape. The aircraft movement introduces asymmetry, which can be significant when there is a low ratio of aircraft height to distance travelled during measurement.

Second, linear error propagation is used to trace the initial Poisson counting error through the various processing operations applied to a gamma-ray spectrum. The method returns estimates of relative error for surveys flown at different heights and with different isotope concentrations and background contamination. The optimal Wiener filter for deconvolution of radiometric data is derived and used to estimate the spatial resolution of radiometric surveys. The analysis reveals that radiometric surveys are usually adequately sampled along-lines but under-sampled across them.

Third, I develop a general methodology which I call the Arbitrary Basis Function (ABF) framework, that unifies many different types of interpolation method (radial basis functions, splines, kriging and sinc functions). An ABF consists of an optional polynomial and a sum of translations of a fixed basis function multiplied by a set of weights. The ABF framework can accommodate anisotropy and can be used for exact or smooth interpolation. Further, it causes the interpolated surfaces to inherit certain desirable properties from the basis function.

The principal impediment for practical

application of the method is the computer memory and time required to solve the resulting large and dense matrix equations. A three-step process for reducing these requirements is proposed, and the first step (an iterative matrix solver) is implemented on a small radiometric survey.

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Next, I derive the Fourier transform of an ABF surface. Traditionally, the Fourier transformation required to deconvolve radiometric data would be calculated by a Fast Fourier Transform of an imaged version of the data. This raises several important considerations (edge mismatch, survey gaps, quadrature approximation, etc.) regarding the relationship between the continuous transformation of theory and the discrete version of practice. The ABF approach, by contrast, can be used to overcome many of these problems, as it defines an interpolant that extends over all space. Further, the Fourier transform of an ABF surface can be calculated exactly on an arbitrary grid, and has a particularly simple form that facilitates computation and subsequent analysis.

Lastly, I show that convolution or deconvolution of an ABF surface gives another ABF surface with the same set of weights but a different basis function.

This means that the deconvolution of

radiometric data can be entirely encapsulated by the new basis function, thus obviating the need to calculate the Fourier transform of the data. The major difficulty with the method is calculating the new basis function, which can be complex for singular bases, such as splines. Spline deconvolution with a radially symmetric Wiener filter is successfully implemented and an approximate method developed for the non-radial case.

ABF can be used for interpolation, Fourier transformation, and convolution/deconvolution of any type of data regardless of the sampling configuration (regular or irregular, sparse or dense, etc.). This thesis demonstrates the potential of the ABF methodology on a small dataset, in this case a subsection of the Jemalong radiometric survey. Application of the ABF framework to larger surveys would require assembling several fast algorithms that currently exist in isolation.

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ACKNOWLEDGEMENTS The greatest source of encouragement, advice and inspiration I received during this thesis was from Garry Newsam of DSTO. He showed a keen interest in a PhD project for which he received no official recognition and that was peripheral to his major field of study. All the material from half way through Chapter 4 to the end of the thesis has its roots in ideas I discussed with Garry. Thank you very much for all your assistance.

During the course of my PhD I was very fortunate to be associated with CSIRO Land and Water, who provided my with computer support and an office at Black Mountain in Canberra. Alan Marks in particular was always ready with a word of encouragement and I benefited tremendously from his endless source of contacts in the geophysics field. Alan always told me that once I got a PhD I’d have an opinion on everything and no friends. I’m waiting to see what happens. While at CSIRO I worked under the supervision of Brian Tunstall. He was always prepared to spend time discussing all sorts of different issues and I learnt a lot from our many lengthy conversations. I was sorry to see Brian leave CSIRO before I finished the thesis.

Thank you to everyone else at CSIRO including

Andrew Bell, Chris Barnes, Peter Fitch, Geoff Pickup, Alex Held and Guy Byrne.

Thank you to my principal supervisor, Brent Jacobs, who took over the helm after my initial supervisor, Craig Pearson, left the department. Brent was always ready to give advice and provided some very constructive reviews of several of my chapters. Steve Roberts became my associate supervisor for the last year of my PhD, and I benefited from our many sessions with marker pen and white board. Steve also reviewed a lot of the mathematical material in the thesis. Thank you to my two honours supervisors, Malcolm Sambridge and Brian Kennett who kindly reviewed some of the chapters in the thesis and were always ready to provide advice.

The Australian Geological Survey Organisation flew the Jemalong radiometric survey specifically for my PhD research. Thank you to all in the airborne mapping group and Brian Minty in particular. Brian was always prepared to assist with advice on every aspect of gamma-ray spectrometry. Figures 1, 2 and 3 in Chapter 2 were reproduced with permission from an electronic version of Brian’s thesis. He contributed the spectral components for calculating live-time (Appendix C), calibration data (Appendix D) and a Monte Carlo code for assessing radon error (Appendix E).

The inspiration for Chapter 5 was provided by Rick Beatson who I conversed with extensively by E-mail and who financed a one month trip to visit him in Christchurch. I also have Rick to thank for

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introducing me to Matlab, which is a fantastic environment for testing new ideas and methods. Discussions over E-mail with Jens Hovgaard resulted in the geometrical detector model derived in Chapter 3. He also supplied the Monte Carlo modelling results that were used to validate the model.

In December 1996 I met John Turner, who was about to commence his Honours year at the University of Canberra. We decided to merge our projects together, with John concentrating on the soil mapping and myself on the geophysics. Some very fruitful sessions of fieldwork followed during which we had an excellent time. Thanks for everything John, not the least for those 500 odd soil horizons you analysed back in the laboratory (sorry I didn’t use them in the thesis!).

Dan Carmody, Grant Koch and Stephen Griffin also assisted with the fieldwork and their contributions are gratefully acknowledged. Dan was always ready and willing to provide some much needed moral support during the PhD. No offence Dan, but I think you have the largest repertoire of bad jokes of anyone I know (and Alan Marks agrees)! I learnt a lot from many lengthy phone conversations with Stephen Griffin, including the practicalities of geophysical surveys, the utility of Fourier transforms, and perhaps most importantly, famous Australian Rugby Union players.

On a personal level, I would like to thank my two good friends, Jim Dunlap and Alfredo Camacho. I enjoyed the morning sessions at the ANU Jim with Gym. Alfredo provided ample inspiration for me to finish, as I bet him that I would be first to hand in the thesis. Unfortunately, I lost by about a year at which point the main incentive to finish was to avoid the painful “Have you handed in yet”?

I would like to extend my appreciation to the Land and Water Research and Development Corporation for a PhD scholarship and also their support and encouragement.

Thank you to the Jemalong-Wyldes Plains Steering Committee, and Rei Buemer and Ian Smith in particular, for funding the soil sampling used in Supplement 1. During the course of our fieldwork in the Jemalong-Wyldes Plains Bob and Jan McPhillamy very kindly allowed us to stay in a cottage on their property.

Thank you to them, Lindsay Ball and David Pengilly for letting us survey their

paddocks on our bizarre quad-bike set-up.

Finally, but at the top of my list, I would like to thank my wife, Janee for all her support and encouragement, especially during the last few months while I was writing up. The writing of a thesis can be a difficult time and her understanding during this time helped me enormously.

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TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Quantifying the influence of survey geometry on spatial resolution and assay uncertainty

........................................................... 3

2.2 Implementation of improved methods for interpolation, Fourier transformation and deconvolution of radiometric data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Thesis layout

........................................................... 5

CHAPTER 2 GEOPHYSICAL CONSIDERATIONS IN SOIL MAPPING USING AIRBORNE GAMMA-RAY SPECTROMETRY . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Fundamentals of gamma-ray spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Natural sources of gamma rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Radiochemistry of potassium, uranium and thorium . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Geochemistry of potassium, uranium and thorium . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Disequilibrium in uranium and thorium decay series

. . . . . . . . . . . . . . . . . . . . . 14

2.2 Sources of background radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Interactions with matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Interference with the signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Detection of gamma rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Airborne gamma-ray surveying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 The effect of aircraft height on the measured count rates . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Calibration of radiometric surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Processing of gamma-ray surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Soil mapping using airborne gamma-ray spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Existing techniques for soil mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Soil information required for land evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Parametric soil survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Previous soil mapping using gamma-ray spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Discussion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Problems with current mapping techniques: A case study 5.1 Soil sampling stratified by airborne radiometrics

. . . . . . . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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6 Geophysical Considerations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.1 Improving the low signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Assay uncertainty and spatial resolution

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.3 Interpolation of data to a regular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4 Compensation for the smoothing effect of height . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 CHAPTER 3 A MODEL OF AIRBORNE GAMMA-RAY SPECTROMETRY . . . . . . . . . 46 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 Modelling the gamma ray signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1 A general mono-energetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Simplifying the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 Derivation of existing results

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 A geometrical detector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 The solid angle of a rectangular detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Incorporating variation in the efficiency of a rectangular detector

. . . . . . . . . . . . . . . . 57

3.3 Comparison of modelled results to experimental data . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 The point spread function in airborne gamma ray spectrometry

. . . . . . . . . . . . . . . . . . . . . 63

4.1 The point spread function of a 16.8 litre airborne detector . . . . . . . . . . . . . . . . . . . . . . 64 5 Comparison to Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6 Contributing area calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7 Incorporation of the platform velocity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 CHAPTER 4 RESOLUTION AND UNCERTAINTY IN AIRBORNE GAMMA-RAY SURVEYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2 Propagation of errors during processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 Background to error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 Tracing the error introduced by each processing step

. . . . . . . . . . . . . . . . . . . . . . . . . 81

2.3 Assay uncertainty under different conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Spatial resolution of radiometric surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2 The Optimal Wiener filter

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3 Estimating the auto-correlation of the signal and noise . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4 Spatial resolution of the Jemalong survey

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5 Influence of assay uncertainty on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.6 Influence of aircraft movement on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . 107

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3.7 Effect of aircraft height on spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 CHAPTER 5 A GENERAL FRAMEWORK FOR INTERPOLATION AND SMOOTH FITTING OF GEOPHYSICAL DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 Exact interpolation using Arbitrary basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3 Solution of the interpolation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.1 Computational cost

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2 Iterative methods for solving the matrix equations

. . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.2.1 Converting the interpolation equations to a positive-definite form by QR factorisation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.2.2 Lanczos iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.3 Preconditioning

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4 Matrix-vector product

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Generality of the Arbitrary Basis Function framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.1 Choice of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.1 Radial and non-radial basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.1.2 Thin-plate splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.3 Tension splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.4 Kriging

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.2 Smooth fits to geophysical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.1 Regularising the solution with GCV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2.2 Smoothing by using less nodes than data constraints 5 Interpolation and smooth fitting of the Jemalong survey

. . . . . . . . . . . . . . . . . . . . 134

. . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.1 Exact interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.1 Exact thin-plate splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.2 Tension spline fit

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 Smooth fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.2.1 Smoothing spline fits to the thorium data

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.2.2 Smooth fitting using the sinc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.2.3 Spatial prediction by kriging 5.2.4 Concluding remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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CHAPTER 6 THE FOURIER TRANSFORM OF AN ARBITRARY BASIS FUNCTION SURFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2 The Fourier transform

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

2.1 Definition of the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2.2 Discretisation and the 1-D Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2.3 Fourier transforms of 2-D geophysical data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3 Fourier transforms and Arbitrary Basis Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.1 The Fourier transform of an Arbitrary Basis Function . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2 The Fourier transform of a sinc interpolant

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.3 The Fourier transform of a spline interpolant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4 The Fourier transform of the Jemalong radiometric data

. . . . . . . . . . . . . . . . . . . . . . . . . . 167

5 Effect of basis function and solution method on the Fourier transform

. . . . . . . . . . . . . . . . 172

6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 CHAPTER 7 CONVOLUTION AND DECONVOLUTION OF ARBITRARY BASIS FUNCTION SURFACES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2 Basis transformation: An exact method for filtering ABF surfaces . . . . . . . . . . . . . . . . . . . 178 3 Weight transformation: An approximate method for filtering ABF surfaces . . . . . . . . . . . . . 181 4 Application of the ABF methodology to a simple one-dimensional example . . . . . . . . . . . . 184 4.1 Convolution with the sinc function as a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.2 Convolution of curves defined by a cubic spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5 Extension to two-dimensional convolution and deconvolution . . . . . . . . . . . . . . . . . . . . . . 202 5.1 Thin-plate spline Basis transformation with radially symmetric filters . . . . . . . . . . . . 203 5.2 An approximate method for thin-plate splines and multiquadrics . . . . . . . . . . . . . . . . 208 5.3 Extending Weight transformation to splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6 Effect of ABF parameters on the deconvolution of radiometric data . . . . . . . . . . . . . . . . . . 210 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 CHAPTER 8 DISCUSSION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 2 Summary of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 2.1 Modelling the gamma-ray signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 2.2 Spatial resolution, assay precision and Wiener filtering . . . . . . . . . . . . . . . . . . . . . . . 217 2.3 The ABF approach to interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 2.4 Fourier transforms, convolution and deconvolution of ABF surfaces . . . . . . . . . . . . . 220 2.5 Integration of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

x

3 Implications and directions for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.1 Soil mapping implications

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

3.2 Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 3.3 Application of the ABF methodology to other types of data . . . . . . . . . . . . . . . . . . . . 225 REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

APPENDIX A: EFFECT OF DEPTH VARIATIONS IN CONCENTRATION AND ATTENUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 A1 Does soil density affect the radiometric signal?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

A2 Effect of depth variations in attenuation and isotope concentration . . . . . . . . . . . . . . . 236 APPENDIX B: CALCULATION OF DETECTOR EFFICIENCY

. . . . . . . . . . . . . . . . . 238

APPENDIX C: CALCULATION OF LIVE-TIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 APPENDIX D: CALCULATION OF THE ERROR IN THE AIRCRAFT AND COSMIC BACKGROUNDS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

APPENDIX E: CALCULATION OF THE RADON BACKGROUND UNCERTAINTY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 APPENDIX F: EFFECT OF HEIGHT CORRECTION ON ASSAY PRECISION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 APPENDIX G: THE LANCZOS METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 G1 The conjugate gradients algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 G2 Selection of the smoothing parameter by generalised cross validation . . . . . . . . . . . . . 251 G3 Smoothing by least-squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 APPENDIX H: THE FOURIER TRANSFORM OF AN ABF EXPANSION

. . . . . . . . . . 255

APPENDIX I: AN APPROXIMATION TO THE FOURIER TRANSFORM ABOUT THE ORIGIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 APPENDIX J: CALCULATION OF A FILTER BASIS FOR CUBIC AND THIN-PLATE SPLINES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

J.1 1-D application using the cubic spline

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

J.2 A cubic spline basis without cancelling discontinuities . . . . . . . . . . . . . . . . . . . . . . . 260 J.3 2-D application using the thin-plate spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 APPENDIX K: CONVOLUTION OF POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . . . 263

xi

LIST OF TABLES AND FIGURES CHAPTER 2 Table 1:

238

Table 2:

232

............................................................

U decay series showing half lives and modes of decay Th decay series showing half lives and modes of decay

. . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . 13

Table 3: Mass and linear attenuation coefficients of the main gamma rays detectable at airborne altitude

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Table 4: Mean, minimum, maximum and standard deviations for the line data processed using standard 4-channel techniques on the raw and NASVD cleaned spectra

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 1: Line spectra for potassium and the uranium and thorium decay series

. . . . . . . . . . . . 11

Figure 2: Gamma-ray flux simulated at 300 m height for potassium, uranium series in equilibrium and thorium series in equilibrium

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 3: Observed gamma ray spectra for potassium, uranium series in equilibrium and thorium series in equilibrium

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 4: Reduction in the observed signal with height of detector above the ground

. . . . . . . . 23

Figure 5: Radius of circle that contributes 90% of the signal recorded at the detector as a function of detector height

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 6: Standard 4 channel processing of the 100 m elevation potassium, uranium, thorium and total count data

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 7: NASVD processing of the 100 m elevation potassium, uranium, thorium and total count data

CHAPTER 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

.............................................................

Table 1: Table of mathematical symbols

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Table 2: Linear attenuation coefficients of NaI

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Table 3: Values of physical parameters used in the comparison of the Monte Carlo simulations with the geometrical detector model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 1: Geometry for setting up the mathematical model Figure 2: Schematic of a four litre detector

. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 3: Geometry for solid angle calculations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Figure 4: Contour and profile plots of the solid angle of a 4.2-l rectangular detector Figure 5: Geometry for detector efficiency calculations

. . . . . . . . . 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 6: Contour and profile plots of the total-count efficiency for a 4.2-l rectangular detector for a gamma ray energy of 2.6 MeV

. . . . . . . . . . . . . . . . . . . . . . . . . . 61

xii

Figure 7: Contour of the detector response of a 4.2-l rectangular detector for a gamma ray energy of 2.6 MeV

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 8: The detector sensitivity and PSF for the geometrical and uniform models at 25 m elevation for a 2.6 MeV gamma ray incident on a 4.1-l crystal

. . . . . . . . . . . . . . . . 62

Figure 9: Contour and profile plots of the PSF of the geometrical detector model for a 2.6 MeV photon incident on a 16.8-l crystal at 60 m elevation

. . . . . . . . . . . . . . . . . . . . . 64

Figure 10: Comparison of Monte Carlo simulations to the geometrical detector model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 11: Comparison of the predicted contributing areas of a 16.8-l airborne system for the geometrical and uniform detector models

. . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 12: Contour and profile plots of the point spread function for a 2.6 MeV gamma ray at 60 m elevation with an aircraft velocity of 60 m/s

. . . . . . . . . . . . . . . . . . . . . 71

Figure 13: Ratio of the peak of the PSF incorporating velocity to the peak of a stationary PSF

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 14: Frequency domain representation of the PSF for thorium

CHAPTER 4

. . . . . . . . . . . . . . . . . . . . 73

............................................................

Table 1: Definition of symbols used in this chapter

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Table 2: Various calibration constants for the AGSO aircraft

. . . . . . . . . . . . . . . . . . . . . . . . . 82

Table 3: Aircraft and cosmic backgrounds calculated for the AGSO aircraft Table 4: Variation in radon background with height over Lake Hume, NSW Table 5: Calculated standard deviations of background errors

. . . . . . . . . . . . . . . 84 . . . . . . . . . . . . . . 85

. . . . . . . . . . . . . . . . . . . . . . . . . 86

Table 6: Standard deviations of the measured count rate, the background correction and background corrected count rate for average crustal concentrations

. . . . . . . . . . . . . . . 87

Table 7: Coefficients from Equation (37) for the change in mean-square-error and the half point of the resolution function

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table 8: Change in spatial resolution from using the NASVD by cluster method developed by Minty and McFadden (1998)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Table 9: Critical and recommended sampling rates for the Jemalong survey.

. . . . . . . . . . . . . 111

Figure 1: Observed and calculated dead-time over the calibration range at Albury, New South Wales

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Figure 2: Variation of assay uncertainty with height for potassium, uranium and thorium

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Figure 3: Change in mean squared error for the synthetic 1-D radiometric survey Figure 4: Resolution curves for the synthetic 1-D radiometric survey

. . . . . . . . . . 97

. . . . . . . . . . . . . . . . . . . 98

Figure 5: Resolution functions for the synthetic 1-D radiometric survey

. . . . . . . . . . . . . . . . . 99

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Figure 6: Power spectra calculated for the Jemalong radiometric survey Figure 7: Normalised power spectral densities for the Jemalong survey Figure 8: Resolution curves for the Jemalong survey

. . . . . . . . . . . . . . . . 102 . . . . . . . . . . . . . . . . 104

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 9: Effect of assay uncertainty on the maximum rate of change of the mean-square error and the half-point of the resolution curve Figure 10: Effect of aircraft movement on spatial resolution

. . . . . . . . . . . . . . . . . . . . . . 106

. . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 11: Resolution curves for potassium at heights of 30, 60, 90, 120 and 150 m Figure 12: Change in frequency and sampling cut-offs with aircraft height

CHAPTER 5

. . . . . . . . 108

. . . . . . . . . . . . . . 109

.............................................................

Table 1: Mathematical symbols used in Chapters 5, 6 and 7

. . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 1: Computational scaling for direct solution by LU decomposition on an 200 MHz Ultra-Sparc with 380 Megabytes of RAM

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 2: The thin-plate spline, the multiquadric , the inverse multiquadric the Gaussian , the sinc function and a tension spline

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 3: Distribution of the data used for the example interpolations

. . . . . . . . . . . . . . . . . . 137

Figure 4: Grey-scale images of the exact thin-plate spline interpolation of total count and thorium

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5: Thin plate and tension spline fits to the total count data, both along an acquisition line and between two acquisition lines

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 6: Plots of interpolated thorium along an acquisition line

. . . . . . . . . . . . . . . . . . . . . . 141

Figure 7: Contour plots of the thorium data interpolated by thin-plate splines in an , subsection of the survey area

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 8: Contour plots of the thorium data interpolated by sinc functions in an , subsection of the survey area

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Figure 9: Kriging semi-variogram after removal of a linear trend from the thorium data

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 10: GCV measure (Equation 30) versus the ratio of the nugget effect to the height to sill

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Figure 11: Contour plots of the thorium data interpolated by kriging in a 3 by 3 km2 subsection of the survey area

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Figure 12: Grey-scale images of the exact and LSQR thin-plate spline surfaces over the entire thorium data set

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 13: Three-dimensional view of the exact and LSQR thin-plate spline surfaces over the entire thorium data set

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xiv

CHAPTER 6

............................................................

Table 1: Several commonly used basis functions and their Fourier transforms Figure 1: Sinc and cubic spline fits to the Bessel function of order zero

. . . . . . . . . . . . 163

. . . . . . . . . . . . . . . . 163

Figure 2: Potassium, uranium and thorium (4.0 to 13.9 ppm eTh) thin-plate spline surfaces interpolated to a 25 m grid cell

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Figure 3: Calculated power spectra of potassium, uranium and thorium surfaces

. . . . . . . . . . 169

Figure 4: Fourier amplitudes of the weights for the potassium, uranium and thorium thin-plate spline surfaces

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Figure 5: Power spectra for a 3 by 3 km2 subset of the Jemalong survey for exact thin-plate spline, exact multiquadric, and GCV and LSQR thin-plate splines

CHAPTER 7

. . . . . . . . . . 173

............................................................

Figure 1: Sinc interpolation applied to convolution with a Gaussian PSF

. . . . . . . . . . . . . . . 187

Figure 2: Sinc interpolation applied to Wiener deconvolution with a Gaussian PSF Figure 3: Sinc high and low pass filters Figure 4: Sinc low pass derivative filter

. . . . . . . . 189

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Figure 5: Sinc interpolation and the derivative of the deconvolved response Figure 6: The cubic spline applied to convolution of the Bessel function

. . . . . . . . . . . . . . . . 195

Figure 7: The cubic spline applied to deconvolution with a Wiener filter Figure 8: Cubic spline, Wiener filter applied to noisy convolution samples Figure 9: Cubic spline and high pass filter with a cut-off of 5 cycles.

. . . . . . . . . . . . . 192

. . . . . . . . . . . . . . . 196 . . . . . . . . . . . . . . 197

. . . . . . . . . . . . . . . . . . 198

Figure 10: Negative of remainder component for cubic spline interpolation and a low pass derivative filter

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Figure 11: Remainder component for the derivative of the deconvolved response for cubic spline interpolation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Figure 12: Contribution of second and remainder terms to Gaussian convolution with the cubic spline

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Figure 13: Comparison of the basis function for exponential damping the natural logarithm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Figure 14: Wiener filter for deconvolution of thorium and near origin polynomial approximations with and without exponential damping

. . . . . . . . . . . . . . . . . . . . . . . . . . 204

Figure 15: Effect of the parameter damping parameter on the remainder component

. . . . . . . . 205

Figure 16: The summation of the second and remainder components for Gaussian damping

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Figure 17: Original and deconvolved surfaces for GCV thin-plate spline fit to Thorium with Gaussian damping applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xv

Figure 18: Interpolated surfaces for the second and remainder terms with Gaussian damping

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Figure 19: Deconvolution of thorium in a 3 by 3 km subsection of the Jemalong survey for the exact thin-plate spline and the exact multiquadric

. . . . . . . . . . . . . . . . . . . 212

Figure 20: Deconvolution of thorium in a 3 by 3 km subsection of the Jemalong survey for the GCV and LSQR thin-plate splines.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Figure 21: Difference between deconvolved surface for the exact thin-plate spline, exact multiquadric and GCV thin-plate spline deconvolution.

CHAPTER 8

. . . . . . . . . . . . . . . . . . . . 214

.............................................................

Figure 1: Exact (top) and GCV (bottom) thin-plate spline surfaces interpolated to a 25 m grid cell

APPENDICES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 ............................................................

Figure C1: Observed and modelled spectra at 60 m elevation over the Albury calibration range

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Figure D1: Standard deviations of the error in the calculated aircraft and cosmic backgrounds

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Figure E1: Theoretical noise free spectra for low and high radon conditions at an altitude of 90 m for a high potassium concentration of 4%

. . . . . . . . . . . . . . . . . . . . . . . . 246

Figure F1: Effect of deviations from nominal altitude for potassium, uranium and thorium

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Figure F2: Size of height correction for potassium, uranium and thorium

. . . . . . . . . . . . . . . 249

CHAPTER 1

INTRODUCTION The most exciting phrase to hear in science, the one that heralds the most discoveries, is not 'Eureka!' (I found it!) but 'That's funny...' Isaac Asimov

1 INTRODUCTION Soils affect land productivity and susceptibility to degradation, making acquisition of soil information important for land use management. Most agricultural areas in Australia have been cleared which creates difficulties for traditional soil surveys based on interpretation of aerial photography, terrain and native vegetation. Airborne gamma-ray spectrometry is a remote sensing technology that is being used to supplement, or even replace, traditional soil mapping methods. It measures radioactive emissions from unstable potassium, uranium and thorium isotopes in the near surface. The distribution of these elements in the landscape is controlled by many of the same factors important in determining the distribution of soils. However, convenience and cost dictate that gamma-ray data are collected at safe flying altitudes by aircraft traversing widely spaced (100-400 m) flight-lines. From these noisy, blurred, and irregularly sampled data, estimates of potassium, uranium and thorium distribution are made on a regular grid. Existing methods for doing this for gamma-ray spectrometry are unsatisfactory and lack mathematical rigour. Therefore, the aim of this thesis is to develop improved and more rigorous methods for creating gamma-ray images

Gamma-ray data are usually collected along with magnetic data in an airborne geophysical survey and have predominantly been used for mineral and petroleum exploration (e.g. Dickson and Scott, 1992; Dentith et al., 1994; Saunders, et al., 1994). However, gamma-ray image quality has often been poor and skilled interpreters scarce, so that the data were frequently archived before being effectively utilised.

Recent developments in multichannel processing techniques (Minty, 1996;

Hovgaard, 1997; Hovgaard and Grasty, 1997; Minty and McFadden, 1998) have significantly improved image quality and, combined with favourable applications in land resource assessment, geological mapping and mineral exploration (e.g. Willocks and Sands, 1995; Bierwirth, 1996; Wilford et al., 1997), have resulted in a renewed interest in gamma-ray surveys (e.g. Gunn, 1998).

2

Even with multichannel processing techniques there are still significant challenges in creating an accurate image of radioelement distribution from an airborne gamma-ray survey. In particular:

Ÿ

Gamma-ray images are created by interpolation between widely spaced flight lines. The spatial resolution and assay uncertainty (how precisely the concentrations are measured) of the resultant images depends on the sampling density, noise levels in the data, and on the interpolation algorithm used.

Ÿ

Gamma-ray data can be noisy, so that a surface passing through each datum may not be ideal. In that case a decision needs to be made regarding the compromise between data fidelity and smoothness.

Ÿ

A gamma-ray detector cannot be focused like a camera, so that each measurement is a complex integration of gamma emission beneath the aircraft (e.g. Duval, 1971). The characteristics of the measurement footprint depend on the shape of the detector, its height above the ground and the aircraft movement during the integration time.

Ÿ

The flight-line spacing and aircraft height required to achieve a desired spatial resolution are unknown and difficult to quantify.

Ÿ

The removal of the smoothing caused by measurement can be achieved by deconvolution (Craig et al., 1998). However, the measuring process attenuates high frequencies in the data, so that the deconvolution involves a delicate compromise between reconstructing valid signal and magnification of unwanted noise.

Ÿ

One method for deconvolution is Wiener filtering (Wiener, 1949), which requires calculation of the Fourier transform of the data. This is usually achieved by the Fast Fourier Transform algorithm (Cooley and Tukey, 1965) applied to an interpolated image of the data. The Fourier transformation raises several issues related to the relationship between the continuous model of theory and the discrete version of practice.

2 THESIS OVERVIEW

Soil mapping by airborne gamma-ray spectrometry usually proceeds by reference to images of radioelement distribution derived from noisy, blurred and irregularly sampled radiometric data. At present, the influence of the survey geometry (detector height, flight line spacing, background radiation etc.) on the quality of these images is poorly understood. Additionally, the methods used to estimate ground concentrations of potassium, uranium and thorium distribution are unsatisfactory. Therefore, the aims of this thesis are:

3

1. To quantify the influence of the survey geometry on the spatial resolution and assay uncertainty of maps of potassium, uranium and thorium distribution.

2. To increase the spatial resolution and accuracy of geochemical maps by implementing improved methods for interpolation, Fourier transformation and deconvolution of radiometric data.

2.1 Quantifying the influence of survey geometry on spatial resolution and assay uncertainty

The spatial resolution and assay uncertainty of a survey are important to quantify as they determine the suitability of gamma-ray images for a given purpose.

Knowledge of the uncertainty and

resolution of a survey flown under different conditions can be used to:

1. Assess whether a pre-existing survey is suitable for a given application; and, 2. Plan an appropriate sampling regime (height and line spacing) for an intended application.

Gamma-ray detection can be modelled as a Poisson process, so that estimates of the uncertainty in observed counts can be obtained from the mean count rates (e.g. Tsoulfanidis, 1995). However, these measurement errors are modified by the requirements for background removal, spectral stripping and height correction. Estimates of the assay uncertainty can be obtained by tracing these various sources of error by linear error propagation.

The spatial resolution of potassium, uranium and thorium images, determines the minimum length scales over which soil variation may be mapped by airborne gamma-ray spectrometry. One measure of spatial resolution is the smallest wavelength (or frequency) feature that can be reliably determined. The approach adopted here, is to estimate this wavelength by analysing the action of the optimal Wiener filter for the deconvolution of radiometric data. The principal behind the method, is that at some point in the Fourier domain, the Wiener filter will stop reconstructing signal and will begin to attenuate both signal and noise. This point will approximate an upper limit on the maximum resolvable frequency because any higher frequencies in the data will be eliminated by the deconvolution. This critical frequency, can then be compared to the Nyquist frequency imposed by the sampling rate. The analysis would determine whether gamma-ray surveys are over-, under- or critically-sampled.

4

2.2 Implementation of improved methods for interpolation, Fourier transformation and deconvolution of radiometric data

Aerial radiometric surveys are collected with different data densities along and across lines and with an uneven spacing between measurements. The data are interpolated to a regular grid for the purposes of visualisation, deconvolution and interpretation. However, current geophysical practice for gridding radiometric data is unsatisfactory (e.g. Craig et al., 1998; Billings and Fitzgerald, 1998). Inspection of the applied mathematics, terrain analysis and geostatistics literature reveals a number of alternative methods that are encompassed within a common algorithmic framework, which I term Arbitrary Basis Functions (ABF). Examples include: the dual formulation of kriging (e.g. Matheron, 1980; Cressie, 1993), tension splines (Mitas and Mitasova, 1993), radial basis functions (Powell, 1992; Hardy, 1990) and smoothing splines (Hutchinson and de Hogg, 1985; Wahba, 1990). The ABF approach is flexible, as from within a common algorithmic base, these diverse interpolation methods can be accommodated. Furthermore, the interpolated surfaces are independent of the cell size (e.g. unlike the minimum curvature algorithm of Briggs, 1974) and they allow considerable control to be placed on the variation of the surface between data points.

Additionally, the surfaces can be

constrained to exactly honour the original data, or can be allowed to smoothly interpolate. Further, the amount of smoothing can be chosen in a rigorous statistical manner by generalised cross validation (e.g. Wahba, 1990; Sidje and Williams, 1997).

The principal disadvantage of ABF

interpolation is the computational cost and memory required to solve the resulting matrix equations, a problem partially addressed in this thesis.

The footprint of a gamma-ray measurement increases with survey height. Therefore, the higher a gamma-ray survey the more blurred the resultant images of potassium, uranium and thorium concentration. Deconvolution can be used to partially remove the smoothing effects of height and detector movement. However, the current method for doing this (Craig et al., 1998) neglects the detector shape and the movement of the aircraft during the integration time. Additionally, the forward and inverse Fourier transforms required by the method, are calculated by Fast Fourier Transforms (FFTs) of imaged data. For aerial surveying, the imaged data may not be very accurate representation of the underlying signal as the images were created by interpolation between noisy data that were irregularly sampled. Additionally, the FFT is only an approximation to the continuous Fourier transform and raises several other important issues, such as edge mismatch, irregular survey shape, grid size and quadrature approximation (e.g. Cordell and Grauch, 1982).

The ABF approach can be used to solve many of the problems with Fourier transformation of imaged data. ABFs specify an interpolant that is defined over all of space, which minimises complications

5

due to edge mismatch and survey gaps. Furthermore, the Fourier transform of an ABF can be calculated exactly on an arbitrary grid. Thus, the only error in computing it is the approximation error incurred by expressing an arbitrary surface as an ABF interpolant.

The Fourier transform of an ABF has a particularly simple form that facilitates its computation and subsequent analysis. In particular, the convolution or deconvolution of an ABF surface is another ABF surface, albeit with a new basis function. The effect of the convolution or deconvolution is embodied in the new basis function so that there is no need to calculate the Fourier transform of the data. Additionally, the implied forward and inverse Fourier transformations are exact, so that there is no error incurred in calculating the new surface. In circumstances where the new basis function cannot be calculated, approximate methods can be used that retain many of the benefits of the ABF approach. At present, the main drawback of the ABF method is the computational burden imposed by large geophysical surveys.

The development of improved methods for processing irregularly sampled geophysical data has much wider implications than the creation of gamma-ray images. The framework could be used to visualise and process any type of irregularly sampled data, including airborne magnetic, electromagnetic and terrain surveys.

3 THESIS LAYOUT

The thesis is organised as follows.

In Chapter 2, I review the fundamentals of gamma-ray

spectrometry before turning to the processing operations applied to the raw spectra. A critique of existing techniques of soil mapping is given and the specific information requirements for users of soil maps are established. Previous work using airborne gamma-ray spectrometry for soil mapping is then reviewed. A case study in the Jemalong-Wyldes Plains, New South Wales, is used to show that many applications have been conducted without a full appreciation of the geophysical issues involved in the creation of a gamma-ray image. With the knowledge gained from an examination of this case study, six specific objectives that would enable gamma-ray images to be used more effectively for soil mapping are developed.

In Chapter 3, the issue of modelling gamma-ray spectrometry is considered. The existing literature is reviewed and the deficiencies of current models are highlighted; specifically, that they ignore the rectangular shape of modern airborne detectors and the movement of the aircraft during the integration time. In the next section, the detector shape is incorporated, by modelling the change in solid angle and thickness for sources at different distances and orientations from the detector. The

6

modelled results are then validated by comparing them to experimental data gathered using a small 4.2 litre crystal and Monte Carlo modelling of a 16.8 litre airborne crystal. The implications of incorporating detector shape on the calculated resolution of a gamma-ray survey are then examined, followed by formulation of a simple method for incorporating the aircraft velocity.

In Chapter 4, I consider the problem of calculating the assay uncertainty and spatial resolution of gamma-ray surveys and also derive the optimum Wiener filter for deconvolution of noisy gamma-ray data. In the first section, linear error propagation is used to trace the influence of the various processing operations (such as background removal, spectral stripping, height correction etc.), on the assay uncertainty. The variation in assay uncertainty over crust with standard concentrations of potassium, uranium and thorium, at different heights and with different radon backgrounds is then considered. In the second section, the optimum Wiener filter for deconvolution of gamma-ray data is derived and used to infer the spatial resolution of geochemical maps under the same conditions considered for assay uncertainty. Finally, some observations regarding current survey practice are made.

Chapter 5 marks a slight change in emphasis of the thesis away from airborne gamma-ray spectrometry itself, towards general techniques for processing of irregularly sampled geophysical data.

This chapter presents a general framework (based on Arbitrary Basis Functions) for

interpolation and smooth fitting of geophysical surveys. ABF unifies many diverse methods (such as smoothing splines, kriging, tension splines and radial basis functions) into a common mathematical framework. A three step process is proposed that minimises the computational time and memory required to solve the large, and typically dense, matrix equations that arise. Only the first step is actually implemented (an iterative solution method) and its application is illustrated on the total-count band of a radiometric survey. Variations on the iterative method which enable a smooth interpolant to be constructed are then described and applied to smooth interpolation of thorium concentration.

Chapter 6 addresses the problem of calculating the Fourier transform of an ABF surface. In the first section the relationship between the continuous and discrete Fourier transforms is examined, particularly in relation to finite domain, edge mismatch, sampling rate and quadrature approximation. The Fourier transform of an ABF surface is then derived and the transformation of surfaces defined by sinc functions and splines are considered in more detail. The application of the technique to a radiometric survey is then compared to a traditional grid based approach. Lastly, the influence of the solution method (exact or smooth) and the basis function on the calculated power spectrum are considered.

7

In Chapter 7, I consider two different methods for convolution and deconvolution of ABF surfaces. The first is exact and involves transforming the basis function so that Fourier domain operations can effectively be applied without calculating the Fourier transform of the data. The second method is approximate and uses a common basis function, with the transformation being applied to the weights. The next section gives examples of the application of the method to a simple one-dimensional example, firstly with the sinc function and then with the cubic spline. The one-dimensional spline method is then extended to two-dimensional deconvolution of radiometric data. The final section considers the influence of the solution method and basis function on the deconvolved surface.

In Chapter 8, the principal results of the thesis are reviewed and an example deconvolution is presented that integrates the methods developed in each chapter. The implications of this thesis to soil mapping and the processing of irregularly sampled data are then discussed.

8

CHAPTER 2

GEOPHYSICAL CONSIDERATIONS IN SOIL MAPPING USING AIRBORNE GAMMA-RAY SPECTROMETRY The Universe is full of magical things patiently waiting for our wits to grow sharper. Eden Phillpots

CHAPTER OBJECTIVES Review the fundamentals of airborne gamma-ray spectrometry and discuss their importance for soil mapping.

1 INTRODUCTION

Land evaluation involves assessing the suitability of land for human use for activities such as agriculture, forestry, engineering, hydrology, regional planning, and recreation.

Invariably a

particular land use depends, not on a single natural resource attribute, but on an aggregate of attributes. The processes of mapping and inventory of all the factors relevant to land evaluation (soils, climate, topography, vegetation, fauna, groundwater etc.) is collectively called land resource assessment.

Within Australia there is a growing need for high quality land-resource assessment (e.g. Young et al., 1990). This growth derives from increasing demands being made on biophysical resources for food, fibre, timber and minerals, waste disposal, urban development and recreation. These land uses should be balanced with ecological considerations, especially the need to preserve remnant vegetation and endangered species of flora and fauna (e.g. ESDWG, 1991). A further reason for the growth of land resource assessment is that poor management, often through lack of adequate information and/or knowledge of biophysical resources, has been held responsible for many of the environmental problems facing Australia (e.g. Young et al., 1990). Examples include soil structural decline, soil acidification, soil salinisation, soil erosion and eutrophication of water resources following application of fertiliser.

9

Soil mapping is an integral part of land resource assessment as soils affect the productivity, susceptibility to degradation and ease of modification and management of land for specific uses. Existing techniques for soil mapping usually involve interpretation of aerial photography, combined with knowledge of correlations between soils, native vegetation and terrain (e.g. Gunn et al., 1988). However, the majority of Australia’s agricultural land is in areas with little relief, where much of the native vegetation has already been cleared (e.g. ESDWG, 1991). In these areas application of existing techniques is difficult. Recently, airborne gamma-ray spectrometry has been proposed as a remote sensing technology capable of distinguishing soil variation in the absence of relief and native vegetation (e.g. Cook et al., 1996; Bierwirth, 1996; Gourlay et al., 1996). Gamma-ray data are usually collected along with magnetic data in an airborne geophysical survey and have been used previously for mineral and petroleum exploration (e.g. Dickson and Scott, 1992; Dentith et al., 1994; Saunders, et al., 1994), geological mapping (e.g. Kovarch et al., 1994) and soil moisture investigation (Carroll, 1981).

In the first section of this chapter the fundamentals of gamma-ray spectrometry are reviewed, beginning with the natural sources of gamma radiation (potassium, uranium and thorium) and their radiochemical and geochemical behaviour.

Issues concerning sources of background radiation,

interactions of gamma-rays with matter, interference and detection of gamma-rays are then described. The next section of the chapter addresses aerial surveying of gamma-radiation, including the effect of aircraft height, calibration procedures and the processing pathway for converting the raw 256-channel spectra into estimates of potassium, uranium and thorium concentration.

Supported by this background material, the use of gamma-radiation for soil mapping is described. Existing techniques for soil survey are reviewed, their deficiencies in areas with little topographic variation are noted, and the need for an alternative technique is established. Airborne gamma-ray spectrometry is then presented as a promising solution and the existing literature dealing with its application is reviewed. This reveals a general lack of knowledge concerning the importance of the geophysical principles behind gamma-ray surveying. A validation exercise of a soil map derived from radiometric data (within the Jemalong-Wyldes Plains of New South Wales) highlights this observation. The case study provides a convenient starting point for a discussion on the importance of geophysical issues in soil mapping. The discussion establishes a framework for the remainder of the thesis.

10

2 FUNDAMENTALS OF GAMMA-RAY SPECTROMETRY

Many isotopes are unstable and undergo radioactive decay through one of three mechanisms; alpha, beta or gamma decay. Alpha decay results in the emission of an alpha particle which is a positively charged helium nucleus. Alpha particles are highly ionising and most are absorbed by only a few centimetres of air (e.g. Tsoulfanidis, 1995). Beta decay can occur through one of three mechanisms: electron emission, positron emission or electron capture (each also results in the emission of a neutrino). Beta particles are more penetrating than alphas but most are stopped by a meter or so of the atmosphere. Gamma decay occurs when a nucleus undergoes a transition from an excited state to a lower energy state, the difference in energy being emitted in the form of a gamma ray (some energy is lost in recoil of the nucleus but it is generally negligible). Gamma rays can penetrate tens of centimetres of rock and hundreds of meters of air making them inherently suitable for remote sensing type applications.

2.1 Natural sources of gamma rays

Potassium, uranium and thorium are the only naturally occurring radioisotopes with gamma ray emissions of sufficient energy and intensity to be measurable at the altitudes flown by survey aircraft (hereafter referred to as “airborne altitude”). Potassium is a major component of the Earth’s crust (2%), while uranium (2.7 ppm) and thorium (8.5 ppm) are present in much lower concentrations (e.g. Wedepohl, 1969). Their low abundances are offset by their higher gamma-ray activities with the unit mass gamma emission of uranium and thorium 1300 and 2,800 times greater, respectively, than potassium. Potassium is measured directly by gamma emission while uranium and thorium are measured indirectly through gamma emissions in their decay series.

2.1.1 Radiochemistry of potassium, uranium and thorium

Unless otherwise indicated the material described in the following three paragraphs has been obtained from Lederer and Shirley (1978). Potassium contains 0.012% of the unstable isotope 40K which has a half life of 1.3 % 10 9 years. It decays through one of two pathways with 89% via beta emission to 40Ca, the other 11% forming 40Ar by electron capture and gamma ray emission at 1.46 MeV (Figure 1). As

40

K occurs as a fixed

proportion of naturally occurring potassium, the intensity of gamma emission is diagnostic of total potassium.

11

Relative Intensity

K-40 (1.460)

(a)

0

K line spectrum

1

2

3

Bi-214 (0.609)

(b)

0

1

B1-214 (2.204)

Bi-214 (1.765)

U line spectrum Bi-214 (1.120)

Relative Intensity

Bi-214 (0.352)

Energy (MeV)

2

3

0

Ac-228 (0.911)

Th line spectrum

1

2

Tl-208 (2.614)

(c) Tl-208 (0.583)

Relative Intensity

Pb-212 (0.239)

Energy (MeV)

3

Energy (MeV)

Figure 1: Line spectra for (a) potassium and the (b) uranium and (c) thorium decay series. Reproduced from Minty (1996). Uranium exists naturally as the two radioisotopes 8.8 % 10 8 years).

U (half life 4.5 % 10 9 years) and

238

235

U (half life

235

U comprises only 0.72% of natural uranium and emits gamma rays with too low

an energy and intensity to be measurable at airborne altitude.

238

U decays through a complex decay

series with multiple alpha, beta and gamma emissions to stable 206Pb (e.g. Lederer and Shirley, 1978;

12

and Table 1). Late in the decay series

214

Bi decays through one of two daughters to

210

Pb and in the

process can emit gamma rays with several different energies (Figure 1). The emissions at 0.609 and 1.76 MeV are of most interest in airborne surveying. Uranium concentration is only indirectly measured by the gamma emission of

214

Bi and is therefore reported as gamma equivalent uranium,

eU. Isotope U-238 Th-234 Pa-234m U-234 Th-230 Ra-226 Rn-222 Po-218 Pb-214

Half-life 4.51 x 109 y 24. 1 0 d 1.17 m 2.47 x 105 y 7.5 x 104 y 1602 y 3.82 d 3.05 m 26.8 rn

Bi-214

19.9 m

Po-214 Pb-210 Bi-210 Po-210

1.64 us 22 y 5.01 d 138.40 d

Pb-206

-

Decay α β, γ β, γ α, γ α, γ α, γ α, γ α, β β, γ α, β, γ (0.609, 1.12, 1.764) α, γ α, β, γ α, β α, γ -

Table 1: 238U decay series showing half lives and modes of decay. Minor alternative modes of decay are emitted. Units are s for seconds, m for minutes, h for hours, d for days and y for years. Bold-faced numbers in the second column are the energies (in MeV) of the primary gamma-rays emitted by the U decay series. Thorium occurs naturally as the unstable isotope 232Th with a half life of 1.41 % 10 10 years and decays through a series of daughters to stable

208

Pb (e.g. Lederer and Shirley, 1978; Table 2). Members of

the thorium decay series emit multiple gamma rays up to an energy of 2.62 MeV (Figure 1). The 2.62 MeV gamma emission that occurs in the beta decay of

208

Tl to

208

Pb is readily observed at airborne

altitude. Gamma emission is therefore an indirect indicator of the concentration of thorium, with reporting units expressed as gamma equivalent thorium, eTh.

2.1.2 Geochemistry of potassium, uranium and thorium

Information on the concentrations of radio-elements in different minerals and rock types and their weathering behaviour can be found in Wedepohl (1969) and Galbraith and Saunders (1983) with specific Australian contexts in Dickson and Scott (1992). The solubility of uranium and thorium under different chemical conditions is addressed in Langmuir (1978) and Langmuir and Herman (1980).

13

Isotope Th-232 Ra-228 Ac-228 Th-228 Ra-224 Rn-220 Po-216 Pb-212 Bi-212 Po-212 (64%)

Half-life 1.41 x 1010 y 6.7 y 6.13 h 1.9 10 y 3.64 d 55.3 s 0. 15 s 10.6 h 60.6 m 304 ns

Tl-208 (36%)

3. 1 m

Pb-208

-

Decay α β, γ β, γ α, γ α, γ α, γ α β, γ α, β, γ α β, γ (2.62 MeV) -

Table 2: 232Th decay series showing half lives and modes of decay. Where an isotope can decay via different daughters the percentage frequency of each mode is indicated. Units are s for seconds, m for minutes, h for hours, d for days and y for years. Potassium in rocks is concentrated mainly in potassium feldspars and micas. Its distribution in weathered rocks and soils is determined by the break-up of these host minerals. Potassium is soluble under most conditions and during weathering is either lost into solution, taken up in potassium bearing clays such as illite or absorbed in small quantities onto montmorillonite or to a lesser extent kaolinite (e.g. Wedepohl, 1969). There is a general tendency for the potassium concentration to decrease as rocks weather (e.g. Dickson and Scott, 1992).

Uranium and thorium may be present in rocks as uraninite (oxide), uranothorite (silicate), in accessory minerals such as monazite, zircon, sphene and xenotine or as trace elements in the major rock forming minerals. Many of the main thorium bearing minerals (monazite, sphene, zircon) are resistant to weathering so it tends to concentrate in residual profiles (i.e. in stable or eroded areas where soil weathers faster than it erodes).

Under oxidising conditions uranium exits in the

hexavalent state and forms carbonate, phosphate or sulphate complexes that can be very soluble (Langmuir, 1978). Uranium under reducing conditions, exists in the tetravalent state and is insoluble. Thorium occurs in the tetravalent state and is insoluble except at low pH, or in the presence of organic compounds such as humic acids (Langmuir and Herman, 1980). Uranium and thorium released by weathering may be retained by clays or iron oxides, or transported out of the system as colloidal material (or in dissolved form if the chemical conditions are suitable).

Different rock types contain different amounts of potassium, uranium and thorium (e.g. Galbraith and Saunders, 1983). For example, granites typically have much higher concentrations of all three

14

radioisotopes than quartz sand. The concentrations of the radioisotopes change during weathering. When weathering rates exceed erosion the potassium concentration tends to decrease, while the uranium and thorium concentration increases, as they are mainly hosted in resistant minerals (Dickson & Scott, 1992; Wilford, 1992, 1995). If erosion equals or exceeds the weathering rate, then the regolith concentrations are similar to the underlying bedrock (Wilford, 1991, 1995). However, radium is soluble and can concentrate in areas of groundwater discharge, causing higher equivalent concentrations for uranium and (rarely) thorium. In summary, the distribution of potassium, uranium and thorium in the regolith reflects Ÿ

Parent material;

Ÿ

Weathering;

Ÿ

Redistribution through leaching and groundwater discharge.

2.1.3 Disequilibrium in uranium and thorium decay series

When all the isotopes in the decay chain between 238U and 214Bi or between 232Th and 208Tl are formed at the same rate as they decay, the system is said to be in secular equilibrium. The gamma equivalent concentration is then an accurate measure of the uranium or thorium concentration. If one or more of the daughter products is formed at a rate different to which it decays, the system is in disequilibrium and gamma equivalent may be a poor measure. Disequilibrium occurs when one or more of the daughter isotopes are added or removed from the system by physical or chemical means. A general rule of thumb is that it takes approximately five half lives for the system to return to secular equilibrium (assuming the system is not disturbed further). Therefore, disequilibrium can occur when processes alter the activity ratio of an isotope to its daughter on time scales less than five times its half life. There are several positions in the Uranium decay series where disequilibrium can occur including (e.g. Dickson, 1995): Ÿ

Deposition or leaching of uranium relative to its daughter nuclides during soil formation on time scales of 0.5 Million years.

Ÿ

Deposition of 226Ra (half life 1600 years) from groundwater into surface discharge zones within the past 5000 years.

Ÿ

Emanation of radon gas

222

Rn (half life 3.8 days) from within soil pores due to changing

atmospheric and moisture conditions.

Dickson (1995) analysed 445 soil samples from around Australia and found a mean 238U/226Ra activity ratio of 1.1:1. More soils where found to be

238

U rich than

238

U poor, although the extent of

disequilibrium was found to be relatively small. A more serious problem is the disequilibrium caused by the varying rates of escape of radon from the soil surface (see sections 2.2 and 2.4).

15

The 2.62 MeV gamma emission of

208

Tl in the thorium decay series is a measure of

220

Rn gas which

fortunately has a short half life of 60 seconds. All daughters between 232Th and 220Rn have half-lives of less than 6 years and, therefore, secular equilibrium is reached in less than fifty years. Thorium series disequilibrium is rare and when it occurs in Australia is usually related to deposition of

228

Ra

around groundwater discharge zones (e.g. Dickson and Herzeg, 1992).

2.2 Sources of background radiation

There are several sources of background radiation that interfere with measurements of the radiation emitted from potassium, uranium and thorium in the soil. Ÿ

Atmospheric radon:

222

Rn is a gaseous intermediatory in the Uranium series with a half life of

3.8 days. It decays to

218

one of two pathways,

214

Po (half life 3.05 minutes) and then to

Pb (26.8 minutes) or

218

214

Bi (half life 19.7 minutes) by

At (2 seconds). The radioactive decay of

214

Bi

results in several gamma rays that are detectable at airborne altitude, including 352 keV, 609 keV, 1120 keV, 1765 keV and 2204 keV. Because radon is a gas it can diffuse out of the soil and through cracks or fissures in rocks and accumulate in the atmosphere. Its daughters, particularly 214

Bi and 214Pb, attach to airborne aerosols and dust and at times can cause significant interference

in the uranium channel, particularly when atmospheric inversion traps them close to the ground (Gold et al., 1964; Charbonneau and Darnley, 1970; Darnley and Grasty, 1970; Grasty, 1979). Ÿ

Cosmic rays: High energy gamma rays of cosmic origin interact with particles in the upper atmosphere causing a spectrum of background gamma rays that can further interact with the detector or aircraft.

Ÿ

Aircraft and spectrometer: The aircraft and equipment contains traces of radioactive material that contribute to the background radiation.

Ÿ

Nuclear fallout: The fallout from atmospheric explosions and accidents causes some background (particularly in the northern hemisphere) principally from

137

Cs with a gamma ray energy of 662

keV.

2.3 Interactions with matter

Gamma-rays interact with matter primarily by one of three mechanisms; the photoelectric effect, Compton scattering and pair production: Ÿ

Photoelectric effect: An incident photon is completely absorbed by a bound electron of an atom. The electron is ejected with an kinetic energy equal to the difference between the incident photon and the binding energy of the electron.

The probability of the interaction (known as the

photoelectric cross-section) decreases with increasing gamma ray energy but increases with

16

atomic number (e.g. Tsoulfanidis, 1995). The photoelectric effect is the dominant form of interaction at energies up to a few hundred keV. Ÿ

Compton scattering: The incident photon is scattered by an electron with a partial energy loss. When the scattering occurs with an electron that can be considered essentially free (binding energy small relative to energy of photon), the energy loss can be quite substantial and is converted into the kinetic energy of the electron. The energy loss depends on the scattering angle and is a maximum when the photon is scattered through 180o. Coherent scattering occurs when the photon interacts with a bound electron and does not result in appreciable energy loss. The Compton scattering cross- section is independent of the atomic number of a medium and depends only on the electron density.

Ÿ

Pair production: If the photon has an energy in excess of the rest mass of an electron-positron pair (1.02 MeV), pair production is possible. The process occurs in the presence of the Coulomb field of the nucleus and results in the annihilation of the photon and the creation of an electron-positron pair. The positron will eventually lose its kinetic energy and come to rest in the field of an electron at which point they will annihilate with each other and form two gamma rays of energy 0.511 MeV. The pair production cross-section increases with both atomic number and energy and is the dominant mode of scattering at large energies (e.g. Tsoulfanidis, 1995).

The total cross-section for a medium is a sum of the individual cross sections for the photoelectric effect, Compton scattering and pair production. For the gamma ray energies of most interest in airborne surveying, Compton scattering is the dominant mode of interaction in the material (air, water, rock, soil) between source and detector (e.g. Heath, 1964; Tsoulfanidis, 1995). Therefore, the main factor influencing the attenuation will be the electron density of the medium between source and receiver. The electron density (and hence attenuation) varies little with rock composition and, therefore, only depends on the soil/rock density (Kogan et al., 1969).

Table 3 lists the cross sections in the form of mass and linear attenuation coefficients for gamma ray energies of 0.606 (radon), 1.46 (potassium), 1.76 (equivalent uranium) and 2.62 MeV (equivalent thorium). The linear attenuation coefficient, l is obtained from the mass attenuation coefficient, l m by multiplying by the density of the medium, q. For a parallel beam of gamma rays with initial intensity, I(0), the number that remain after traversing a thickness Dx of material with a linear attenuation coefficient, l, is given by I(Dx ) = I(0 ) exp(−lDx )

(1)

17

The length that results in reduction of the gamma ray intensity by one-half is known as the half thickness and is obtained simply from the above equation

Dx =

Isotope Radon (0.609 MeV) Potassium (1.46 MeV) Uranium (1.76 MeV) Thorium (2.6 MeV)

Mass attenuation coefficient. (cm2/g) Air Rock/Soil 0.0801

0.0891

0.0526

0.0528

0.0479

0.0482

0.0391

0.0396

log 2 0.693 l j l

(2)

Linear attenuation coefficient (m-1) (and half thickness) Air Regolith Regolith (STP) (1 gm/cm2) (2.8gm/cm2) 0.0104 8.91 24.9 (67 m) (7.8 cm) (2.8 cm) 0.0068 5.28 14.78 (102 m) (13.1 cm) (4.7 cm) 0.00619 4.82 13.5 (112 m) (14.4 cm) (5.1 cm) 0.00506 3.96 11.1 (137 m) (17.5 cm) (6.2 cm)

Table 3: Mass and linear attenuation coefficients of the main gamma rays detectable at airborne altitude. The half widths are also shown. The half thickness for air and extreme soil/rock densities of 1.0 and 2.5 gm/cm3 are listed in Table 3 for gamma ray energies of 0.609, 1.46, 1.76 and 2.62 MeV. They show that the half thickness in air is on the order of 100 m, while within rocks and soils it is generally less than 15 cm. This implies that the majority of the gamma ray signal that reaches the surface will be due to material in the top half meter or less of soil/rock (e.g. Grasty, 1979).

Interactions of gamma rays with matter between the source and the detector result in a loss of energy and a smearing of the theoretical line spectrum of Figure 1. This process has been modelled by Kirkegaard and Lovborg (1974) with simulated spectra at 300 m above flat, uniform, half-spaces of potassium, uranium and thorium shown in Figure 2. Energy loss mainly by Compton scattering results in a continuum of energies with a maximum at the most energetic photon. Emission peaks from gamma decays within the uranium and thorium decay series are clearly seen to stand out above the Compton continuum.

Intensity (photons/sec/cm2/10keV)

18

10

(a) K flux 1

0.1 0

1

2

3

Intensity (photons/sec/cm2/10keV)

Energy (MeV) 100

(b) U flux

10

1

0.1

0.01

0.001 0

1

2

3

Intensity (photons/sec/cm2/10keV)

Energy (MeV) 100

(c) Th flux

10

1

0.1

0.01 0

1

2

3

Energy (MeV)

Figure 2: Gamma-ray fluence rate simulated at 300 m height for (a) potassium, (b) uranium series in equilibrium and (c) thorium series in equilibrium. From Minty (1996), with data originally extracted from Kirkegaard and Lovborg (1974).

19

2.4 Interference with the signal

There are several environmental factors that can interfere with the gamma-ray fluence rate at the detector, including: Ÿ

Barren overburden: Such as the parna (wind blown sand) that covers parts of Australia can act to significantly reduce the flux of gamma rays emitted from the surface.

Ÿ

Rain and soil moisture: Water within the soil increases the attenuation of gamma rays and cause the fluence rate of potassium and thorium to be deceased, generally by 10% for every 10% increase in soil moisture (Kogan et al., 1969). For uranium the effect is more complicated because radon escapes more readily from drier soils (Megumi and Mamuro, 1973). In wetter soils the lower escape rate can result in a build-up of radon in the near surface and an apparent increase in uranium content (e.g. Tanner, 1964; Lovborg, 1984; Grasty, 1997). During rainfall the uranium signal can also increase by more than 2,000%, because daughter products of radon disintegration, attach to airborne aerosols and dust and can precipitate with the rain drops (Charbonneau and Darnley, 1970).

Ÿ

Atmospheric effects: Pressure, temperature and humidity effect the density or air between source and detector and hence the attenuation of the signal.

Temperature inversion can act to

concentrate radon daughter particles close to the ground. Ÿ

Vegetation: Dense vegetation can cause some shielding of radiation and may also contribute to the gamma ray flux through radioisotopes within the biomass. In heavily forested areas, tree trunks can collimate the flux of gamma rays (Kogan et al., 1969; Travassos and Pires, 1994). .

2.5 Detection of gamma rays

The typical detection equipment used for aerial surveying consists of a sodium iodide crystal connected to a spectrometer by a photo-multiplier-tube. The sodium iodide is doped with a small amount of thallium which cause it to emit sparks or scintillations when a gamma ray passes through it. The scintillation process is complex and won’t be described in detail here (see Tsoulfanidis, 1995). Briefly, interaction of an incident photon causes an excited state in the crystal which decays with the emission of a photon in the visible part of the electromagnetic spectrum.

The amount of light emitted by a scintillation detector is very small and must be amplified by a photo-multiplier-tube before it can be recorded. A photon incident on the cathode of a photomultiplier-tube causes the emission of an electron which is accelerated by a voltage difference to a dynode. This is coated with a material that emits secondary electrons if an electron impinges upon it. These secondary electrons then accelerate to a second dynode where they cause further emissions and

20

Intensity (cps/chan/% K)

12

(a) 10

K spectrum

8 6 4 2 0 0

1

2

3

Energy (MeV)

Intensity (cps/chan/ppm eU)

4.5

(b)

4 3.5 3

U spectrum

2.5 2 1.5 1 0.5 0 0

1

2

3

Intensity (cps/chan/ppm eTh)

Energy (MeV) 2.5

(c) 2

Th spectrum 1.5 1 0.5 0 0

1

2

3

Energy (MeV)

Figure 3: Observed gamma ray spectra for (a) potassium (b) uranium series in equilibrium and (c) thorium series in equilibrium. From Minty (1996). so on in a cascading process down the tube. The electrons are finally collected at the end of the tube, at which point, their number has been amplified by about 106. The output of the photo multiplier is converted to a voltage pulse whose height is proportional to the energy of the light emitted by the

21

scintillometer, which, in turn, is proportional to the energy of the incident gamma ray. For a sodium iodide crystal the relationship between pulse height and gamma ray energy is linear down to about 400 keV. The spectrometer analyses the pulse heights and assigns the energy to one of (usually) 256 energy bins covering the spectrum from 0 to 3 MeV.

The effect of the detection system on the measured spectrum is considerable, and includes Compton scatter escape, escape peaks, accidental summing and annihilation peaks (e.g. Heath, 1964). The effect of the detector will not be discussed in detail here except in the context of detector efficiency and resolution. Detector efficiency is the fraction of gamma rays of a particular energy that enter the detector and cause a measurable event. It depends on the photon energy and, within the range of interest for airborne surveying, decreases with increasing energy. A related concept is the peak efficiency which is the fraction of gamma rays that are recorded with the correct energy within the limits imposed by the detector resolution. The detector resolution is a measure of the system’s ability to separate two photons that are closely spaced in energy. Statistical fluctuations in the processes occurring in each of the steps following the initial event within the detector cause the emission peaks to be smeared into approximately Gaussian shapes. For a sodium iodide crystal the resolution is usually expressed as the ratio of the full width at half maximum to the peak position of the 662 keV 137

Cs peak. Resolutions of around 7% are usually achieved in practice (e.g. Tsoulfanidis, 1995).

Spectra from pure sources of potassium, uranium and thorium recorded by simulating 100 m of air are shown in Figure 3 (from Minty, 1996). The detector has caused an obvious smearing of the distinct peaks that were present in the theoretical spectra of Figure 2. Interaction cross-sections are higher at lower energy. Further, higher energy photons may need to undergo two or more scattering events before depositing all their energy in the detector and on occasions the scattered photons may escape detection. These two effects result in proportionally more events at lower energy than indicated by the theoretical spectra of Figure 2.

3 AIRBORNE GAMMA-RAY SURVEYING

Aerial gamma ray data are collected by a helicopter or fixed wing aircraft that flies transects across the area of interest at approximately constant ground clearance and with a regular separation between survey lines. The aircraft usually flies at ground clearances of between 60 to 120 m, although in some cases helicopter surveys may be flown as low as 25 m.

At the other extreme, rugged

topography may require a higher clearance than 120 m. The line spacing is dictated largely by the survey height and cost constraints and generally lies within the range of 100 to 400 m. Extremely low elevation surveys may have closer line spacing to ensure adequate coverage. Additionally, many

22

reconnaissance grade surveys have been flown with a line spacing of the order of 1600 m. The along line survey density is determined by the aircraft speed and the integration time used by the spectrometer. Typically, the aircraft speed is between 50 - 70 m/s and the integration time is usually one second, giving along line samples spaced between 50 and 70 m.

Tie lines are flown

perpendicular to the acquisition lines and are usually spaced ten times further apart.

The detection equipment used in an airborne survey consists of a least one (and usually two or even three) 16.8 litre crystal packs connected to a 256 channel spectrometer (e.g. Minty, 1997). Each crystal pack contains four 4.2 litre (10.2 x 10.2 x 40.6 cm3) sodium iodide crystals laid side by side within a protective casing. The gamma ray data are usually collected commensurate with a magnetic survey. Pressure, temperature, ground clearance, barometric altitude and positional information are also measured on a regular basis.

3.1 The effect of aircraft height on the measured count rates

The height of the aircraft significantly influences the count rates recorded and the spatial resolution achieved by an aerial survey. By making several simplifying assumptions (such as flat uniformly radioactive ground, detector response independent of direction etc.) it is a relatively straightforward matter to derive a model of the measurement process (e.g. King, 1912; Grasty, 1979; and see Chapter 3). The analysis reveals that the count rate recorded at an altitude, h, through air with a linear attenuation coefficient, l a , is reduced relative to its strength at ground level by I = E 2 (l a h )

(3)

where E 2 ($ ) is the exponential integral of the second kind (Abramowitz and Stegun, 1972) and gives an approximately exponential decay in count rate with height (Figure 4). The curves shown in Figure 4 were calculated using the linear attenuation coefficient of air at standard temperature and pressure (Table 3). The thorium gamma-ray has a higher energy and hence lower attenuation coefficient causing the count rate to decrease less rapidly with the height of the detector.

A gamma-ray spectrometer cannot be focused like a camera and the signal recorded represents a complex spatial average extending over a relatively large area (theoretically infinite). The proportion of the total signal contributed by a circle of radius r centred below the aircraft has been shown to be (e.g. Grasty, 1979),

23

C(r ) = 1 −

E 2 la h2 + r2 h E 2 (l a h ) h 2 + r2

(4)

An indication of the change in spatial resolution with height can be obtained by calculating the radius of the circle that contributes 90% of the signal (Figure 5). The calculations indicate that the footprint for a radiometric survey is quite large and increases significantly with height. As the thorium gamma-ray is more penetrating than those from potassium or uranium the footprint for thorium is larger at a given height. 1.0

Proportion of counts at ground level

Potassium Uranium Thorium 0.8

0.6

0.4

0.2

0.0 0

50

100

150

200

Height of detector (m)

Figure 4: Reduction in the observed signal with height of detector above the ground. 400 Potassium Uranium Thorium

350

Radius of circle (m)

300 250 200 150 100 50 0 0

50

100

150

200

Height of detector (m)

Figure 5: Radius of circle that contributes 90% of the signal recorded at the detector as a function of detector height.

24

The equations in the previous two paragraphs show that the count rate decreases while the spatial footprint increases with height.

The model used neglected the shape of the detector and the

movement of the aircraft during the integration time. These two factors can have a significant influence on both count rate and footprint and are addressed in Chapter 3.

3.2 Calibration of radiometric surveys

Any type of gamma ray survey, whether on the ground or in the air, requires calibration of the measuring system. This allows the raw count rates to be corrected for background interference and to correct for the complex effect of the detector and material between the source and detector on the gamma ray energy distribution. It also allows surveys flown at different times, heights or with different equipment to be meaningfully compared. Recommendations for calibration are given in IAEA (1991) and Grasty and Minty (1995). For an airborne survey the following four calibration procedures need to be carried out: Ÿ

Cosmic and equipment: The count rate due to cosmic rays increases exponentially with altitude, whereas the aircraft background is constant. By flying calibration flights at a minimum of two altitudes (Grasty and Minty, 1995 recommend 250 m and 2250 m) the relative contributions of these two sources of background can be determined.

The component spectra may be

contaminated by radon but this is not problematic as the contamination is in the form of a radon spectrum and can be removed during radon correction. Ÿ

Radon: A radon spectrum can be determined in any of three ways, (i) repeat flights at a constant altitude of at least 800 m, (ii) repeat measurements over a survey test line and (iii) a single flight at least 800 m above the ground. With the first two, any differences between flights will generally be due to radon. For the third method, the difference between the observed and cosmic plus background spectra is due to radon. Radon is removed from a survey by either using upward looking detectors or the spectral ratio technique of Minty (1992).

To complete the radon

calibration for both techniques, the relative contributions of the ground components of the thorium and uranium windows to the radon window need to be determined. This can be achieved by methodology described in IAEA (1991). Ÿ

Pad calibrations: The count rates in any energy window are contaminated by contributions from the other energy windows. Predominantly, this occurs because Compton scattering in the source, air, aircraft and detector cause many gamma rays to lose energy. To correct for this spectral contamination a calibration is conducted over a minimum of four pads with known concentrations of potassium, uranium and thorium. From this calibration six stripping ratios can be determined as described in Grasty and Minty (1995). Below I will use the terminology Th into U to mean Th counts in the U window per Th count in the Th window, with analogous meanings

25 for expressions such as U into K. There are three forward stripping ratios, α (Th into U), β (Th into K) and γ (U into K) and three reverse stripping ratios a (U into Th), b (K into Th) and g (K into U). The reverse stripping ratios are generally small with the last two components almost always assumed to be zero. Ÿ

Sensitivity and height attenuation: The count rates recorded by an airborne detector for a given ground concentration depend on a number of factors including the size of the detector and the height of the aircraft above the ground. To determine the sensitivity (number of counts per unit concentration) and attenuation (change in count rate with height) a series of flights at different altitudes is generally flown over a calibration range with known ground concentrations of potassium, uranium and thorium. The attenuation for each energy window is determined by fitting an exponential function to the change in count rate with altitude. The sensitivities at a given height are calculated by dividing the count rate in the given energy window by its ground concentration.

3.3 Processing of gamma-ray surveys

Pre-processing of 256 channel spectra Until recently the standard method for processing radiometric data was to convert the raw 256 channel spectra collected during a survey to four channel count rates (potassium, uranium, thorium and total count) using procedures outlined in the IAEA (1991). Recent developments in multichannel processing allow the 256 channel data to be pre-processed before being subjected to this procedure (e.g. Hovgaard, 1997; Hovgaard and Grasty, 1997; Minty, 1996; Minty and McFadden, 1998). These full spectrum techniques will be briefly discussed in section 6.

Dead-time corrections Each time a gamma ray interacts with the detector there is a finite amount of time where additional gamma rays cannot be detected because the system is busy processing the event. For large volume airborne detectors this dead-time can be significant and needs to be corrected. If there are M i raw counts in a channel and the dead time is t d , the dead-time (or live-time) corrected counts, m i , for a t n second integration time are given by M M m i = t − it = t i n d

where t = t n − t d is the live-time during the integration period.

(5)

26

Energy calibration During the course of a gamma-ray survey the relationship between channel number and gamma ray energy can change due to variations in the temperature and pressure of the detector. To account for this, the Th and K peak positions should be determined periodically (generally of the order of every 1000 seconds) to within one-tenth of a channel. The gamma ray energy and channel position are linearly related from the Th peak at 2615 keV down to the about 400 keV (e.g. Grasty and Minty, 1995; Tsoulfanidis, 1995) so the energy of each channel can be determined by linear interpolation.

Cosmic and aircraft background removal A cosmic spectrum expressing the number of counts in each channel per unit count in the cosmic channel as well as an aircraft background spectrum are determined during calibration. The cosmic background is monitored by recording all gamma rays with energies above 3 MeV in a separate cosmic channel. Grasty and Minty (1995) recommend applying a 10 to 20 point filter to the cosmic channel data before subtracting the aircraft and cosmic backgrounds from each observed spectrum.

Radon background removal There are two methods for estimating the contamination of the spectra; upward looking detectors (IAEA, 1991) or the spectral ratio technique (Minty, 1992). Most Australian contractors use the spectral ratio technique which relies on the observation that the ratio of the keV and 1764 keV will be larger for atmospheric

214

Bi photo peaks at 609

214

Bi (from emanation of radon gas), than for

214

Bi

in the soil. This occurs because the lower energy gamma ray is attenuated much more rapidly with height than the lower energy one (half thickness in air of 67 m compared to 112 m; Grasty, 1979).

To calculate the radon correction, four calibration constants are needed. They are cˆ1 , which is the ratio of radon counts in the low energy uranium window (centred about 0.609 MeV) to radon counts in the standard uranium energy window (centred about 1.76 MeV); cˆ2 , ratio of U counts in the low energy to standard uranium windows, cˆ3 , ratio of Th counts in low energy uranium window to the thorium window and cˆ4 , ratio of K counts in low energy uranium window to K counts in the potassium window. Note the use of the ‘hats’ over each calibration constant to distinguish them from the cosmic background. To calculate the number of counts in the radon window, L ob , the background Compton continuum must first be subtracted by linear (or exponential) extrapolation from channels away from the photo peak. The radon background is calculated after removal of the aircraft and cosmic backgrounds, but before spectral stripping. With unstripped potassium, uranium and thorium count rates of K, U ob and Th the radon background is given by (modified from Minty, 1992)

27

rU =

L ob − U ob (cˆ2 − cˆ4 c − cˆ3 a ) − Th(cˆ3 − acˆ2 − (b − ac)cˆ4 ) − Kcˆ4 (cˆ1 − cˆ2 )

(6)

where the a, b, c and a are the stripping constants.

The total background (b i ) in any given channel is a linear summation of the aircraft (a i ), cosmic (c i ) and radon (r i ) contributions: bi = ai + ciC + ri

(7)

where C is the count rate in the cosmic channel.

Reduction to 4 channel count rates Once the background corrections have been applied the data are reduced to standard 4 channel count rates in the following energy windows (IAEA, 1991); Ÿ Potassium:

1370-1570 keV,

Ÿ Uranium:

1660-1860 keV

Ÿ Thorium:

2410-2810 keV

Ÿ Total count:

410-2810 keV

Let J be an index of all channels within the j-th energy window. Then the measured count rate in that energy window, m j , is given by mj = S mi

(8)

icJ

Note, that if the boundary of the energy window falls somewhere within a channel bin, the summation for the extreme left and right channels may include only part of these channels. For the purposes of analysis I will also carry through the individual background corrections, b j = a j + c j C + r j , into the standard three channel processing operations that are now applied.

Calculation of effective height The count rates observed at airborne altitude depend on the density of air between the ground and the detector, which is affected by the pressure and temperature. For surveys flown at constant height, the effect of temporal variations in pressure is quite small (<2% change in assigned concentration of K at 100 m elevation; Grasty and Minty, 1995). Variations in pressure due to changes in altitude have far

28

more effect, with pressure having a negative exponential relationship with height. To correct for temperature and pressure variations the observed radar altitude, h, should be converted to an effective height, h e , at standard temperature and pressure by:

he = h %

273 % P T + 273 1013

(9)

where T is the temperature in degrees Celsius and P is barometric pressure in millibars. The effective height is used to correct the stripping ratios and for height attenuation.

Spectral stripping The count rates in any energy window are contaminated from gamma rays with energies outside the window, w j , that have either had their energies reduced by scattering or effectively increased by the finite energy resolution of the spectrometer. The net count rate in the j-th window is then given by nj = mj − bj − wj

(10)

The contamination is determined by the stripping ratios of the system which comprise three principal stripping ratios a (Th counts in U), b (Th counts in K) and c (U counts in K) and generally one reverse stripping ratio, a (U counts in Th). The other two reverse stripping ratios (i.e. K counts in U and K counts in Th) are negligible and are usually assumed to be zero. Due to scattering in the air the three principle stripping ratios a (Th into U), b (Th into K) and c (U into K) increase with height. The reverse stripping ratio is small and not corrected for variations in height. The net count rates (after stripping and background correction) in the potassium, uranium and thorium windows can then be shown to be given by (e.g. Lovborg and Mose, 1987) n Th = k 1 (m Th − b Th − a(m U − b U ))

(11a)

n U = k 1 (m U − b U − a(m Th − b Th ))

(11b)

n K = (m K − b K − k 1 [k 2 (m U − b U ) + k 3 (m Th − b Th )])

(11c)

where k 1 , k 2 and k 3 are derived from the stripping constants of the spectrometer:

k1 =

1 ; k 2 = c − ab; k 3 = b − ac; (1 − aa )

(12)

29

This form of the spectral stripping equations facilitates the error propagation analysis conducted in Chapter 4.

Height correction An attenuation coefficient for each energy window, l j , is determined during calibration and models the approximately exponential decrease in stripped count rates with heights. Given a stripped count rate of n j (h e ), at an effective height h e , the corrected count rate, n j (h o ), at the nominal survey altitude, h o , is given by: n j (h o ) = n j (h e ) exp(−l j (h o − h e ))

(13)

Conversion to ground concentrations The sensitivity of the spectrometer at the nominal survey altitude, s j , can be used to convert the dead-time, background, stripped and height corrected count rate, n j (h o ), to ground concentration of potassium uranium or thorium, f j , by

fj =

n j (h o ) sj

(14)

The concentrations are generally reported in %K, ppm eU and ppm eTh.

Levelling of line data Changes in soil moisture, vegetation thickness and the radon content of rocks and soils cause unwanted variation in the radiation both along- and across- lines. This can cause noticeable banding parallel to the acquisition lines. There are several techniques commonly used to remedy these errors including tie-line levelling, use of between channel correlation information (Green, 1987) and micro-levelling (Minty, 1991). However, Craig et al. (1998) found serious deficiencies with some conventionally levelled surveys (valid signal along with noise was removed) and suggested an alternative approach. The alternative uses a moving average of the line means to derive a correction for each line.

Interpolation to a regular grid After the above processing steps, estimates of the potassium, uranium and thorium concentrations and the total count activity are available at each of the measurement points. As the pilot inevitably drifts off the intended path and the data sampling is usually different in the along- and across-line directions, the data are then interpolated to a regular grid. General practice is to interpolate to a grid

30

spacing of one-quarter to one-fifth of the line spacing using one of several commercially available interpolation packages. The interpolation aids both visualisation and interpretation and also enables manipulation within widely available image processing and GIS software. Recently this image manipulation has extended to deconvolution, which is a procedure that attempts to remove the blurring imposed by the aircraft altitude (Craig et al., 1998).

4 SOIL MAPPING USING AIRBORNE GAMMA-RAY SPECTROMETRY

Now that I have reviewed the fundamentals of airborne gamma-ray spectrometry I turn to the issue of applying the technique to soil mapping.

4.1 Existing techniques for soil mapping

Until recently, Australian approaches to soil survey have been mostly qualitative.

Existing

information on geology and terrain are used to supplement air-photo interpretation and field investigation to create a categorical soil map (e.g. Gunn et al., 1988). The soil mapping units are portions of the landscape that are as homogeneous as possible in terms of the soil characteristics that are relevant to the purpose and scale of a survey. Soil mapping is not the delineation of the spatial extent of soil taxonomic classes which are groupings and labelling of soils based on the properties of a soil profile (e.g. Northcote, 1979; Isbell, 1996). Ideally, soil mapping units are comprised of one soil-type that recurs in a predictable pattern throughout the landscape. In some cases two or more different kinds of soil may occur together in such intricate patterns that separating them would take undue effort. In that case the soil mapping unit is a soil complex. The soil mapping units should be taken from the country in the sense of Butler (1980), whereby a representative range of soils within the survey area is examined and a classification scheme is constructed based around the properties of these soils. It is only at a later stage of the survey that the soil mapping units are related to standard classification schemes.

Existing techniques for soil survey rely heavily on air-photo interpretation, or in some instances analysis of satellite imagery. On aerial photography, soil landscapes with different sets of attributes are discriminated by differences in colour, tone and texture. However, only the surface of the soil is directly visible and then only when the vegetation is sparse or absent. Therefore, differences in soil properties are generally inferred indirectly from land form and patterns of vegetation. Land forms can be identified on aerial-photographs by their structure, relief, and drainage pattern under stereo viewing and reflect the underlying rocks or materials, the history of weathering, erosion and deposition. When combined with geological information they indicate the nature and mode of soil

31

formation. In undisturbed landscapes the local distribution of native vegetation often correlates closely with land form elements and soil properties, particularly with respect to water supply (Gunn et al., 1988). Where patterns of vegetation are associated with soil properties in a consistent manner, the vegetation can be used to identify different soil mapping units. For areas with little topographic variation, vegetation is often the most important factor reflecting differences in soil properties. Undisturbed native vegetation provides the best discrimination because in disturbed areas, the vegetation may reflect human impact.

A disadvantage of air-photography and satellite imagery is that some cultural and natural effects (such as burn off patterns, different land use histories, different crops etc.) can obscure variations caused by soil changes. Soil mapping is then very difficult, especially in disturbed areas with low relief, as occurs in the majority of Australia’s agricultural land. Airborne radiometrics maps the distribution of potassium, uranium and thorium (and also the total radioactivity) in the top 30 cm or so of soil. The distribution of these elements in the landscape is controlled by many of the same factors important in determining the distribution of soils. This suggests that radiometrics may provide an effective method for stratification of field sampling, and for spatial prediction of soil properties, especially in areas of low relief.

4.2 Soil information required for land evaluation

The dominant paradigm for land evaluation has been based on transfer by analogy (Nix, 1968). All occurrences of a particular class of land mapped during a survey are assumed to respond similarly under similar use.

The success of transfer by analogy depends on the classification scheme

developed and on the how well the parameters used to define the initial experimental outcome were controlled. However, there are significant difficulties in developing effective classification schemes that are appropriate for many users in diverse environments (e.g. McKenzie and Austin, 1993).

An alternative paradigm is process modelling whereby land performance (in terms of productivity, need for fertiliser inputs etc.) is predicted using a simulation model with individual land characteristics as inputs (e.g. Bouma, 1989).

A major requirement for process modelling is

parametric soil survey whereby predictions of individual soil characteristics, rather than soil types, are required.

Providing predictions of individual soil properties allows the requirements for

information of diverse groups of users to be accommodated. For example, a soil survey may be used by farmers for determining irrigation potential and crop suitability; by engineers for planning of buildings and roads; and by land managers for determining appropriate changes in management and regulation.

32

4.3 Parametric soil survey

One method for parametric survey is to assume that the sampled profiles in a soil mapping unit are representative of the properties over the whole unit. However, soil mapping units are generally heterogeneous. Additionally, the assumption that soil properties vary in concert is usually not explicitly tested (Butler, 1980). It may apply in some landscapes but not in others (Webster and Butler, 1976). Furthermore, the specification of soil mapping units is usually based on subjective decisions, and in reports it is often difficult to distinguish between interpretation and evidence (Austin and McKenzie, 1988). The results of the survey may not be amenable to rigorous checking, repetition or rational criticism by other workers, unless the survey is repeated.

The above disadvantages can be overcome by using quantitative methods that can be rationally criticised, checked or extended. When there is a high density of observations, a suitable, objective quantitative method is a surface fitting technique, such as kriging, applied directly to the sampled soil properties. Kriging uses a model of the spatial correlation between individual soil properties to interpolate between observations (e.g. Webster, 1985; McBratney and Webster, 1986).

The

technique can work well when there are few major discontinuities, and has been extended to include correlation with more readily observed environmental variables through co-kriging (e.g. Stein et al., 1988; Odeh et al., 1991).

A second quantitative approach, more applicable to larger scale surveys with a lower density of observations, is based on environmental correlation (McKenzie and Austin, 1993).

Statistical

analysis is used to develop statistical models between soil properties and more readily observed environmental variables important to pedogenesis. Many of these variables are related to terrain and can be derived efficiently and objectively from digital elevation models (e.g. Gessler et al., 1995).

Good quality information on terrain is difficult to obtain over areas with little topographic variation, as is typical of the majority of Australia’s agricultural land. Further, in these areas the correlation between soil properties and landscape features may be more difficult to quantify. In such situations, an alternative environmental variable is required that is both important in pedologic development and is relatively easy to record. Airborne radiometrics may provide such an alternative variable, as the distribution of potassium, uranium and thorium in the soil, are controlled by factors important in soil development. Further, the information can be obtained remotely and provided in a digital image to facilitate analysis.

33

In summary, current techniques for soil mapping may be ineffective in areas with little topographic variation, which are representative of the majority of agricultural land in Australia.

Airborne

radiometrics may overcome these deficiencies in two respects. 1. Provide a stratification of the landscape for map production and sample design of field surveys. 2. Act as an environmental variable in areas with low relief.

4.4 Previous soil mapping using gamma-ray spectrometry

Airborne radiometrics may have been applied to soil mapping over 20 years ago as indicated by Russian data on the correlation between radiometric signals and Great Soil Group (Kogan et al, 1969). The technique appears to have first been used in Australia around 1989 by Brian Tunstall and Alan Marks (Marks, pers. comm.) who attempted to map an area based on interpretation of reconnaissance grade (1600 m line spacing) radiometric data.

Tunstall and co-workers (including Tunstall et al., 1998, Gourlay et al., 1996) generally use a classification of radiometric data to stratify field sampling. They advocate using only radiometrics for stratification so that individual data layers (such as soils, terrain, vegetation etc.) are independent. Tunstall considers that the independence of data layers is important when analysing data-sets on multiple attributes. To apply the technique, individual classes are randomly sampled and statistical analysis is used to combine the field sampling with the classification. Categorical data are converted to numerical form and simple averaging is used to determine the average value of each soil property in each class. Predictions of individual soil properties are provided by assuming that all occurrences of a class have the mean value of the property for that class.

Cook et al., (1996) used airborne radiometrics to map the spatial distribution of soil material within a dissected lateritic landscape in Western Australia. They found that it could successfully discriminate between granitic outcrops, ferricrete and associated material, depositional (or partially eroded) material derived from granite and residual sands. However, they could not identify doleritic dykes in the area which were too narrow to be detected at airborne altitudes. Cook et al., (1996) concluded that the technique had considerable promise for soil mapping, but mostly in conjunction with additional information such as topography and other remotely sensed data. The specific weaknesses they identified were (i) a low detected radiometric signal may represent either a material with low concentrations of isotopes, or a strongly emitting material that has the signal attenuated by overlying material or water; (ii) the potassium signal relates to total K and not plant available K and (iii) radiometrics provides limited direct information on pedological alteration.

34

Bierwirth (1996) conducted an extensive study of the Wagga Wagga area of New South Wales with the aim of investigating the usefulness of airborne radiometrics for mapping soil properties. Bierwirth found that radiometrics were a valuable tool for soil survey that generally needed to be combined with topographic information and traditional soil survey to realise their true potential. He found the radiometric images could be used to map several properties that were unable to be mapped by other remotely sensed data. The relationship of radiometric signal and soil properties were found to depend on geology and geomorphic history. In erosional areas the signal related to parent material and weathering; in residual soils the potassium appeared to relate to soil pH, and thorium to clay content and type; in alluvial areas all isotopes were thought to be associated with clay or silt particles and therefore reflect texture and related properties.

In a related application, considerable success has been achieved in using airborne radiometrics, combined with Landsat TM, for regolith mapping (e.g. Wilford, 1992, 1995; Wilford et al., 1997). Potassium, uranium and thorium concentrations were found to respond differently to different styles of in-situ weathering, reflecting underlying lithology, time and geomorphic processes.

Active

depositional environments with minimal weathering were distinguishable from less active environments where pedogenic processes had modified the radio-element chemistry. The relative rates of regolith formation and removal could be mapped, and in places, the degree of regolith development could be determined. Interpretations were best made within major geological groups since the relationship between gamma-ray emission and weathering styles was often specific to particular bedrock types. Lastly, Wilford (1995) noted the potential of radiometrics for mapping nutrient status, as mature, highly weathered soils identified by the radiometrics, had low exchangeable cation and organic matter content.

The literature reviewed above indicates that the same radiometric signal can occur in different lithologies that are subjected to different styles of weathering and erosion.

Therefore, any

investigation of statistical relationships between soil properties and radiometrics should occur within areas with similar geomorphic history. This suggests that parametric soil survey should first proceed by partitioning the landscape into areas with similar parent material and styles of weathering/erosion. Only then could radiometrics be used to extrapolate soil properties.

A suggested methodology for soil survey is (generalised from that given in McKenzie and Austin, 1993). Ÿ

Initial stratification of landscape into areas with distinct geomorphic and pedogenic processes by reference to the radiometric data supplemented by aerial photographs and/or satellite imagery and

35

by any additional data available on geology, topography etc. Preliminary field sampling may also be used to assist in the stratification. Ÿ

Field sampling either by transects across areas with the steepest environmental gradients (e.g. McKenzie and Austin, 1993), and/or by random sampling within the classes defined by the provisional stratification.

Ÿ

Exploratory data analysis and refinement of initial stratification.

Ÿ

Development of statistical models between soil properties and environmental variables (radiometrics, topography etc.) within each class in the stratification.

Ÿ

Spatial extrapolation of properties using the statistical models.

Ÿ

Validation and testing of the accuracy and reliability of the map products.

4.5 Discussion

The studies reviewed above have had mixed success in applying radiometrics to soil mapping. This variable success is due to applications in different environments, using different methodology and with mixed quality radiometric data. In particular, none of the studies had the advantage of using the recently developed techniques for multichannel processing (Minty, 1996; Hovgaard, 1997; Hovgaard and Grasty, 1997). Many were conducted without a full appreciation of the geophysical principles behind radiometrics, which provide insights into both the limitations of the method and the most effective strategies for the intended purpose. Before delving further into these issues, I will present the results of a validation exercise in the Jemalong-Wyldes Plains of New South Wales. This involved testing the predictions of a parametric soil map developed by Gourlay et al. (1996) from interpretation of 400 m line spacing radiometric data. The failure of the map to reliably predict soil properties forms a convenient starting point for a discussion of the importance of geophysical issues in soil mapping.

5 PROBLEMS WITH CURRENT MAPPING TECHNIQUES: A CASE STUDY

The Jemalong-Wyldes Plains are an area of extensive alluvial deposition within the Lachlan Catchment, NSW, Australia.

Irrigation, with water diverted from the Lachlan River, has been

conducted in the area for almost 50 years. However, in the early nineties it was discovered that there was a rising ground water mound threatening nearby Lake Cowal with salinisation. The lake is ephemeral, an important bird habitat and covers parts of several dry land farms. In response to this threat, it was decided to develop a Land and Water Management Plan for the Jemalong-Wyldes Plains area.

The object of the plan was to assess the impact of irrigation practices on the

groundwater mound and develop appropriate management actions to eliminate/minimise the problem.

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Central to the success of the plan was good quality information on soils, as soil characteristics largely determine the amount of water that enters the water table. Existing soils information for the area was limited to a geomorphologic map (Kelley, 1971) and a 1:250,000 soil landscape map under development (King, 1998). The flatness of the terrain and lack of obvious geological patterns made application of existing techniques difficult. It was therefore decided to commission a soil survey (Gourlay et al., 1996) based on interpretation of airborne radiometric data that had recently been flown over the area by AGSO. However, after the soil survey was completed there were serious questions raised regarding the validity and accuracy of the soil property maps (McGowan, 1996). Partly, this concern was due to the survey being the first detailed application of radiometrics over a large area and partly to questionable methodology.

It was decided to test the map in an area of approximately 120 km2 felt to be representative of the soils expected in the irrigated parts of the Jemalong-Wyldes Plains. Its location was motivated by a detailed radiometric survey flown by AGSO over an area covered by an AIRSAR (airborne synthetic aperture radar) image acquired in 1993. The site was some 50 km east of Forbes along the Lachlan Valley Way, near the small settlement of Warroo (see Figure 1 in Supplement 1).

The validation exercise formed the basis of an unpublished report (Billings and Turner, 1997) commissioned by the Jemalong-Wyldes Plains Steering Committee. The report is reproduced as Supplement 1 and describes the sampling strategy, field methodology, analysis of soil samples and statistical analysis of the data. Here, I provide a brief summary of the methodology and the results.

5.1 Soil sampling stratified by airborne radiometrics

The soil sampling for the validation exercise was conducted within an area of 120 km2 that had been flown by AGSO in December 1995 (see Figure 1 in Supplement 1) at 100 m line spacing and 60 m elevation. At the time that the fieldwork program was being devised the only radiometric processing available was the standard 4-channel technique as recommended by the IAEA (1991). Additionally, there were problems in the format of the survey data which prevented the survey being levelled correctly. These two factors, coupled with a small variation in uranium concentration across the survey area meant that the uranium band had very little spatially coherent signal and was severely contaminated by noise (Figure 6b). Therefore, the uranium band was not used in the design of the field survey.

A classification (see Supplement 1) of the 100 m line spacing 60 m elevation

potassium, thorium and total count bands (Figure 6) was used to provide the basis of a stratified random sampling.

The total count band is not generally used for geological or soil mapping

37

Potassium

Uranium

Total count

Figure 6: Standard 4 channel processing of the 100 m elevation potassium, uranium, thorium and total count data, with a normalised stretch applied to the data. No levelling has been applied to the line data.

38

applications. However, it can provide very useful information on the distribution of soils, partly because it has such a high signal to noise ratio compared to the other bands.

5.2 Summary of results

Gourlay et al. (1996) derived their soil map by first classifying radiometric data acquired by the Australian Geological Survey Organisation in 1993 at 100 m elevation with a 400 m flight line spacing. The classification was based on an iterative procedure that used information on both the spatial and spectral relationships between classes. Each of the classes in their resultant 22 class radiometric map were sampled at several different locations (ranging from 4 to 11 replicates in each class) with a total of 176 profile observations. Simple averaging was then used to determine the soil properties in each class. From this soil map, several maps of soil properties were produced (pH, texture and salinity) by assuming that every occurrence of the class had the mean value of the soil property under consideration.

For the 61 sites we sampled (see Supplement 1) statistical analysis was used to test the predictions of the Gourlay et al. (1996) maps of A1 and B2 horizon texture (converted to numerical form) and pH, and the salinity of the B2 horizon. Two analyses were performed for each soil property under consideration. The first was a direct regression of the Gourlay et al. predicted property against our measured property and tests the reliability of their predictions.

The sampling and synthesis

techniques used by Gourlay et al. assume that a one to one relationship exists between radiometric class and soil properties. Over the area of the whole survey this assumption may be invalid, but within a smaller area it may apply. The second analysis (an analysis of variance) was designed to test whether the Gourlay et al. (1996) classes could be used to group our measured soil properties in a consistent manner. Because our survey covers a relatively small proportion of the total area, this would show whether the Gourlay et al. classes are able to discriminate local soil variation.

The measured and predicted texture and pH for the A1 and B2 horizons and salinity in the B2 are given in Supplement 1. The correlation between measured and predicted values for all five properties is extremely poor (refer Figure 3 in Supplement 1). When I regressed the predicted soil properties against the measured properties, the maximum R2 for any property was less than 0.03. A perfect fit between properties would have R2=1, a very good fit would have say R2> 0.8, and a poor fit would have R2<0.5. Gourlay et al.’s predictions thus have an very poor fit to the measured soil properties.

For the texture of the A1 and B2 I analysed how often Gourlay et al. (1996) predicted the correct texture class. I found that for the A1 the prediction was correct at 20% (12/61) of sites compared to

39

38% (23/61) for the B2. The higher success rate for the texture of the B2 may reflect that 79% of Gourlay et al.’s predictions and 50% of our measurements had a light medium clay texture (texture class = 14). This appears to be the dominant B horizon texture encountered in the Jemalong-Wyldes Plains (King, 1998).

The results indicate that the Gourlay et al. radiometric classes cannot be used to extrapolate texture, pH and salinity within the area analysed. This could be due to many (or all) of the Gourlay et al. soil classes being composed of one or more soil types. Therefore, the average soil properties within a class may not relate very well to the actual soil properties measured in that class. Even though the Gourlay et al. maps are unable to reliably predict soil properties over the whole survey area they may possibly be used to discriminate local differences in soils.

To assess this possibility I performed an analysis of variance on Gourlay et al’s. radiometric classes (as factors) against our measured soil properties. As our survey area represents a small proportion of the total survey area, this analysis would show whether radiometric classes have relatively consistent soil properties in a local area. No significant differences were found between the Gourlay et al. radiometric classes for the pH of the A1 and B2 horizons and the texture and salinity of the B2 horizon. The texture of the A1 was the only soil property with a statistically significant (at the 5% significance level) relationship with Gourlay et al. class. However, the variation within each class was relatively large with an adjusted R2 = 0.16. Therefore, even for the texture of the A1 the local soil discrimination of the Gourlay et al. classes is relatively poor.

Gourlay et al. (1996) were unable to effectively map soil properties due to a number of reasons. Several relate to relatively poor survey design and analysis, such as the location of the majority of sample sites along road verges, a simplistic treatment of the soil profile and the use of a linear scale to calculate the average texture of each class (see Supplement 1). However, several reasons illustrate some of the difficulties that one may encounter when using airborne radiometrics to map soil, as discussed in the next section.

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6 GEOPHYSICAL CONSIDERATIONS

I turn now to the geophysical issues that influence the suitability of radiometric data for soil mapping applications.

6.1 Improving the low signal-to-noise ratio

Radiometric data processed using the standard 4-channel method, can give very noisy estimates of potassium, uranium and thorium concentrations. For example, the uranium band for the 60 m elevation Jemalong survey is so noisy that there is little obvious spatially coherent signal in the processed image (Figure 6b). The thorium image is slightly better (Figure 6c), but when used to construct a classification, the noise would cause many pixels to be assigned to incorrect classes. Additionally, efforts to extrapolate soil properties by environmental correlation would be hindered.

Recently, multichannel processing techniques have been developed that can exploit the information contained in (virtually) the entire 256-channel spectrum. These include, noise adjusted singular value decomposition (NASVD; Hovgaard, 1997; Grasty and Minty, 1997; Minty and McFadden, 1998), statistical noise reduction (SNR; Green et al., 1988; Dickson and Taylor, 1998), spectral components (Minty, 1996) and Spectra-Plus (see Minty, 1998). The NASVD and SNR algorithms are similar to a principal component transformation as they decompose the spectra into components, except the ordering is based on signal rather than variance. The two techniques differ in the way that signal is separated from noise.

The “cleaned” spectra are then processed using the standard 4-channel

technique. In contrast, the spectral components and Spectra-Plus algorithms fit component spectra directly to the raw 256-channel data. They differ in the way the component spectra are calculated. The spectral components technique uses six analytically determined K, U and Th spectra. For each radioelement there is one base spectra and one modelling variation with height. The Spectra-Plus technique uses height dependent spectra calculated directly from a Monte Carlo model.

For the Jemalong survey, the standard 4-channel processed data (without any levelling applied to the line data), interpolated to a 25 m cell size are shown in Figure 6. The potassium and total count bands look reasonable, while the thorium and, especially, uranium bands look very noisy. To investigate the improvement that can be achieved by a multichannel technique, Hovgaard applied his NASVD technique to the survey. The first four spectral components along with the mean spectra were used to construct cleaned spectra for the survey. These were then processed using exactly the same parameters as the standard 4-channel method and the unlevelled data interpolated to the same 25 m grid using the same method (Figure 7; Table 4). The visual improvement in the data is

41

Potassium

Uranium

Pot

Figure 7: NASVD processing of the 100 m elevation potassium, uranium, thorium and total count data. No levelling was applied to the line data. The colour lookup tables are identical to this used in Figure 6.

42

substantial, especially for uranium. Additionally, the numerous levelling errors apparent in the original total count band are eliminated, while there is a considerable reduction in noise for thorium. The effect on potassium is quite small, which is expected as it (usually) has a low error when processed using the standard 4-channel method. The improvement is more than just visual, as the statistical summary of the line data in Table 4 indicates.

Isotope

%K ppm eU ppm eTh TC (cps)

Min. 0.57 -1.22 4.37 1282

Raw Spectra Max. Mean 1.86 1.15 5.45 1.85 17.3 10.4 2631 1999

Std. dev. 0.16 0.82 1.58 147

Min. 0.54 0.36 4 1327

NASVD cleaned spectra Max. Mean Std. dev. 1.84 1.15 0.15 2.76 1.87 0.19 13.9 10.4 1.02 2604 2009 136

Table 4: Mean, minimum, maximum and standard deviations for the line data processed using standard 4-channel techniques on the raw and NASVD cleaned spectra. 6.2 Assay uncertainty and spatial resolution

Radiometric data are collected along lines that can be a substantial distance apart. Coverage of the regions between lines is achieved because radiometrics is a smoothing process that integrates a large region beneath the aircraft. The size of the measurement footprint is determined by the aircraft height and movement during the integration time.

Current methods (e.g. Grasty, 1979) for

calculating the footprint (and removing its influence, e.g. Craig et al., 1998) usually neglect the shape of the detector (Grasty, 1975 and Tewari and Raghuwanshi, 1987 are exceptions) and the aircraft movement. We therefore come to the first objective of the thesis which is addressed in Chapter 3.

Objective 1: Develop a model of gamma-ray spectrometry that accounts for the shape of the detector and the movement of the aircraft during the integration time.

The assay uncertainty of a survey (ratio of standard deviation to mean value) is important to quantify as it influences the suitability of the data for the intended purpose. For example, the uranium channel in the radiometric data interpreted by Gourlay et al. (1996) had a very low uranium content (average ~ 2 ppm eU) and large error (standard deviation 1.15 ppm eU), and hence a large uncertainty. In such cases error in the uranium concentration may cause many pixels to be assigned to the wrong class, degrading the quality of soil property maps derived from the classification. Lovborg and Mose (1987) present an error propagation analysis for collection of radiometric data using a portable

43

spectrometer and estimate assay uncertainty under a range of different conditions. The generalisation of this technique to airborne surveys is the second objective and is addressed in Chapter 4.

Objective 2: Use error propagation to determine the expected assay uncertainty for a given survey.

Cook et al. (1996) were unable to map some narrow doleritic dykes from interpretation of radiometric data. Gourlay et al. (1996) used a 50 m pixel size for their radiometric images which were obtained from data collected with lines 400 m apart. They claim that it could map soil variations at scales of less than 100 m, which may have been possible along-lines but almost certainly not across-lines. Current estimates of the spatial resolution involve calculating the footprint of a single measurement (e.g. Pitkan and Duval, 1980). However, the size of the footprint is difficult to quantify (it is theoretically infinite) and only gives an indication of the size of the area that contributes to the signal, not the spatial resolution that can be achieved. This leads to our third objective which is addressed in Chapter 4.

Objective 3: Develop a method for estimating the spatial resolution for potassium, uranium and thorium for a given survey.

6.3 Interpolation of data to a regular grid

During the course of a survey, the aircraft speed fluctuates slightly and the pilot drifts off course, causing the separation between measurements to be irregular. A radiometric image is obtained from interpolation of the line data usually by some type of local operator (nearest neighbour; bi-cubic spline etc.) followed by application of a smoothing kernel (Briggs, 1974; minimum curvature). The pixel size in the grid, usually one-quarter to one-fifth of the line spacing, can exert a strong influence on the final result. For example, radiometric data collected at 400 m line-spacing radiometric data would usually be gridded with an 80 m pixel. Therefore, in the direction of the lines, the pixel size is slightly larger than the acquisition spacing (60-70 m), while across-lines there are roughly 5 pixels between each measurement point.

In such instances, the reliability of the assigned potassium,

uranium and thorium concentrations may be suspect in the regions furthest from any measurement points (maximum ~200 m), although radiometrics is a smoothing process so that rapid changes in count rates would not be expected. However, there is considerable dissatisfaction in the quality of current gridding algorithms (e.g. Cordell, 1992; Billings and Fitzgerald, 1998).

Alternative interpolation methods are required that honour the original data, are independent of the grid spacing and which allow greater control in specifying how the surface changes between grid

44

points. Methods based on radial basis functions (Powell, 1992), tension splines (Mitasova and Mitas, 1993) and kriging (Matheron, 1980; Cressie, 1993) are promising in this regard.

They are

encompassed within a common mathematical formulation which I term Arbitrary Basis Functions (ABFs). Currently, application of these techniques to large geophysical problems is infeasible because the algorithms required to fit the surfaces have extremely large computational and memory overheads. Addressing these issues is the subject of Chapter 5 and leads to the fourth objective.

Objective 4: Implement alternative methods for interpolating the line data to a regular grid that honour the original data, are independent of the cell size and can be applied to large datasets.

Even with multichannel processing techniques, radiometric data can still be very noisy (especially for uranium) and an exactly interpolating surface may not be appropriate. Ideally the suite of techniques should include a facility for fitting smooth surfaces to the data (fifth objective, Chapter 5).

Objective 5: Implement alternative methods for interpolating the line data to a regular grid that smoothly fit the original data , are independent of the cell size and can be applied to large datasets.

6.4 Compensation for the smoothing effect of height

Radiometric data gives a smoothed version of the variation of radioisotopes at ground level, with the extent of the smoothing determined by the aircraft height and the distance travelled during the integration time. For example, at 100 m elevation, the 90% contributing area for thorium is ~20 ha (calculated by Equation 4). Therefore, each measurement is a complex integration of a relatively large region beneath the aircraft. Parametric soil survey requires removal, as much as possible, of the smoothing effects of aircraft height and movement by deconvolution. This has already been achieved by Craig et al. (1998) through Wiener filtering of radiometric data defined on a regular grid. The model used by Craig et al. (1998) neglected the shape of the detector and the aircraft movement during the integration time, and was applied to a gridded version of the data. Implementation of a more rigorous method for deconvolution is the sixth objective with is addressed in Chapters 6 and 7.

Objective 6: Implement a method for direct deconvolution of the line data using a model that accounts for detector shape and aircraft movement.

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7 DISCUSSION In this chapter we have reviewed the fundamentals of airborne gamma-ray spectrometry and discussed their implications for soil mapping. Analysis of the literature established that radiometrics is a promising technique for soil mapping.

The data are usually delivered to soil mapping

practitioners as four images; potassium, uranium, thorium and total count. For these radiometric images to be properly utilised, a good understanding of the way they are created is required. The burden is on the geophysicist to ensure that they deliver the highest quality radiometric images given the constraints of the data. These include the irregular sample spacing, the often high noise levels and the blurring of spatial detail caused by the aircraft height and practicalities of gamma-ray detection.

The remainder of this thesis is dedicated to addressing these geophysical issues.

From the

perspective of a soil scientist, the work clarifies the constraints on map precision and resolution imposed by the radiometric data. From the perspective of a geophysicist, it addresses the central issue of creating images of radioelement concentration from noisy, blurred and irregularly sampled data.

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CHAPTER 3

A MODEL OF AIRBORNE GAMMA-RAY SPECTROMETRY1 As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein

CHAPTER OBJECTIVES Develop a model of gamma-ray spectrometry that accounts for the shape of the detector and the movement of the aircraft during the integration time.

1 INTRODUCTION

Each radiometric measurement represents a complex average of a relatively large region (the contributing area), areas closer to the aircraft contributing more than those further away. Hence, individual features on the ground appear blurred in the aircraft image. This blurring of spatial detail can be partially removed by deconvolution, the quality of the result depending on the noise level within the data, the sampling density and the mathematical model describing the smoothing process (the point spread function, PSF). Gunn (1978), Craig (1993) and Craig et al. (1998) present a PSF for deconvolving airborne radiometric data that neglects aircraft movement and assumes that the detector response is non-directional. However, during a one-second measurement, the aircraft may move about 70 m, which can significantly influence the contributing area given that aircraft altitudes generally lie between 50 and 120 m. In addition, the rectangular shape of modern detectors leads to significant differences in detector sensitivity depending on the angle of incidence of the gamma ray. In this chapter, I derive a model for the detector based on geometrical arguments, and show how to include the aircraft movement in a simple way. The application of the model to both resolution analysis and deconvolution is addressed in Chapters 4 and 7, respectively.

Gamma rays interact with matter through three principal mechanisms: the photoelectric effect, Compton scattering and pair production. For the gamma ray energies of most practical interest in 1

The material presented in this chapter is to be published, as Billings and Hovgaard, 1998. Modelling detector response in airborne gamma-ray spectrometry: Geophysics (accepted).

47

airborne radiometrics, the Compton effect dominates the attenuation (e.g. Grasty, 1979; and section 2, Chapter 2). In the Compton effect a gamma ray collides with an electron, losing some of its energy and deflecting at an angle to its original trajectory. This scattering occurs in the earth, in the column of air between source and receiver, and in the detector itself. Several authors (Spencer and Fano, 1951; Fano, 1953; Gregory, 1960; Davis and Reinhart, 1957, 1962) have modelled this process by using a two- component system of primary unscattered radiation linked with a “build-up” factor to account for scattered radiation. The build-up factor has a polynomial form, with the coefficients determined by experiment. The main disadvantage of this method is that the coefficients are valid only for the source/detector geometry used in the experiment.

A comprehensive solution of the full gamma ray transport problem is complex. Many gamma ray energies from the U and Th decay series have to be included, and each has its own characteristic emission intensities and attenuation coefficients. Beck and de Planque (1969) and Kirkegaard (1972) solved the Boltzmann transport equation for a uniformly radioactive absorbing and attenuating earth overlain by a layer of non-radioactive air. Their calculations give the gamma ray fluence rate at the detector and need to be convolved with the detector response function in order to simulate the observed spectrum. This detector response function is extremely complicated, and according to Grasty (1979), can be determined only by a combination of Monte Carlo simulations and experiments.

The main disadvantage of any Monte Carlo based method is the amount of computational effort required. For example Monte Carlo simulations, presented later in the chapter, are used to model the count rates in each energy window on a 40 % 40 grid. For each grid point, 10 000 gamma rays need to be traced, making a total of 16 000 000 tracings for the whole grid. In contrast, simulating a gamma ray survey by modelling only unscattered radiation (a mono-energetic approach) would require only a few multiplications, divisions and function evaluations for each grid cell. The mono-energetic approach is around five orders of magnitude faster than Monte Carlo and hence more suitable for routine processing. However, at present it cannot deliver very high accuracy due to the difficulty of incorporating the detector response. This deficiency is addressed in this chapter.

The chapter is organised into six sections:

1. A mono-energetic model of gamma ray transport is presented and several results obtained by earlier authors are derived within this formulation.

2. A geometrically derived model for the angular variation in sensitivity of a rectangular detector is developed and compared with previously collected experimental data (Tewari and Raghuwanshi, 1987).

48

3. The point spread function of airborne gamma ray spectrometry is presented. 4. The model is compared to Monte Carlo simulations. 5. An analytical approximation to the geometrical detector model is derived and contributing area calculations are made.

6. Lastly I examine how to include the effect of the aircraft velocity. Symbol (x, y, z ) (q, w ) r t , r a and r e r h h w l a , l e and l d n de A and e (a, b, c ) W e T and e P g I p m Capital &

Meaning Cartesian co-ordinates. Plane polar co-ordinates. Total distance, distance in air and distance in earth. Distance from detector to surface point directly above source. Height of detector above the ground. Angle in a vertical plane relative to (0, 0, 1 ). Angle in a horizontal plane relative to (1, 0, 0 ). Linear attenuation coefficients in air, earth and detector. Number of γ-rays emitted per unit second per unit volume. Detector sensitivity. Area and efficiency of a spherical detector. Dimensions of a rectangular detector. Solid angle. Total and peak efficiencies. Detector photo-fraction (ratio of peak-to-total efficiency). Gamma rays per second recorded by detector Point spread function of a stationary detector. Point spread function of a moving detector. Indicates non-dimensional co-ordinates (except l h G). Indicates Fourier transformation.

Table 1: Table of mathematical symbols.

2 MODELLING THE GAMMA RAY SIGNAL

Modelling gamma ray detection by a mono-energetic approach considers only primary unscattered radiation. This approach compares favourably to the processing guidelines of the IAEA (1991) and the more recent Australian guidelines of Grasty and Minty (1995). Although 256-channel spectra are initially collected these are cleaned by the removal of cosmic, atmospheric radon and aircraft background spectra and then converted to count rates in the standard energy windows (section 3.3, Chapter 2). The processing guidelines are comparable to a mono-energetic approach because they are designed to achieve the most accurate estimates of the count rates due to primary unscattered radiation arising from within the Earth’s near surface.

49

Z X Y (0,0,0)

Figure 1: Geometry for setting up the mathematical model.

2.1 A general mono-energetic model

With the geometrical orientation of Figure 1, and the mathematical symbols listed in Table 1, a general mono-energetic model of the gamma ray intensity, I, recorded by a detector at height h is I = ° dx ° dy ° dz n d e exp(−l a r a ) exp(−J ) / 4o r 2t , − º < x, y < º; − º < z < s(x, y )

(1)

where the factors in Equation (1) have the following interpretation: Ÿ

n = n(x, y, z ) is the number of gamma rays emitted per second per unit volume at the point (x, y, z )

Ÿ

d e = d e (x, y, z, h ) is the detector sensitivity for a detector at height h, for gamma rays emitted at

the point (x, y, z ). Ÿ

r t = r t (x, y, z, h ), r a = r a (x, y, z, h ) and r e = r e (x, y, z, h ) are, respectively, the total distance, the

distance in the air and the distance in the earth that the gamma rays traverse between source and detector. Ÿ

l a is the linear attenuation coefficient of the air. It depends on the air pressure and temperature

but can be considered constant over the altitude of a radiometric survey. The factor exp(−l a r a ),

50

gives the proportion of gamma rays that have not undergone scattering in the column of air between the source and detector. Ÿ

J = J(x, y, z, h ) = ° l e (t )dt, 0 [ Èt È [ r e , where l e is the linear attenuation coefficient within the

earth which may not be constant. The integral is along the straight line connecting the source to the detector and is truncated where this line intersects the Earth’s surface (Figure 1). If the attenuation is assumed constant with depth and varies only slowly with (x, y ), then J j l e (x, y )r e . Ÿ

1/4or 2t is the inverse square law that accounts for geometrical dispersion. As the gamma rays

propagate in all directions away from the source, the number (neglecting attenuation) that pass through a given area perpendicular to the propagation direction decreases in proportion to 1/r 2t . Ÿ

s(x, y ) represents the deviation in height of the earth's surface over the mean value within the

survey area.

Many different versions of Equation (1) have appeared in the literature, with different assumptions about the distribution of the isotopes, the variation in attenuation and the form of the detector response (King, 1912; Godby et al., 1952; Purvis and Foote, 1964; Darnley et al., 1968; Kogan et al., 1969; Duval et al., 1971; Clark et al., 1972; Grasty, 1975; Grasty et al., 1979; Pitkin and Duval, 1980; Tewari and Raghuwanshi, 1987).

2.2 Simplifying the model

Equation (1) is relatively complicated and difficult to apply in its present form. Fortunately, the attenuation coefficients of rocks and soils are quite large for all three radioisotopes. This fact results in the absorption of almost all gamma rays emitted below about the top 50 cm of the Earth's surface (e.g. Kogan, et al., 1969; Grasty, 1979). Except for ground based surveys, the height, h, is much greater than 50 cm. Hence, Equation (1) can be simplified by noting that r t i r a i r (i.e. the total distance, the distance in air and the distance from the detector to the surface point directly above the source are equivalent) and r e = zr/h. Equation (1) then becomes I =

° dx ° dy

d e exp(−l a r )

° dz n exp(−J )

/ (4o r 2 )

(2)

where the integration limits are the same as Equation (1), but notice that the depth and horizontal parts of the integration have been separated.

The general form of gamma ray emission, transport and detection shown above is three dimensional. The standard practice is to integrate out the depth component by assuming that topographic variations

51

are minimal and that the attenuation coefficient and isotope concentration are constant with depth. With these assumptions J = ° l e dt, 0 [ t [ zr/h = l e zr/h, and° dz exp(−J ) = h/l e r which then implies that, I = ° dx

° dy

h n d e exp(−l a r )/ (4o l e r 3 )

(3)

Notice, that Equation (3) contains the term n/l e = n(x, y )/l e (x, y ) which is the ratio of the concentration to the linear attenuation coefficient. The attenuation coefficient is linearly dependent on density which appears to suggest that the number of gamma rays emitted will also depend on the density. However, n is the concentration per unit volume which is also linearly dependent on the density, causing a cancellation of terms (see Appendix A). Therefore, I can assume the linear attenuation coefficient is constant and only need to consider variation in n.

The full 3-D problem can be reduced to a much simpler two-dimensional problem, involving the convolution of the ground isotopic concentration, n(x, y ), with a factor related to the measurement process.

Non-dimensionalising the parameters using X = x/h, Y = y/h, G e = l e h, G a = l a h,

N = nh 3 and D e = d e /h 2 I find that the gamma ray intensity at an arbitrary position (X, Y ) is I (X, Y) = ° dX Â ° dY Â N(X Â , Y Â )P(X − X Â , Y − Y Â ), −º < X, Y < º

(4)

The factor P(X, Y ) in Equation (4) is the so-called point spread function (PSF), P(X, Y) = D e exp(− G a R ) / 4o G e R 3

(5)

where R 2 = X 2 + Y 2 + 1. I define G a as the survey parameter, as only it and the form of the detector response, D e , affect the shape of the PSF. The only other free parameter, which I will here define as the Earth parameter, is G e . It affects only the magnitude of the PSF, not its shape. The point spread function shows how an ideal point source (here an infinitely long, vertically oriented rod with constant attenuation and radioactivity) would be smoothed by the measurement process.

2.3 Derivation of existing results

Under the additional assumption that the radio element concentration is laterally constant (i.e. N(X, Y ) = N o ) Equation (4) can be simplified further. Change to polar co-ordinates (q, w ), and

introducer = 1 + q 2 , and their normalised equivalents, Q and R (which will be in multiples of

52

height). The variable Q is the horizontal distance from the point directly underneath the aircraft to the point in question, while R is the distance to the aircraft in a direct straight line. Then for the signal, from a circle of radius Q, centred directly below the aircraft I find I(Q) = N o / (4oG e )

° ¹u exp(−G a u ) G(u) / u 2 , 1 [ u [ í(Q 2 + 1 )

(6)

where G(u ) = ° D e (í(u 2 − 1 ) cos w, í(u 2 − 1 ) sin w ), 0 [ w [ 2o

(7)

is the detector sensitivity. Equations (6) and (7) separate the transmission and detection aspects of the problem. Assuming the detector response is uniform, and using the normalised diagonal distance, R, in place of the horizontal distance, Q, I derive the exponential integral solution, I(Å(R 2 − 1 )) = (Ae/ 2G e )[E 2 (G a ) − E 2 (G a R ) / R ]

(8)

which was first obtained by King (1912). Here E 2 (x ) is the exponential integral of the second kind and Ae is the non-dimensionalised detector response (area multiplied by efficiency). Using the detector model of Grasty (1979), D e (R) = Ae(a + b/R), I can integrate Equations (6) and (7) analytically to yield the solution obtained by Tewari and Raghuwanshi (1987): I(Å(R 2 − 1 )) = (Ae/ 2G e )[a E 2 (G a ) − a E 2 (G a R )/R + b E 3 (G a ) − b E 3 (G a R )/R 2 ]

(9)

where E 3 (x ) is the exponential integral of the third kind.

3 A GEOMETRICAL DETECTOR MODEL

Existing mono-energetic models of gamma ray spectrometry have assumed that the detector has either spherical (e.g. King, 1912; Godby et al., 1952; Purvis and Foote, 1964; Darnley et al., 1968; Kogan et al., 1969; Duval et al., 1971; Clark et al., 1972; Grasty, 1975; Pitkin and Duval, 1980) or cylindrical symmetry (Grasty et al., 1979; Tewari and Raghuwanshi, 1987). However, most crystals in use today for airborne surveys are rectangular so one would expect rectangular symmetry in a detector model. For example, a commonly used airborne system consists of four 10.2 % 10.2 % 40.6 cm3 (4.2-l) crystals laid out side by side to make effectively one crystal of dimension 10.2 % 40.6 %

53

40.6 cm3 (16.8-l). A detector model can be derived by considering the change in detector geometry

with incidence and azimuth angle. Two factors with a significant influence on the number of gamma rays recorded by the detector are:

1. the solid angle of the detector as seen by the source; 2. the thickness of the detector in the direction of the source. The solid angle influences the number of gamma rays that enter the detector, while the thickness of the detector determines the proportion recorded. For the hypothetical case of a spherical detector, both of these factors are independent of the incidence angle of a gamma ray.

For a vertical

cylindrical detector, the solid angle and thickness depend only on the angle to the vertical, not the azimuthal angle. A rectangular detector has a much more complicated relationship, dependent on both vertical and azimuthal angle. For example, the solid angle is large and thickness small directly along the short axis of a 10.2 % 10.2 % 40.6 cm3 detector, while the opposite applies along the long axis of the detector (Figure 2).

Figure 2: Schematic of a four litre detector emphasising the variation in thickness and area at different incidence and azimuthal angles. 3.1 The solid angle of a rectangular detector

Gamma rays, emitted from a point source are assumed to have an isotropic distribution. The solid angle of the detector, as seen from the source, determines the proportion of gamma rays that are emitted in the direction of the detector. For a source point X, placed at a distance h vertically above one corner of a rectangular surface of dimensions a % b (Figure 3), the solid angle W xy , of the surface is given by (Tsoulfanidis, 1995): W xy = arctan ab/ h a 2 + b 2 + h 2

2 /4o j ab/(4oh 2 ) + O([ab/h 2 ] )

if h >> a and b, as occurs in airborne surveying and I have used arctan x j x for x small.

(10)

54

Figure 3: Geometry for solid angle calculations. The schematic on the left shows a source point, X, a vertical distance, h, above one corner of a rectangular face(a % b ), of the detector. The schematic on the right shows a 2-D view with the source not directly above the detector. This causes the detector face to present an oblique projection to the source. The notation O([ab/h 2 ] 2 ) indicates that the accuracy of the approximation depends on the squared ratio of the area of the detector face to the square of the height. The far-field approximation in Equation (10) indicates that curvature can be neglected in the calculation of the solid angle when h is remote relative to the dimensions of the detector. The solid angle, is then just the area of the base of the detector, divided by the surface area of a sphere with radius h. As the vertical distance, h, between the source and detector increases the solid angle decreases proportional to 1/h 2 . When the source is not vertically above a corner of the rectangle, the detector has an oblique projection to the source (Figure 3). For a source at (x, y, h ) the projected area of the rectangle is ab cos h = ab(h/r ), where r = Å(x 2 + y 2 + h 2 ). Combining this reduction in projected area with the r 2 decrease in solid angle I find, W xy = ab/(4ohr 3 ) = ab/(4oh 2 R 3 )

where I have now introduced non-dimensional co-ordinates.

(11)

Equation (11) can be used to calculate

the solid angle of the base of the detector. Depending on where the source is located, one or both vertical sides of the detector may be visible to the source. The solid angles of these sides can be obtained from Equation (11) by symmetry, giving for the total solid angle, W = abc(X/a + Y/b + 1/c )/(4oh 2 R 3 )

(12)

Note, that the non-dimensionalised solid angle, for a fixed detector size, depends only on 1/h 2 , so the shape of the falloff with (X, Y ) is independent of detector height. Additionally, modelling the

55

detector response by the solid angle explicitly incorporates geometrical dispersion and accounts for an additional factor of R 2 = X 2 + Y 2 + 1 from the gamma ray model presented in Equation (5).

The detector response is a two dimensional function. However, displaying three dimensional mesh diagrams of the response tends to obscure the azimuthal dependence. Therefore, throughout this chapter either contour plots or profiles (as a function of horizontal distance) will be shown at particular azimuths. Unless otherwise stated, all curves will be normalised to be unity at the origin. Additionally, I assume the detector has the geometry of Figure 2, with the long axis of the detector orientated in the X-direction.

Figure 4 shows contours and a series of profiles of the solid angle of a detector with a 4.2-l crystal pack. Inspection of the contour plot reveals considerable anisotropy in the solid angle, which arises from both variation in the detector shape and the change in projection of each detector side with orientation. Figure 4a plots the maximum contributions to the solid angle for each of the detector sides (i.e. for profiles along either the X- or Y-axes). The solid angle of the base is a maximum directly underneath the aircraft and immediately begins to decrease due to both an increase in distance between source and detector and a decrease in projected area. The solid angle of both vertical sides is zero at the origin because the gamma rays are parallel to the surfaces giving a projected area of zero. As the source point moves away from the origin, the projected area of the vertical sides begins to increase and the solid angle starts to grow. At a certain point the increase in projected area is overshadowed by the increase in distance and the solid angle then begins to decrease. The base contributes the largest proportion of the solid angle until the horizontal distance and height are equal at which point the large vertical side begins to contribute a larger proportion. The small vertical side has 1/4 the area of the large vertical side and hence, in an equivalent configuration, contributes only 1/4 the solid angle.

For a profile along the X-axis only the base and the small vertical side are visible to the source and the solid angle is relatively low (“X-axis” in Figure 4c). In contrast, for a profile along the Y-axis, both the base and the large detector side contribute and the solid angle is large (“Y-axis” in Figure 4c). When the profile is at an angle to the X- and Y-axes, all three sides contribute to the solid angle, in varying proportions (e.g. “Diagonal” in Figure 4c). For example, starting with a profile along the X-axis and rotating it towards the Y-axis, the solid angle of the large vertical side will initially be small. As the profile approaches the Y-axis the contribution from the large vertical side will increase while that from the small vertical side will decrease.

56

Notice that the maximum solid angle does not occur directly under the aircraft or along a profile parallel to the X- or Y-axes. Transform the (X, Y ) co-ordinates to polar co-ordinates (Q, w ) to give, X = Q cos w and Y = Q sin w, and introduce G = 4oWh 2 /abc. These substitutions reduce Equation (12)

to G = [(Q/a ) cos w + (Q/b ) sin w + 1/c ]/(Q 2 + 1 )

3/2

(13)

(a)

2.5 0.20

2.0

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1.5

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Y - Horizontal distance (units of height)

3.0

1.0 0.60 0.80

0.5

0.40

1.00

0.20

0.0 0.0

(b)

0.5

1.0

1.5

Normalized solid angle

0.8 Normalized solid angle

1.0

2.0

2.5

3.0

(c)

X - Horizontal distance (units of height) 1.2 Horizontal base Small vertical side Large vertical side 1.0

0.6

0.4

X-axis Diagonal Y-axis

0.8

0.6

0.4

0.2 0.2

0.0

0.0 0.0

0.5

1.0

1.5

2.0

2.5

Horizontal distance (units of height)

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Horizontal distance (units of height)

Figure 4: Contour (a) and profile (b and c) plots of the solid angle of a 4.2-l rectangular detector. In (b) I show the solid angles contributed by each side along a profile parallel to the X- or Y- axes. In (c), the total solid angle is shown along the X- and Y-axes and also long the dotted line shown on the contour plot. Differentiating with respect to w and setting the result to zero, it is easy to show that G and hence W has a maximum when tan w = a/b.

For the 4.2-l crystal a/b = 4 and hence w j 76 o .

Thus, the

57

maximum value is much closer to the Y-axis in agreement with Figure 4(i). For a given value of w the maximum value of G and hence W, does not occur directly below the aircraft, Q = 0.

For

example, the maximum solid angle occurs at Q = 0.08, 0.28 and 0.29 for w = 0 o , 90 o and (w max =) 76 o , respectively.

In angle terms, these maximums are at vertical angles of

h = 4.7 o , 15.7 o and 16.0 o , respectively.

For a 16.8-l airborne crystal both horizontal sides are equal, a = b, and the vertical side has one quarter the length of the base, c = a/4. The solid angle is then symmetrical about the diagonal profile of Figure 4 (i.e. w = 45 o ), and the shape of the X- and Y-axes profiles are the same as the X-axis profile of the 4.2-l crystal (i.e. Q = 0.08 and h = 4.7 o ). Therefore, the solid angle of a 16.8-l airborne crystal is more symmetrical and has a less distinct maximum than a single 4.2-l crystal.

3.2 Incorporating variation in the efficiency of a rectangular detector

The detector model presented above assumes that the detection of gamma rays is independent of the angle of incidence. However, the detection of a gamma ray is dependent on the path length through the crystal, which changes with both incidence and azimuth angle and with the point of entry into the detector. Photons incident upon the long axis of a 4.2-l detector will much more likely be detected than photons incident upon the short axis (see Figure 2). If the linear absorption coefficient is l d , and the path length in the detector is L, then the total count efficiency is given by e T = 1 − exp(−l d L)

(14)

For example, using an attenuation coefficient of 14.85 m-1 in NaI (Table 2) for a 2.62 MeV photon (thorium), the total count efficiency is 99.8% along the long axis and only 78% along the shorter axis. Equation (14) gives the proportion of gamma rays of a particular energy that interact in some way with the detector. Some photons will be completely absorbed, while others will deposit only some of their energy in the detector and continue with a reduced energy. The scattered gamma ray may then either escape from the detector, or suffer further interactions. If the gamma ray is completely absorbed in the detector by one or more interactions it will be recorded at approximately the correct energy (as the NaI crystal has a relatively low energy resolution). However, if part of the gamma ray energy escapes the detector, then it will be recorded with reduced energy. For modern detectors consisting of several crystals (e.g. a 16.8-l airborne system), multiple interactions occurring nearly simultaneously in adjacent crystals, will be accepted as the same event. If the multiple interactions are recorded in non-adjacent crystals, then the event will be rejected. However, it is unlikely that a

58

photon will interact in one crystal, pass through a second and then interact with a third. Therefore, a multiple crystal system can be approximately modeled as a single composite crystal system.

Potassium (1.46 MeV) Uranium (1.76 MeV) Thorium (2.6 MeV)

18.39 16.95 14.85

Table 2: Linear attenuation coefficients of NaI for 1.46 MeV potassium, 1.76 MeV uranium and 2.62 MeV thorium gamma rays (Lide and Frederikse, 1994). The detector photo-fraction, or peak-to-total efficiency, g, gives the proportion, of the total count rate, for events that are recorded at the correct energy. The photo-peak count efficiency, e P is then related to the total count rate by the equation: e P = ge T

(15)

The detector photo-fraction depends on both the incidence and azimuthal angles and also on the point of entry into the detector. Here I make the simplifying assumption that, g is constant with incidence and azimuthal angle, after the photo-fraction has been averaged over all points of entry into the detector. This assumption is supported by Monte Carlo simulations presented later in this paper and also in Allyson and Sanderson (1998). They found that the photo-fraction varied only slightly with azimuthal and incidence angle.

To incorporate the effect of variation in efficiency into the detector model assume, once again, that the source is remote relative to the dimensions of the detector. This means that all gamma rays from a particular point source will strike any part of the detector at approximately the same angles of incidence and azimuth. In contrast, if the source were very close to the detector, the angles would vary significantly depending on where the gamma ray strikes the detector.

At a given incidence and azimuthal angle the efficiency depends on where the photon strikes the detector. The mean efficiency for each detector side can be calculated by averaging the efficiency over all parts of that detector side. For example, for a gamma ray incident on the bottom side of the detector (at angle (h, w ) see Figure 5) the average total count efficiency is given by −1 e T = 1 − (ab ) ° ° exp[−l d & r d (x, y, h, w )] ¹x¹y, 0 [ x [ a, 0 [ y [ b

(16)

59

where a and b are the dimensions of the bottom side of the detector and the function r d (x, y, h, w ) gives the path length of a photon that hits the detector at position (x, y ) at angle (h, w ). The integral can be split into pieces and evaluated analytically. The partitioning of the integral is based on where photons would exit the detector if they weren't recorded. For example, in Figure 5 (which shows the detector upside down), region 1 is all those parts of the detector that have photons exiting out of the top of the detector, regions 2 and 5 those that exit out of one side, and regions 3 and 4 those that exit out of the other side. The calculation of the region boundaries and the integral over each of the regions is straightforward to solve analytically (see Appendix B).

Equation (16) can be used to calculate the average efficiency for each of the three sides of the detector. To calculate the average efficiency for the whole detector, weight these with their solid angles. Given solid angles of W side , and total-count efficiencies of e Tside , for the average total count efficiency I find e T = (e Txy 1/c + e Txz Y/b + e Tyz X/a )/(1/c + Y/a + X/a )

(17)

Figure 5: Geometry for detector efficiency calculations with the detector of dimension(a, b, c ), shown upside down. For a photon incident at the angle (h, w ) the dotted lines on the face of the detector in the perspective plot, delineate boundaries where the photons exit out of different sides. After extension of the vertical and horizontal lines on the plan view, the regions are labelled 1 to 5. The algebraic formulas for e T at a general position (X, Y, 1 ) are complicated as contributions from five different regions (Figure 5) for three different sides of the detector need to be considered. Temporarily restricting attention to the case Y = 0 (the case X = 0 can be obtained by symmetry), the contribution from the vertical side perpendicular to the Y-axis is zero. Furthermore, the contributions

60 from regions 2, 4 and 5 are zero for the remaining two sides and assuming cX < a the contribution from region 1 for the vertical side is zero (Figure 3). I find (Appendix B), e T = 1 − 2X/R + [l d (a − cX ) − 2X/R ] exp(−l d cR ) /[l d (cX + a )]

(18)

In contrast, if cX m a, the contribution from region 1 is zero for the base side and I find e T = 1 − 2X/R + [l d (cX − a ) − 2X/R ] exp(−l d aR/X ) /[l d (cX + a )]

(19)

The higher the energy of the gamma ray the lower is l d and hence, the greater the variation in the exponential term with X and R, which implies that the variation in efficiency for thorium will be greater than for either potassium or uranium. Figure 6 shows the significant asymmetry that occurs in the total count efficiency for thorium for a 4.2-l crystal (results for K and U are similar but show slightly less asymmetry). At all azimuths the efficiency decreases for sources at slightly non-vertical incidence. Even though the path length within the main body of the detector increases as cR = c sec h, the non-vertical incidence means that many gamma rays can travel very short paths near the ends of the detector. For the profile along the X-axis in Figure 6, this initial reduction is soon compensated by the large length of the crystal in the X-direction (four times the length in the vertical direction). For the Y-axis the efficiency continues to decrease until the horizontal distance equals the height, before it begins a slow growth that asymptotically approaches the value underneath the aircraft. This variation is due to the horizontal and vertical sides having exactly the same length. If the ordinate in Figure 6b was h in place of Y = tan h, the Y-axis efficiency would be symmetrical about h = 45 o .

Notice, that the variation in efficiency (Figure 6) partly compensates the variation in solid angle (Figure 4), with large efficiencies generally implying small solid angles and vice-versa.

This

relationship is intuitively expected, as where a large face occurs on the 4-2-l crystal it is generally associated with a small thickness and vice versa.

The entire detector effect is given by combining the solid angle and efficiency equations, D e = W $ e P = (gabc/4oh 2 ) e Txy 1/c + e Txz Y/b + e Tyz X/a /R 3

(20)

Equation (20) implies that the detector effect for a rectangular crystal can be split into three separate contributions, namely the base and the two vertical sides. Each has its own solid angle and efficiency which are then related by a linear sum. The overall detector effect is also dependent on the cube of the non-dimensionalised distance which accounts for both geometrical dispersion and obliqueness.

61

(a)

(b) 0.85

2.5

X-axis Diagonal Y-axis

0.80 0.65

Total-count efficiency

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0.75 0.70 0.80

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X - Horizontal distance (units of height)

0.5

1.0

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2.0

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Horizontal distance (units of height)

Figure 6: Contour (a) and profile (b) plots of the total-count efficiency for a 4.2-l rectangular detector for a gamma ray energy of 2.6 MeV. The net results of incorporating the solid angle and non-uniform efficiency are shown in Figure 7, for detection of 2.62 MeV, thorium photons by a 4.2-l crystal. The response is more symmetrical than either solid angle or efficiency alone, a consequence of their compensating effects. Additionally, the distinct maxima and minima that occurred for both solid angle and efficiency are absent. Notice, that the diagonal profile now exceeds the profile along the Y-axis (compare Figure 8a to Figure 4), until about 0.8h, at which point the curves cross.

2.5

2.0

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Y - Horizontal distance (units of height)

3.0

1.0 0.20 0.60

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0.80 0.40 1.00

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

X - Horizontal distance (units of height)

Figure 7: Contour of the detector response of a 4.2-l rectangular detector for a gamma ray energy of 2.6 MeV.

62

As long as the survey height is greater than a few meters (so that the far field approximation can be used), the shape of the non-dimensionalised detector response is independent of the height, and is only dependent on the relative dimensions of the detector (a, b, c ). In particular as a = 4c for both the 4.2 and 16.8-l crystals their detector profiles along the X-axis are identical. For the rest of this chapter, the detector model presented in Equation (20) will be referred to as the geometrical detector model.

3.3 Comparison of modeled results to experimental data

Tewari and Raghuwanshi (1987) repeated an experiment by Grasty (1975) to determine the variation with incidence angle in the response of a rectangular 4.2-l crystal. The experiment consisted of placing a thorium source at a given distance directly below the detector and rotating the crystal around the Y-axis (using the geometry of Figure 2). Tewari and Raghuwanshi (1987) then fitted an equation of the form a + b cos h for 0 [ h [ o/2 to the experimental data, and, when normalised, found coefficients of a = 0.39 and b = 1 − a = 0.61. The variable h is the angle to the vertical unit vector (0, 0, 1 ). Figure 8a compares this empirically determined cosine model with the detector response

function derived here, plotted as a function of horizontal distance (i.e. tan h). The two models show excellent agreement for the angular sensitivity parallel to the X-axis of the detector.

This

corresponds to the experimental set up used by Tewari and Raghuwanshi (1987) and gives us confidence in the general validity of our model. 1.0

(a)

0.9 0.8

(b)

X-axis Diagonal Y-axis Cosine

0.9 Normalized point spread function

Normalized detector response

1.0

X-axis Diagonal Y-axis Cosine

0.7 0.6 0.5 0.4 0.3 0.2

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.1

0.1

0.0

0.0 0.0

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1.5

2.0

Horizontal distance (units of height)

2.5

0.0

0.5

1.0

1.5

2.0

Horizontal distance from plane (units of height)

Figure 8: The (a) detector sensitivity and (b) point spread function for the geometrical and uniform models at 25 m elevation for a 2.6 MeV gamma ray incident on a 4.1-l crystal. The“X-axis”, “Y-axis” and “Diagonal” curves are profiles of the geometrical model in the same configuration as Figure 7. The curve marked “Cosine” is a profile of the cosine model (which is independent of orientation).

2.5

63

The geometrical detector model is able to capture more of the complexity related to the shape of the detector. The cosine model gives a poor representation of the change in sensitivity along the diagonal or Y-axis profiles. In both cases the sensitivity first increases as the angle of incidence increases, and then decreases. This is due to the area and thickness the detector’s vertical side being the same as the area(40.6 % 10.2 cm2) and thickness (10.2 cm) of the detector’s base. In contrast, for the X-axis, even though the thickness (40.6 cm) is large, the side area (10.2 % 10.2 cm2) is very small, leading to a low solid angle. The detector response is then dominated by the bottom side of the detector, and the presence of the photo multiplier tubes (which would decrease the flux further) on the small detector side would have little effect.

4 THE POINT SPREAD FUNCTION IN AIRBORNE GAMMA RAY SPECTROMETRY

The results presented in the previous section show that the detector response depends strongly on the angles of incidence and azimuth, and that a cosine model is a poor approximation to a rectangular detector. Thus far I have only considered the detector response and have yet to incorporate the other terms in the point spread function. The PSF is given by combining the general PSF model of Equation (5) with the detector model of Equation (20): P(R, w) = (gabc/4oh 2 ) e Txy 1/c + e Txz Y/b + e Tyz X/a exp(−G a R)/G e R 4

(21)

Note the use of the total count efficiency e T , in place of the peak efficiency e P , and the associated inclusion of the peak-to-total ratio, g. The attenuating effects of the column of air between source and receiver, exp(−G a R), result in a decrease of the gamma ray flux with increasing angle of incidence (i.e. increasing radial distance from the detector). In addition, attenuation within the earth results in a cos h (or equivalently 1/R) dependence of the emission of gamma rays in the direction of the detector, while the detector response varies as 1/R 3 . These effects further decrease gamma ray fluence rate at the detector as the incidence angle increases. The PSF for the geometrical and cosine detector models are shown in Figure 8b, for 2.62 MeV photons incident on a 4.2-l crystal at 25 m elevation. The shape of the PSF is very similar to the shape of the detector response function (Figure 8a) at any given azimuth. However, the difference between different azimuths is reduced. This is especially apparent at large distances from the detector, and is due to the attenuation within the column of air between source and detector becoming more significant than the azimuthal difference in detector sensitivity.

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4.1 The point spread function of a 16.8 litre airborne detector A commonly used airborne detector system consists of a four-pack of 10.2 % 10.2 % 40.6 cm3 crystals to give a total volume of 16.8-l. To a close approximation, this system may be treated as a single crystal with dimensions 10.2 % 40.6 % 40.6 cm3. In many acquisition systems there are two or even three 16.8-l systems. For example, the Australian Geological Survey Organisation (AGSO) currently operates two sets of 16.8-l crystal packs. The crystal packs are spatially separated and operate independently of each other. Therefore, the two-pack system can be modeled as two independent single-pack systems. As the pack separation is very much smaller than the survey height the point spread function of the combined two-pack system is, approximately, double the magnitude of the PSF for a single pack.

The degree of asymmetry in a 16.8-l system is considerably less than for a single 4.2-l crystal, as the response is dominated by the base of the detector (Figure 9, Th at 60 m elevation). The base has four times the area of either vertical side and is oriented perpendicular to the direction of maximum gamma ray flux. The base has a much thinner width (10.2 cm) than the vertical sides (40.6 cm), which, for a 2.62 MeV gamma ray, cause a 25% decrease in efficiency. However, the solid angle is significantly larger and outweighs this decrease in efficiency.

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Figure 9: Contour and profile plots of the point spread function of the geometrical detector model for a 2.6 MeV photon incident on a 16.8-l crystal at 60 m elevation. The curves marked “X-axis” and “Diagonal” are profiles of the geometrical model, while the curve marked “Uniform” is a profile of the uniform model.

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Figure 9b shows the PSF of the geometrical detector model in two configurations, compared with the PSF of a uniform detector model (the Gunn, 1978; and Craig et al., 1998 model). The geometrical model is normalised to be unity at the origin, while the uniform model is normalised to the geometrical model by equating their integrals over an infinite horizontal plane. Both models would then predict exactly the same count rates when surveying over a flat, homogeneous half-space. The uniform detector model predicts a much lower peak in the PSF, which would lead to overestimation of the area that generates a given percentage of the signal. That is, a uniform model underestimates the spatial resolution of an airborne detector system. This implies that, if a survey was designed around the assumption of a uniform detector model, the areas between flight lines may be under-sampled.

5 COMPARISON TO MONTE CARLO SIMULATIONS

The shape of the PSF predicted by the geometrical model is dependent on the detector response and the survey parameter, G a . This was defined as the product of the attenuation in the air with the detector height (l a h, see Equation 5). Monte Carlo simulations, supplied by Hovgaard (pers. comm.), were compared to the geometrical detector model for three, wide-ranging values of G a (Figure 10). They corresponded to thorium at a survey height of 50 m (G a = 0.25 ), uranium at 80 m(G a = 0.5 ) and potassium at 120 m (G a = 0.82). Other physical parameters used in the modelling comparisons are given in Table 3.

Element: Gamma ray Energy (MeV): Survey height (m): Mass attenuation coefficient of soil (cm2/g): Mass attenuation coefficient of air (cm2/g): Linear attenuation coefficient NaI (cm-1): Concentration of isotope: Gamma rays per second per gram (γ /(sg)): Soil density (g/cm3): Air density (g/cm3): Air pressure (Pa):

Thorium 2.62 50 0.0396 0.0391 0.1485 10 ppm 0.015 2.0 0.001293 101 325

Uranium 1.76 80 0.0482 0.0479 0.1695 5 ppm 0.010 2.0 0.001293 101 325

Potassium 1.46 120 0.0528 0.0526 0.1839 2% 0.066 2.0 0.001293 101 325

Table 3: Values of physical parameters used in the comparison of the Monte Carlo simulations with the geometrical detector model. For each grid point (on a 40 % 40 grid), the Monte Carlo simulation involved tracing 10 000 gamma rays within the cone of the solid angle of the detector (total of 16 000 000 gamma rays traced). Only primary (unscattered) gamma rays were traced all the way to the detector. Within the detector, all gamma rays were traced until they were either completely absorbed or escaped the detector.

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Figure 10: Comparison of Monte Carlo simulations to the geometrical detector model. From top to bottom the comparisons are Th at 50 m, U at 80 m and K at 120 m. The figures on the left show the raw Monte Carlo peak and total count rates against the geometrical detector’s predicted total count rate. On the right the scaled Monte Carlo peak count rate is compared to the geometrical and uniform detector models.

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Hovgaard’s Monte Carlo code is very similar to that described in Allyson and Sanderson (1998). The main difference is in the method for tracing the path of the photon to the detector. Hovgaard’s code models only those photons that reach the detector without scattering and hence only sends photons out in a direction that intersects the detector.

For the Monte Carlo simulations, two sets of results were produced; the total count rate and the peak count rate. The geometrical model evaluates the total count rate and calculates the peak count rate by multiplying by the detector photo-fraction, which is assumed to be constant with incidence and azimuthal angle. This assumption is supported by analysis of the Monte Carlo simulations on the 40 % 40 grid. The average values of the detector photo-fraction were 0.637 ! 0.006 for K, 0.606 ! 0.006 for U and 0.520 ! 0.016 for Th, where the errors are the calculated standard deviations.

Note, that the values quoted here would be slightly higher than the real values, as the Monte Carlo model does not take account of the escape of bremsstrahlung. This process would contribute only a small, constant factor independent of the incidence and azimuthal angles.

As the energy of the gamma ray increases, the photo-fraction decreases. This implies that (as one would expect) higher energy photons that have suffered one scattering event in the crystal are more likely to escape the detector than scattered low energy photons.

The higher variation in the

photo-fraction calculated for Th may indicate that the photo-fraction becomes more variable the higher the gamma ray energy. Hovgaard (pers. comm.) speculates that this results from the generally higher energy of scattered thorium gamma rays. When the scattered gamma ray energy is low there are only two ways that they are likely to escape detection:

1. by reflection when they interact close to the crystal surface. 2. if they pass close to the edge or corner of the crystal and are scattered outwards. The effect of (i) for both high and low energy photons will be almost independent of direction and where the photon strikes the crystal. The effect of (ii) will be very small for low energy photons as the volume where it takes place is small relative to the total volume of the crystal. In contrast, the volume where (ii) is important is larger for higher energy gamma rays. The angle and location of the incident photon are then likely to have a larger effect on the probability of escape after a scattering event.

The left hand column of Figure 10 shows a comparison of the raw Monte Carlo results to the raw values produced by the geometrical model. For all three simulations, the total count rates are similar, with the Monte Carlo method predicting slightly higher count rates. The maximum differences between the two models occur at the origin and are 4.2% for Th, 6.0% for U and 3.0% for K. The

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consistent under-prediction of the geometrical model may be due to slight differences in the parameterisation of the two models.

To calibrate either model with real data would involve flying a calibration line over an area with known and relatively uniform radioactivity. The predicted infinite-source yield of the model would then be compared to the observed count rates with the ratio of the two determining any unknown constants (g/G e for the geometrical model; see Equation 21). This suggests that the appropriate method to compare the Monte Carlo peak-count-rate simulation with the geometrical model would be to equate their infinite-source yields. They would then give exactly the same count rates when flying over flat, uniformly radioactive and attenuating ground. When the geometrical model is scaled the results are excellent (right hand column of Figure 10). The two models are in very close agreement, with maximum errors (occurring at the origin) of 3.7% for Th, 1.0% for U and 1.3% for K. Apart from the slight mismatch directly beneath the aircraft, the results are nearly identical. The slightly poorer fit for thorium may be due to the higher variation in the peak-to-total ratio as discussed earlier in this section.

As a means of comparison, the curves that would be obtained by calibrating a uniform model to the Monte Carlo results are also shown (RHS of Figure 10). The correspondence between the two is extremely poor, with maximum errors of 20% for Th and 12% for both U and K.

In conclusion, the agreement between the Monte Carlo results and the geometrical model are excellent. This verifies the general validity of the modelling approach and justifies the assumption that the photo-fraction is (at least approximately) constant. The Monte Carlo results also show that the assumption of uniform detector sensitivity is inadequate.

6 CONTRIBUTING AREA CALCULATIONS

One measure of the spatial resolution of a radiometric survey is the contributing area which is defined by reference to an imaginary homogeneous half space (e.g. Pitkan and Duval, 1980). Consider the counts per second recorded by the detector, I(Q ), from a circle of radius Q, centred directly beneath the aircraft. The contributing area is then defined through the ratio, C(Q ) = I(Q )/I(º ),

(22)

where the factor, I(º ), is the infinite source yield (the signal recorded from a circle with infinite radius). From this definition one can define, for example, the 90% contributing area, which is the

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area that contributes 90% of the total signal. Inferences about spatial resolution can be obtained by investigating how C(Q ) changes with increasing radius. For example, if C(Q ) for a given Q is larger for one system over another, then the spatial resolution is higher.

The contributing area of the uniform detector model can be obtained from the exponential integral solution of Equation (8) and is given by, C[í(R 2 − 1 )] = 1 − E 2 (G a R)/[R E 2 (G a ) ]

(23)

where I am now using the diagonal distance, R 2 = Q 2 + 1. The contributing area for the geometrical model could, in principle, be expressed by an analytical formula. However, on attempting this approach, one quickly finds that the algebra involved is both tedious and excessive. A simpler method is use numerical integration to obtain estimates of I(Q ) and I(º ) . For a 16.8-l crystal, these can be obtained efficiently by assuming the point spread function, P(R ), is radially symmetric (which is approximately the case), to give  I(Q ) = ° 2oQ  P[í(Q Â2 + 1 )]¹Q , 0 [ Q  [ Q

(24)

The 2oQ Â factor in the above equation comes from the transformation of plane Cartesian co-ordinates to plane polar co-ordinates, and the radial symmetry of P(R ). The integral can be approximated accurately as the point spread function decays rapidly, P(R )~ exp(−G a R )/R 4 .

Figure 11 compares the contributing area curves for the uniform and geometrical detector models for a survey parameter of G a = 0.3 (Th at 60 m elevation). The uniform detector model, as mentioned in the last section, tends to overestimate the size of the contributing area. It predicts that 90% of the signal comes from a circle with radius 3.1 times the height, compared to 2.6 times the height for the geometrical detector model. In terms of area, the 90% contributing area predicted by the uniform model is 10.9 ha compared to 7.6 ha for the geometrical model, a difference of 42%. As the survey height increases the difference between the two models decreases. At 120 m, the radius for the uniform model is 2.3 times the height compared to 2.0 times the height for the geometrical model. The 90 % contributing area is then 18.7 ha compared to 24.6 ha, a difference of 32%. The energy of the gamma ray also significantly influences the contributing area. For example, for the 1.46 MeV potassium gamma ray at 60 m elevation the 90% contributing areas of the geometrical and uniform models are circles with radii 2.3 and 2.8 times the height (6.0 and 8.6 ha), an area difference of 43%. At 120 m elevation, the circles have radii 1.8 and 2.1 times height (14.9 and 19.2 ha), a reduced area difference of 28%.

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Figure 11: Comparison of the predicted contributing areas of a 16.8-l airborne system for the geometrical and uniform detector models. 7 INCORPORATION OF THE PLATFORM VELOCITY

The point spread functions calculated so far can be viewed as either representing a stationary system or the instantaneous response of a moving system. If, during the accumulation period, the detector moves only a small distance relative to its height above the earth, a stationary PSF may be an adequate approximation. However, the stationary model would be a poor approximation if the detector moved a large distance during this time. The instantaneous response then needs to be integrated over the count accumulation time. For airborne systems, the counting time is generally one second and the platform velocity is of order 60-70 ms-1.

This can be quite a significant

movement, granted that survey altitudes generally lie between 50-120 m. Some highly detailed surveys have been flown as low as 25 m in elevation with 40 m spacing between samples. Therefore, the detector movement could potentially exert a strong influence on the shape of the PSF.

If the stationary PSF is given by P, the count time by t seconds and the velocity by v m/s in the positive X-direction (non-dimensionalised to T = t/1 and V = v/h), then the PSF of the moving detector, M, is given by: M(X, Y) = (1/T ) ° P(X + t, Y)¹t, − VT/2 [ t [ VT/2

(25)

The integral could be numerically evaluated using a quadrature scheme such as Simpson’s rule. However, Equation (25) can be solved using Fourier transforms by casting it as a convolution

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between the stationary PSF and a rectangular function of width VT centred at the point X. In the Fourier domain, the model incorporating velocity is just the Fourier transform of the stationary model multiplied by the sinc (= sin ox/ox) function: M & (u x , u y ) = P & (u x , u y ) sinc(Vu x )

(26)

where the star notation indicates the Fourier transform and (u x , u y ) are frequency co-ordinates. The space domain representation of the model incorporating movement can then be found by inverse Fourier transformation of the RHS of Equation (26). The simple form of the model in the Fourier domain would be particularly advantageous for deconvolving a radiometric survey (e.g. Craig et al., 1998). (a)

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Figure 12: Contour (a) and profile (b) plots of the point spread function for a 2.6 MeV gamma ray at 60 m elevation with an aircraft velocity of 60 m/s. The curves marked “X-axis” and “Y-axis” are profiles along the X- and Y-axes, while the curve labelled “Diagonal” is a profile along the dotted line shown on the contour plot. The aircraft movement decreases the maximum of the PSF and introduces asymmetry (Figure 12). The PSF is elliptical in shape with the sensitivity greatest along (X-axis) and minimum perpendicular to the direction of travel (Y-axis). The effect of velocity is dependent on the ratio of the height of the survey relative to the distance travelled during the integration time, h/vt. When this ratio is low, the peak of the PSF is markedly depressed and the asymmetry (defined by the ratio of the half-widths perpendicular and parallel to the direction of travel) is significant (Figure 13). As this ratio increases, the magnitude of the peak approaches that of the stationary response and the asymmetry decreases. For example, at a constant velocity of 60 m/s the maximum of the PSF is reduced by 12% at 60 m

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elevation and by only 4% at 120 m in elevation. The asymmetry in the PSF is also considerably reduced with the ratio of half widths increasing from 0.89 at 60 m to 0.97 at 120 m.

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Figure 13: Ratio of the peak of the PSF incorporating velocity to the peak of a stationary PSF, calculated for Th assuming v = 60 m/s and an integration time of one second. The ratio of the widths at half maximum along the directions parallel and perpendicular to the direction of travel are also shown. A brief note on the calculation of the Fourier transform, P & (u x , u y ), of the stationary point spread function. One advantage of the Craig et al. (1998) PSF is that its Fourier transform can be calculated by a series expansion (Craig, 1993). The equivalent calculation for the geometrical PSF must proceed by a numerical approximation. Fortunately, the rapid falloff of the model with distance means that it can be calculated accurately and very rapidly by a Fast Fourier Transform algorithm (Cooley and Tukey, 1965) or, assuming radial symmetry, a fast Hankel transform (e.g. Siegman, 1977). For example, Figure 14a shows the falloff curves for thorium at heights of 30, 60, 90, 120 and 150 m calculated by a Hankel transform. As the height increases, the count rate decreases and the high frequency content within the PSF progressively reduces. As discussed in the next chapter these two factors strongly influence the noise levels and spatial resolution of a radiometric survey.

The movement of the aircraft causes a more rapid falloff in the direction of travel and has no effect perpendicular to that direction (Figure 14b).

The sinc function passes through zero when its

argument is one, i.e. when u x = 1/(vt ), which implies that these frequencies will be absent from the survey. This is to be intuitively expected as these frequencies will always be sampled at the same point in their cycle.

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7 h = 30 m h = 60 m h = 90 m h = 120 m h = 150 m

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Figure 14: (a) Frequency domain representation of the PSF for thorium at 30, 60, 90, 120 and 150 m elevation. Calculations were made using the fast Hankel transform of Siegman (1977) which assumes the PSF is radially symmetric. (b) Thorium at 60 m elevation with an aircraft movement of 70 m during the integration time (the same configuration as the Jemalong survey). 8 DISCUSSION

The results presented in this chapter show that a geometrically based model of primary unscattered radiation can be used successfully to simulate gamma ray surveys. The model calculates the total count rate by considering the characteristics of a detector (the solid angle, thickness and attenuation coefficient) that influence the number of interactions within its volume. The total count rate is converted to a peak count rate by multiplying by the detector photo-fraction that was shown from Monte Carlo modelling to be relatively constant with incidence and azimuthal angles. The variation in the photo-fraction appears to increase slightly the higher the gamma ray energy.

Building sodium iodide crystals is an expensive business. Therefore, determination of the detector dimensions that would maximise either count rates or spatial resolution should first proceed by numerical calculation. The geometrical model could be used to investigate rapidly the effect of changing the detector dimensions. However, as the dimensions of the detector change, the peak-tototal ratio may vary. Therefore, one may find that a certain detector geometry has a very high total count rate but very poor spectral resolution. The peak count-rate of any favourably shaped detectors would need to be determined by Monte Carlo simulations. This two-step approach would enable many alternatives to be considered quickly.

The geometrical model is also general enough to

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calculate the response of other detector shapes. This would involve modifying the solid angle and efficiency equations, which would be relatively straightforward for regular geometrical shapes.

The relative dimensions of a rectangular crystal determine the degree of asymmetry in the detector response.

For a detector that has a square horizontal side larger than either vertical side the

asymmetry will be weak, as there is little change in solid angle or thickness with azimuthal angle (cf. results for 16.8-l).

However, for a detector with a rectangular horizontal shape, significant

asymmetry can be present (cf. results for 4.2-l crystal). This is predominantly due to variations in the solid angle contributed by the vertical sides rather than variations in the solid angle of the base side. In fact, if the horizontal side is square but the size of the vertical sides are of the order of the bottom side, then asymmetry may also be significant. This arises because the vertical sides then contribute a significant and varying portion of the solid angle. For a rectangular base, there are two axes of symmetry, while for a square base there are four. The presence of photo multiplier tubes on one detector side would halve the number of symmetry axes due to their influence on the flux of gamma rays that reach the detector. However, where the vertical side is relatively small compared to the horizontal, the photo multipliers would have little effect, as the flux on the side would be low. This is the case for both the 4.2-l (as the tube is on the small side of the crystal) and 16.8-l detector systems. The model is dependent on a multiplicative constant (ratio of g/G e , Equation (21)) that would need to be determined by calibration. The procedure for calibration would involve measuring the count rates at the survey height over an area with known and relatively uniform radioactivity as already recommended as standard calibration practice (e.g. IAEA, 1991; Grasty and Minty, 1995). The unknown constant would be given by the ratio of the airborne count rates to the measured isotopic composition of the soil, divided by the infinite source yield. Once calibrated the model could be used for resolution and deconvolution calculations, applications which are addressed in Chapters 4 and 7 respectively.

The geometrical detector model indicates that the spatial resolution of a rectangular detector is higher than that predicted by assuming the detector is spherical. For example, the 90% contributing area for Th at 60 m elevation predicted by the uniform model is 10.9 ha compared to 7.6 ha for the geometrical model. Therefore, surveys designed around the assumption of uniform sensitivity may cause an under-sampling of the signal between flight lines. The use of the uniform model for deconvolution would introduce substantial modelling error into the process. The PSF can also be significantly affected by the aircraft velocity, which causes the contributing area of the detector to be elliptical rather than spherical in shape. This effect would tend to be greater the lower the altitude of

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the survey. As such low altitude surveys are costly and designed to maximise spatial resolution, the assumption of a stationary response for the deconvolution may substantially degrade the results of the reconstruction.

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CHAPTER 4

RESOLUTION AND UNCERTAINTY IN AIRBORNE GAMMA-RAY SURVEYS1 Why shouldn’t truth be stranger than fiction? After all fiction has to make sense. Mark Twain.

CHAPTER OBJECTIVES Develop methods for estimating the assay uncertainty and spatial resolution of a radiometric survey and derive the optimum Wiener filter for the deconvolution of gamma-ray data.

1 INTRODUCTION

The Macquarie Dictionary defines resolution as the act or process of distinguishing between the individual parts of an (object or) image. When applied to an airborne gamma-ray survey there are two aspects to resolution; 1. Spectral (or detector) resolution: The smallest detectable change in spectrometer output, which reflects the ability of the spectrometer to distinguish individual photo-peaks. 2. Spatial resolution: The minimum distance between two sources that can be separately identified, or (in Fourier terms) the maximum frequency that can be resolved.

The spectral resolution can be critical when using gamma-ray spectrometry to find and identify stolen radioactive sources (e.g. Nikkinen et al., 1995) or when monitoring radioactive fallout from an accident (e.g. Toivonen, 1995). For soil/geological mapping the main effect of imperfect spectral resolution is a broadening of the 1.46 MeV potassium, 1.76 MeV uranium and 2.62 MeV thorium photo-peaks. To counteract this effect, the IAEA (1991) recommends that broad energy windows (given in section 3.3, Chapter 2) be used to measure potassium, uranium and thorium count-rates. When the detector is functioning correctly, and the thorium resolution (ratio of full-width-at-half- maximum to peak-position) is at its recommended value of 7% or better, the imperfect spectral resolution is of little concern (e.g. Grasty and Minty, 1995). Therefore, the spectral resolution will not be considered further in this thesis. 1

Calibration data obtained from the Australian Geological Survey Organisation and Minty (1996).

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In airborne radiometrics the assay uncertainty (also called assay precision by Lovborg and Mose, 1987) is defined as the ratio of the standard deviation to the mean value. It is a measure of the relative error in the concentrations of potassium (%K), uranium (ppm eU) and thorium (ppm eTh).

The spatial resolution and assay uncertainty determine how suitable a survey is for a given purpose. For example, a high altitude (say 100 m) survey with a wide line spacing will be of limited value for high precision agriculture, where within paddock soil differences are important. The same survey may be quite suitable for broad scale mapping of soils in an area with significant geological variation, but unsuitable in an alluvial environment where subtle variations in radioelement concentrations may be important.

Knowledge of the uncertainty and resolution of a survey flown under different

conditions can be used to: 1. Assess whether a pre-existing survey is suitable for a given application; 2. Plan an appropriate sampling regime (height and line spacing) for an intended application. However, at present there is a poor understanding of how the spatial resolution and assay uncertainty are affected by the aircraft height, background contamination and radiometric characteristics of an area. These issues are addressed in this chapter. Note, that the resolution and uncertainty of the three radioisotopes will be expected to differ as each has different background characteristics and detector response.

The uncertainty in recorded count rates can be obtained from the mean value as detection of gamma radiation is a Poisson process (e.g. Tsoulfanidis, 1995), which has variance equal to mean value. These errors are then modified by the requirements for background correction, spectral stripping, height correction etc. The assay uncertainty can be estimated by tracing errors through each of the processing steps using the rules of linear error propagation. Such an analysis has previously been performed for portable spectrometers by Lovborg and Mose (1987). The approach can be generalised to the airborne case but requires that several additional factors are considered (e.g. three components to background estimation, height correction etc.).

A radiometric detector cannot be focused like a camera and the pixel size in an image does not reflect the underlying spatial resolution. The pixel size is usually dictated by convention to be one-quarter to one-fifth of the across line spacing (e.g. Pettifer, 1995). On the other hand, the spatial resolution is determined by a number of factors including: 1. The sampling density: It is usually greater (often by a factor of 5 or more) along lines than across lines, which can cause the spatial resolution to be different in the two directions.

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2. The aircraft height: Controls the size of the footprint of a single measurement (e.g. Figure 4, Chapter 2), and hence the amount of blurring in a radiometric image. 3. Detector movement during the integration time: Causes a further blurring of spatial detail. 4. The assay uncertainty: To partially remove the blurring caused by the aircraft height and detector movement the radiometric data need to be deconvolved (e.g. Craig et al., 1998) with the point spread function developed last chapter. The quality of the result, and hence the spatial resolution achieved, is dependent on the noise levels (viz assay uncertainty) within the data and the shape of the point spread function.

Previous work on the issue of spatial resolution has usually involved estimating the detector footprint. The percentage overlap of adjacent measurements can then be used to specify maximum line spacings at a given survey altitude (e.g. Pitkan and Duval, 1980). The analysis ignores the noise levels in the data and does not give explicit predictions of spatial resolution. One way to measure spatial resolution is by the maximum frequency that can be resolved by a radiometric survey. As the data are collected almost on a regular grid with spacing (Dx, Dy ), the maximum frequencies, (u c , v c ), are specified by the Nyquist relationship, u c =

1 2Dx

and v c =

1 2Dy .

The actual maximums may be lower

than these sampling cut-offs, as the blurring imposed by aircraft height and movement attenuates the high frequency content of the signal. Therefore, at some point in the frequency domain the noise levels may dominate the signal, causing an effective frequency cutoff. If this cutoff could be estimated then; (i) the spatial resolution of an existing survey could be appraised; and (ii) the Nyquist relationship could be used to specify an appropriate sampling regime for a proposed survey. Note, that if the noise cutoff is greater than the sampling cutoff (i.e. under-sampling), the higher frequency content is aliased (folded) into the lower frequency parts of the spectrum (e.g. Bracewell, 1986).

The Poisson nature of radiation detection means that the assay uncertainty and, hence, the spatial resolution, depend on the concentration of radioelements. Therefore, the resolution and uncertainty will vary over different parts of the survey area and will also be effected by changes in background radiation. From the perspective of survey design and analysis, the important consideration is the resolution and uncertainty that can be achieved under average (e.g. standard crustal concentrations of potassium, uranium and thorium, average radon contamination) or extreme (e.g. low radon, high radon) conditions.

The chapter is organised as follows. In the first section a method is developed for the estimating the assay uncertainty of a survey flown under specified conditions (e.g. height, background, isotope concentration etc.). The next section considers the problem of estimating the spatial resolution of a

79

survey, while the last discusses the implications of the previous two sections to survey design. The definitions of each of the symbols used in this chapter are given in Table 1.

2 PROPAGATION OF ERRORS DURING PROCESSING

Grasty and Minty (1995) and Minty (1996) have both considered the contributions of various sources of error to the uncertainties in assigned concentrations of potassium, uranium and thorium. Minty (1996) also presents a method for estimating fractional errors by analysing the deviation about a recursively filtered version of the data (Tammenmaa et al., 1976). The method can be used to estimate the assay uncertainty of an airborne survey that has already been processed. The approach I take here is to estimate the error contributed by each process and then combine the various errors together using linear error propagation.

2.1 Background to error propagation

The gamma ray field at a fixed location with constant background undergoes statistical fluctuations. These are related to the random nature of radioactive decay and detection which follow a Poisson distribution (e.g. Tsoulfanidis, 1995). The Poisson distribution has a variance equal to the mean value and for counts exceeding 20 (typical of airborne surveys), is almost identical to the Gaussian distribution (e.g. Dickson, 1980). The equality of mean and variance implies that estimates of the uncertainty in the measured counts can be obtained from the mean. For detection of M counts, the relative error (defined as the ratio of the standard deviation over the mean) is approximately given by 1/ M . Therefore, the number of gamma rays detected has a significant influence on the error. For

example the relative errors for 10, 100, 1000 and 10000 counts are 32%, 10%, 3.2% and 1% respectively. The uncertainty in the measured count rate is not the only factor that influences the error in the assigned concentrations of potassium, uranium and thorium.

The issue is complicated by the

requirements for live-time correction, background removal, spectral stripping and height attenuation (see section 3.3 in Chapter 2). The simplest way to estimate the uncertainties in the elemental concentrations would be to trace the errors by a Monte Carlo analysis (e.g. Keilis-Borok and Tanovskya, 1967). In some instances I adopt this approach, but for clarification of the important components, I prefer to explicitly trace the errors using linear error propagation.

Note, that

uncertainty introduced by variations in soil moisture, topography and disequilibrium in the uranium and thorium decay series, will not be considered.

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Symbol f(x ) f F(u ) x or E[x ] a b c a e(x ) and E(u ) y e (x ) and Y e (u ) y g (x ) and Y g (u ) la r(x ) a j and a i bj c j and c i cˆk C f˜(u ) and F˜(u ) f(u ) and F(u ) fj g˜(x ) and G˜ (u ) g(x ) and G(u ) h and h o i j J k(x ) and K(u ) k 1 , k 2 and k 3 mj nj p(x ) and P(u ) q rj r sj t tb u = (u, v ) wj w(x ) and W(u ) x = (x, y ) z

Description Fourier transform pair Expected value of the variable x Stripping ratio (Th counts in U window) Stripping ratio (Th counts in K window) Stripping ratio (U counts in K window) Stripping ratio (U counts in Th window) Noise Auto-correlation and power spectral density of the noise Auto-correlation and power spectral density of the signal Linear attenuation coefficient in air Standard deviation of the variable x Aircraft background count rate in window j or channel i Background count rate due to all sources Cosmic count in window j or channel i per cosmic count in cosmic channel Radon calibration coefficients (k = 1, 2, 3 and 4 ) Count rate in cosmic channel Observed ground isotopic composition Ground isotopic composition observed under error free conditions Concentration of j-th element (%K, ppm eU or ppm eTh) Deconvolved ground isotopic composition Actual ground isotopic composition Aircraft height and nominal survey altitude Index for i-th channel in spectrum (i = 1, ...256 ) Index for j-th window (j = 1, ..., 4 ) Mean-squared error Resolution functions Constants derived from stripping coefficients Measured count rate in window j Net count rate in window j Point spread function and its Fourier transform (the transfer function) Radial frequency coordinate q 2 = u 2 + v 2 Radon background in window j Radial spatial coordinate r 2 = x 2 + y 2 Broad source sensitivity for window j Live-time Background estimation time for portable spectrometer Frequency coordinates Interfering count rate in window j Optimum Wiener filter Spatial coordinates Assay precision

Table 1: Definition of symbols used in this chapter.

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Consider some functional combination, f(x, y ), of the two independent random variables (x, y ). Assume that f(x, y ) can be expanded in a Taylor series about the mean values, x and y, ¹f(x, y ) ( ¹f(x, y ) ( 2 2 f(x, y ) = f(x, y ) + x − x) + y − y ) + O([x − x ] ) + O([y − y ] ) ¹x ¹y

(1)

Linearised error analysis requires that, in the region of interest, f(x, y ) can be approximated by the linear terms. Depending on the form of f this would usually imply that there is some limit to the size of the deviations from the mean value, and hence some limit to the size of the errors that can be propagated. Using only the linear terms the variance of f can be approximately calculated by (e.g. Tsoulfanidis, 1995),

r 2 (f) = r 2 (x) ¹f ¹x

2

+ r 2 (y) ¹f ¹y

2

(2)

where r 2 (x) and r 2 (x) are the variances and the derivatives are assumed to be evaluated at the mean values of x and y, i.e at x and y. Certain special cases of this general formula will be used many times in the following analysis including, r 2 (x + y) = r 2 (x) + r 2 (y)

(3)

for addition of independent variables, r 2 (x $ y) = y 2 r 2 (x) + x 2 r 2 (y)

(4)

r 2 (x/y ) = r 2 (x )/y 2 + r 2 (y )x 2 /y 4

(5)

for their multiplication and

for their division.

2.2 Tracing the error introduced by each processing step

Each of the processing steps in a gamma-ray survey (described in section 2.2 of Chapter 2) can influence the final error in the assigned concentrations of potassium, uranium and thorium. In this section I analyse the contribution to the error of each individual processing operation. Some of the

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calculations are involved and to avoid obscuring the major points these are relegated to a series of appendices. All results in this chapter are calculated assuming the calibration constants of the AGSO aircraft at the time it flew the Jemalong survey (Table 2). Parameter Stripping constants α β γ a Sensitivities K (γγ/s per %K) U(γγ/s per ppm eU) Th (γγ/s per ppm eTh) Height correction K (m-1) U (m-1) Th (m-1) Radon constants cˆ1 cˆ2 cˆ3 cˆ4

Base value

Increase with height (h)

0.30987 0.38308 0.83815 0.06

0.00049 0.00065 0.00069 -

13.55 -1.66 0.51

247 e −0.011741h 18.8 e −0.007912h 10.6 e −0.008705h

0.00943 0.01150 0.00747 1.940 0.650 0.010 -0.018

Table 2: Various calibration constants used by the AGSO aircraft when it flew the Jemalong survey (aircraft and cosmic background are given in Table 3). Live-time correction Detection of gamma rays is a Poisson process which implies that estimates of the variance of the measured counts (K, U and Th gamma-rays plus background) can be readily obtained as the variance is equal to the mean. Therefore, for M j mean observed counts the variance will be M j . Given a live-time of t seconds, the variance for the count rate, m j = M j /t, will be

r 2 (m j ) =

M j tm j m j = 2 = t t2 t

(6)

A commonly used spectrometer is the Exploranium GR-820 which has a dead-time overhead per integration of 10 ms, coupled with a certain dead-time per recorded pulse which I assume is constant with count rate. This is a good approximation except at very high count rates (such as within the cosmic widow during cosmic calibration) when the dead-time per unit pulse can change (Cox, pers. comm.). However, the dead-time per pulse is not expected to vary substantially with the typical count rates observed during an airborne survey.

83 0.11 Observed deadtime Calculated deadtime

Dead time (seconds)

0.10

0.09

0.08

0.07

0.06 1250 1500

1750 2000 2250

2500 2750

3000 3250

Total count (γ/s)

Figure 1: Observed and calculated dead-time over the calibration range at Albury, New South Wales. The live-time can have a significant influence on the propagation of errors and it is important to be able to obtain an accurate estimate. The calculation of the live-time requires an estimate of the total number of gamma rays of all energies incident on a detector. One way to achieve this goal is to construct a series of component spectra (Minty, 1996), obtain an estimated spectrum by a linear summation and then calibrate the observed live-time over a series of calibration flights to the number of counts in the modelled spectra. This procedure is described in Appendix C, where it was found that excellent agreement between modelled and observed live-time (or dead-time) could be achieved (Figure 1). Note, that if the assay uncertainty of a pre-existing survey (or part thereof) is being estimated, the average values of the live-time and observed count rates can be used in Equation (6).

Energy calibration (and reduction to four channel count rates) For a correctly functioning spectrometer that is calibrated on the order of every 1000 seconds, accurate energy stabilisation can be easily achieved and the error in the windowed count rates will be small (Grasty and Minty, 1995).

Background removal For each energy window j, estimates of the uncertainty in both the aircraft, r 2 (a j ) and cosmic background, r 2 (c j ) , per unit count in the cosmic channel, can be obtained during calibration as outlined in Appendix D (Table 3). The total cosmic background is obtained as a multiplication of the unit cosmic spectrum with the number of counts in the cosmic channel, C. For the j-th energy window this gives a count rate of c j C which, by the multiplication rule of error propagation, gives a 2

combined variance of c 2j r 2 (C ) + C r 2 (c j ).

The detection of cosmic rays follows a Poisson

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distribution which implies that the variance of the cosmic channel count rate is equal to the mean. Therefore, the variance of the cosmic background in window j is given by C c 2j + Cr 2 (c j ) . The cosmic counts vary slowly and are usually averaged over a 10 or 20 second window which would reduce the uncertainty. For example, a 10 second window reduces the uncertainties from 9.1% to around 2.9%.

Window K U Th

Aircraft background aj r(a j ) r(a j )/a j 13.5 0.11 0.8% 2.8 0.07 2.5% 0 0.08 -

Cosmic background Cc j r(Cc j ) r(Cc j )/Cc j 6.80 0.62 9.1% 5.30 0.48 9.1% 6.17 0.56 9.1%

Table 3: Aircraft and cosmic backgrounds calculated for the AGSO aircraft (assuming 120 cps in the cosmic window). Estimation of the variance of the radon background removed by the spectral ratio technique, is difficult to determine. The principal difficulty is obtaining an estimate of the uncertainty in the count rate of the low energy

214

Bi window, L ob (Equation 6, Chapter 2). This depends on how easily the

curve representing the Comptom continuum can be determined. Therefore, rather than using linear error propagation I use a Monte Carlo simulation procedure developed by Minty (1996), and described further in Appendix E. It involves the construction of a theoretical spectra followed by the repeated addition of Poisson noise and the recalculation of the radon background.

The Monte Carlo procedure uses the same component spectra as that used for live-time corrections (see Appendix C). A theoretical spectra can then be calculated by using these spectral components along with the concentration of K, U and Th (and their live-time corrected sensitivities), the aircraft height, the counts per second in the U channel due to radon and the radon integration time (the number of seconds spectra are accumulated between each radon correction). This last factor can significantly influence the error in the calculated radon background.

Therefore, I adopt the

recommendations of Grasty & Minty (1995) and always use a 200 second integration time.

Finally, note that I assume that low and high radon conditions are essentially independent of altitude. Thus a low radon concentration at 30 m has exactly the same radon counts in the uranium window as a low radon concentration at 150 m. This assertion, while not always the case, is probably valid when the radon is not strongly layered. For example, calibration data obtained over Lake Hume in New South Wales by AGSO on a low radon day (Table 4) show little variation with increase in altitude (see Minty, 1996, p 127). I assume that 10 cps in the uranium window represents a low radon day, while 40 cps represents a high radon day. Of course in certain situations both higher and

85

lower counts are possible, but according to Minty (pers. comm.) these values represent ‘typical’ highs and lows encountered under Australian conditions.

Height (m) 56.8 79.4 103.5 134.4 158.6 184.9 214.8

K (cps) 26.8 27.6 28 28.8 28.4 28.4 28.6

U (cps) 19.2 19.9 20.4 20.7 20.7 21 21

Th (cps) 6.8 7 7.1 7.6 7.7 7.9 7.7

Table 4: Variation in radon background with height over Lake Hume, NSW (data supplied courtesy of The Australian Geological Survey Organisation). Using the addition rule of error propagation given at Equation (3) the variance of the total background for the window, j, is given by r 2 (b j ) = r 2 (a j ) + C c 2j + Cr 2 (c j ) + r 2 (r j )

(7)

The estimated standard deviations of each of these components of the total background for a height of 60 m, typical crustal concentrations of 2% K, 2 ppm eU and 8 ppm eTh, a typical cosmic channel count rate of 120 cps (and a one-second window) and both low and high radon conditions (10 and 40 cps in U window) are shown in Table 5. For potassium and uranium, the errors due to both the cosmic and aircraft backgrounds are very low compared to the errors associated with the calculation of the radon background, while for thorium the errors for all three components are very small. Therefore, aircraft and cosmic background errors can effectively be ignored for all three windows. Note, that even for uranium, the radon error is little effected by an increase in the radon concentration. However (as shown in section 2.3), while increased radon has little impact on the absolute error of the radon correction, it has a substantial impact on the relative error for uranium.

Window K U Th

Aircraft (cps) 0.11 0.07 0.08

Cosmic (cps) 0.62 0.48 0.56

Radon error (cps) Low radon High radon 1.86 1.94 2.36 2.45 0.24 0.25

Total error (cps) Low radon High radon 1.97 2.04 2.41 2.50 0.62 0.62

Table 5: Calculated standard deviations of background errors assuming a height of 60 m and typical crustal concentrations of 2 %K, 4 ppm eU and 8 ppm eTh, and both low and high radon conditions (10 and 40 cps in U window respectively).

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The errors calculated by Equation (7) do not account for any errors in the radon calibration constants. These are difficult to quantify and may result in a larger error in the radon correction, than given by Equation (7), especially after height correction (e.g. Minty, 1996). Equation (7) therefore, establishes a lower bound on the background error.

Calculation of effective height Uncertainty in the effective height influences the processing at two stages. The first is during spectral stripping, where the principal stripping ratios are adjusted for effective height using the conversion factors recommended by the IAEA (1991) and listed in Table 2. The second is for height correction where the counts are adjusted using the exponential model of Equation (13), Chapter 2.

Inaccuracies in pressure and temperature calculation are unlikely to significantly influence the calculation of effective height. For example, a temperature change of 3 oC or a pressure change of 10.1 millibars at standard temperature and pressure is required to change the effective height by 1%. The main factor that influences the error in effective height is the altimeter. The manufacturers specifications on the altimeter used by the AGSO aircraft (Collins ALT-50 radar altimeter) claim an accuracy of ± 2%. At 60 m elevation this results in a ± 1.2 m error in altitude, rising to ± 3 m at 150 m elevation. However, much larger errors in effective height, of the order of 20 m or greater, can occur in practice. This generally occurs when the altimeter bounces off the tops of the trees rather than the ground. Additionally, within dense forests the large amounts of biomass above ground can significantly increase the amount of gamma ray attenuation and cause an increase in effective height (e.g. Bierwirth et al., 1998). The effect of errors in effective height are addressed in the sections on spectral stripping and height correction.

Spectral stripping Applying the error propagation formulae to Equations (11), Chapter 2, for spectral stripping and assuming that the stripping ratios have negligible error, we obtain

r 2 (n Th ) = k 21

m Th 2( 2 mU 2( ) ) t + r b Th + a t + r bU

(8a)

r 2 (n U ) = k 21

mU 2( 2 m Th 2( ) ) t + r bU + a t + r b Th

(8b)

m m m r 2 (n K ) = tK + r 2 (b K ) + k 21 k 22 tU + r 2 (b U ) + k 23 tTh + r 2 (b Th )

(8c)

87

where t is the live-time, m j is the observed count rate, r 2 (b j ) is the variance of the background in window j (either K, U or Th) and k 1 , k 2 and k 3 were defined at Equation 12, Chapter 2. These equations were previously given in Lovborg & Mose (1987) for calculating the error when assaying with a portable spectrometer.

For that mode of survey, the standard method for background

estimation is to count for a long period of time, t b , over a large water body near the survey area. The variance of the background can then be obtained in the same way as the measured count rate and is given by r 2 (b j ) = b j /t b (Equation 6). In most cases the variance of the measured count rates, r 2 (m j ), substantially exceeds the variance of the background correction, r 2 (b j ). For example, Table 6 shows the standard deviations of the measured count rate, the background correction and the result after combining the two sources of error, for average crustal concentrations and a height of 30 m with low radon (low ratio of b j /m j ) and 150 m with high radon (high ratio of b j /m j ). For both conditions the background contributes only a small correction to the combined standard deviations. Therefore, in most cases its contribution can be ignored. Additionally, radon errors are long wavelength and would typically be removed by cross-over ties or micro-levelling. However, uncertainty in the background correction can be important when the data are height corrected (e.g. Appendix F and Minty, 1996).

Window K U Th

30 m; Low radon conditions

150 m; High radon conditions

Poisson

Background

Total

Poisson

Background

Total

22.48 8.63 9.32

1.84 2.24 0.61

22.55 8.92 9.34

14.12 8.48 6.38

1.65 1.99 0.6

14.22 8.71 6.41

Table 6: Standard deviations (in cps) of the measured count rate, the background correction and background corrected (but not stripped) count rate for average crustal concentrations. Results for 30 m elevation with low radon and 150 m elevation with high radon are given. Including uncertainty in the stripping ratios has little impact on the variance of the net count rates. Errors in the stripping ratios caused by inaccuracies in calculating effective height are negligible (less than one-eighth of a percent per meter of error in effective height, using the parameters in Table 2). The main source of uncertainty in the stripping ratios is the systematic error from the pad calibration (or a change in values with time). The calculated standard deviations from calibration of the AGSO spectrometer (from the program PADWIN; e.g. Grasty and Minty, 1995) were around 2%. Propagating errors through Equation (7) including this uncertainty resulted in only a very slight increase in the variance for uranium (generally less than 0.1% increase in relative error), with negligible impact for potassium and thorium. The error in a (Th into U) has the largest impact and

88

will only be significant for cases where the thorium concentration is extremely high relative to uranium.

Height correction The corrected count rate, n j (h o ), at the nominal survey altitude, h o , is related to a stripped count rate of n j (h e ), at an effective height h e , by the exponential model of Equation (13), Chapter 2. If h e = h o + Dh, Equation (2) can be used to show that the variance after height correction, is

approximately given by r 2 n j (h o ) = exp(2lDh ) r 2 n j (h e ) + n j (h e ) l 2j r 2 (Dh ) + Dh 2 r 2 (l j ) 2

(9)

where r 2 (Dh ) is the variance of the error in calculating the effective height and r 2 (l j ) is the error variance of the attenuation coefficient. From this equation I can calculate the relative error after height correction,

r n j (h o ) = n j (h o )

2

 r n j (h e )    + l 2j r 2 (Dh ) + Dh 2 r 2 (l j ) ( ) n h j e  

(10)

From Equation (10) we see that there are two types of influence on the error contributed by height correction: 1. The datum is acquired at a different effective height, h e , than the nominal height, h 0 , which means the error associated with that measurement will be different. If h e > h o , the relative error will be increased while, conversely, if h e < h o the relative error will be decreased. 2. Errors in the calculation of effective height and/or the attenuation coefficient. Given the size of errors in effective height discussed in a previous section (< 5%), and even allowing for a 5% uncertainty in the attenuation coefficients, the errors introduced by these factors will generally be small, except possibly for large height corrections (> 25 m).

Note that the exponential is only modelled with less than 5% error with a first order Taylor expansion in the range -0.29 to 0.35. Therefore, the error propagation in Equations (9) and (10) will only be valid for height corrections of 25 m or less (using typical attenuation coefficients given in Table 2).

While the height correction can have a large influence on the error of any given measurement, its effect over the survey as a whole is usually small (see Appendix F), except possibly in areas with rugged topography. The reason for this is that the aircraft is just as often above the nominal altitude

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(larger error) as it is below (smaller error) and the two effects tend to cancel each other out. Note, that height correction can have a larger effect if the background has been improperly removed (Appendix F and Minty, 1996).

Conversion to ground concentrations The last processing step is the conversion of the background corrected, stripped and height corrected count rates, n j to ground concentrations, f j using the appropriate sensitivity factors, s j (Equation 14, Chapter 2; f j = n j /s j ). The variance of the assigned concentrations can be obtained by using the division rule of error propagation (Equation 5),

r 2 (f j ) =

2 2 r 2 (n j ) n j r (s j ) + s 4j s 2j

(11)

The relative error of the assigned concentrations, can be obtained by dividing the square root of the above equation by f j = n j /s j to give

zj =

r(f j ) = fj

r(n j ) nj

2

+

r(s j ) sj

2

(12)

where z j is the assay uncertainty. The errors in the sensitivities of the AGSO system were calculated by Minty (1996), for effective heights between 50 and 215 m, and range from 1.2% to 1.9% for K, 3.2% to 6.4% for uranium and 0.8% to 0.9% for thorium.

The conversion to ground sensitivity does not affect the relative noise level in the data because every measurement in the survey is converted using the same sensitivity. Its effect will be to introduce a systematic error into the ground concentrations that may become important if the survey is meshed with another flown at a different height or with a different aircraft.

Impact of full spectrum processing Full spectrum processing techniques such as NASVD (Hovgaard, 1997; Hovgaard and Grasty, 1997) and MNF (Green et al., 1988) can significantly reduce the noise in a radiometric survey. The noise reduction results from the use of information outside the standard 4-channel windows. Hovgaard (pers. comm.) estimates that NASVD causes a 1.2, 2.7 and 1.7 fold decrease in the variance for potassium, uranium and thorium respectively. The actual reduction in variance depends on the survey, but these numbers are at least a rough guide. Minty and McFadden (1998) found even larger reductions in error (defined by maximum deviation) of 2.4, 3.4 and 2.5 fold for potassium, uranium

90

and thorium, respectively, by applying NASVD to individual clusters of spectra with similar spectral shape (compared to 1.2, 1.4 and 1.4 fold reductions for the standard NASVD method).

2.3 Assay uncertainty under different conditions

In summary, the main factors that influence the propagation of errors in an airborne gamma-ray survey are; 1. the characteristics of the measuring system, such as the stripping constants, sensitivities, attenuation coefficients (values for the AGSO aircraft are given in Table 2); 2. the count rate in the uranium channel due to atmospheric radon; 3. the potassium, uranium and thorium concentrations; 4. the survey height; 5. the live-time which is a function of the previous variables combined with the integration time. Errors in background correction, effective height, stripping ratios and height correction generally have little effect. However, note that height correction can have a significant impact on the uranium channel when surveying in an area with rugged topography.

There are two contexts in which the error propagation analysis can be applied. The first is for a pre-existing survey where many of the above parameters (e.g. live-time, cosmic and radon background, radioelement concentration) can be estimated by their mean value over the whole, or parts of, the survey. The second is for survey design (or experimentation) where the interest lies in the uncertainty under different user-specified conditions. To calculate the expected error for either of these applications, I wrote a Matlab (MathWorks, 1998) program. It reads in the characteristics of the aircraft (1 and 2 above), and accepts as command line input, the isotope concentrations (f j ), the survey height (h ), the number of counts in the Uranium window due to radon (r U ) and either the live-time (t ) or the integration time (t n ). The program then calculates 1. the background count rate (Equation 7, Chapter 2), b j = a j + c j C + r j ; 2. the stripped count rates in the potassium, uranium and thorium windows (Equation 14, Chapter 2), n j = s j f j ; 3. the measured count rates (by inverting the equation for spectral stripping given at Equation 11, Chapter 2; let this inverse be expressed by A −1 ), m j = A −1 n j + b j ; 4. the error in the radon background estimation (r(r j ), Appendix C), and (if it were not input) the live-time (Appendix A); 5. the errors (r(n j ) in cps), of the stripped potassium, uranium and thorium count rates (Equation 8); 6. the assay uncertainty by z j = r(n j )/n j .

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For the AGSO system I used this program to calculate the relative error assuming typical concentrations of 2% K, 2 ppm eU and 8 ppm eTh, for a range of heights between 30 and 150 m, and for radon conditions of 10 and 40 cps in the uranium window (Figures 2a-c). I also calculated the relative errors for a high thorium concentration of 16 ppm eTh, to illustrate the effect of spectral stripping on the error budget.

16

125 8 ppm eTh 16 ppm eTh

8 ppm eTh 16 ppm eTh

100 Uranium relative error (%)

Potassium relative error (%)

14

12

10

75

50 8

6

25 30

60

90

120

150

30

60

Survey altitude (at STP)

90

120

150

Survey altitude (at STP)

25.0 8 ppm eTh 16 ppm eTh

Thorium relative error (%)

22.5

20.0

17.5

15.0

12.5

10.0

7.5 30

60

90

120

150

Survey altitude (at STP)

Figure 2: Variation of assay uncertainty with height for potassium, uranium and thorium for standard concentrations of 2 %K, 2 ppm eU and either 8 or 16 ppm eTh. For each thorium concentration, results for both low (10 cps in U) and high (40 cps in U) are shown. The high radon curve always appears above the low radon curve. Starting with thorium, the radon concentration has little effect on the relative error, especially when the thorium concentration is high (Figure 2c). This is to be expected as the number of counts in the thorium window due to radon is one-tenth the value in the uranium window (Table 2). Coupled with

92

the generally higher count rates in the Th window over U and the low value of the reverse stripping ratio, a, this keeps the relative error down. For the standard earth model with 8 ppm eTh, the relative error starts at ~14% at 30 m, rising to ~24% at 150 m. With higher thorium (16 ppm eTh) the relative error ranges from 9.6% at 30 m and low radon to ~16% at 150 m and high radon.

For uranium the relative errors are much larger, even at 30 m with low radon background (Figure 2b). In many cases the relative error for uranium exceeds the average stripped uranium count rate. For 8 ppm eTh, the low radon range is 36-87% and high radon range, 44-113%. For 16 ppm eTh the low radon range is 44-106% and high radon 50-124%. The high relative errors for uranium are due to three factors: 1. low count rates due to low concentrations of uranium in typical crustal material; 2. the presence of atmospheric radon; 3. the large number of thorium gamma rays scattered into the uranium window.

The situation for potassium is more favourable than either thorium or uranium (Figure 2a). Both increasing radon and thorium have little influence on the potassium relative error. For 8 ppm eTh and low radon the relative error ranges from 6.3-13.0%, and is only marginally effected by higher radon, with a range of 6.7-14.6%. With a high thorium concentration of 16 ppm the low radon range is 6.7-14.0% and the high radon range is 7.1-16.0%. The relative errors for potassium are low due to the high count rates typically observed in the potassium window. For example, with average crustal concentrations, the background corrected and stripped count rates observed at 60 m elevation by the AGSO system would be around 272 cps for K, 20 cps for U and 54 cps for Th.

An application of the method to the estimation of the assay uncertainty of an existing survey is given in section 3.4. 3 SPATIAL RESOLUTION OF RADIOMETRIC SURVEYS

Now that we are able to estimate the noise in a radiometric survey, we turn to the estimation of the spatial resolution. One method to infer the resolution is to assume that once the data have been collected and processed using standard techniques (section 3.3, Chapter 2), they will be deconvolved by a Wiener filter (Wiener, 1949; Craig et al., 1998). The effect of this filter is dependent on the shape of the point spread function (PSF) presented in the last chapter, and on the signal-to-noise ratio. The idea behind the method is that at some point in the frequency domain, the Wiener filter will begin to severely attenuate both signal and noise. While the exact criterion used to determine the cutoff is subjective, this point can be used to infer an upper limit (u c ) on the frequencies resolvable in

93 an airborne survey. The Nyquist relationship (Dx = 1 ) can then be used to specify an appropriate 2u c sampling regime.

3.1 Theoretical background

In the last chapter I derived a model (the so called point spread function, PSF) of airborne gamma-ray spectrometry based on geometrical arguments. The shape of the PSF is determined by the nominal survey altitude, h 0 , the attenuation of gamma rays in the atmosphere, l a , and the movement of the aircraft, vDt, during the integration time, Dt. The PSF defines a model that can be used to convert an arbitrary distribution of ground sources, g(x ), to count rates, f(x ) at the nominated survey altitude. Assuming small variations in survey altitude and topography, they are related via the convolution equation f(x ) = ° ‘2 g(x − y )p(y )dy

(13)

Equation (13) has a very simple form in the Fourier domain, F(u ) = G(u )P(u )

(14)

where I adopt the convention that upper and lower case functions, f(x ) f F(u ) denote a Fourier transform pair and u = (u, v ) is a frequency coordinate. Assuming x and y are in meters u and v will be spatial frequencies with units of m −1 . Note, that to simplify matters, f(x ) should first be converted to ground concentration, so that f(x ) and g(x ) have the same units and P(0 ) = ° ‘2 p(y )dy = 1. The effect of measurement is to smooth the true ground distribution by the point spread function. In an airborne survey, samples of f(x ) are recorded and, therefore, F(u ) can be estimated. However, g(x ) and, hence, G(u ) are required. A simple way to estimate G(u ) is by spectral division,

G(u ) =

F(u ) P(u )

(15)

The problem with this equation is that F(u ) is not known exactly and at high frequencies it is dominated by noise. As P(u ) decreases rapidly with frequency the denominator of Equation (15) can be very small at high frequencies, resulting in considerable amplification of noise. To minimise this

94

contamination, the deconvolution should occur with a filter, W(u ), derived from both the PSF and knowledge of the noise levels, G(u ) = F(u )W(u )

(16)

For frequencies where the signal-to-noise ratio is high the filter should behave as 1/P(u ), while where low it should approach zero.

3.2 The Optimal Wiener filter The observed response, f˜(x ), consists of the signal, f(x ), and an unwanted noise component, e(x ), f˜(x ) = f(x ) + e(x ) e F˜(u ) = F(u ) + E(u )

(17)

From these noisy observations it is desired to extract an estimate, g˜(x ) = °ℜ 2 f˜(y )w(x − y )dx

(18)

that minimises the mean squared error between estimated and actual isotope concentration, J =E[g˜(x ) − g(x )]

2

(19)

where E[$ ] is the expected value. Equation (18) is the space domain version of the filter given at Equation (16). To derive the optimum filter, some assumptions need to be made regarding the characteristics of e(x ) and g(x ). If they are both assumed to be independent random variables that are wide-sense stationary (constant mean and a covariance that depends only on the distance between two points) with zero mean, they are described by their spatial auto-correlations (or covariances as the two are identical when the mean is zero), y e (x ) =E[e(y )e(y + x )]

for the noise and

(20)

95 y g (x ) =E[g(y )g(y + x )]

(21)

for the concentration. The condition that g(x ) is stationary with zero mean doesn’t create a problem as its mean value, or any polynomial trend, can (and should) be removed and calculated independently (see Appendix K). When a k-degree polynomial required to achieve stationarity, g(x ) is viewed as an intrinsic random function of order k (e.g. Matheron, 1973) and y g (x ) is known as a generalised covariance function.

With the above assumptions, some tedious algebraic manipulation, followed by Fourier transformation, can be used to show (e.g. Helstrom, 1967), that the expected mean-squared error is, 2  P & (u )Y g (u ) Y e (u )Y g (u )  J = °ℜ 2  W(u ) − G(u ) +  du ( ) Gu G(u )  

(22)

where the asterisk denotes complex conjugation, Y e (u ) and Y g (u ) are the Fourier transforms of the auto-correlations (the so-called power spectral densities, e.g. Wiener, 1949; Papoulis, 1991) and G(u ) = P(u ) 2 Y g (u ) + Y e (u ). The PSF is real and even which implies that its Fourier description is

also real and even and hence, P & (u ) = P(u ). Differentiating with respect to W(u ), it is straightforward to show (e.g.. Helstrom, 1967) that the mean-squared error will be a minimum when

W(u ) =

P(u )Y g (u ) Y g (u ) P(u ) 2 + Y e (u )

(23)

This optimum Wiener filter has a mean-squared error given by substituting Equation (23) into (22), which zeros the first term, to give

J = °ℜ 2

Y g (u )Y e (u ) du Y g (u ) P(u ) 2 + Y e (u )

(24)

If g(x ) and e(x ) are Gaussian, the spatial covariances of Equations (20) and (21) are seen as a-priori information that can be combined by Bayes’ rule to give a posterior probability density function. The mean-squared error of Equation (19) is then the log-likelihood, and its minimisation is equivalent to maximising the posterior probability (e.g. Helstrom, 1967).

96

The presence of the noise term in Equation (24) means that convolution with the PSF, followed by deconvolution with the Wiener filter, will not return the original input. The extent that the result will agree with the input can be determined by cascading the two filtering operations together,

K(u ) = P(u )W(u ) =

Y g (u ) P(u ) 2 Y g (u ) P(u ) 2 + Y e (u )

(25)

The filter K(u ) has a maximum of one and shows to what extent frequencies are attenuated by the successive operations of convolution and deconvolution. When there is no noise, Y e (u ) = 0, hence K(u ) = 1 and all frequencies are returned unaffected.

The inverse Fourier transform, k(x ), of

Equation (25) is a resolution function (e.g. Parker, 1994) and shows how a delta-function input is affected by convolution with P(u ) followed by deconvolution with W(u ). For the zero-noise case, the delta-function input would be regained, while in the general case the output will be a smooth function with a width determined by the signal-to-noise ratio.

Equations (24) and (25) are the principal tools used in this thesis for investigating the spatial resolution. They are related as J = ° ‘2 Y g (u ) 1 − K(u ) du

Note, that we will be more interested in J u (u ) =

(26)

¹J(u ) = Y g (u ) 1 − K(u ) , which shows how rapidly ¹u

the mean-square error increases with increasing frequency.

The variation of J u (u ) and K(u ) with frequency is governed by both the signal-to-noise ratio and the strength of P(u ) at any given frequency. To illustrate how these quantities can be used to infer spatial resolution I consider a simple 1-D analogue of a radiometric survey. For the PSF I use the falloff curve for thorium illustrated previously at Figure 14, Chapter 2, and normalise so that P(0 ) = 1. I choose the power spectral density of the signal to be Y g (u ) = f 2 u −2 , which effectively

assumes that the signal contains discontinuities (Bracewell, 1986 and next section). I assume the noise is white (see next section) which implies that its power spectral density is constant with frequency Y e (u ) = r 2 . With these assumptions I find

J u (u ) =

and

r2 P(u ) + u 2 r 2 /f 2 2

(27)

97

K(u ) =

P(u ) 2 P(u ) 2 + u 2 r 2 /f 2

(28)

To assess how the ratio of f 2 /r 2 effects Equations (27) and (28) I arbitrarily normalise to the frequency 0.02 m-1 and define R = 2500f 2 /r 2 . Figure 3a shows how Equation (27) varies with frequency for R =1, 2, 4, 16 and 64, or in decibel terms R =1, 3, 6, 12 and 24 dB. The curves all increase to a maximum at which point they decay towards zero at an asymptotic rate of O(u −2 ). The Wiener filter represents the optimal compromise (in a least-squares sense) between reconstructing valid signal and attenuating unwanted noise. The presence of noise at all frequencies means that whenever the Wiener filter amplifies signal it also amplifies noise. At low frequencies this noise contamination is outweighed by the benefits of reconstructing the signal. As the frequency increases the PSF decreases causing the signal strength to reduce, while the noise level remains the same so that the penalty for reconstruction increases. The maximum in Figure 3a represents the point at which the amplification of noise becomes more important than the reconstruction of signal. At higher frequencies the Wiener filter effectively stops reconstructing signal and begins to attenuate both signal at noise. The maximum point is one subjective measure of the spatial resolution and changes only slowly with increase in signal-to-noise ratio (Figure 3b). Fitting a model of log 10 (R ) to the maximum point I find, u max = 0.0063 + 0.0028 log 10 (R ).

The logarithmic dependence of the

spatial resolution on the signal-to-noise ratio has two important implications. Firstly, increasing the signal-to-noise ratio by a factor of two, results in only an additive increase in the maximum frequency. Secondly, the estimates of signal and noise don't have to be overly accurate to obtain sensible estimates of the spatial resolution. (a)

0.015 R = 1 (0 dB) R = 2 (3 dB) R = 4 (6 dB) R = 16 (12 dB) R = 64 (24 dB)

6.0

5.0

Ju(u)

4.0

3.0

2.0

1.0

(b) Calculated Logarithmic fit

0.014 Frequency (m-1) where J'(u) is maximum

7.0

0.013 0.012 0.011 0.010 0.009 0.008 0.007

0.0

0.006

0.000

0.005

0.010 Frequency (m-1)

0.015

0.020

0

5

10

15

20

25

Signal-to-noise ratio (dB)

Figure 3: (a) Change in mean squared error, J u (u ), for different values of R for the synthetic 1-D radiometric survey. (b) Frequency where J u (u ) is maximum.

30

98 Turning now to K(u ), Figure 4a shows how it varies for R =1, 2, 4, 16 and 64. For low frequencies K(u )~1 and frequencies are largely unaffected by the filter (in signal processing literature this would

be called the pass band, e.g. Kuc, 1988). As the frequency increases the filter begins to attenuate signal (the transition band) and at some point the filter effectively eliminates all signal (the stop band). Measures of spatial resolution obtained from K(u ) are subjective, one example is the point, u 1/2 , where K(u ) = 12 . This is equivalent to the point where the signal-to-noise ratio is unity (i.e.

where Y e (u ) = Y g (u ) P(u ) 2 ). Figure 4b shows how u 1/2 changes with increasing signal-to-noise Again it is logarithmic, u 1/2 = 0.0055 +0.0024 log 10 (R ) which implies that multiplicative

ratio.

improvements to the signal-to-noise ratio only additively change u 1/2 . (a)

0.013 R = 1 (0 dB) R = 2 (3 dB) R = 4 (6 dB) R = 16 (12 dB) R = 64 (24 dB)

0.8

K(u)

0.6

0.4

(b) Calculated Logarithmic fit

0.012 Frequency (m -1) where K(u)=1/2

1.0

0.011 0.010 0.009 0.008 0.007

0.2 0.006 0.0

0.005

0.000

0.005

0.010

0.015

0.020

0

-1

Frequency (m )

5

10

15

20

25

30

Signal-to-noise ratio (dB)

Figure 4: (a) Resolution curves for different values of R for the synthetic 1-D radiometric survey. (b) Frequency where K(u ) = 12 . Figure 5 illustrates how the resolution functions, calculated by inverse Fourier transformation of K(u ), change for R = 1, 4 and 64. These show how point sources are smeared by the successive

operations of convolution and deconvolution. As the signal-to-noise ratio increases the width of the resolution function decreases while its height increases. In the limit of infinite signal-to-noise ratio the resolution functions become delta functions.

3.3 Estimating the auto-correlation of the signal and noise

I turn now to the estimation of the auto-correlations of the signal and noise. Considering the noise first, it will be assumed to be a white noise process, which has an auto-correlation given by

99 y e (x ) = r 2 d(x )

(29)

where d(x ) is the Dirac delta function. The power spectral density is constant with frequency, Y e (u ) = r 2

(30)

0.025 R = 1 (0 dB) R = 4 (6 dB) R = 64 (24 dB)

0.020

k(x)

0.015

0.010

0.005

0.000

-0.005 -500

-250

0

250

500

Distance (m)

Figure 5: Resolution functions for three values of R for the synthetic 1-D example. The auto-correlation function specified by Equation (29) has been chosen for its simplicity. The true noise process would be more complex and, in particular it would: 1. be coloured rather than white (i.e. the auto-correlation would be non-zero away from the origin); 2. depend on position (i.e. would not be stationary); 3. depend on the signal strength, g(x ).

These conditions arise for two reasons. Firstly, the noise has a strong Poisson component whose variance depends on the concentration. Secondly, the background corrections will change over the course of the survey. This effect would be most noticeable for uranium and to a lesser extent potassium.

However, as the example in the previous section illustrated, changes in the

signal-to-noise ratio have only a small effect on the spatial resolution. Therefore, we can obtain a sensible estimate of the spatial resolution by using the techniques developed in section 2 for estimating the noise under typical conditions (e.g. standard crustal concentrations of 2% K, 2 ppm eU and 8 ppm eTh and average radon conditions).

100

I turn now to the estimation of the auto-correlation of the concentration. Leaving aside issues related to its magnitude, Fourier theory can be used to constrain the shape of the power spectrum. Specifically, if a two-dimensional function and its first n − 2 derivatives are continuous, then the Fourier transform behaves as O(Èu È −n ) as Èu È d º (e.g. Bracewell, 1986). This means that the expected smoothness of g(x ) can be used to constrain the shape of the power spectrum. For example, if g(x ) is continuous but has discontinuous first derivative the transform will decay as O(Èu È −3 ), if the second derivative is discontinuous the decay will be O(Èu È −4 ) etc.

When G(u ) decays as

−n −2n O(Èu È ) it implies that Y g (u ), which is a measure of power spectral density, decays as O(Èu È ).

At this point it was intended to use the results of a detailed ground based survey to estimate the form of the auto-correlation. The survey was conducted within the Jemalong-Wyldes Plains using a 4.2 litre crystal mounded on a quad-bike. Line spacing was either 10 or 20 m and samples were collected at 3 m intervals along each line. Unfortunately, there were electrical problems with the crystal system which made energy calibration difficult and which adversely affected the instrument live-time. Therefore, the survey could not be used for its intended purpose.

An alternative technique is to estimate the auto-correlation from an existing airborne survey. This, in many ways, is similar to the procedure for general deconvolution: the original signal needs to be extracted from observations that have been smoothed by the measurement process and corrupted by noise. Up to this point, the description is this chapter has been cast in general terms. From this point onwards I concentrate on the Jemalong radiometric survey and derive results that strictly apply for that survey only. Therefore, the conclusions drawn are tentative and their extrapolation to other areas should be treated with caution. Ideally, we would like to build up a library of power spectra from different surveys and from these determine what shape and magnitude the power spectral density might take in different areas.

Power spectra were calculated for the Jemalong survey using the methodology outlined in Chapter 6. Briefly, a thin-plate spline surface was fit to the data, and the Fourier transform calculated exactly using a formula derived in Appendix H. The transform was calculated for 100 radial frequencies between 0 and 0.01 m-1 at 90 angles between 0 and 180o (total 9,000 frequencies). For each radial frequency the average power was then calculated which reduces the uncertainty in the power spectrum caused by noisy observations. The across-line spacing of the survey was 100 m (frequency cutoff 0.005 m-1) and along-line interval 60 m (0.0083 m-1). Therefore, all frequencies above 0.0083 m-1 are more a function of the interpolation method than data variation, while those values between 0.005 and 0.0083 m-1 are probably suspect. The Fourier transform of a thin-plate surface grows as

101 −2 O(Èu È ) near the origin, where it is unbounded. Therefore, all frequencies below 0.00025 and

above 0.0083 m-1 were not considered in the following analysis.

The observed spectrum can be decomposed into a signal and a noise component, F˜(q, h ) = G˜ (q, h )P(q, h ) + E(q, h )

(31)

where (q, h ) are radial frequency coordinates. The point spread function is approximately radially symmetric, so that the radially averaged power spectrum can be written as E[ F(q ) 2 ] = P(q ) 2 E[ G(q ) 2 ] +E[ E(q) 2 ]

(32)

where the signal and noise are assumed uncorrelated so that their cross product is zero. I hypothesise that the signal follows a power law, i.e. E G˜ (q )

2

= f 2 q −2n , where f and n are to be determined.

Note, that q −2n d º as q d 0, but this doesn’t create a problem as the signal dominates noise at low frequencies and can be assumed effectively infinite. Indeed if it is felt to be a problem, we can specify a minimum frequency either by the size of the largest feature in the survey, or the spatial extent of the survey, and set the power spectrum to a constant value or decrease it towards zero. At low frequencies the noise term can be ignored and after taking logarithms of both sides of Equation (32) I find, log[F(q )] − log[P(q )] + n log[q ] = log(f )

(33)

For a given n, we can solve for log(f ) by calculating the average value of the left-hand-side of the above equation.

Figures 6a-c show the power spectra for potassium, uranium and thorium compared to models with n = 0, ½ and 1, plotted on a log-linear scale. The spectra for the three radioisotopes are very similar

in shape. For each a covariance model with n = 0 gives a very poor fit at all frequencies. For n = 1 the fit is good down to about 0.004 m-1 at which point it decays faster than the observed spectrum. For these higher frequencies an intermediate value of n =½ gives the best fit. This may indicate a change in behaviour of the signal, although at these high frequencies we expect the power spectrum to comprise a large component of noise. Therefore, I tentatively conclude that E G˜ (q )

2

= f 2 q −2

fits all three bands of the Jemalong survey (with f 2 = 45.8, 41.2 and 299 for potassium, uranium and thorium, respectively). This is equivalent to assuming that isotope concentrations are discontinuous,

102

not necessarily at the finest scale, but at least within distances substantially less than the size of the measurement footprint.

0

(a)

0

(b)

Power spectrum n=0 n = 0.5 n=1

Power spectrum n=0 n = 0.5 n=1

-20 Power (dB)

Power (dB)

-20

-40

-40

-60

-60

0.000

0.002

0.004

0.006

0.008

0.010

0.000

0.002

-1

Frequency (m )

0.006

0.008

0.010

Frequency (m-1)

(c)

0

0.004

Power spectrum n=0 n = 0.5 n=1

Power (dB)

-20

-40

-60 0.000

0.002

0.004

0.006

0.008

0.010

-1

Frequency (m )

Figure 6: Power spectra calculated for the Jemalong radiometric survey for (a) potassium, (b) uranium and (c) thorium. Best fitting power laws with n = 0, ½ and 1 are also shown (Note the curves shown are actually P(q ) 2 f 2 q −2n ). At this point we should make a clear distinction between the spatial expectation used to estimate G˜ (q )

2

in the last paragraph and the ensemble average implied by Equation (21) for the

auto-correlation function and its power spectral density. For the latter, the expectation operates on multiple realisations of the random process assumed to underlie g(x ).

If we assume that the

auto-correlation is spatially ergodic (e.g. Papoulis, 1991), then the power spectral density, Y g (q ), can be estimated from the spatial average since

103

E G˜ A (q ) Y g (q ) = lim A dº A

2

(34)

where the A subscript implies that G˜ A (q ) was calculated from a region or subset of the plane, A, with area A . The Jemalong radiometric survey covered an irregularly shaped area and the spacing between measurements varied. With average spacings of Dx (100 m) and Dy (70 m) and N (18824) points the area is A = NDxDy = 1.32 % 10 8

(35)

3.4 Spatial resolution of the Jemalong survey

For the Jemalong data considered in Chapter 2, the standard 4-channel processing gave mean potassium, uranium and thorium concentrations of 1.15 %K, 1.87 ppm eU and 10.4 ppm eTh, an average radon background of 10 cps in the uranium channel and a live-time of 920 milliseconds. Applying the method developed last section for error propagation, I estimate the assay uncertainties as 11.2, 49 and 13.4% for potassium, uranium and thorium respectively.

For the j-th radioelement, the signal-to-noise ratio is,

SNR j (q ) =

f 2j P j (q ) r 2j q 2

2

(36)

To allow the quantities for the three radioelements to be meaningfully compared I divide both the top and bottom of the above equation by the square of the average isotope concentration f j so that

SNR j (q ) =

fj/ f j

2

P j (q )

2

(37)

2

rj/ f j q2

and the power spectral densities are scaled to

Y g (q ) =

f 2j 2

f j q2

and Y e (q ) =

r 2j 2

fj

(38)

104

The normalisation reveals the unfortunate aspects of surveying for uranium (Figure 7a). Firstly, the signal strength of the uranium band is the lowest of the three isotopes. Secondly, noise in the uranium band is an order of magnitude larger than it is for potassium or thorium. The low signal and high noise mean that uranium has a very poor signal-to-noise ratio (Figure 7b). Potassium, as expected, has the most favourable signal-to-noise characteristics, with the situation for thorium only slightly worse. (a)

(b)

0

40

Potassium Uranium Thorium

Potassium Uranium Thorium

20 Noise (U)

-40

Noise (K and Th)

Signal-to-noise ratio (dB)

Normalised power (dB)

-20 0

-20

-40

-60 -60

-80

-80

0.000

0.005

0.010

0.015

0.000

0.005

-1

Spatial frequency (m )

0.0010

0.015

-1

Spatial frequency (m )

Figure 7: For the Jemalong survey: (a) Normalised power spectral densities for the noise and concentration (after the latter has been multiplied by P(q ) 2 ). (b) Signal-to-noise ratio. A plot of the change in the mean-square-error with frequency, J q (q ), further emphasises the relative situations of the three isotopes (Figure 8a). For uranium, the curve very rapidly approaches a maximum at a frequency of 0.0018 m-1. For thorium, the maximum frequency is increased to 0.0059 m-1 and for potassium to 0.0072 m-1. These frequency cut-offs correspond to sampling rates of 69 m for potassium, 278 m for uranium and 85 m for thorium. A similar situation prevails for the resolution function, K(q ) (Figure 8b), where q 1/2 for potassium, uranium and thorium is 0.0062, 0.0025 and 0.0052 m-1 or Dx =80, 203 and 96 m, respectively. Note, that the q 1/2 points in Figure 8b are the same as the zero crossings of the signal-to-noise ratio curves in Figure 7b.

For the next three sections, I assume that the signal and noise characteristics are the same as the Jemalong survey and investigate the effect on the spatial resolution of the assay uncertainty, the aircraft height and the movement during the integration time.

105

(a)

(b)

0.30

1.0 Potassium Uranium Thorium

0.25

Potassium Uranium Thorium 0.8

0.20

K(q)

Jq(q)

0.6 0.15

0.4 0.10

0.2

0.05

0.00

0.0

0.000

0.005

0.010

0.015

0.000

Spatial frequency (m-1)

0.005

0.010

0.015

Spatial frequency (m-1)

Figure 8: For the Jemalong survey, (a) J q (q ) and (b) K(q ), for potassium, uranium and thorium 3.5 Influence of assay uncertainty on spatial resolution

To investigate how the spatial resolution changes with assay uncertainty, I calculated the frequencies where J q (q ) was a maximum and K(q ) = 1/2, for multiple assay uncertainties between 0.001 and 1.0, assuming a survey altitude of 60 m (Figure 9). A value of r/ f = 1 corresponds to the unfortunate situation where the uncertainty equals the mean value, a circumstance that can occur for uranium under high radon conditions (section 2.3). The spatial resolution changes linearly with the logarithm of assay uncertainty except at high values, where there is a slight divergence for both the mean-square-error and resolution curves. Coefficients for best fitting models of the form

q c = a o + a 1 log 10

r(f ) f

(37)

where determined by regression within the frequency ranges where the relationship was linear (Table 7). The logarithmic behaviour re-emphasises that estimates of the signal-to-noise ratio don’t have to be overly accurate to return sensible estimates of the spatial resolution. For example, doubling the potassium assay uncertainty causes only a linear decrease of 0.0014 m-1 in the frequency cut-off calculated as K(q ) = 1/2. At 60 m elevation for the Jemalong survey, this results in a change of 23 m in Dx from 80 m up to 103 m. Similarly, reducing the assay uncertainty by one-half causes a 15 m decrease down to 65 m.

These calculations imply that the 1.1, 1.65 and 1.3 fold reductions

(Hovgaard, pers. comm.) in assay uncertainty expected from using the standard NASVD technique

106

will have little impact on the spatial resolution. The effect of the 2.4, 3.4 and 2.5 fold reductions expected from the Minty and McFadden (1998) method for, respectively, potassium, uranium and thorium, will be more significant (Table 8). The largest effect is for uranium, where the predicted spatial cutoff is reduced by almost a half from 203 m down to 110 m. (a) 0.012 Potassium Uranium Thorium

0.012

0.010

0.008

0.006

0.004

Potassium Uranium Thorium

0.008

0.006

0.004

0.002

0.002

0.000 0.010

(b)

0.010 Frequency (m-1) where K(q)=1/2

-1

Frequency (m ) where Jq(q) is maximum

0.014

0.000 0.100

1.000

0.010

Assay uncertainty

0.100

1.000

Assay uncertainty

Figure 9: Effect of assay uncertainty on (a) the maximum rate of change of the mean-square error and (b) the half-point of the resolution curve.

Element Potassium Uranium Thorium

Mean-square error maximum Intercept (a 0 ) Slope (a 1 ) 0.0017 -0.00573 0.00014 -0.00580 0.00085 -0.00577

Resolution Intercept (a o ) Slope (a 1 ) 0.0017 -0.00479 0.00068 -0.00466 0.0011 -0.00473

Table 7: Coefficients from Equation (37) for the change in mean-square-error and the half point of the resolution function.

Element Potassium Uranium Thorium

Standard 4-channel method Space cutoff (m) q 1/2 0.0062 80 0.0025 203 0.0052 96

NASVD by cluster Space cutoff (m) q 1/2 0.008 62 0.0046 110 0.007 71

Table 8: Change in the K(q ) = ½ measure of spatial resolution from using the NASVD by cluster method developed by Minty and McFadden (1998). The calculations assume the assay uncertainties are reduced by 2.4, 3.4 and 2.5 fold for potassium, uranium and thorium, respectively.

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3.6 Influence of aircraft movement on spatial resolution

Aircraft velocity has no effect across lines, while along lines it causes a more rapid falloff in the PSF with increasing frequency (see Figure 14 in Chapter 3). With a movement of Dx the increase in falloff rate is proportional to sinc(uDx ), which implies that at u = 1/Dx the PSF passes through zero. Practically, this is of little concern as this frequency is double the Nyquist sampling cut-off, 1/(2Dx ).

Figure 10 shows the effect of velocity on K(u, v ) (which is no-longer radially symmetric) for potassium from the 70 m movement between adjacent samples in the Jemalong survey. At large assay uncertainties the movement has little effect on the frequency cut-off which reflects that at low frequencies the movement only slightly decreases the PSF. At smaller assay uncertainties, the frequency cut-offs are larger and lie within regions where the movement has a more substantial effect on the PSF. Correspondingly, the cut-offs diverge from the zero velocity case. For the Jemalong survey the velocity decreases the frequency cut-offs from 0.0062 to 0.0058 m-1 for potassium, 0.0247 to 0.0242 m-1 for uranium and 0.00523 to 0.00494 m-1 for thorium. These correspond to sampling cut-off changes of 80 to 86 m, 203 to 207 m and 96 to 101 m. (a) 1.0

0.012

(b) Across lines Along lines

Across lines, K(0,v) Along lines, K(u,0) 0.010 Frequency (m-1) where K(q)=1/2

Resolution curves

0.8

0.6

0.4

0.2

0.008

0.006

0.004

0.0

0.002

0.000

0.005

0.010

0.015

Spatial frequency (m-1)

0.010

0.100

1.000

Assay precision

Figure 10: Effect of aircraft movement (70 m at 60 m elevation) on (a) the resolution curve of potassium for the Jemalong survey (11.2% assay uncertainty) and (b) the frequency cutoff as the assay uncertainty changes. 3.7 Effect of aircraft height on spatial resolution

The aircraft height influences the spatial resolution in two ways. The first is to change the number of recorded counts which, in turn, alters the assay uncertainty. The second is to modify the footprint of

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a single measurement. As the height is increased this causes a more rapid falloff in frequency of the PSF (Figure 14, Chapter 3). As the aircraft height varies the recorded count rates change approximately as E 2 (l a h ), where l a is the linear attenuation coefficient in air for the given gamma-ray. The power spectral density of the signal was recorded at 60 m elevation with the normalisation that P(0 ) = 1. The PSFs at other heights were normalised to this value by multiplying them by E 2 (l a h )/E 2 (60l a ).

The resolution functions for potassium that would be obtained at different heights over the Jemalong area are shown in Figure 11. At 30 m elevation there is significant signal passed at high frequencies, with a cutoff of 0.0117 m-1. As the height is increased the high frequency content is progressively attenuated and by 150 m elevation the cutoff has been reduced to 0.0023 m-1. Increasing height also reduces the difference (in both a relative and absolute sense) between thorium and potassium (Figure 12a). This occurs because thorium has a lower attenuation coefficient so that (i) the number of counts recorded decreases less rapidly and (ii) the PSF decays slower with increasing frequency. Note, that if the frequency cut-offs were only dependent on the observed count rate (i.e. the assay uncertainty) they would change at approximately the same rate as the infinite source yield, E 2 (l a h ). The actual cut-offs decay faster than this (Figure 12 a) which highlights the importance of the more rapid falloff in the PSF with increasing height. Finally, fitting a straight-line curve to the potassium sampling cutoff in Figure 12b indicates that a rough rule of thumb for minimal sampling rate in metres is −8 + 1.44h. 1.0 h = 30 m h = 60 m h = 90 m h = 120 m h = 150 m

0.8

K(u)

0.6

0.4

0.2

0.0 0.000

0.005

0.010

0.015

Spatial frequency (m-1)

Figure 11: Resolution curves for potassium at heights of 30, 60, 90, 120 and 150 m.

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0.012

(a)

250 Potassium Thorium Uranium E2(µah)

200 Nyquist sampling cutoff (m)

Frequency (m -1) where K(q) = 1/2

0.010

(b)

0.008

0.006

0.004

150

100

50 0.002

Potassium Uranium Thorium 0

0.000 30

60

90 Height of survey (m)

120

150

30

60

90

120

150

Height of survey (m)

Figure 12: Change in (a) frequency and (b) sampling cut-offs as the height of the aircraft varies between 30 and 150 m elevation. 4 DISCUSSION

In this chapter I developed a method for estimating the assay uncertainty that involved tracing the various sources of error in a radiometric survey using linearised error analysis. The most significant source of error was from the Poisson nature of radiation detection coupled with, for potassium and uranium, the contamination of the windowed count rates by thorium gamma-rays. The Poisson error depends on the number of gamma-rays detected which in turn depends on the aircraft height, the radioelement concentrations, the integration time (live-time) and the background contamination. Other sources of error generally contributed little to the overall error budget, even for uranium under conditions of high radon. However, high radon could have a large effect on the relative error (viz assay uncertainty) for uranium, and to a lesser extent, potassium. Error in the radon background correction was found to be significant when large height corrections were required. Under most circumstances the height correction does not contribute a significant amount to the overall error budget.

An indication of the assay uncertainty at different heights can be obtained by assuming typical crustal concentrations of the radioisotopes (2 %K, 2 ppm eU and 8 ppm eTh) and average radon (25 counts per second in the uranium window). This indicates that for heights of 30, 60, 90, 120 and 150 m, respectively, the assay uncertainty for potassium is 6.5%, 7.9%, 9.5%, 11.5 and 13.7%; for uranium 40%, 49%, 61%, 77% and 100%; and for thorium 14%, 15.9%, 18%, 20.7% and 23.7%. These

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numbers are based on standard 4-channel processing, with multichannel techniques expected to reduce the uncertainties.

In the second part of the chapter we concentrated on developing methods for inferring spatial resolution. Our technique was to assume that once the data have been processed using standard methods, they will be deconvolved by a Wiener filter. The action of the filter at any given frequency depends on the shape of the PSF, the power spectrum of the signal and the noise levels in the data. For the Jemalong survey the signal was found to follow a power law with its amplitude decaying at O(q −1 ).

The noise was assumed white with its strength calculated by the methods used for

determining the assay uncertainty. With these assumptions we found that the assay uncertainty only weakly affected the spatial resolution. This implied that standard NASVD technique would only marginally improve the spatial resolution of a radiometric survey, with the NASVD by cluster method having a more substantial influence. Incorporating aircraft movement only had a significant impact when the assay uncertainty was low. It had no effect across-lines and caused a decrease in the frequency cutoff along-lines. Aircraft height had a very significant effect on the spatial resolution, with the reduction is resolution occurring faster than the decay in infinite source yield.

The

difference between the resolution calculated for potassium and thorium decreased with increasing height, due to the latter’s more penetrating gamma-rays.

I know draw some tentative (as they apply only for the signal model for the Jemalong survey) conclusions about current radiometric survey practice using the K(q ) =½ criterion. The spatial resolution for uranium is generally very poor and for thorium it is slightly worse than for potassium. Therefore, if potassium is adequately sampled, then so will uranium and thorium. The spatial resolution predicted by the methods in this chapter are only rough estimates. To ensure adequate coverage some degree of over-sampling would be desirable, which I will here assume is 1.5 times the critical sampling rate. I also adopt a conservative approach and use the assay uncertainties expected from processing with a standard 4-channel method.

The Jemalong radiometric survey was flown at 60 m elevation, with the potassium assay uncertainty estimated at 10.9%. This gave expected frequency cut-offs of 0.0062 m-1 across lines and 0.0058 m-1 along lines. These correspond to critical sampling rates of 80 m and 86 m respectively. With 1.5 fold over-sampling the recommended sampling rates are 53 m across-lines and 57 m along-lines. Therefore, for the Jemalong survey the signal was under-sampled across-lines (100 m) and adequately sampled along-lines (70 m), although we would recommend slightly finer along-line sampling.

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Table 9 shows how the sampling rates would change if the Jemalong survey were flown at a different height, but with the same along-line sample rate, isotope concentrations and background characteristics.

Note, that if a finer along-line sample rate were chosen the critical sampling

along-lines would decrease. As the ratio of movement to height decreases the difference between along- and across-line sampling rates diminishes, which is apparent in the table.

Height 30 60 90 120 150

Critical sampling (m) Across-lines Along-lines 50 62 80 86 107 111 134 136 160 161

Recommended sampling (m) Across-lines Along-lines 33 41 53 57 71 74 89 90 107 107

Table 9: Critical and recommended sampling rates for the Jemalong survey if it were flown at different heights. Except at very low elevations the aircraft line-spacing of existing surveys is unlikely to be less than 100 m. This implies that the both critical and recommended across-line sampling rates are unlikely to be achieved in practice, except possibly at very low elevations, where much closer line spacings are used. Along-lines the critical sampling rates should be easily achieved (assuming low altitude surveys are flown at less than 60 m/s) , while the recommended rates would usually be achieved at heights greater than about 60 m.

In conclusion, the analysis indicates that surveys flown at heights of 60 m or greater are under-sampled across-lines and over-sampled along lines (a situation that also prevails for aeromagnetics).

There is little that can be done to remedy the situation without significantly

increasing the cost of geophysical surveys by decreasing the line spacing. The analysis is perhaps most pertinent for low altitude surveys, which are costly and where the recommended sampling rates are not significantly different than current practice (e.g. 30 m elevation, 40 m sampling along lines, 25 to 50 m spacing between samples).

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CHAPTER 5 A GENERAL FRAMEWORK FOR INTERPOLATION AND SMOOTH FITTING OF GEOPHYSICAL DATA Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell

OBJECTIVES Implement alternative methods for interpolating line data to a regular grid that are independent of the grid cell size, can be applied to large datasets and used to either exactly or smoothly fit the data.

1 INTRODUCTION

The objective of an airborne geophysical survey is to collect data along transects with regular spacing along and between transects. However, inevitably the speed fluctuates and the aircraft drifts slightly off course. The result is that the sample spacing in both the along and across line directions can vary substantially. Added to this irregularity is the tendency to collect the data at a higher density (by an order of magnitude or more for magnetics) along lines than between. After preliminary processing the data are then interpolated onto a regular grid, generally with an equal spacing in both directions. The gridding occurs for two reasons; (1) visualisation and; (2) to simplify and speed up subsequent processing operations.

A popular algorithm used for the gridding is the minimum curvature

algorithm of Briggs (1974). However, increasingly attention has been drawn to the deficiencies of current geophysical gridding practice ... an area that has seen no definitive improvement in over 20 years (Craig, et al., 1998, p 17).

Gridding has an extensive literature and it is not the purpose here to conduct a thorough review (see Foley and Hagen, 1994 and references therein). The aim is to investigate a number of alternative gridding practices that have developed in the geostatistical, terrain modelling and applied mathematics areas, and to show how they can be unified in a common algorithmic framework. They include the dual formulation of kriging (e.g. Matheron, 1980; Cressie, 1993), tension splines (Mitasova and Mitas, 1993), radial basis functions (Powell, 1992; Hardy, 1990) and smoothing

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splines (Hutchinson 1993; Wahba, 1990). Each of these interpolating or smoothing operations can be cast as the sum of a set of weights times translates of a fixed basis function plus an optional low order polynomial. Solution of the resulting equations in each case reduces to the same standard form with slight variations. I term this general framework for interpolation and smoothing, Arbitrary Basis Function Fitting. The close connections between kriging and splines have been explored previously by, for example, Matheron (1980), Dubrule (1984), Wahba (1990) and Hutchinson and Gessler (1994), while the connection between kriging, radial basis functions and partial differential equations was examined by Horowitz et al. (1996).

ABF’s are well suited to geophysical applications because;

1. Many different interpolation methods can be encompassed within a common framework. 2. When using an ABF to interpolate a geophysical survey the resulting grid will inherit certain desirable properties from the basis function. For example, interpolating with a thin-plate spline in 2-D will result in the smoothest possible surface (defined in terms of second order derivatives) that passes through all the data points. On the other hand, when interpolating with the sinc function the surface will be band limited, which means it has no signal above a certain cut-off frequency.

Finally, interpolating with a kriging semi-variogram will give a surface that

approximates the best linear unbiased estimate that can be obtained from the data.

3. The basis functions do not have to be radial which means that anisotropy related to different sampling densities can be accommodated

4. The results of the interpolation are unaffected by the cell size chosen for the interpolated image. 5. The equations can be manipulated to give a smooth (i.e. non-exact) fit to noisy geophysical data, with the amount of smoothing determined by generalised cross validation (e.g. Wahba, 1990).

6. An ABF expansion is effectively a continuous model of the data that can be feed through subsequent processing operations, including Fourier transformations and deconvolution. This means that there is no need to explicitly interpolate the data onto a regular grid for later processing, although of course one would generally still interpolate in order to visualise the data.

The main impediment to the application of ABFs to geophysical data has been the solution of the matrix equations that arise. With N data points, direct solution gives a computational scaling of O(N 3 ) with O(N 2 ) memory requirements.

Once N gets beyond a few thousand direct solution on

even high end work stations becomes problematic. There are essentially three key steps in reducing this computational cost to manageable levels: (1) use of iterative matrix methods such as conjugate gradients (Hestenes and Stiefel, 1952; Golub and Van Loan, 1996) to decouple the O(N 3 ) dependence into a factor O(N ) related to the number of iterations and a factor O(N 2 ) for the matrix

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vector product required at each iteration; (2) decrease the number of iterations required by preconditioning the matrix system (Dyn et al., 1986); and (3) reduce the cost of a single matrix vector product by fast algorithms such as the fast moment method of Beatson and Newsam (1998a).

In this chapter, I address only the first of these issues here by showing that iterative methods based on the Lanczos Method (Lanczos, 1950) can reduce the computational cost on a relatively small geophysical survey (~6,000 points) to manageable proportions. Applications to larger surveys will be possible in the near future. For example, Beatson and Powell (1993) developed a fast method for interpolation by thin-plate splines, while Horowitz et al., (1996) have discussed an alternative strategy that involves finding a differential operator equivalent of the basis function. The system can then be reduced to a partial differential equation and solved rapidly using a multigrid technique. Lanczos methods applied to the symmetric matrix system Ak = f, are used to generate an orthonormal basis to the system’s Krylov subspace. At the k-th iteration the basis of the Krylov subspace, K k , consists of the initial residual, r o = f − Ak o , and a k − 1 length sequence of applications of the matrix A, i.e. K k = span{r o , Ar o , ..., A k−1 r o }. Successive approximations to the solution are obtained from

within the Krylov subspace. Lanczos methods are the preferred iterative method as they only require the action of the matrix, A, on a vector k (although in certain situations they also require the application of A T on a vector where the T denotes the transpose). This means that fast methods for computing the matrix-vector product can be used in place of direct evaluation.

Different Lanczos methods differ in the type of matrix equations that they can solve, and on how the iterate, k k , is calculated at each step.

In this chapter I consider three types of problems which are

best solved by different Lanczos methods. For exact interpolation I use the conjugate gradients method, which requires that the matrix A is symmetric positive definite. A QR factorisation by Householder reduction (e.g. Golub and Van Loan, 1996) of any polynomial constraints can be used to transform the ABF equations into a positive definite form. For smooth fitting there is either a rectangular least squares problem that can be solved by the LSQR method of Paige and Saunders (1982), or a regularisation parameter to choose by generalised cross validation. For the latter I use the Fast GCV algorithm of Sidje and Williams (1997).

The chapter is organised as follows. In the first section I present the equations that arise in ABF fitting and discuss algorithms for their solution.

I then illustrate the generality of ABF’s by

examining the different interpolating and smoothing operations that are encompassed by the framework. Finally I give some examples of the application of the methodology to an ~6,000 point airborne radiometric survey.

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Symbol f(x ) f F(u ) x or E[x ] a b c a e(x ) and E(u ) y e (x ) and Y e (u ) y g (x ) and Y g (u ) lj r(x ) a j and a i bj c j and c i cˆj C = Q T2 AQ 2 f(u ) and F(u ) g(x ) and G(u ) h(x ) and H(u ) J(s ) p(x ) and P(u ) q(x ) and Q(u ) q r rk s(x ) and S(u ) u = (u, v ) w(x ) and W(u ) x = (x, y )

Description

Fourier transform pair Expected value of the variable x Stripping ratio (Th counts in U window) Stripping ratio (Th counts in K window) Stripping ratio (U counts in K window) Stripping ratio (U counts in Th window)

Noise Auto-correlation and power spectral density of the noise Auto-correlation and power spectral density of the signal Attenuation coefficient for window j Standard deviation of the variable x Aircraft background count rate in window j or channel i Background count rate due to all sources Cosmic count in window j or channel i per cosmic count in cosmic channel Radon calibration coefficients Matrix transform

Linear-space-invariant filter Penalty functional Polynomial in original ABF surface Polynomial in transformed ABF surface Radial frequency coordinate q 2 = u 2 + v 2 Radial spatial coordinate r 2 = x 2 + y 2

Residual after k-iterations of a Lannzos procedure Arbitrary Basis Function approximation Frequency co-ordinates Optimum Wiener filter Spatial co-ordinates

Table 1: Mathematical symbols used in Chapters 5, 6 and 7.

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2 EXACT INTERPOLATION USING ARBITRARY BASIS FUNCTIONS

Consider the problem of interpolating a d dimensional geophysical dataset with M observations, f m = f(x m ), at the locations x m : m = 1, ... , M . This thesis is predominantly concerned with 2-D

interpolation and smooth fitting, although I maintain a fairly general notation to indicate that other dimensions, particularly 1-D and 3-D, are quite suitable. In the ABF framework one has a fixed, typically radially symmetric, basis function v(x), from ℜ ‘ d to ℜ, a set of N weights k n : n = 1, ... , N defined at the N nodes or centres x˜ n : n = 1, ... , N and an optional k degree polynomial, p(x ). An ABF expansion then has the form s(x ) := p(x ) + S n=1,N k n v(x − x˜ n )

(1)

Note, that each of the bold face x’s in the above equation are vectors, i.e. x = (x 1 , ... , x d ). I initially consider exact interpolation which must have N = M and generally has x˜ n = x n , which will be assumed at present (later I will consider smooth fitting, i.e. N < M and x˜ n ! x n ). The weights, k n , and the polynomial coefficients of the approximation are determined by requiring that s(x ) satisfy the interpolation conditions s(x n ) = f n , for n = 1, ..., N

(2)

When polynomial terms are present there are additional conditions that the weights must satisfy. For ) polynomial terms. Let {p i : i = 1, ..., K} be a basis for all a k degree polynomial there are K = ( k+d d

polynomials in d dimensions with degree at most m, and let the polynomial coefficients be a i : i = 1, ... , K .

For example, in two dimensions with k = 1, implies that K = 3,

p(x ) = a 1 + a 2 x + a 3 y, and thereforep 1 (x ) = 1, p 2 (x ) = x and p 3 (x ) = y. A system of equations with a

unique solution can be formed by requiring that the weights satisfying the following boundary conditions: N

S k n p i (x n ) = 0, for all i = 1, ..., K n=1

(3)

The conditions specified in Equations (2) and (3) give a linear system of N + K simultaneous equations in N + K unknowns (as N and M were assumed equal). They may be written in matrix form as

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A&k& =

A P PT 0

k a

=

f 0

= f&

(4)

where A mn = v(x m − x n ), P mj = p j (x m ), k = (k 1 , k 2 , ..., k N ) , a = (a 1 , a 2 , ..., a K ) and f = (f 1 , f 2 , ..., f N ) . T

T

T

When I refer to the whole matrix in Equation (4) I will use A & , similarly k & for the ABF weights and the polynomial coefficients together and f & , for the data plus the additional zeros. The size of the domain over which the nodes are distributed can influence any ill-conditioning in the matrix system of Equation (4). To avoid unnecessary ill-conditioning the domain should be scaled so that it lies within the square [−2, 2 ]x[−2, 2 ]. Note, that for some basis functions (in particular the splines) the system of Equation (4) can be converted to a form where the conditioning is independent of data scaling (see section 3.2 and Appendix A).

The matrix in Equation (4) is symmetric but in general is not positive (or negative) definite (i.e. it does not satisfy the conditions k &T A & k & > 0 for all k & `‘ ℜN+K ). The solvability of this matrix system depends on the non singularity of the matrix, which in a radial basis function setting have been shown to hold for a wide choice of functions v($ ) and polynomial degrees k, with only very mild restrictions on the distributions of the centres (e.g. Micchelli, 1986; Powell, 1987). One sufficient condition is that the ABF part of the matrix in Equation (4) is conditionally positive (or negative) definite. That is,k T Ak > 0 (or k T Ak < 0 ) for all k such that P T k = 0. These conditions are also a requirement for a valid semi-variogram or covariance function in kriging, which implies that the kriging equations are always non-singular. The conditional positive (negative) definite requirement can be used to convert Equation (4) to a positive definite form.

3 SOLUTION OF THE INTERPOLATION EQUATIONS

3.1 Computational cost

Before investigating the generality of the ABF framework I first discuss computational issues that preclude naive solution of the above set of equations. Many ABFs (such as spline and kriging bases) are non zero everywhere and do not vanish outside some small neighbourhood, while most geophysical data is collected with irregular spatial sampling. Thus the matrix in Equation (4) usually has full rank, non-zero entries, and no special structure such as being banded or Toeplitz; that is it has none of the structures normally exploited in calculating fast solutions of linear systems. Thus, solution of the interpolation equations by direct, or even simple iterative methods requires O(N 3 )

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operations and O(N 2 ) storage. The symmetry in the system can be used to reduce the operation count by up to a half, but even symmetric solvers will still be impracticable when N is large.

The O(N 3 ) scaling means that ABF type interpolations have generally been restricted to at most a few thousand points, and have often used a restricted number of nodes to further reduce the computational load (e.g. Bates and Wahba, 1982; Hutchinson, 1993). I conducted some numerical experiments to illustrate the computational scaling of a direct method on a fast 200 MHz dual processor Ultra-Sparc 2 computer with 380 Megabytes of RAM. This is typical of current high end work stations. The algorithm I chose for the direct solution was LU decomposition (e.g. Golub and Van Loan, 1996). I first formed the matrix given in Equation (4) and then inverted it, ensuring that the computation was done within internal memory. For problems of size, 100, 200, 500, 1000 and 2000 the average timings just for the inversion (i.e. not including the formation of the matrix) over five runs each were 0.029 sec, 0.194 sec, 7.78 sec, 120 sec and 1060 sec respectively. I then fitted a function of the form N 3 to these points and obtained the computational scaling illustrated in Figure 1. The results show

that it would take at least an hour to do a 3,600 point interpolation, a day for 8,600 points, a week for 16,600 points, a year for 62,000 points and a decade for 133,000 points. These are very optimistic estimates (!) as the memory requirements scale asO(N 2 ), which means the machine’s RAM capacity would very rapidly be exceeded. Once the calculations required disk access the time required would increase substantially.

The rough calculations in the last paragraph show that direct methods of solution on even high end work stations are unfeasible for N greater than a few thousand. There are three key steps involved in reducing the computational cost of ABF fitting:

1. Using an iterative matrix solver in place of a direct method. This effectively decouples the O(N 3 ) operation count of direct solution into two factors O(N 3 ) i O(N )O(N 2 ). The O(N ) expresses the dependence of the convergence of the iterative method on the size of the problem, while the O(N 2 )is for the matrix-vector multiplication needed at each iteration.

2. Decoupling the O(N ) dependence of the number of iterations from the problem size by preconditioning. With an appropriate preconditioner the dependence can often be reduced to O(1 ) (e.g. Dyn et al., 1986). That is, the number of iterations is independent of the problem size.

3. Reducing the O(N 2 ) operation count in the matrix vector product by hierarchical tree structures and approximation techniques (e.g. Greengard and Rohklin, 1987; Beatson and Newsam, 1998). In recent times considerable efforts have been invested in each of these three steps. In this thesis I address only the first of the three in detail, but will give some indication of the state of developments in the other two.

119 1011

Time (seconds)

109

Decade Year

107 Week 105

Day

103

103

104

105

106

Number of points

Figure 1: Computational scaling for direct solution by LU decomposition on an 200 MHz Ultra-Sparc with 380 Megabytes of RAM. 3.2 Iterative methods for solving the matrix equations

In its current form Equation (4) can be solved by Generalised Minimisation of Residuals (GMRES, Saad and Schultz, 1986), which is a very stable and reliable iterative method that only requires the action of the matrix on a vector (i.e. the matrix is not explicitly required). However, the conditioning of the system (and hence, the number of iterations required) can vary substantially with any scaling applied to the position of the data. Additionally, as the iteration proceeds, a progressively greater number of basis vectors are required to obtain the next estimate. This means that to minimise storage on large problems the method has to be restarted which can significantly slow the convergence (e.g. Turner and Walker, 1992). Conjugate gradients is an iterative method that only requires information from the last one (or possibly two) basis vectors and therefore does not have the same storage problems. However, it requires that the matrix system is symmetric-positive definate. Equation (4) is symmetric but is only positive definate for certain bases (e.g. Gaussian and sinc functions) when the polynomial terms are absent.

3.2.1 Converting the interpolation equations to a positive-definite form by QR factorisation

Within the context of spline interpolation, it has been known for some time (Bates et al., 1987) that a QR factorisation (e.g. Golub and van Loan, 1996) of the polynomial matrix can be used to convert Equation (4) to a positive-definite form. However, many implementations involve the formation of a large and dense matrix (Q 2 in the next paragraph) followed by a double matrix-matrix multiplication (e.g. Bates et al., 1987). Beatson (pers. comm.) suggested an alternative computational method that

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is fast to calculate and apply and noted that, for many basis function, the QR factorisation causes the conditioning to be invariant to any data scaling.

The method described by Beatson, uses a QR-factorisation based on Householder reflections (e.g. Golub and van Loan, 1996). A Householder matrix, H, is symmetric, orthogonal (H T H = I) and is specified by a single vector, v, through the equation H = I − 2vv T /v T v. Householder reflections are usually used to zero selected elements within a given column of a matrix. Applied to the N % K polynomial matrix P the QR-factorisation method generates K Householder matrices, H 1 ...H K , such that H K ...H 1 P = R 0

u P = H 1 ...H k R 0

(5)

where R is a K % K upper triangular matrix. Let Q = H 1 ...H k , then the first K columns, Q 1 , span the column space of P, so that P = Q 1 R, while the remaining N − K columns, Q 2 , span the null space of P T . Therefore, the polynomial constraints can be made automatic by requiring that k = Q T2 l for some l c ‘ N−K . The original system given in Equation (4) may then be written as AQ 2 l + Pa = f

(6)

Pre-multiplying both sides of the equation by Q T2 and noting that Q T2 P = 0, I find (Q T2 AQ 2 )l = Q 2 f u Cl = z

where C = Q T2 AQ 2 and z = Q 2 f .

(7)

Since A is positive- (negative-) definite over the space of k’s

eliminated by P T , the identity l T (Q T2 AQ 2 )l = k T Ak shows that the matrix C is positive- (negative-) definite. Thus, Equation (7) is suitable for solution by the conjugate gradient method. It remains to find the polynomial coefficients. From Equation (6) Pa = f − Ak, and the identity P = Q 1 R plus the orthoganality of Q 1 imply that Ra = Q T1 (f − Ak )

(8)

This is an upper triangular K % K matrix equation that is trivial to solve for the unknown polynomial coefficients.

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The advantage of the Householder approach is, that neither the matrix Q 2 , nor the product, C = Q T2 AQ 2 , have to be explicitly formed and stored. The matrix Q 2 is the last N − K columns of the

matrix, Q, which is constructed from a sequence of Householder matrices. The action of Q on a vector can be evaluated by calculating a matrix-vector multiplication for each Householder matrix. Due to their special structure each Householder matrix-vector multiply can be calculated very quickly in about 4NK flops (e.g. Golub and van Loan, 1996). As there are K Householder matrices, the matrix-vector multiplication of Q, or of Q T , can be calculated in approximately 4NK 2 flops. The calculation of Q 2 l can be achieved by stacking l with K leading zeros, to form l  and then calculating Ql  . Similarly, matrix-vector multiplication with Q T2 can be achieved through that of Q T by ignoring the first K rows of the resultant vector. Therefore, the QR-factorisation parts of the matrix-vector multiplication, Cl, can be applied rapidly at each iteration by first calculating m = Q 2 l, followed by j = Am and finally Q T2 j.

3.2.2 Lanczos iterative methods

I describe the Lanczos method in more detail in Appendix B and here only outline it’s implementation as the conjugate gradients method. Excellent references to the material presented here are Kelley (1995) and Golub and Van Loan (1996).

Given a symmetric matrix C and a starting vector r, the Lanczos process is a method for generating a sequence of vectors v i and scalars such that C is reduced to tridiagonal form. After k steps of the Lanczos method there is an orthogonal matrix V k = (v 1 , ..., v k ) such that V Tk CV k = T k

(9)

where T k is k % k and tridiagonal. The eigenvalues of T k are progressively better estimates of the eigenvalues of the matrix C. Additionally, when applied to the matrix system Cl = z, with a starting vector r o = z − Cl o the orthonormal basis, V k , spans the Krylov subspace, K k K k = span r o , Cr o , ..., C k−1 r o

(10)

The iterate at the k-th step, l k , is then calculated as lk = lo + Vkyk

(11)

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where y k `K k . Different implementation of Lanczos differ in how the y k in Equation (11) is calculated at each step. There are four key reasons why the Lanczos method applied to a N % N positive definite matrix system is an efficient iterative scheme:

1. The iterates, l k , converge to the solution in at most N iterations. However, an acceptable solution is often found with many times less than N iterations depending on the condition number of the matrix (the ratio of the largest to smallest eigenvalues).

2. The orthogonal vector v k+1 that extends V k to V k+1 can be found by referring to only the most recent updates. Therefore, there is no need to store the entire basis V k .

3. After k steps the vector y k in Equation (11) is obtained as the solution to the tridiagonal positive definite equation T k y k = Èr o Èe 1 ; e 1 = (1, 0, ..., 0)

(12)

The solution can be obtained very efficiently by using short term recurrences based on the Cholesky decomposition of T k .

4. The matrix C only enters the procedure through the matrix-vector products Cv k . This means that C does not need to be explicitly stored if fast methods can be used in place of direct evaluation: it

also allows considerable flexibility in preconditioning.

3.3 Preconditioning

The condition number of a symmetric matrix is defined to be the magnitude of the ratio of the largest and smallest eigenvalues, and ranges from 1 to ∞. It influences the rate of convergence of the iterative methods, with large condition numbers implying slow convergence. Many of the matrices that occur in ABF interpolation have very large condition numbers, indicating that convergence may be a problem. Where possible, some type of preconditioning is applied to the matrix system in order to speed up convergence.

The idea behind, for example, left preconditioning of a matrix system Ak = f is to find a matrix B, such that the transformed system BAk = Bf has a much lower condition number than the original system. One way to achieve this goal is to find B j A −1 , which makes the transformed system close to diagonal and hence easier to solve. To minimise computation one usually requires that the action of B on a vector, i.e. Bk can be rapidly calculated, which typically means that B must be sparse. In a

radial basis function setting preconditioners for multi-quadric and thin-plate splines are discussed in

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Dyn et al. (1986). Rick Beatson has developed a preconditioner for thin-plates and multi-quadrics that has been used successfully on geophysical data (Beatson, pers. comm.).

3.4 Matrix-vector product Fast methods for the evaluation of the matrix vector product Ak, where A mn = v(Èx m − x n È), have been around for over ten years and include tree codes (e.g. Appel, 1985) and fast multipole methods (e.g. Greengard and Rohklin, 1987; Beatson and Newsam, 1992). These algorithms significantly reduce the computational cost when N is large. The main idea behind fast methods is to lower the operation count required by using hierarchical tree structures and approximation techniques. This results in the computational effort scaling as O(N log N ) or even O(N ). The algorithms also need substantially less than the O(N 2 ) memory required to store the matrix.

Fast multipole methods are becoming increasingly popular for matrix vector products of interest. They are based on finite expansions that approximate the far field influence of a given cluster of centres as a sum of truncated Laurent series expansions of v. Their main disadvantage is that each time a new v is encountered a new series of expansions needs to be derived and the software has to be modified. Beatson and Newsam (1998) have developed an alternative fast algorithm for one dimensional evaluation (termed the fast moment method) that does not suffer these disadvantages. The algorithm is adaptive and can be applied in any case where the function v is smooth over the interval of interest. Currently the algorithm is being generalised to 2-D fast evaluation.

The adaptive nature of the Beatson and Newsam (1998) algorithm means that it can be applied to any suitably smooth function. These include the host of radial basis functions, the tension splines of Mitas and Mitasova (1993) and all examples of semi-variograms that I have seen in the kriging literature (e.g. Journel and Huijbregts, 1978).

4 GENERALITY OF THE ARBITRARY BASIS FUNCTION FRAMEWORK

The ABF framework allows many different types of interpolatory and fitting operations to be applied within the one generic algorithm base. In this section I discuss several aspects of this freedom with regard to the choice of basis function, the number and location of the nodes relative to the sample points of the data and regularising a fit to include smoothness constraints.

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4.1 Choice of basis

One of the most advantageous characteristics of the ABF approach is that a particular basis function can be selected to impose some desired property on the surface. For example, using a thin-plate spline in two dimensions the surface will be the smoothest possible (in terms of second order derivatives) that passes through all the data points (Duchon, 1976). Alternatively, one could use the sinc function, in which case the surface would be band-limited (i.e. no frequency content above a certain cut-off), or a kriging basis which would give the best linear unbiased estimate (e.g. Matheron, 1980; Cressie, 1993). Here I briefly review some of the commonly used basis functions.

4.1.1 Radial and non-radial basis functions In radial basis function interpolation, the term v(x − x n ) in Equation (1) is replaced by v(r ) = v(Èx − x n È)where È$ È is the Euclidean norm. Examples of functions classically regarded as

RBFs are: the linear spline, r; the cubic spline, r 3 ; the thin-plate spline, r 2 log(r ); the Gaussian, +1/2 −1/2 exp(−cr 2 ); the multiquadric, (r 2 + c 2 ) ; and the inverse multiquadric, (r 2 + c 2 ) , where c is a

positive constant. Three dimensional perspective plots of these functions are shown in Figure 2. Inspection of the graph reveals that several basis functions (the splines, and multiquadric) are unbounded as r approaches infinity. This counterintuitive behaviour has been shown to led to very effective approximation strategies (e.g. Powell, 1992). In fact, basis functions that are bounded at infinity can have worse approximation properties.

Multiquadrics have been widely applied in the natural sciences including geophysics (see the review of Hardy, 1990). The Gaussian kernel appears frequently in the electrical engineering literature where it is often coupled with neural networks (e.g. Cha and Kassam, 1996). I consider splines in the next section.

In many geophysical problems, such as airborne surveys, the data sampling in one direction is significantly different from that in others. If any smoothing or scaling parameter is included in a basis it should be allowed to vary with direction. Therefore, the ABF framework needs to be able to handle non-radial basis functions. In many cases a simple linear scaling of one direction can convert the basis into an effectively radial form. In the unscaled original coordinate space the basis is elliptically symmetric. Another type of symmetry that arises is when the basis is the sinc function. In that case Shannon’s sampling theorem states that, for example, an appropriately sampled 2-D band limited function can be expressed as a tensor products of sinc functions; i.e. sinc(x/Dx ) sinc(y/Dy )

125

where Dx and Dy are the sample spacings in the x and y directions respectively. Both the above types of symmetry can be simply dealt with by fast evaluation algorithms.

3.0

9 8

2.5

7 6

2.0

5 1.5

4 3 2

1.0

2

1

1

2 1

0.5

0 0

-1 1 0 -1 -2

0

0.0 1

-1

-1

0 -1

-2

-2

5

-2

1.00

4

0.75

3 0.50 2 2 1

2

0.25

1

1

0

0 1

0

0.00 1

-1

0 -1

-1

-2

-2

-1

0 -2

-2

12

1.0

10 0.5 8 6

0.0

2

-0.5 1 0

-1.0 1

-1

0 -1 -2

-2

4

2 1

2 0

0 1

-1

0 -1 -2

-2

Figure 2: From top left to right the thin-plate spline, the multi-quadric (c = 0.1), the inverse multi-quadric (c = 0.1) the Gaussian (c = 1), the sinc function (Dx = Dy = 0.5 ) and a tension spline (w = 1000 ).

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4.1.2 Thin-plate splines

Many interpolants with their associated basis functions, including thin-plate and tension splines and kriging semi-variograms, arise naturally as the solution of certain variational problems. These arise through defining a reasonable penalty functional J(s ) (almost always a quadratic function of s), and then choosing the interpolant to be the solution of the constrained optimisation problem min(s ) : J(s ) subject to s(x n ) = f n : n = 1, ..., N

(13)

For quadratic functionals, this problem can usually be solved exactly and yields an ABF solution.

As already noted, choosing J(s ) to emphasise smoothness gives rise to some particularly useful and well-performing families of spline interpolants. Duchon (1976) characterised the interpolants associated with a general family of smoothness measures. In particular he showed that in 2-D the thin-plate spliner 2 log r is the basis function associated with the second order smoothness penalty 2 2 2 J(s ) = ° ‘2 (¹ 2 s/¹x 2 ) + 2(¹ 2 s/¹xy ) + (¹ 2 s/¹y 2 ) ¹x¹y

(14)

Similarly higher order smoothness can be enforced giving other basis functions; for example if J(s ) is a sum of squares of fourth order derivatives then the basis function is r 4 log r. The presence of polynomial terms in smoothing splines is a natural consequence of the form of the smoothness constraint in Equation (14): the derivatives annihilate constants and linear polynomials, so they can be included without affecting the smoothness measure.

Smoothing splines have been used extensively for interpolating and smoothing data (see Wahba, 1990 and references therein). Geophysicists would be familiar with Equation (14) which also occurs in the minimum curvature algorithm of Briggs (1974) that has been used so widely for interpolation. Minimum curvature is essentially a discrete approximation to the continuous thin-plate spline interpolant.

4.1.3 Tension splines

When fitting a thin-plate spline to 2-D data which contains regions with rapid change of gradients, the thin-plate’s stiffness can cause overshoots to occur (e.g. Mitas and Mitasova, 1993). When such overshoots are deemed problematic alternatives to standard splines can be derived that include additional derivative constraints than those shown in Equation (14). Mitasova and Mitas have

127

developed a formulation which they termed completely regularised splines (CRS). This arises from a modification of the usual spline constraint (Equation (14) in 2-D) that uses derivatives of all orders, with a variable weighting for each derivative order that depends in a regular way on a generalised tension parameter, d:

J(s ) = S B a °ℜ 2 a

¹as ¹x a x ¹y a y

2

¹x¹y

where a = (a x , a y ), a = a x + a y and B a = a !d −2 a / a x !a y !( a − 1 )!

(15)

if a ! 0 andB a = 0 otherwise.

The a summation is inferred to go from 0 to º, with a x and a y taking on every value such that a = a x + a y . When d is small the higher order derivatives terms dominate and the surface resembles a

thin-plate spline; conversely when it is large the lower order derivative terms are the most significant and the surface resembles a membrane.

Mitas and Mitasova showed that the function that minimise J(s ) subject to interpolating the data can be cast as an ABF expansion with the basis function, 2 v(Èx È) = 2 ln((d/2 )Èx È) + E 1 ((d 2 /4 )Èx È ) + c

(16)

where E 1 is the exponential integral function of the first kind, and c = 0.5772.. is Euler’s constant. At the origin the logarithmic term and Euler’s constant are exactly cancelled by the exponential integral so that v(0 ) = 0. In summary, the tension splines have a similar derivation to thin-plate splines and result in an analogous system of linear equations.

4.1.4 Kriging

Kriging is a geostatistical technique that has its roots in the estimation of ore grade at an arbitrary location from discrete samples in an ore body (e.g. Krige, 1951, Matheron, 1963). It has since been vigorously adopted by practitioners from many other fields (see Cressie, 1990), and is suited to spatial prediction in many geophysical contexts.

The derivation of the kriging equations is different than for the splines but results in an analogous set of simultaneous equations. In kriging it is assumed that the data are realisations of a stochastic process, which in universal kriging (also called kriging with a trend) typically takes the form, f(x n ) = p(x n ) + z(x n )

(17)

128

The z(x n ) are realisations of the stochastic process which we will here assume is stationary with zero mean, while the p(x n ) is a low order polynomial drift in the data. The stationarity assumption implies that, Var[z(x + h ) − z(x) ] = 2V(h ),

(18)

where Var[x ] = E[x 2 ] − E[x ] 2 , is the variance, E[x ] is expected value of x and V(h ) is defined as the semi-variogram (2V(h) is defined as the variogram).

To allow for possible anisotropy, the

semi-variogram depends on the vector h but in many cases an isotropic model V(h ) = V(Èh È) is used. In what follows I will assume the geostatistical model defined by Equation (17) but I should point out that it is not the only type of model considered by geostatisticians (e.g. Cressie, 1993). Indeed the ABF framework is able to encompass the more general case where z(x ) is a so called generalised increment (Matheron, 1973; and see later).

The kriging estimate, s(x ), at an arbitrary location x, is the best linear unbiased estimate obtained from a weighted sum of the original data, s(x ) = S n=1,N k n (x )f(x n )

(19)

This is a linear inverse problem with the weights chosen to ensure the estimate is unbiased and optimal. The solution is unbiased only if the expectation is zero, i.e. E[s(x ) − f(x )] = 0 which is equivalent to the interpolation condition satisfied by splines. The optimal solution is found by minimisation of the penalty function, J(s ) = Var[s(x) − f(x) ]

(20)

The J(s ) here is the equivalent in kriging of the penalty functions for thin-plate and tension splines given in equations (14) and (15) respectively.

The formal equivalence between splines and kriging has been known for some time (Matheron, 1980) and has been examined extensively in the literature (e.g. Dubrule, 1984; Wahba, 1990; Hutchinson and Gessler, 1994). Matheron (1980) showed that the optimality condition in Equation (20) implies that the universal kriging estimate s(x ) is given by

129 s(x ) := p(x ) + S n=1,N k n V(x − x n )

(21)

where the polynomial, p(x ), corresponds to the order of drift of the data. This is analogous to the ABF expansion in Equation (1) with the basis function v(x ) replaced by the semi-variogram V(x ). The requirement that the solution is unbiased leads to an equivalent set of conditions on the weights as Equation (3) for ABFs. Solution of the kriging equations is then equivalent to Equation (4) for ABF expansions, which in the kriging literature is referred to as the dual form of kriging. The semi-variograms are always selected to satisfy a conditional negative semi-definite constraint, which is sufficient to ensure that the linear system has a unique solution (Myers, 1988).

The dual form of kriging defined above is often presented with a “generalised covariance function” in place of the semi-variogram (e.g. Matheron, 1980). This makes the matrix system positive definite (instead of negative definite) and is generally the path taken when one wants to use a spline function in a kriging context (e.g. Dubrule, 1984; Wahba, 1990; Hutchinson and Gessler, 1994).

The

stationary error assumption of Equation (17) is usually replaced with a more general assumption based on intrinsic random functions of order k, IRF-k (e.g. Matheron, 1980). The basic idea is to linearly filter the drift terms from the data, as these can create problems in estimating the semi-variogram by traditional methods. An IRF-0 is equivalent to the stationarity condition in Equation (17) with the generalised covariance function, K(h ), and the semi-variogram simply related by K(h ) = −V(h ). When k > 0 the relationship between the variogram and the generalised covariance function is more complicated (e.g. Cressie, 1993).

Often the semi-variogram is defined as V(h ) = c o d(h) + V o (h )

(22)

where V o (h ) is continuous, d(h ) is the kronecker delta with d(h) = 0 if h ! 0 and d(0) = 1, and c o is the so called nugget effect. A non-zero nugget variance arises from both random variation due to measurement error and fine scale variability in the surface not modelled by the semi-variogram. With a zero nugget the kriged surface gives an exact interpolation. A c o ! 0 effectively adds a non-zero constant to the diagonal of the matrix, A, in Equation (4), and the surface no longer passes through the data points. Thus including a nugget effect effectively smoothes the data rather than interpolating it: this issue is explored further later in the chapter

The variogram encapsulates the likely connections between values of the field at locations separated by a given distance (or lag). One of the main challenges in successful application of kriging is

130

selecting a variogram that adequately models the actual change in variance with distance, and which also ensures the negative definite constraint is satisfied (e.g. Journel and Huijbregts, 1978). I do not address this issue here (although a simple method for fitting a variogram is used the results section), but give some indication of the character of the variograms as this is important from a computational viewpoint.

Variograms fall into two main categories, transitive, in which the variance increases with increasing lag distance to some maximum value after which it remains constant, and unbounded, where the variance increases without limit. The transitive models require the specification of three parameters; (i) the range which is the lag distance after which the variance remains constant; (ii) the sill which is the maximum variance; and (iii) the nugget variance which is the value of the variogram at the origin. The predicted values of a kriged surface (but not their estimated errors) are unaffected by a linear scaling of the nugget and sill variances. The most common models (which are truncated in the transitive model at the range value) are sums of odd powers, logarithms, exponentials, or Gossips, or various combinations of the above (e.g. Journel and Huijbregts, 1978; McBratney and Webster, 1986). These variograms are all globally supported which means that the matrix system arising in kriging has no special structure that would allow a fast direct matrix method to be used. I conclude this section by noting that the Beatson and Newsam (1998) fast method for the matrix-vector product is applicable to all these variograms. However, a performance penalty would apply if there were a derivative discontinuity at the range value in a transitive model.

4.2 Smooth fits to geophysical data

In most geophysical applications the data is not known precisely and an interpolant passing exactly through each data point may not be ideal. The ABF framework encompasses three different methods for generating a smooth surface:

(i).

Introduce regularisation with Generalised Cross Validation (GCV e.g. Wahba, 1990) to choose the smoothing parameter.

(ii). Use fewer nodes (the xˆ n in Equation (1)) than data points (the x m in Equation (1)). (iii). Only iterate conjugate gradients to a partial solution. In principle (i) is the best founded approach, especially when fitting arbitrary functions of unknown variation. It is based on a clear statistical model of the data (signal plus noise) and does a provably good job of estimating the noise (e.g. Wahba, 1990). In (ii), having many more data points then nodes means that the noise levels do not have to be explicitly estimated as in GCV. They are

131

obtained implicitly through the least squares equation that arises. The least rigorous approach is (iii) which is based on the observation that the terms in an eigenvector expansion of the solution that are associated with large eigenvalues usually define the smooth or low frequency components of the surface while the terms associated with the small eigenvalues define the rough or high frequency components (e.g. Dyn et al., 1988).

Conjugate gradients tends to determine the low order

coefficients in the eigenvector expansion first, which means that it generates a good smooth approximation to the overall surface in the early iterations and only fits the fine details in subsequent iterations. The technique can work well in practice, especially if the convergence rate is monitored and good stopping criteria are selected. However, I will not consider it further here.

Although (i) is more rigorous in practice (ii) may be more applicable in geophysics for two reasons. Firstly, there is often a-priori information on the spatial bandwidth of the data from which means there may be a ‘natural’ nodal spacing. For example, in the last chapter I showed that the height of the detector above the ground plus the noise level imposes an upper limit of the spatial frequencies that can be resolved in an airborne gamma-ray survey. Secondly, movement of the nodes onto a regular grid can significantly simplify subsequent processing operations (see Chapters 6 and 7).

4.2.1 Regularising the solution with GCV

The standard model underlying methods for automatic smoothing of data is to assume that a datum f n can be decomposed into a signal component y n and a noise component e n , i.e. fn = yn + en

(23)

where the e n are assumed to be independent and normally distributed but with an unknown variance. Recall that the general interpolation process as in Equation (13) involves minimising a penalty function subject to exactly fitting the data. If the data points are known to be noisy, there is little point in exactly interpolating them: therefore one instead seeks to trade off minimising the penalty function against the mean square error in fitting the data. This leads to minimising the following regularised least squares problem min(s ) : S i=1,N (f i − s(x i )) + mJ(s ) 2

(24)

where the parameter m governs the trade-off between goodness of fit and smoothness (in some general sense). The problem thus reduces to deciding on what is a sensible or optimal value for m and then solving Equation (24).

132

Taking the second problem first, one obvious way of at least approximately solving Equation (24) is to substitute the general ABF expansion of Equation (1) into Equation (24). If the penalty functional J(s ) is quadratic in s, this gives the discrete regularised least squares problem, T min(k, a ) : (f − Pa − Ak ) (f − Pa − Ak ) + mk T Wk

(25)

where W is a positive definite matrix. Indeed it can be shown that if the ABF is the natural expansion associated with the exact interpolation in Equation (13), then the solution to Equation (25) is also the exact solution to Equation (24). Moreover the matrix W is identical with the matrix A: this follows since the spline ABF is the Green's function of the operator defining the penalty functional (Duchon, 1976; Mitas and Mitasova, 1993). Taking derivatives of Equation (23) with respect to k and a, and rearranging terms, the solution of the above equation satisfies the matrix system (e.g. Wahba, 1990) (mI + A ) P PT 0

k a

=

f 0

(26)

Thus apart from the addition of the positive constant m to the diagonal, the matrix system is identical to that for exact interpolation given in Equation (4). By applying the Householder QR-factorisation the system can be converted to a positive definite form and solved by conjugate gradient iteration, sinceQ T2 (A + vI )Q 2 = Q T2 AQ 2 + vQ T2 Q 2 = C + mI, because Q 2 is orthogonal (Q T2 Q 2 = I). Although the derivation and interpretation are different, an exactly analogous expression to Equation (26) occurs in kriging where it is known as the dual form (e.g. Matheron, 1980). Relative to the definition of the generalised covariance function in Equation (22), the correspondence can be made explicit by separating the nugget effect from the definition of v(x ) and identifying c m = c o +o c 1

(27)

where, c o is the nugget variance and c o + c 1 is the height to the sill (e.g. Hutchinson, 1993). This result expresses the well known fact that the kriging equations depend only on the ratio of the nugget variance to the sill height, i.e. they are unchanged by a linear scaling of the generalised covariance.

133

I now turn to the problem of estimating m. In kriging the natural interpretation of m as defining the nugget effect means that it can be determined in the process of estimating the complete semi-variogram: this is usually done by restricted maximum likelihood (Zimmerman, 1989). In general interpolation the usual approach is to adopt the interpretation of Equation (23) and to seek a way of estimating the unknown variance of the independent noise process e n : this is equivalent to varying m. In this thesis I propose to do this by generalised cross validation.

GCV is essentially a bootstrap measure of the predictive error of a surface. It is determined by removing each data point in turn, determining the residual at the data point of a surface fitted to the remaining data points, and then summing the squares of the residuals multiplied by appropriate weights. Fortunately, it can be shown that for splines the GCV measure reduces to the following simple expression (Craven and Wahba, 1979)

G(m ) =

(N − K ) z T (C + mI ) −2 z −1 2 tr(A + mI )

(28)

where C = Q T2 AQ 2 and z = Q 2 f have been transformed by a QR-factorisation of the polynomial matrix and tr(C + mI ) denotes the trace (the sum of the elements on the diagonal). G(m ) can still be difficult to calculate in general, especially the trace term in the denominator. Girard (1987) and Hutchinson (1989) independently developed stochastic approximations of G(m ) that are well suited to large problems. Hutchinson’s estimate has the form

G(m ) l

(N − K ) z T (C + mI ) −2 z 2 −1 u T (C + mI ) u

(29)

where u is a vector with random entries drawn with equal probability from −1, 1 . The objective is to find the m that minimises Equation (29).

Until recently, the most efficient approach for estimating the GCV was to reduce A to tridiagonal form using Householder reductions (Hutchinson and de Hogg, 1985). The tridiagonal decomposition is O(N 3 ) but once obtained can be used to rapidly evaluate the GCV for multiple smoothing parameters. Sidje and Williams (1997) , however, have just developed an efficient computational −2 2 procedure for GCV based on the Lanczos method that is O(N 2 ). Using h(m ) = z T (C + mI ) z/Èz È and −1 2 g(m ) = u T (A + mI ) u/Èu È converts Equation (29) to

134

G(m ) l

(N − K ) Èz È 2 h(v ) (Èu È 2 g(v) ) 2

(30)

The key idea behind the Sidje and Williams algorithm is to approximate the matrix inverses in h(v ) and g(v ) by their tridiagonal Lanczos representation (see Appendix B).

This requires two

separate Lanczos procedures applied to A with one initiated with the vector f and the other by the vector u. The algorithm is very efficient because, given h k (v ) and g k (m ), the estimates at the next iteration,h k+1 (v ) and g k+1 (m ), can be calculated by short term recurrences. Furthermore, once good estimates of h k (v ) and g k (m ) have been obtained for one m they can be rapidly generated for an arbitrary m. Generally, this does not require any further Lanczos iterations unless the estimates h k (v ) and g k (m ) at the new m are not accurate enough, in which case the Lanczos procedures resume.

The final part of the algorithm involves calculating the solution l opt , and hence k opt = Q 2 l opt , that corresponds to the optimal smoothing parameter m opt . This can be obtained efficiently by a recurrence formula that involves all the Lanczos basis vectors that defined h(m ). The basis vectors are either stored along the way, or can be regenerated when needed. In both cases there is no need for extra matrix vector products.

Finally since GCV is based on the statistical model in Equation (22), it also provides an estimate of the standard error in the fitted surface (Wahba, 1990; Hutchinson and Gessler, 1994). This can be compared with that obtainable from fitting a nugget effect in kriging; however I will not pursue these issues further here.

4.2 Smoothing by using less nodes than data constraints

The points at which the function is known (the x m in Equation 1) and the nodes that define the ABF approximation (the x˜ n ) do not have to coincide. In magnetic surveys the along line spacing is of the order of 10 m or less, while the across line spacing is generally between 50 and 400 m. The substantial over sampling along lines means that not all data constraints need to be used as nodes to define a representative surface. Selection of the defining nodes has been discussed in the spline literature by, for example, Hutchinson and de Hogg (1985) where it is referred to as knot selection. It is possible to devote considerable effort to deriving “optimal” knot locations, although in many practical situations good results are obtained as long as there are enough knots (Hutchinson, 1997).

One situation where the number and spacing of nodes can be decided on before hand with some confidence is in sinc interpolation of data known to be band-limited. If the data is known not to

135

contain frequencies above a certain cut-off value f c , then it can be appropriately interpolated by sinc functions on a regular grid of nodes where the inter-nodal spacing Dx is related to the cut-off frequency by the well-known Nyquist criterion f c = 1/(2Dx ). Least squares fitting in this fashion effectively carries out a low pass filtering of the available data. Geophysical applications of this form of interpolation occur where convolutions occurring in the measuring process and noise in the data impose effective frequency cut-offs (see Chapter 4). Such situations occur in airborne magnetic and radiometric surveys where high frequencies are effectively damped by the measuring process to below the instrument noise level, and so are just not present in the data. However some care must be taken when using sinc interpolation in this way as there can be severe boundary effects.

When there are less nodes then data constraints, the matrix in Equation (4) is non-square and one has a least squares problem in place of the exact equation for interpolation. With M data samples, x m , and N nodes, x˜ n , the least squares problem is, 2 minÈf − Ak − Pa È subject to: P˜ T k = 0 k,a

(31)

where the terms have the same interpretation as Equation (4), except P is M % K with P mj = p j (x m ) and P˜ is N % K with P˜ nj = p j (x˜n ). The QR-factorisation can again be used to enforce the polynomial

constraints but this time requires two separate factorisations as P and P˜ are different matrices (see Appendix A for more details). This reduces Equation (31) into the form of a standard least squares problem minÈz − Cl È 2 , where C = Q T2 AQ˜ 2 , z = Q T2 f , k = Q˜ 2 l and Q 2 and Q˜ 2 are the relevant parts of the QR-factorisations of P and P˜, respectively. Note, that once again the matrix C does not need to be explicitly formed, as matrix-vector multiplication with both Q T2 and Q˜ 2 can be rapidly achieved at each iteration. Once the ABF weights have been found the polynomial coefficients can be obtained by Equation (8).

Conjugate gradients requires the matrix to be symmetric positive definite and cannot be used to solve rectangular systems. One way around this difficulty is to multiply the equation by C T and then solve the normal equation C T z = C T Cl. This has the unfortunate consequence that the condition number of the system is squared, which significantly slows convergence. A more efficient Lanczos based approach can be developed by recognising that the residual at the least squares solution, r = z − Cl ls , satisfies C T r = 0. This converts the problem into an augmented, symmetric matrix equation

I C CT 0

r l

=

z 0

(32)

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Application of the Lanczos approach to Equation (32) results in a special structure that is efficiently exploited by the LSQR algorithm of Paige and Saunders (1982). On successive applications of the Lanczos iteration the I and 0 parts of the matrix are unchanged, but the C and C T parts reduce to upper bidiagonal form. Both generation of the next Lanczos vector and solving for the iterate [r k l k ] can be accomplished by short term recurrences (see Appendix B). The principle disadvantage of the LSQR algorithm is that it requires the action of C T as well as C, which means that two matrix vector products are required per iteration. However, one can still reduce the cost of a single iteration by using fast matrix methods, as calculation of C T takes approximately the same computational effort as C. Additionally, several overheads required in the fast calculation of Cl can be used without

modification in the calculation of C T r.

5 INTERPOLATION AND SMOOTH FITTING OF THE JEMALONG SURVEY

To illustrate the generality of the approach and the performance of the Lanczos algorithms I applied the techniques to the Jemalong radiometric survey. The Lanczos algorithms with the direct method for the matrix-vector product are relatively computationally efficient as long as the data can be stored in internal memory. I ran the simulations on an Ultra-Sparc 2 with 380 megabytes of RAM, which means I could store a 7,000 x 7,000 matrix in memory (using double precision). Therefore, I used only a subset of 30 of the 90 acquisition lines of the survey, and also slightly truncated the survey at its southern end (Figure 3). This gave a total of 5,795 points which could easily be stored in internal memory. Using a direct method for a problem of this size would take around 7 hours for the inversion alone (i.e. not including the formation of the matrix and the interpolation onto a regular grid). The co-ordinates were first scaled so that the maximum distance between any two points was [ 1, which prevents the basis functions from growing too large. To solve the interpolation equation the matrix of Equation (4) was generated and stored in memory, which meant that a single matrix-vector product could be calculated relatively rapidly (around 1 to 2 seconds on our work station).

5.1 Exact interpolation

5.1.1 Exact thin-plate splines

I concentrate on fitting just two bands, total count (which has a very high signal to noise ratio, SNR, and is very smooth) and thorium (which has a lower SNR and consequently a much rougher surface). Initially both surveys were exactly interpolated using the thin plate spline. The convergence to the

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tolerance threshold occurred rapidly for both bands taking 333 and 342 iterations for total count and thorium respectively. The fitted surfaces in both cases were accurate to greater than eight significant figures indicating that the procedure could have been stopped earlier. The total times (including formation of the matrix) for convergence were around 20 minutes for total count and 21 minutes for thorium. This is over 20 times less than the time predicted for solution by LU decomposition.

The thin-plate spline fitted surfaces for total count and thorium are shown in Figure 4. The total count surface because of its much higher SNR is much smoother than the thorium surface. It appears that an exact interpolant is ideal for total count, but probably not for thorium.

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For total count, the remaining issue is whether to use the thin-plate spline fit, which is as smooth as possible in terms of second order derivatives, or instead a tension spline, which uses additional

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derivative constraints. A tension spline may be favoured if there are regions of rapid change in the data, as a spline, because of its stiffness, may exhibit overshoots in these regions. With the tension spline, the unattractive feature is the determination of the tension parameter which currently needs to be chosen empirically. One way to achieve this relatively rapidly, is to select a small representative region of the survey and chose an appropriate value by trial and error. Care must be taken to ensure that any scaling applied to the co-ordinates in the trial area are identical to that for the whole survey, as the tension parameter is scale dependent.

Figure 4: Grey-scale images of the exact thin-plate spline interpolation of total count (left-hand side) and thorium (right-hand side).

139 Using this trial and error procedure I found a tension parameter of d = 1, 000 was adequate. The CG algorithm applied to the whole dataset reached convergence very rapidly, requiring only 107 iterations. I do not show the full 2-D gridded surface as at the scale used the difference between it and the thin-plate spline interpolant is relatively subtle. Therefore, I show profiles of the tension spline and thin-plate spline interpolants firstly along one of the acquisition lines and then slightly offset from an acquisition line (Figure 5). A tension spline with too high a tension (d = 10, 000) is also included for comparison. Along the acquisition line, the tension spline with d = 1, 000 and the thin plate spline interpolant give similar curves (Figure 5a). As expected the tension spline in regions of rapid change gives a ‘tighter’ curve with higher second derivatives. The curve with too high a tension contains ‘bumpy’ artefacts especially at the survey boundary and after regions of rapid change in gradient. Away from the acquisition line the tension and thin-plate spline curves differ much more substantially (Figure 5b).

The differences are especially apparent within the two

prominent troughs on the right hand side of Figure 5b. The thin-plate is much stiffer than the tension spline and changes more slowly, which along this profile appears to cause overshoots. However, only a 2-D profile of a 3-D surface is shown and the change in shape of the surface in the other directions needs to be considered. Inspection of the raw data revels that the nearest neighbour of the point at the bottom of the deepest trough (at a level of 1520 counts/sec for the spline) is 1406 counts/sec. Therefore, it appears that the profile is a slice of the surface oriented at an angle to the direction of maximum change. (a)

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impose, and hence which basis function is the most appropriate. The example presented above shows that the basis function selected can have a very strong influence, especially away from the data constraints.

5.2 Smooth fits

The exact interpolation of the thorium data using the thin-plate spline looks relatively noisy, indicating that an exact fit may not be ideal. Here I briefly present the results of the several methods discussed earlier applied to smoothing this data within the ABF framework.

5.2.1 Smoothing spline fits to the thorium data

For splines I applied two smoothing methods to the thorium data; regularisation with the parameter chosen by generalised cross validation (the Fast GCV algorithm) and implicit smoothing by using less nodes than data constraints (the LSQR algorithm). The GCV application involved estimating the scalars h(m ) and g(m ) that occur in Equation (30) by the Lanczos procedure for a single m. Convergence occurred relatively quickly taking a total of just under 350 iterations. The Lanczos procedure was then continued for an extra 50 iterations to ensure that the Lanczos iterations would not need to be restarted when h(m ) and g(m ) were calculated for a different m. A one-dimensional golden section search (Whittle, 1971) was then used to find the m that minimised the GCV within the range [0, 0.5 ]. This optimal smoothing parameter was found to be m opt = 1.3e − 04.

The second smoothing example used a regular grid of nodes with a 100 m spacing to give a total of 4495 nodes compared to 5795 data constraints. The solution was found by the LSQR algorithm which was slow to converge. Just under 1300 iterations were required to reach a tolerance threshold, selected asÈA T rÈ/ÈA ÈÈr È < e, where e = 10 −7 . The norm ÈA T r È is zero at the least squares solution, while ÈA È is an estimate of the Frobenius matrix norm (Paige and Saunders, 1982), and Èr È is the residual norm. Each of these norms can be estimated during the LSQR iterations.

The GCV and LSQR thin-plate spline fits together with the exact thin-plate spline fit are shown along part of a flight transect in Figure 6a. Notice that the GCV and LSQR fits are qualitatively similar and effectively capture the medium (3 to 5 data points) and larger scale trends along the transect. Both appear to disregard the influence of the significant outliers at about 6500 m (11.7 ppm eTh) and 7200 m (8.3 ppm eTh), and interpolate smoothly through the obvious data scattering between 9 000 and

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1000 meters. Inevitably the smoothing process filters out signal as well as noise. However, in the previous chapter I estimated the standard deviation of the noise for this survey to be 1.4 ppm eTh. As most of the residuals are bounded above by this number, it appears (qualitatively) that the GCV and LSQR methods are not 'over-smoothing' the data. (b) 13

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Figure 6: Plots of interpolated thorium along an acquisition line; (a) Exact thin-plate spline, GCV thin-plate and LSQR thin-plate with 100 m nodal spacing; (b) LSQR thin-plate and sinc function both with 100 m nodal spacing; and (c) GCV thin-plate spline and GCV kriged surface with a linear trend. A second comparison between the smooth and exact fits was made by investigating contour plots of the surfaces in a small 800 m by 800 m region (Figures 7a-c). The exact thin-plate spline surface is complicated and contains many small “bulls-eye” shaped contours centred about data constraints. These artefacts are generally indicative of outliers and occur because the surface is constrained to be an exact interpolant. The “bulls-eye” artefacts are absent in both the GCV and LSQR surfaces which

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are significantly less complex. In this view the GCV and LSQR surfaces again appear similar, although there are subtle differences in the size of the peak at approximately 200 m East, 280 m North and the shape of the trough centred around 600 m East, 500 m North. (a) 800

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Figure 7: Contour plots of the thorium data interpolated by thin-plate splines in an 800 m % 800 m, subsection of the survey area; (a) Exact thin-plate spline; (b) GCV thin-plate spline; (c) LSQR thin-plate spline with 100 m nodal spacing. The solid circles in each figure are the data sampling locations. 5.2.2 Smooth fitting using the sinc function

The sinc is an attractive choice as a basis function as it imposes an upper limit on the frequency content of the interpolated surface. Due to their periodicity sinc interpolants should always be defined on a regular grid of points. The spacing of the grid points, Dx and Dy, defines the frequency

143 cut-offs by the Nyquist relationship, f x = 1/(2Dx ) and f y = 1/(2Dy ). I choose to interpolate with the sinc function defined on the same 100 m spacing regular grid as the thin-plate spline example. This gives a frequency cut-off of 0.005 m-1 in both directions, compared to the Nyquist sampling frequencies of 0.0071 and 0.005 m-1 in the along and across line directions, respectively. In the along line direction the sinc function is acting as a low pass filter, while across lines the frequency cut-off corresponds with the sampling frequency.

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Figure 8: Contour plots of the thorium data interpolated by sinc functions in an 800 m % 800 m, subsection of the survey area; (a) LSQR sinc function with 100 m nodal spacing; (b) LSQR sinc function with 150 m nodal spacing and 10 extra grid points around the boundary. The solid circles in each figure are the data locations. The LSQR algorithm applied to the sinc data converged much faster than the thin plate spline fit taking only 14 iterations to reach the same tolerance threshold. This occurs because the matrices that arise in sinc interpolation generally have much lower condition numbers than those occurring in spline interpolation. Further, the closer the data distribution approximates a regular grid the lower the condition number for a sinc interpolant. The reason is that for sinc interpolation of regularly sampled data, the weights and function values are identical; i.e. the matrix A is the identity.

Along a flight transect the sinc interpolant appears to give a reasonable fit which results in less noise removal than the equivalent thin-plate spline fit (Figure 6b). However, when a contour plot of the surface (Figure 8a) is constructed a considerably different picture emerges. Near the data constraints the surface appears to be reasonable, but in between there are substantial circular artefacts. They arise because of the finite size of the domain over which the nodes are distributed. The sinc’s slow falloff results in these boundary effects extending a considerable distance into the interior of the

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domain. Fortunately, Newsam and Beatson (1998) show that it is possible to eliminate these end effects using an adaptation of the fast moment method algorithm of Beatson and Newsam (1998).

One can also go a long way towards eliminating the boundary effects by extending the distribution of nodes past the domain of the data. To illustrate this idea I used a regular grid with a 150 m spacing, with the grid extending 1500 m (10 nodes) past the survey boundary in each direction. The LSQR algorithm converged rapidly taking only 10 iterations.

The interpolated surface appears to be

sensible and essentially free of boundary artefacts (Figure 8b). It models the larger scale structure in the thorium surface but its shape is different than either of the smoothing spline surfaces (Figures 7b and c). An attribute of this sinc interpolated surface is that it contains no frequency information above 0.0033 m-1.

5.2.3 Spatial prediction by kriging

To obtain a reliable kriged surface the semi-variogram first needs to be estimated.

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When I calculated a semi-variogram by this discrete approximation to Equation (33), the variance at large lags was found to increase linearly, implying that there was a linear (or possibly higher order) drift in the data. I then fitted a linear surface to the data by least squares, removed the linear trend and then reapplied Equation (33).

The resultant semi-variogram appeared to oscillate about a

constant factor at large lags (Figure 9). On repeating the procedure with a quadratic surface I found that the variogram changed little, so inferred that there was only linear drift in the data. I then least square fitted a (transitive) spherical semi-variogram,

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to the first 13 samples in Figure 9, and found coefficients of c o = 0.28, c 1 = 0.22 and a = 1750 m (Figure 9). The parameter a is the range, c o is the nugget variance and c o + c 1 is the height-to-sill. This method for semi-variogram estimation can potentially introduce bias and a more rigorous alternative is to use primary increments (e.g. Cressie, 1993).

However, I wished to avoid

complicating this illustrative example and therefore chose the much simpler method used here.

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Hutchinson (1993) advocates and alternative approach which is to fix the range parameter and determine the ratio of the nugget variance to the height-to-sill by GCV. Applying the Fast GCV

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algorithm to the kriging equations, I found a quadratic variation in the GCV measure (Figure 10) and a minimum atv = 0.19. This is a substantially smaller smoothing parameter than that predicted by directly fitting the semi-variogram. When the two kriged surfaces are compared, it is apparent that the larger smoothing parameter from directly fitting the variogram results in a much smoother surface than the GCV kriged surface (Figure 11). Given the difficulties in accurately estimating the nugget variance and sill directly from the variogram, the more rigorous approach afforded by GCV is to be preferred. 0.37

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The GCV kriged surface is shown compared to the GCV thin-plate spline curve along a data transect in Figure 6c. The curves are similar although the kriged surface tends to have slightly lower residuals (i.e. removes less “noise”). This is particularly apparent around the outliers at 6 500 m (11.7 ppm eTh) and 7 200 m (8.3 ppm eTh) and the obvious data scattering between 9 000 and 10 000 meters. The kriged surfaces also tends to display more structure than either of the smoothing spline surfaces (compare Figure 11b with Figures 7b and 7c), as evidenced by the rougher contours of the kriged surface. However, the surfaces (especially for the GCV spline and kriging examples) are very similar on a large scale and only appear to differ over relatively small distances.

In principle, if the data is indeed well modelled as a spatial random field with a spherical semi-variogram and if the parameters in this semi-variogram have been successfully estimated, then the kriged surface is "more accurate" than a spline interpolation; it is the best linear predictor that can be obtained from the data. In practice, there is usually little difference between the kriged surface and a thin-plate spline fit. Moreover, Hutchinson (1993) has shown that the same is true of fits to simulated data: even when they are simulated using the same assumptions underlying the variogram that is used to krige the surface. Therefore, as Hutchinson (1993) observed, since it is difficult to decide whether the kriging assumptions are indeed met, and since fitting the spline requires significantly less calculation and fitting (it does not require estimating the semi-variogram), there is much to be said for using thin-plate smoothing splines in place of kriging.

5.2.4 Concluding remarks

I conclude this section by illustrating the visual improvement to the whole thorium image that results by using smooth fitting rather than exact interpolation (Figures 12 and 13). Only the LSQR thin-plate spline image is shown as at this scale is very similar to the GCV thin-plate spline and kriging images (not shown). Notice that the smoothed image effectively models most of the visible structure in the exactly interpolated image and appears to eliminate the “salt and pepper” noise apparent in the exact surface. From an applications viewpoint the smoothed image would be much easier to interpret than the exact image. Inevitably, in regions of rapid change of the thorium concentration, some smoothing of ‘signal’ may occur, making sharp boundaries appear more gradual. However, the sensor height and non-directional nature of the detector cause radiometric images to be blurry representations of the true radio-element distribution (Chapters 3 and 4). Therefore, there are few, “real”, sharp edges to be blurred by a GCV or LSQR fit. The removal of the blurring caused by the measurement process is addressed in the next two chapters.

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Figure 12: Grey-scale images of the exact (left) and LSQR (right) thin-plate spline surfaces over the entire thorium data set. Both images are shown with the same linear stretch applied to the data.

6 DISCUSSION

In this chapter I have presented a general approach for interpolation and smooth fitting of geophysical data. Naive solution of the resulting equations is O N 3  in computation and 

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To reach an acceptable solution none of the iterative schemes (with the exception of the LSQR spline example) required a significant number of iterations, relative to the problem size. Therefore the cost lies somewhere in the lower part of the range O(N 2 ) to O(N 3 ). Clearly, for the approach to be feasible for interpolation of large geophysical surveys, the fast method for matrix-vector multiplication of Beatson and Newsam (1998) and efficient preconditioners would need to be used to further reduce the computational burden. Further discussion on these issues is deferred to the final chapter (Chapter 8) of this thesis.

Figure 13: Three-dimensional view of the exact (left) and LSQR (right) thin-plate spline surfaces over the entire thorium data set. Data are often interpolated onto a regular grid for the purpose of visualisation. When the ABF matrix system has been solved, the data can be interpolated onto any set of points using a single matrix-vector product. One challenge in visualisation of large data sets is accommodating the change in scale as the user zooms from a general overview of the whole survey area, to a more detailed view of a specific area or feature. In these two cases the appropriate grid spacings vary substantially. The ABF approach is ideal in this respect as, with a fast method for the matrix-vector product, only certain data structures inherited from the weights need to be stored (which may require much less memory than storing the weights) and different views of the surface can then be rapidly generated at any given scale.

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The ABF approach is based on a rigorous theoretical framework. If the user wants the interpolated data to exhibit some desirable property they can enforce it by choosing the appropriate basis function. In the results sections I considered the interpolation of the total count band using both thin-plate and tension splines. While both exactly interpolated the data there were significant differences in the surfaces away from the data points. The thin-plate spline gave a surface that minimised second derivatives, while the tension spline minimised a constraint that included derivatives of all orders. The difficulty with the tension spline approach is that there is a scale dependent parameter to choose. I would therefore recommend the thin-plate spline for exact interpolation where overshoots are not deemed problematic.

Geophysical data is inevitably noisy and an interpolant passing through the data points may not be ideal. The ABF framework offers considerable flexibility in the smoothing method, as we illustrated with the thorium band. The most rigorous of these is the GCV which can be calculated efficiently using the Lanczos based approach of Sidje and Williams (1997). The GCV was successfully applied to the selection of the smoothing constraint for the thin-plate spline and the nugget effect in kriging. Both of these gave smoothly varying surfaces with the noise removal based on a rigorous statistical basis.

The second mode of smoothing considered was using less nodes than data constraints. The resulting least squares equations could be solved efficiently by the Lanczos based approach of Paige and Saunders (1982). While not as rigorous as the GCV approach the method appeared to effectively estimate the noise levels in the data for both the sinc function and the thin-plate spline. For the sinc function the data should be interpolated from an ABF defined on a regular grid that is larger than the data domain otherwise artefacts can be introduced into the surface. The interpolated surface is band limited, implying that it has no frequency information above a certain cut-off inherited from the nodal spacing.

The benefits of moving the nodes onto a regular grid may outweigh the rigorous approach afforded by GCV. In the next two chapters I examine how ABFs can be efficiently utilised in subsequent processing operations such as Fourier transforms and deconvolution. In some cases movement onto a regular grid considerably speeds up the additional processing.

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CHAPTER 6

THE FOURIER TRANSFORM OF AN ARBITRARY BASIS FUNCTION SURFACE [about Fourier] It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius. Rudoph E. Langer

CHAPTER OBJECTIVES Develop methods for calculating the Fourier transform of surfaces defined by Arbitrary Basis Function expansions.

1 INTRODUCTION

The Fourier transform is a linear transformation that provides the passage between two alternate descriptions of a function. In modelling physical systems the original function exists in the time or space domain (the latter in this thesis), while the transformed function lies in the frequency domain. The frequency domain description is a decomposition of the original function into sinusoidal and cosinusoidal components of different frequencies. When these components are superimposed via the inverse Fourier transform the original space domain description of the function is regained.

The principal utility of Fourier theory is that certain operations that are difficult in the space domain are very simple in the frequency domain. An example is the convolution of two functions to form a new function, a process that occurs extensively in the modelling of many physical systems. In the space domain, calculation of the convolved function at every point involves an integral over infinite limits. In the frequency domain the convolution reduces to a multiplication of the Fourier transforms of the two original functions. The convolved response is then obtained as the inverse Fourier transform of this product.

In many practical situations one does not have complete knowledge of the function under study, only a finite number of discrete samples. In that case the continuous Fourier transform is replaced by its

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discretised version, the discrete Fourier transform (DFT). The application of Fourier techniques to discretely sampled data began to proliferate in the 1960’s due principally to two major advances. The first was the availability of digital computers with both increased speed and reduced cost and size. The second was the development of the Fast Fourier Transform (FFT) algorithm by Cooley and Tukey (1965). This algorithm drastically reduced the computational cost of a DFT on uniform grids from O(N2) operations to O(N log N ) operations. Increases in the speed of computers then made application of the DFT to larger and larger geophysical surveys possible. Today, application of Fourier theory to the processing of geophysical data is commonplace (e.g. Blakely, 1995).

In geophysics, Fourier transforms are required to calculate the depth to basement from airborne magnetic surveys (e.g. Blakely, 1995), to downward and upward continue magnetic surveys (e.g. Henderson and Zietz, 1949), to calculate components of potential field measurements in specified directions and in reduction to the pole for magnetic surveys (e.g. Ervin, 1976). The principal interest in Fourier theory in this thesis is that under certain simplifying assumptions (flat topography, minimum deviation from nominal survey height etc.), airborne gamma ray surveys can be modelled as a convolution between the ground isotopic concentrations of potassium, uranium and thorium and the point spread function developed in Chapter 3. The inverse process of deconvolving the observed count rates to return estimates of the isotopic concentrations can be efficiently solved using Fourier theory (see Chapter 4).

The high speed of the FFT and the ease of discrete Fourier domain operations can mask a very important point: the DFT is only an approximation to the continuous Fourier transform. Geophysical data such as airborne magnetic, radiometric and electromagnetic surveys are collected at an irregular spatial sampling interval. In order for the FFT algorithm to be used the data must first be interpolated onto a regular grid, raising a number of important issues (e.g. Cordell and Grauch, 1982). Firstly, a choice of interpolation method needs to be made which can have a significant influence on the calculated Fourier transform. Secondly, the FFT algorithm requires that the data fill an entire rectangle. If the original survey covers an irregularly shaped area the gaps need to be filled with synthetic data. Thirdly, the FFT of a finite number of samples has an implied periodicity causing the Northern most grid cells to be neighbours to the Southern most grid cells, and likewise for the Eastern and Western edges. To prevent artefacts in the Fourier domain resulting from this implied periodicity, some type of windowing or domain extension needs to be applied so that the opposite edges match. Forthly, the DFT is a trapezoidal quadrature approximation to the continuous Fourier transform which may not be the most accurate quadrature scheme.

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The ABF framework developed in the last chapter can be used to solve many of the problems with Fourier transformation described in the last paragraph. In particular, 1. The ABF provides a natural interpolant that is defined over all of space. 2. The Fourier transform of an ABF has a particularly simple expression that facilitates its calculation and further analysis. 3. The formal expression for the Fourier transform of an ABF can be calculated exactly on an arbitrary grid. Thus, the only error incurred in computing it is the approximation error between the original function and the ABF interpolant. No numerical quadrature errors are introduced and the issue of choosing appropriate quadrature points and weights is avoided. 4. The convolution or deconvolution of an ABF is another ABF (see Chapter 7).

The chapter is organised as follows.

In the next section the continuous and discrete Fourier

transforms are presented and several problems related to the use of the DFT in a one-dimensional context are highlighted. Attention then turns to the two-dimensional DFT and the current practice of its application to irregularly sampled geophysical data. The ABF framework discussed in the last chapter is then presented as a method that can overcome many of the deficiencies of the existing methodology. Examples are presented of the calculation of the Fourier transform for the Jemalong radiometric survey using a standard grid based approach and the new technique based on the ABF framework. Lastly, the influence of the basis function and solution method on the calculated Fourier transform are examined.

2 THE FOURIER TRANSFORM

2.1 Definition of the Fourier transform

In one dimension the Fourier transform, F(u ), of a function f(x ) defined over the continuous domain of x c (−º, º ) is given by F(u ) = ° −º f(x ) exp[−2oiux ]dx º

(1)

where u is a frequency domain transform of x. If x is a space domain co-ordinate in meters then u will be in cycles per meter, while if x is a time domain co-ordinate with seconds as units, u will be in cycles per second. The inverse Fourier transform can be used to recover f(x) from F(u) and is defined similarly,

154 f(x ) = ° −º F(u ) exp[2oiux ]du º

(2)

Verification that Equations (1) and (2) do indeed form a transform pair can be obtained by substituting one equation into the other. This process implies that for the transforms to exist both f(x) and F(u) must (i) be absolutely integrable and (ii) have finite discontinuities, (e.g. Bracewell, 1986). However, the class of functions that have Fourier transforms can be made more general than this by considering appropriate convergent sequences of functions (e.g. Bracewell, 1986). Each member of the sequence satisfies the two conditions given above but the limit of the sequence may not which leads to the concept of a generalised function (e.g. Gelfand and Shilov, 1964). An example of a generalised function is the Dirac delta function or impulse function, d(x ), which can be defined as the limit of the sequence a −1 exp(−ox 2 /a 2 ) as a approaches zero. The Dirac delta function is zero everywhere except at the origin where it has a pulse of infinite height but infinitesimal width such that its integral over the real line is unity. In two-dimensions, the Fourier transform F(u ) = F(u, v ), of a function f(x ) = f(x, y ), defined over the entire real plane,

2

ℜ ‘ , is given byy

F(u ) = °ℜ2 f(x ) exp[−2oiu $ x ]dx = ° −º ° −º f(x, y ) exp[−2oi(ux + vy )]dxdy º

º

(3)

where u = (u, v ) and x = (x, y ) are vectorised frequency and spatial co-ordinates, and u $ x = ux + vy is the dot product. The inverse transform is defined analogously to the 1-D case.

In real world applications it is not possible to continuously measure the function or signal of interest, nor is it possible to measure it indefinitely. Rather, there are a finite number of samples, N, obtained over a finite domain, which generally means that the Discrete Fourier Transform (DFT) must be used. The DFT can be defined on any arbitrary set of points (in either 1-D, 2-D or higher dimensions), x n , and is usually evaluated at N frequency coordinates, u m , F(u m ) = S n=1 w n f(x n ) exp(−2oi u m $ x n ) N

(4)

where the w n are a set of quadrature weights. A reverse transformation can be similarly defined, N f˜(x n ) = S n=1 w m F(u m ) exp(2oi u m $ x n )

(5)

155

Note, that f˜(x n ) has been used to denote the results of the reverse transformation. This indicates that, for an arbitrary set of points, evaluation of Equation (4) followed by (5) may not return the original function values f(x n ). This will occur only if the x n are regularly distributed and the u m are chosen appropriately (see next section). For the general case, the inverse transformation, when F(u m ) is known, has to proceed by solving the matrix equation specified by (4) for the unknown f(x n ).

2.2 Discretisation and the 1-D Fourier transform

Consider the case of regular sampling in 1-D. There are many alternative definitions of the DFT on uniform grids in the literature and I will use a very similar form to that given in Briggs and Henson (1995).

Assume that there are an even number of regularly spaced samples, f n , translated (if

necessary) to the range, x n = nDx, n = −N/2 + 1, ..., N/2. From these N input samples, N outputs, F n , can be calculated on a regular grid of frequency co-ordinates, u n = n/(NDx ), n = −N/2 + 1, ..., N/2. Note, that the highest frequency (the Nyquist cutoff) is determined by the separation of the data samples, 1/(2Dx ), while the smallest non-zero frequency is determined by the spatial or temporal extent of the data sequence, 1/(NDx ). The forward DFT, on the uniform grid, is obtained with uniform quadrature weights (w n = Dx),

F m = Dx

N 2

S f n exp n=− +1 N 2

−2oi mn N

(6)

An important property of the DFT on uniform grids is that it has an inverse which is almost identical to the forward transform, except for the quadrature weights (w m = Du) and sign of the exponent,

f n = Du

N 2

S F m exp m=− +1 N 2

2oi mn N

(7)

To verify that the DFTs are an inverse pair, substitute Equation (6) into (7) to find, N 2

N 2

N

2

2

2 DuDx S f k exp 2oim (n − k ) = DuDxN S d(n − k )f k = f n S N m=− N +1 k=− N +1 k=− N +1 2

where d(n ), is the Kronecker delta (d(0 ) = 1, d(! 0 ) = 0), and I have used DuDx = 1/N.

(8)

The

summation over m reduces to N times the Kronecker delta because the complex exponentials, of the DFT on uniform grids, are orthogonal (e.g. Briggs and Henson, 1995).

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Use of the DFT on uniform grids in place of the continuous definition of Equation (1) raises several issues: 1. Finite number of terms in a finite domain: The continuous Fourier transform involves an integral over an infinite domain, while the DFT is a summation with a finite number of terms over a finite sized domain. This results in an implicit assumption that the input sequence is periodic so that the entry afterf N is f 1 . If these two endpoints do not match then spurious high frequencies are introduced into the spectrum. The result is that after filtering and inverse transformation oscillations, (Gibb’s effects) can be introduced (e.g. Bracewell, 1986; Briggs and Henson, 1995). They are mainly concentrated around the edges but sometimes extend right to the centre of the data sequence. In one dimension there are three different approaches to minimise these edge effects, (i) Synthetically extend the data so that the two ends join smoothly; (ii) Remove a linear trend that passes through the end points in the domain; (iii) Remove the mean value and multiply the data by a window function that approaches zero at the end points. The shape of the window function is determined by its effect on the spectrum (e.g. Harris, 1978). 2. Sampling rate: The sampling rate, Dx, imposes an upper limit on the frequencies that can be resolved through the Nyquist relationship, u max = 1/(2Dx ). If the function being sampled contains higher frequencies than u max contamination of the lower frequency parts of the spectrum by the unsampled higher frequency components will occur. This phenomena is termed aliasing and can only be avoided when the function has no frequency content above a certain cut-off, u c (the function is then termed band limited). The aliasing will then be zero as long as the Nyquist sampling rate equals or exceeds the functions frequency cut-off, i.e. u max m u c . In many practical situations, the Fourier transform approaches but never reaches zero. For example, it is well known that a function or signal with a finite spatial or temporal extent cannot also have a finite extent in the frequency domain (e.g. Bracewell, 1986). If the transform decays faster than u

−1

as u d º then the transform is termed

essentially band limited (Paley and Wiener, 1934; Briggs and Henson, 1995) and a sufficiently small sample spacing can be chosen such that the aliasing error is negligible. More specifically, a function is essentially band limited if there exists positive constants a and b such that F(u ) [ a(1 + u ) −b−1 . 3. Quadrature approximation: The DFT can be interpreted as a quadrature approximation to the continuous Fourier transform for a function with finite spatial extent. Write,

F(u ) = ° x −N/2 f(x ) exp(−2oiux )dx j x N/2

N 2

S

n=− N2

w n f(x n ) exp(−2oiux n )

(9)

157

where w n are a set of weights to make the quadrature as accurate as possible. With regular spacing between quadrature points, equality of endpoints so that f(x N/2 ) =f(x −N/2 ), and the weights constant and equal to the sample spacing (i.e. w n = Dx), Equation (9) is the trapezoidal quadrature formula. With these conditions the formula is exactly analogous to the definition of the DFT on uniform grids given at Equation (6). Therefore, we can interpret a DFT on a uniform grid as a trapezoidal approximation to the continuous Fourier transform. It is relatively straightforward to extend the definition of the DFT to higher order quadrature schemes such as Filon’s or Simpson’s rules (e.g. Briggs and Henson, 1995). For example, Simpson’s rule has (w 1 , ..., w N ) = Dx/3(1, 4, 2, ..., 4, 2, 4, ..., 2, 4, 1 ), and it is only a matter of pre-multiplying each f n by w n before calculating the DFT. However, which quadrature scheme is ‘best’ is difficult to determine because higher order doesn’t always equal higher accuracy and the errors in each scheme depend on the characteristics of the function being approximated (e.g. Briggs and Henson, 1995). 4

Computational cost: With N input samples the Fourier transform is calculated at N frequency

coordinates making a direct application of the DFT O(N 2 ) in complex operations. If N is a factor of 2 then the DFT on a uniform grid can be calculated rapidly in O(N log N ) operations using the Fast Fourier Transform (FFT) algorithm (Cooley and Tukey, 1965). If N is not a multiple of 2, the DFT input sequence is padded with zeros (or synthetically extended as discussed above) to makes its size the nearest factor of 2. Note, that there exists ‘mixed radix’ versions of the FFT algorithm that do not require N to be a multiple of 2. The speed depends on the number of prime factors in N and the algorithm generally works fast as long as N is a product of small prime numbers (e.g. Singleton, 1969). Note further, that additional computational savings can be made if the input data is purely real or imaginary and/or if the input data is symmetric or anti-symmetric.

2.3 Fourier transforms of 2-D geophysical data.

When data are collected with an irregular spatial distribution, as is generally the case in geophysics, the usual procedure is to first interpolate the data onto a denser regular grid and then calculate the Fourier transform via the FFT algorithm. In many airborne geophysical surveys the sample spacing in the along and across line directions may be an order of magnitude different, but the interpolated grid generally has identical spacing. For example, magnetic surveys are usually collected with less than 10 m spacing along lines and 100 m or greater spacing between lines. The interpolated grid is usually chosen to have a regular spacing of between one-quarter and one-fifth of the across line spacing (e.g. Pettifer, 1995) which means that there is an increase in points along at least one direction. Often, there is a decrease in points in the other direction, although this is not always the case (e.g. radiometric survey with 70 m along line and 100 m across line spacing gridded at 100/4 =

158

25 m; the number of points increases in both directions). Note, that the along line direction may be over-sampled, while the across line direction is often under sampled, causing aliasing of high frequencies.

Standard Fourier domain processing of geophysical data then reduces to a problem of computing the forward and inverse transforms of a regular grid of points. A compact notation for the elements of the regular grid is f nm for the n-th row and m-th column in the space domain, and F jk for the j-th row and k-th column in the frequency domain. From a regular N % M grid (assumed even for simplicity), with a spacing of Dx and Dy, the two dimensional forward DFT is defined analogously to the one dimensional case,

F jk = DxDy

N 2

M 2

S S f nm exp −2oi n=− +1 m=− +1 M 2

N 2

jn km + N M

(10)

The inverse, assuming grid spacings in the frequency domain of Du and Dv, is

f nm = DuDv

N 2

M 2

S S F jk exp 2oi j=− +1 k=− +1 N 2

M 2

jn km + N M

(11)

Application of the two dimensional DFT raises the same four questions considered in the 1-D case; namely: (i) finite domain; (ii) sampling rate; (iii) quadrature approximation; and (iv) computational cost. A brief note on this last issue. The 2-D FFT can be calculated by a sequence of 1-D FFTs. Firstly, perform N FFTs of length M on the columns of the grid. Secondly perform M FFTs of length N on the rows of the grid obtained by the first process. The entire 2-D DFT can therefore be calculated withO(MN[log M + log N ]) complex operations.

Again, symmetric, pure real or pure

imaginary inputs can be computed with less computational effort and memory than the general complex case.

Use of the 2-D DFT in a geophysical context raises several additional issues; 1. Interpolation: The interpolation algorithm that is used may not necessarily give a very good representation of the true underlying function or signal, and hence the Fourier transform of the gridded data may be a poor approximation of the true Fourier transform. 2. Finite and possibly irregular domain: Fourier transformation of an image whose opposite edges do not match results in high frequency contamination of the spectrum in the form of a “St-Georges Cross” (high intensities concentrated along the axes of the domain).

Any high pass filtering

159

operations applied to the data (such as upward continuation) result in the amplification of these high frequencies and cause noticeable ringing (undulating patterns) back in the space domain (e.g. Craig, 1995). Methods to deal with edge mismatch have been given by Ricard and Blakely (1988) and Cordell and Grauch (1982). However, many geophysical surveys do not occupy an entire rectangle and often tend to have large gaps with no data. An method to deal with both edge mismatch and survey gaps was given in an unpublished manuscript by Craig and Green (1998). The basic idea is to slightly extend the grid and assume that the domain is wrapped to form a torus. Points on the bottom edge of the grid are neighbours to points on the top edge, while left edge points are neighbours to right edge points. Values are then interpolated (or extrapolated except the distinction blurs because the domain wraps around) onto any unassigned grid points in the synthetic boundary region and the unsampled data gaps using a multigrid technique. 3. Sampling rate: The maximum frequency that can be resolved by finitely sampled data is limited by the Nyquist frequency which, for uniformly spaced data, is one-half of the reciprocal of the sample spacing. While airborne geophysical data is irregularly sampled the data samples are almost distributed on a regular grid, so that approximate Nyquist cut-offs can be specified. The different sample spacings in the along and across line directions gives two separate frequency cut-offs in the two directions, u max = 1/(2Dx ) and v max = 1/(2Dy ). The interpolation to a denser set of points followed by an FFT can introduce spurious high frequency information that is not well constrained by the data. One way to eliminate this contamination is to zero all frequency components above the Nyquist cut-off. A sharp cut-off is undesirable as it may lead to ringing (Gibb’s effects) when inverse transformed back into the space domain so a cosine roll off would be preferred (e.g. Craig, 1995). An alternative to explicit interpolation onto a regular grid is to calculate the Fourier transform directly from the scattered data. Using the DFT in this way is equivalent to uniform weights in the definition given at Equation (4). However, uniform weights are not optimal for scattered data (in many cases they are not optimal for regularly sampled data).

Therefore, accurate Fourier

transformation directly from the data, requires the calculation of an optimal set of quadrature weights. For scattered data, where there is incomplete control over the locations of the samples, this is a nontrivial exercise that has no simple solution (see Press et al., 1992).

A further consideration is that the irregular sample spacing means that the Fast Fourier Transform algorithm cannot be used. For N data points F(u ) is usually estimated at N values of u, which by a discrete Fourier transform (DFT) algorithm requires O(N 2 ) operations. In addition, to calculate the inverse transformation requires the inversion of an N % N matrix, which by a direct method is O(N 3 ) in operations. Fortunately, fast algorithms for both the forward and inverse DFT on scattered 1-D data are now available (e.g. Dutt & Rohklin, 1993; 1995).

Each of the algorithms require

160 O[N log N + N log(1/e )] operations to calculate the Fourier transformations within accuracy e, which is

not much more than the FFT which requires O(N log N ) operations. Generalisation of the methods to 2-D is not straightforward as there are considerable differences in irregular sampling in 1-D compared to 2-D. However, present indications are that it is only a matter of time before fast 2-D methods will become available (e.g. Newsam, pers. comm.).

3 FOURIER TRANSFORMS AND ARBITRARY BASIS FUNCTIONS.

The ABF framework discussed in the previous chapter provides a very flexible set of techniques for interpolation and smooth fitting of irregularly sampled geophysical data. When coupled with Fourier theory it can also be used as a method for solving some of the problems with Fourier transformation discussed in the last two sections.

3.1 The Fourier transform of an Arbitrary Basis Function Given data samples at the discrete locations x n : n = 1, ... , N I can form an ABF approximation, M s(x ) := p(x ) + S m=1 k m v(x − x˜ m )

(12)

where k m : m = 1, ... , M are a set of M weights defined at the M nodes or centres x˜ m : m = 1, ... , M and p(x ) is an optional k degree polynomial.

Depending on how the ABF was defined, the

approximation may either exactly or smoothly interpolate the data, and the nodes may either coincide with the centres, or may be distributed on a regular grid. In any of these cases we can take the Fourier transform of the ABF expansion.

As Fourier transformation is a linear operation the polynomial and summation terms can be dealt with separately.

I show in Appendix H that the summation occurring in Equation (12) is a

convolution between the basis function and the weights, M S(u ) = F(u ) S m=1 k m exp(−2oi u $ x˜ m ) + P(u )

(13)

where S(u ), P(u ) and F(u ) are the Fourier transforms of the ABF expansion, polynomial and basis functions, respectively. In 2-D, u = (u, v ) and x˜ m = (x˜m , y˜m ) are vectorised frequency and spatial co-ordinates, and u $ x˜ m = ux˜m + vy˜m is the dot product. Equation (13) is a powerful result because it shows that if a set of sampled data is approximated by an ABF expansion, the Fourier transform of that approximation is a DFT of the weights multiplied by the Fourier transform of the basis function.

161

Turning now to the polynomial terms, the Fourier transform of a constant is the Dirac delta function which is infinite at the origin and zero everywhere else. Linear monomials are first derivatives of the Dirac delta function, while higher order monomials are higher order derivatives of the Dirac delta function (e.g. Gelfand and Shilov, 1964). In particular,

°ℜ

2

x j y k exp(−2oiu $ x )dx =

¹ j+k d(0 ) ¹u j ¹v k

(14)

Each of the polynomial terms only directly contribute to the Fourier spectrum at zero frequency, but all affect the spectrum about zero indirectly, through their influence on the Fourier transform of the summation (see section 3.3).

There is an alternative derivation of the Fourier transform of an ABF expansion which leads to a good intuitive understanding.

Using the shifting property of the Dirac delta function,

° d(x − y )f(y )¹x = f(x ), I can rewrite Equation (12) as, M s(x ) − p(x ) = S m=1 k m ° ‘d d(x − x˜ m )v(x − y )¹y = °ℜ d k(y )v(x − y )¹y

(15)

where, k(x ) = S M ˜ m ). The far right of Equation (15) is a convolution which converts to m=1 k m d(x − x L(u )F(u ) in the Fourier domain, where L(u ) is the Fourier transform of k(x ). This is given by M M L(u ) = S m=1 k m °ℜ d d(x − x˜ m ) exp(−2oiu $ x )¹x = S m=1 k m exp(−2oiu $ x˜ m )

(16)

by the shifting property of the Dirac delta function. Equation (13) immediately follows. A physical interpretation is that an ABF expansion is a convolution of the given basis function with a series of M impulsive point sources located at the points x˜ m .

There are several important implications of Equations (13) and (14): 1. Fourier transforms of the basis function: Table 1 lists the Fourier transform of several basis functions commonly used for interpolation and smooth fitting, including the sinc function, the Gaussian, the multiquadric and the cubic (1-D) and thin-plate (2-D) splines. The transform of the sinc is a positive constant over a finite interval, while the Gaussian transform has a maximum at zero and then rapidly decays. The behaviour of the multiquadric and splines are significantly different as they have transforms that are unbounded at zero. At zero frequency the Fourier transform is the integral of the function over the whole real line (in 1-D) or real plane (in 2-D). The integrals of the multiquadric, the tension, cubic and thin-plate spline bases are unbounded, which causes their Fourier

162

transform at zero frequency to be unbounded. At first site, their singularity would appear to imply that the approximation of Equation (13) breaks down at the origin. As discussed in section 3.3, this is not necessarily the case, because the behaviour at the origin is tempered by the polynomial term in Equation (12). 2. Finite domain: An ABF expansion defines a continuous model that extends over the entire real line in 1-D or the real plane in 2-D (although if the basis function has compact support so will the model). Generally, the model defined outside the region covered by the data sampling will not resemble the true underlying function. The model values in this region have been extrapolated from the sample values within the domain. For example, Figure 1 shows the summation parts (i.e. with the polynomial term removed) of the cubic spline and sinc function interpolants fitted to 100 equally spaced samples between 0 and 10 of the Bessel function of order zero (Abramowitz and Stegun, 1972). The sinc interpolant displays severe oscillations about the boundary that gradually die down further within the interior of the domain. The cubic spline interpolant is free from these boundary effects and displays linear growth outside the sampling domain. This linear growth causes the cubic spline interpolant to have an unbounded integral and leads to a singularity at the origin in the Fourier domain. However, as discussed in section 3.3 the polynomial terms temper this singularity and in some instances can eliminate it completely. 3. Quadrature approximation and sampling rate: An ABF expansion defines a continuous model of the function or data. That is, we are specifying how the function, or data, changes between neighbouring sample points. When the Fourier transform of the ABF is calculated by Equation (13), this behaviour carries over into the Fourier domain. A quadrature scheme has been created for the approximation of the integral from an arbitrarily distributed set of samples.

This solves the

quadrature weight problem discussed earlier. The quadrature approximation will only be as accurate as the ABF expansion through which it was defined and will depend on the sample spacing between adjacent points and on how accurately the expansion models the variation between samples. If the ABF expansion gives a poor representation of the underlying function then the quadrature approximation will also be poor. However, an advantageous aspect of ABF’s is that they encompass many different basis functions and can either exactly or smoothly fit a set of data. This means that there is considerable control over the characteristics of the quadrature approximation. 4. Computation: If the sample points, x n , are irregularly distributed and the location of the nodes, x˜ m , are chosen to coincide with the samples, then the Discrete Fourier Transform can only be

calculated at better than O(N 2 ) operations if a fast algorithm for DFT’s on scattered data is used.

163

In this chapter, I will concentrate predominantly on two different types of basis functions. Firstly, the sinc function which is ideally suited for geophysical data where the noise imposes an effective frequency cut-off on the extractable signal (such as airborne radiometrics and magnetics e.g. Rauth and Strohmer, 1998). Secondly, spline interpolation which minimises a measure of smoothness on the interpolated surface; the cubic spline in 1-D and the thin-plate spline in 2-D.

Basis function

Spatial representation

Fourier transform

Sinc function

sinc(x/Dx ) = sin(ox/Dx )/(ox/Dx )

1 for u [ 1/(2Dx ) and 0 for u > 1/(2Dx )

Gaussian

exp(−cx

(o/c ) −1/2 exp(−o 2 u 2 /c )

Cubic spline

x

Multiquadric (1D)

(x 2 + c 2 ) 1/2

Multiquadric (2D)

(r 2 + c 2 ) 1/2

Thin-plate spline

r 2 log(r )

o/2(oq )

2 2 ln (dr/2 ) + E 1 (dr/2 ) + c

(2q/d ) 2 exp(−(2q/d ) 2 )

Tension spline

2)

3

3/4(ou )

−4

−2 c/(ou ) K 1 (2ocu ) i 1/2(ou )

o (2c/2oq )

3/2

K 3/2 (2oqc )

−4

Table 1: Several commonly used basis functions and their Fourier transforms. The function K 1 (u ) is a modified Bessel function of first order. For the multiquadrics we show the near origin behaviour by using the small u asymptotic expansion of K 1 (u ) i u −1 and K 3/2 (q ) i q −3/2 . For the tension spline c is Euler’s constant, d is the tension parameter and E 1 (q ) is the exponential integral of the first kind.

1.2 Original function Sinc interpolant Cubic spline

Function value

0.8

0.4

0.0

-0.4

-0.8

-5

0

5

10

15

Ordinate

Figure 1: Sinc and cubic spline fits to the Bessel function of order zero based on 101 equally spaced increments between 0 and 10. The cubic spline is shown without it’s associated polynomial (hence why it doesn’t fit the curve).

164

3.2 The Fourier transform of a sinc interpolant

When using the sinc function for interpolation, there are no singularity concerns at the origin and no polynomial term to consider. However, the simple example in Figure 1 for approximation of a very smooth function, showed that the finite size of the sampling domain caused a second problem: oscillations (Gibbs’ effects) about the boundary. These oscillations have their roots in the implicit periodicity of a finite number of regularly sampled data, the slow, ~1/x, decay of the sinc basis function and Shannon’s sampling theorem (Shannon, 1949). The one-dimensional version of this theorem states that a function band-limited with a maximum frequency u c , can be exactly reconstructed by regularly sampling at the Nyquist rate Dx c = 1/(2u c ) with the sinc function as basis, (x − x m ) º f(x ) = S m=−º f m sinc Dx c

(17)

Notice, that the weight at the point x m is given by the function value, f m , at that point. This occurs because the sinc function is periodic and disappears at every sample point, except x = x m . The unfortunate aspect of the sampling theorem is that the summation has an infinite number of terms. Far from the sampling boundaries, the finite number of terms available has little impact on the sinc interpolant as the contribution from the unsampled points is small. However, near the boundary the unsampled points would contribute substantially and their absence results in the observed oscillations. Note that far from the boundary, is defined as the point at which 1/x is sufficiently small. Since 1/x decays very slowly, the oscillations will be evident at quite some distance from the boundary.

In 1-D some of the ideas discussed in section 2.2 for elimination of boundary effects in the DFT can be used for the sinc function. For example, inclusion of a linear polynomial in the interpolation or reflection of the data can significantly reduce the size of the oscillations about the boundary. In 2-D the situation is more difficult and some type of domain extension and/or windowing suggests itself. When over sampling occurs (i.e. the actual sample spacing Dx is smaller than the critical Nyquist spacing, Dx c ), it is possible to eliminate the boundary effects in a more rigorous manner (Beatson and Newsam, 1998b). The idea behind their technique is to define the sinc interpolant on a regular and effectively infinite series of points with a sample spacing of Dx c (or some estimate of the actual Nyquist if it is not exactly known). Because this spacing is greater than the data spacing, there are less nodes in the ABF expansion that fill the sampled parts of the space domain than there are data constraints. One can use the extra degrees of freedom to specify a series of ‘boundary correction terms’ which summarise the contributions from increasingly larger intervals away from the boundary.

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When the Fourier transform of this compound sinc interpolant is calculated it will also be free of boundary effects, without any need for explicit padding. Note, that by Equations (13) and (17), a sinc interpolant defined on a regular N % M grid of nodes has a Fourier transform given by S f (u ) = DxDy S n=1 S m=1 f nm exp(−2oi u $ xˆ nm ), for u < u c and v < v c N

M

(18)

where the f nm are the values of the interpolated data on the regular grid. With appropriate scaling and translation of coordinates this corresponds to the definition of the 2-D DFT on uniform grids given at Equation (10), and shows that the DFT quadrature approximation is exact (up to truncation error) for band-limited functions that are appropriately sampled.

3.3 The Fourier transform of a spline interpolant

The principal issue with Fourier transformation of a spline is the singularity that can result at the origin of the Fourier domain. In Figure 1 linear growth of approximately equal magnitude was evident on either side of the domain covered by the data samples. This linear growth is in the spline summation itself and is not directly caused by the linear polynomial that supplements the spline. Careful expansion of a cubic spline ABF shows that outside the data domain the spline increases as (! )3

M k m x˜2m S m=1

x, where the ! indicates that the sign of the gradient is different at opposite ends of

the data domain. Similar analysis for the thin-plate spline revels that the dominant behaviour at large distances from the data domain is proportional to

M k m (x˜2m + y˜2m ) S m=1

log Å(x 2 + y 2 ).

To understand how this asymptotic behaviour of a spline influences the singularity at the origin of the Fourier domain I approximate the near origin behaviour using a Taylor series expansion of the exponential function.

I show in Appendix A that, near zero, the 1-D Fourier transform is

approximately given by, S(z ) = F(z/2o )[S k m + iz S k m x m − (z 2 /2 ) S k m x 2m − i(z 3 /6 ) S k m x 3m + O(z 4 ) + iO(z 5 )]

(19)

where z = 2ou and each of the summations are inferred to go from m = 1 to M. As discussed in Appendix A, a similar result holds for the 2-D Fourier transform, except the approximation contains

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contributions from various cross terms (e.g. z x z y and xy).

Notice, that there are really two

approximations; one for the real part of the Fourier spectrum, and the other for the complex part.

Consider, 1-D cubic spline interpolation where the Fourier transform of the basis function behaves as F(z ) i 1/z 4 near the origin.

Standard practice is to supplement the cubic spline by a linear

polynomial which imposes two constraints on the weights, specifically

S k m = 0 and S k m x m = 0.

These constraints eliminate the first two terms in Equation (19) and imply that S f (z ) i 1/z 2 + i/z about the origin. Therefore, for cubic spline interpolation with linear polynomials the real part of the Fourier transform has a second order singularity at the origin, while the complex part has a first order singularity.

In many cases, we are not interested in the Fourier transform as such, but only want to perform some filtering operations before moving back to the real domain. In that case any singularity at the origin, while important, is not a dominant issue. However, if the actual Fourier transform is required for some other reason (e.g. depth to basement calculations where the slope of the radially averaged spectrum is required) the singularity may need to be eliminated. To achieve this aim, in one dimension, the first four terms in the approximation of Equation (19) need to be made zero; i.e. require S k m x tm = 0 for t c (0, 1, 2, 3 ). With these constraints the real part of the spectrum is bounded at the origin and behaves as 1 + z 2 , while the imaginary part is also bounded and behaves as z. In the space domain, this implies that the spline interpolant approaches zero as x approaches infinity. There are two different ways that we can enforce the extra constraints (t = 2, 3); 1. Use two (or more) extra ‘dummy’ nodes in the definition of the ABF. The extra degrees of freedom can be used to enforce the required constraints. When included the nodes cause the ABF expansion to be zero above and below the largest and smallest nodal positions, respectively. This property occurs because a cubic spline involves approximating the curve between two nodes by a cubic polynomial. The only cubic polynomial that approaches zero as x approaches infinity is the zero polynomial, hence the observed compact support. As the nodes force the surface to zero, their position can have a significant influence on the curvature of the surface, especially near the ends of the domain.

If the nodes are placed within the domain or very close to the boundary the

approximation can be very poor and/or the curvature very large near the boundary. The optimal nodal positions would need to be determined by minimising either the spline penalty functional or some measure of error in the new interpolant. The solution to this problem in the general case has no known solution (Beatson, pers comm.). My own empirical experience indicates that a workable solution is to place the nodes at a minimum of the average sample spacing away from the extreme points on either side of the data stream.

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2. Include the x 2 and x 3 monomials in the supplementary polynomial. In this case the constraints immediately apply due to the boundary conditions imposed on an ABF expansion. The cubic spline minimises a smoothness penalty based on second derivatives (the 1-D analogue of Equation (13) given in the last chapter) when supplemented by a constant and linear monomial (which have zero second derivative). Technically, the addition of a quadratic and/or cubic monomial causes the spline smoothness penalty to be unbounded. Practically, this does not create a problem as our interest generally lies in only a finite segment of the real line.

Of the two methods given above the second is to be preferred. The main reason for this is that a cubic spline defined through an ABF expansion with a supplementary linear polynomial is a so called “natural” cubic spline (i.e. s  (x ) = 0 at either endpoint; DeBoor, 1978). The error in the natural cubic spline when fitting a function that has continuous derivatives up to fourth order is at best O(Dx −2 ). A “complete” cubic spline (DeBoor, 1978) is constrained by the same interpolation conditions but with the additional constraint that it matches the derivatives of the data at the end points. It has the more desirable property that the approximation error is O(Dx −4 ). However, the derivatives at the end points are unknown and must be approximated (for example by fitting a cubic to segments of data near the endpoints), so that the full O(Dx −4 ) accuracy cannot be obtained in practice. If the unique cubic polynomial interpolating f(x o ), f(x N ), and approximations to f  (x o ) and f  (x N ) is subtracted from the data before fitting the cubic spline, the interpolant will be compactly supported.

Analogous results to those discussed above hold for the thin-plate spline in 2-D whose basis function 4 also has a fourth order singularity at the origin, F(z ) i 1/Èz È . However, negating the singularity

for the spline involves 10 polynomial terms, 1, x, y, x 2 , xy, y 2 , x 3 , x 2 y, xy 2 and y 3 .

As splines are

usually supplemented by linear polynomials, this requires either seven additional polynomial terms, or a minimum of seven ‘dummy’ nodes. Note, that the only thin-plate spline surface with compact support is zero everywhere. Therefore, the best that can be achieved is a surface that approaches zero far from the region covered by the data.

4 THE FOURIER TRANSFORM OF THE JEMALONG RADIOMETRIC DATA

To illustrate the application of the ABF technique to the calculation of the Fourier transform of geophysical data, I use the Jemalong radiometric survey as an example. Beatson (pers. comm.) fitted an exact thin-plate spline to the potassium, uranium and thorium outputs from the NASVD program of Hovgaard (1997); see Chapter 2. The survey was collected at 100 m line spacing and I first adopted “standard practice” and interpolated each band (by the thin-plate spline) to a 25 m grid in both directions (Figure 2). The grid was rotated so that the columns aligned with the general strike of

168

the survey (16.3o East of North). The thin-plate spline had 18,824 nodes and the interpolated grid was 684 rows by 352 columns giving a total of 240,768 interpolated points. Notice that I have not levelled the line data which causes the obvious North-South stripping in the uranium image.

Figure 2: From left to right; Potassium (0.57 to 1.84 %K), uranium (1.25 to 2.4 ppm eU) and thorium (4.0 to 13.9 ppm eTh) thin-plate spline surfaces interpolated to a 25 m grid cell, with a linear contrast stretch between each image minimum and maximum. The Fourier transform of the interpolated grid was then calculated using the Matlab (Matlab, 1996) two dimensional Fast Fourier Transform algorithm. Each band took less than 30 seconds on a 200 MHz Pentium with 64 Megabytes of RAM (a standard Personal Computer). No attempt was made to join the edges of the grid so that the opposite ends joined seamlessly on a torus (the Craig and Green, 1998 approach). The resulting transforms are shown with logarithmic stretch in the left-hand column of Figure 3. The solid box delineates the Nyquist sampling limits imposed by the survey geometry, which are, respectively, 0.005 and 0.0083 cycles per meter. As the survey was interpolated to a 25 meter cell, the grid contains power above the Nyquist limits. A noticeable feature of each of the three spectra are the high amplitudes along the axes of the frequency domain, the so called “St George's Cross” that result because opposite ends of the grid do no match. This contamination and the numerous horizontal strips are caused by the intermediate grid through which the data must pass en-route to Fourier transformation.

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Figure 3: From top to bottom: the calculated power spectra of potassium, uranium and thorium surfaces. Images in the left hand column were obtained by an FFT on an intermediate grid, those on the right from exact Fourier transformation of the spline surface. The rectangle in the centre of each plot shows the approximate Nyquist limits imposed by the survey geometry. The same logarithmic stretch is applied to both images within a given row. The interpolated grids in Figure 2 contain contributions from the spline summation plus the linear supplementary polynomial. This suggests that a better result may be obtained by ignoring the linear

170

polynomial in the calculation of the discrete Fourier transform. However, on using this approach I found that the resultant power spectra differed little from those shown in Figure 3.

In the right-hand column of Figure 3, I show the results of Fourier transforming the thin-plate spline surfaces directly by the Fourier transform formula given at Equation (13). The logarithmic stretch applied to each image is the same as its equivalent in Figure 3. The calculation required the computation of the DFT of the weights, followed by multiplication with the transform of the thin-plate spline given in Table 1. The resulting power spectra are free of the central cross and the numerous horizontal stripes. Apart from these features they qualitatively resemble the power spectra calculated from the interpolated grid. One could argue that the artefacts in the grid based approach could have been avoided by appropriate extension and seamless joining of the opposite ends of the grid. Granted, but the important point is that appropriate Fourier transformation of the spline surface using Equation (10) does not introduce artefacts. The movement into the Fourier domain is exact in that it solves the quadrature weight problem and avoids discretisation error and edge mismatch.

There are a couple of additional points to be made regarding the Fourier transformation. 1. The horizontal band of increased amplitude in the Uranium spectrum is from line based errors in the Uranium band which was not levelled prior to interpolation. 2. The Fourier transform of the weights is based on 18,824 irregularly spaced nodes. I used a direct calculation to transform these weights to the same 240,768 points used to define the grid based estimate. The Fourier transformation required just over a half hour on an Sun UltraSparc 2 with 380 Megabytes of RAM and a 200 MHz processor, many times that required by the FFT grid based approach (which was calculated on a much slower computer). Note, that usually one would not bother to transform outside the Nyquist interval, and for visual inspection could transform onto a coarser grid. Assuming the Fourier transform was required on the same number of points as there were nodes the calculation would have taken less than three minutes. 3. Note, that the spectra calculated from the ABF approach in Figure 3 all have singularities at the origin. These arise because the thin-plate spline basis is unbounded at the origin and no attempt was made to make the transform finite using the methods outlined in the last section. The singularity only affects the spectrum very close to the origin so these methods would generally not be required. 4. The Fourier transform of an ABF is the DFT of the weights multiplied by the Fourier transform of the basis function. For each of the bands, the DFT of the weights are shown in Figure 4, this time with a linear stretch applied to the amplitudes (a logarithmic stretch conveyed less information). Notice, that the low frequency parts of the spectrum contain small amplitudes, while high amplitudes

171

are mainly concentrated along diffuse horizontal bands at high frequency. This behaviour results from the Èu È

−4

decay in the Fourier transform of the spline, which amplifies the low frequencies and

substantially attenuates the higher frequencies. It also explains the large oscillations that are apparent in an inspection of the weights that define a spline surface. Notice that the transform of the weights appears to show a quasi-periodic behaviour. The easiest way to understand this behaviour is to consider the variation along one-dimension. Write the m-th term along the x-axis in the ABF expansion as x˜m = m˜ Dx, where m˜ is some (possibly large) perturbation to an integer (equal to an integer if the nodal spacing were regular) and the j-th frequency coordinate is given by u j = j/(DxN ) for j = −N/2 + 1, ..., N/2. Then, the formula for the Fourier transformation of the weights may be written as

M L(u j , v k ) = S m=1 k m exp −2oi

jm˜ +y v N ˜m k

Figure 4: Fourier amplitudes of the weights for the potassium, uranium and thorium thin-plate spline surfaces, with a linear stretch applied. The rectangle in the centre of each plot shows the approximate Nyquist limits imposed by the survey geometry.

(20)

172 The Nyquist limit along the horizontal frequency axis is u max = 1/(2Dx ). If we add any even integer multiple of the Nyquist (say 2nu max ) then the first term inside the brackets in the exponential becomes mˆ Dx

j jm˜ + 2n = + m˜n, and the Fourier transform is N DxN 2Dx

M L(u j + 2nu max , v k ) = S m=1 [k m exp(−2oinm˜ ) ] exp −2oi

jm˜ +y v N ˜m k

(21)

If the nodal spacing were regular the m˜ become integers and the exponential multiplier on each weight reduces to unity causing the transform to be periodic. This occurs because all M complex exponentials in Equation (20) have an integer number of oscillations within the Nyquist interval. When the x˜m are irregularly spaced, some (possibly all) complex exponential will not have an integer number of oscillations within the Nyquist interval and the transform is then not periodic. The geophysical data used to create Figure 4 is “almost” regularly spaced, hence the spectrum is almost periodic. 5 EFFECT OF BASIS FUNCTION AND SOLUTION METHOD ON THE FOURIER TRANSFORM

In the last section we saw that exact Fourier transformation of an ABF surface by Equation (13) solved many of the problems inherent with a grid based approach. When using the ABF framework for interpolation and Fourier transformation, a decision needs to be made regarding choice of basis function, the placement of nodes and whether the surface smoothly or exactly interpolates the data. In this section I give a brief qualitative analysis of the effect of these decisions on the Fourier transformation. For the comparisons I selected a 3 % 3 km2 area in the North-east corner of the Jemalong survey (6 - 9 km East, 10 - 13 km North of the local origin in Figure 2).

The analysis was conducted on the

thorium band and involved (i) an exact thin-plate spline fit; (ii) an exact multiquadric fit (c = 100 ); (iii) a GCV thin-plate spline fit; and (iv) a LSQR thin-plate spline fit with 100 m nodal spacing. These surfaces are also used in the next chapter and are therefore shown at Figure 20, Chapter 7. The Fourier transforms are shown in Figure 5, with the same logarithmic stretch applied to each image.

Exact interpolation: Effect of basis function The two surfaces constrained to honour the input data have very similar Fourier transforms within the limits imposed by the approximate Nyquist cutoffs of u max = 0.005 m-1 and v max = 0.0083 m-1 (Figure 5). This is to be expected as the basis function should not significantly influence the shape of an

173

exactly interpolated surface at scales larger than the sample spacing (except at very low frequencies corresponding to the maximum distance between any two points, where the asymptotic properties of the basis function become important). Outside the Nyquist limits the Fourier transforms differ considerably, with the O(q exp[−q ]) behaviour of the multiquadric, causing it to decay much more rapidly than the thin-plate spline, which behaves as O(q −4 ).

Figure 5: Power spectra for a 3 % 3 km2 subset of the Jemalong survey for (from top left to right, then down) exact thin-plate spline, exact multiquadric, and GCVand LSQR thin-plate splines. The same logarithmic stretch is applied to each image. Exact versus smooth interpolation The exactly interpolated images have broader power spectra in the along-line direction (vertical), because the sample spacing is finer than across lines and the surfaces are constrained to honour the data. This broadening of the power spectrum is absent from the transform of the GCV-spline surface

174

which is more symmetrical (Figure 5). In fact, the finer spacing in the along-line direction is not evident from a casual inspection of the power spectrum of the GCV surface.

Placement of nodes The power spectrum of the LSQR spline surface (Figure 5) contains numerous circular features, which arise because the Fourier transform of the weights is periodic. This periodic behaviour is superimposed over the q −4 decay in the transform of the thin-plate spline. The period is the same in the along and across-line directions as the nodes are distributed on a grid with an equal spacing of 100 m in both directions. The example illustrates that there can be disadvantages in placing the nodes on a regular grid. In fact, it also implies that sampling on a regular grid can have drawbacks if the data need to be interpolated to a finer grid. In that case, the weights will be periodic regardless of the solution method. Summary

Ÿ

For exact interpolation the choice of basis function mostly effects the power spectrum outside the approximate Nyquist limits imposed by the survey geometry;

Ÿ

Smooth solution reduces the asymmetry in the spectrum caused by the different sample spacings in the along- and across-line directions;

Ÿ

Placing the nodes on a regular grid introduces circular features into the power spectrum;

Ÿ

The best of the four examples appears to be the GCV thin-plate spline.

6 DISCUSSION

Contrast the ABF technique to a standard method for calculation of the Fourier transform of geophysical data. For the standard method, the data are first interpolated onto a regular grid, with a cell size of approximately one-quarter of the flight line spacing, using a local technique such as minimum curvature (Briggs, 1974). The limits of the grid are then extended slightly and a multigrid technique used to infill these buffer cells along with any survey gaps (Craig and Green, 1998). The infilling is such that if the grid was translated and repeated across the real plane it would be difficult to locate the edges of the grid or the survey gaps by a visual inspection.

The data are then

transformed into the Fourier domain using the FFT algorithm.

The ABF approach involves the selection of a basis function and supplementary polynomial, placement of the defining nodes and a decision regarding whether the data should be exactly interpolated or smoothly fit. The weights that solve the interpolation problem are then found using the techniques described in the previous chapter. The process defines a continuous model of the data

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that can be exactly Fourier transformed using Equation (13). There are no concerns with opposite ends of the survey matching as the model extends over the whole real plane. Additionally, the surface is implicitly interpolated into any survey gaps so there is no need to create synthetic data.

Currently, unless the nodes are distributed on a regular grid, the principal difficulty with the approach is the computational cost associated with the calculation of the DFT. For M irregularly distributed nodes that need to be Fourier transformed onto N points in the Fourier domain the DFT requires O(MN) operations. For example, application to the relatively small Jemalong radiometric survey (M = 18,824 nodes; N = 240,768) required half an hour of CPU time on a fast workstation. As both M and N can be large for many geophysical surveys (M ~ 1,000,000 is not uncommon) timely calculation of the Fourier transform requires a fast algorithm that can be applied to scattered data. Currently, an algorithm does not exist for two dimensional applications. However, I believe this is only a temporary impediment as there are several existing 1-D algorithms (e.g. Dutt and Rohklin, 1993; 1995) that could be extended to 2-D. There is also at least one unprogrammed technique that could be coded for a two dimensional application (Newsam, pers. comm.).

These issues related to computational cost are only pertinent if the Fourier domain description is required explicitly for analysis, such as depth to basement calculations. However, in the majority of cases the Fourier description is only an intermediate step in some type of filtering operation. In the next chapter I show that in many cases it is possible to implicitly move into the Fourier domain, perform filtering operations and then move back into the space domain. This can all be achieved without ever having to explicitly calculate the Fourier transform of anything derived from the data.

The emphasis of this chapter has been on calculating the Fourier transforms of surfaces defined through splines or sinc functions. Any of the other basis functions considered in the previous chapter could be used in the same context. The only criteria is that the Fourier transform of the basis function is known (several are listed in Table 1). One attraction for proponents of kriging is that the basis (equivalently variogram) can be chosen that best models the spatial structure in the data. In order to calculate the Fourier transform of the kriged surface one needs to know the Fourier transform of the variogram. This is usually straightforward as most variograms are constructed from simple functions with well known Fourier transforms (either in the ordinary or generalised sense). Tables of Fourier transforms would be particularly useful in this context (e.g. Oberhettinger, 1990). Otherwise, Christakos (1984) gives several techniques that can be used in the calculation of the Fourier transform of an arbitrary variogram.

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CHAPTER 7

CONVOLUTION AND DECONVOLUTION OF ARBITRARY BASIS FUNCTION SURFACES The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast. Bertrand Russell

CHAPTER OBJECTIVES Develop methods for convolution and deconvolution of surfaces defined by an Arbitrary Basis Function, and apply the methodology to the deblurring of radiometric data.

1 INTRODUCTION

The last chapter developed methods for exact Fourier transformation of surfaces defined by an ABF expansion. This chapter addresses the issue of calculating a new ABF expansion after Fourier domain operations have been applied to the original surface. The ultimate goal is the deconvolution of radiometric data but the techniques developed can potentially be applied to any linear, spatially invariant filter.

In this chapter I show that ABF surfaces can be efficiently manipulated in the Fourier domain. In particular: 1. The convolution or deconvolution of an ABF is another ABF, albeit with a new basis function. In some cases this basis function can be determined analytically from knowledge of the convolution kernel and the original basis function, so obviating the need to calculate the Fourier transform at all. 2. In cases where the new basis function cannot be computed or is not convenient to use, a DFT algorithm can be used to compute a new ABF expansion of the filtered data using the original basis function and nodal positions. This has obvious conceptual and practical advantages: it introduces no new grid points beyond the original data points and it allows the reuse of data structures, etc., already set up for the original ABF expansion.

177

Throughout the last two chapters and continuing with this one, the objective has been to develop a new methodology for processing irregularly sampled geophysical data. Many geophysical surveys are quite large and the pressure to produce results fast is often high. The methods considered in this chapter are global, so that naive implementation would be quite slow. These practicalities of speed and memory are discussed but are not a core focus. The main issues addressed concern how to convolve or deconvolve ABF surfaces. Eventually, I believe that fast methods for matrix-vector multiplication, such as the Beatson and Newsam (1998a) algorithm, and fast DFTs on scattered data, will render feasible application of the methodology to large geophysical surveys.

Continuing the theme of the last chapter I concentrate on the using as a basis either a sinc function or a spline. Again the sinc function provides a convenient link between the standard grid-based methods and those based on ABF surfaces. When the data are regularly sampled there are many similarities between the sinc and grid-based approaches. On the other hand, splines demonstrate the considerable differences between the two approaches. Again there are some difficulties in accommodating the singularity that occurs at the origin in the Fourier transform of a spline surface.

In the first part of this chapter I present the methodology for the exact filtering of an ABF surface (point 1 above). I then discuss a second method which uses the same basis function to define the original and filtered surfaces (point 2 above). However, depending on the basis function this method may only be approximate. The methods are then illustrated on a synthetic one-dimensional dataset. The next section considers how to extend the methodology to two-dimensional applications with singular basis functions (such as thin-plate splines and multiquadrics). The last section considers how the parameters defining the ABF surface (basis function, smooth or exact interpolation and placement of nodes) effect the results of a deconvolution of the Jemalong radiometric survey.

Mathematical preliminaries In many geophysical problems there is a convolution relationship between a set of measured data, f(x ), and some desired quantity, g(x ), g(x ) = ° ‘d h(x − y )f(y )dy

(1)

where h(x ) is known as the ‘convolution kernel’ or ‘transfer function’. This equation reduces to a simple multiplication in the Fourier domain, G(u ) = H(u )F(u )

(2)

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Examples include low, high, band pass and derivative filters, and reduction to the pole and upward continuation of magnetic data. The model specified by Equations (1) and (2) is the definition of a linear, spatially invariant filter (e.g. Kuc, 1988).

In other situations, such as downward continuation of magnetics and deblurring of radiometrics (Chapter 4), the roles of f(x ) and g(x ) are reversed, f(x ) = ° ‘d h(x − y )g(y )dy

(3)

In that case, an estimate of G(u) and hence g(x) can be obtained by spectral division,

G(u ) =

F(u ) j W(u )F(u ) H(u )

(4)

where usually one replaces the reciprocal of the convolution kernel by a Wiener filter, W(u ), which has the property that (see Chapter 4)

W(u ) ~

1/H(u ) for H(u ) large 0 for H(u ) small

Wiener deconvolution is equivalent to convolution with W(u ) as the kernel.

(5)

The techniques

developed in this chapter can be applied to both convolution and deconvolution. I will therefore always express the relationship between F(u ) and G(u ) in the form of Equation (2), and will only explicitly refer to the Wiener filter of Equations (4) and (5) when required.

2 BASIS TRANSFORMATION: AN EXACT METHOD FOR FILTERING ABF SURFACES

In this section I consider the exact Fourier domain processing of an ABF surface by Basis Transformation. Given N discrete data samples, f(x n ), the first step is to construct an ABF expansion, s(x), using the methodology outlined in Chapter 5. This involves the specification of a basis function, w(x ) and order of the supplementary polynomial, p(x ), a strategy for fitting the approximation to the

data (exact interpolation, GCV or implicit smoothing) followed by the solution of the resulting set of simultaneous equations for the weights, k m , at the nodes x˜ m , m = 1, ..., M. The ABF expansion has the form, M s(x ) = S m=1 k m v(x − x˜ m ) + p(x )

(6)

179

Using the results of the last chapter the Fourier transform of this ABF surface can be calculated exactly and is given by M S(u ) = F(u ) S m=1 k m exp(−2oi u $ x˜ m ) + P(u )

(7)

where S(u ), P(u ) and F(u ) are the Fourier transforms of the ABF surface, polynomial and basis function, respectively.

S(u ) is our approximation to the true F(u ). Inserting Equation (7) into the general convolution model

of Equation (2) I find, G(u ) = S(u)H(u) = H(u )F(u ) S m=1 k m exp(−2oi u $ x˜ m ) + H(u )P(u) M

(8)

Inverse Fourier transforming both terms (by a reverse application of the derivation in Appendix H) leads to a new ABF surface, g(x ) = S m=1 k m y(x − x˜ m ) + q(x ) M

(9)

Note that this equation has identical weights to the original surface and a new basis function, y(x ), that is given by y(x ) = ° ‘d F(u )H(u ) exp(2oi u $ x )¹u

(10)

This is a convolution integral between the space domain version of the convolution kernel, h(x ) and the original basis function. y(x ) = ° ‘d h(x − y )v(y )dy

(11)

In retrospect this equation is fairly obvious as the original ABF surface is a convolution of the basis function with the weights,

M k m d(x − x˜ m ), where d(x ) is the Dirac delta function. S m=1

As convolution

is associative the order of the convolutions can be changed and the required result follows.

The remaining term in Equation (9), q(x ), is the result of filtering the polynomial part of the ABF surface. I show in Appendix K, that it is another polynomial with equal or lower degree. The two

180

polynomials are related by an upper triangular matrix equation. For example, for a two dimensional application with a supplementary polynomial of degree 1, p(x ) = a o + a 1 x + a 2 y, and the coefficients of the new polynomial, q(x ) = b o + b 1 x + b 2 y, are given by

b0 b1 b2

where H u (0 ) =

=

H(0 ) 0 0

H u (0 ) H v (0 ) 2oi 2oi H(0 ) 0 0 H(0 )

ao a1 a2

(12)

¹H(0 ) . Note, that for the common case of a real, symmetric point spread function, ¹u

H u (0 ) = H v (0 ) = 0. As higher degree polynomial terms are added higher order derivatives of the

Fourier transform of the transfer function about zero are required (Appendix K).

The implications of the above equations are very significant. They imply that Fourier domain processing of irregularly sampled geophysical data can be achieved without explicitly calculating the Fourier transform of anything derived from the data. The filtering is achieved implicitly through the application of the new basis function on the existing set of weights. Further, the movement into and out of the Fourier domain is exact and avoids the complications associated with the discrete Fourier transform.

Successful application of the technique reduces to a problem of computing the new basis function. In some cases the calculation can be achieved with an analytical formula. Failing this, a numerical calculation would be needed. For the sinc function, the transform of the new basis function is band limited, while for the Gaussian it rapidly decays to zero. Assuming the filter is sufficiently well behaved, numerical solution presents few problems as the product of the transforms will also be well behaved, and the integration only needs to extend over a finite domain. The multiquadric and the splines have singularities at the origin which make the calculation of the new basis function more difficult. Methods for the cubic spline in 1-D for the thin-plate spline in 2-D are described in Appendix J and also in the relevant results sections.

Once the new weights or basis function have been obtained, the next step is to interpolate the filtered surface onto, usually, a regular grid of points.

This interpolation occurs for visualisation and

interpretation purposes, where the number of interpolated points, N, typically exceeds the number of points in the ABF expansion, M. Using a direct method for the calculation would take, O(MN ), operations, which can be prohibitively expensive if either or both of M and N are large. To reduce

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this computational cost a fast method for ABF expansions needs to be used (e.g. Greengard and Rohklin, 1987; Beatson and Newsam, 1998a).

Due to the wide range of basis functions that can be encountered (some that will only be known numerically), the adaptive approach of Beatson and Newsam, (1998a; see Chapter 5), would be required. Their technique generates Chebyshev polynomial approximations to the basis functions and then uses hierarchical tree structures to reduce the computational cost of the above interpolation to around,O(N log M ) operations. The interpolation can only be calculated rapidly if the new basis function is suitable for the fast method. The basic requirement is that the function is smooth away from the origin because then, low degree polynomial approximations can be used to calculate the contributions of points far away in the ABF summation. If there is any difficulty in using the fast interpolation methods, or if the new basis function cannot be calculated, an alternative method for filtering of the ABF surface can be used. This is described in the next section and a second alternative in a subsequent results section.

3 WEIGHT TRANSFORMATION: AN APPROXIMATE METHOD FOR FILTERING ABF SURFACES

For Weight Transformation the filtered response is assumed to be modelled with the same basis function as the original, but with a new set of weights, j m and supplementary polynomial, q(x ), g˜(x ) j S m=1 j m v(x − x˜ m ) + q(x ) M

(13)

The coefficients of the new polynomial are calculated in the same way as the exact method, through Equation (12). The remaining task is to find a set of weights such that the transformed surface matches the exact surface calculated by Equation (9) as closely as possible. Before discussing computational methods there are two important points to emphasise regarding this new method. 1. The method may not be exact: A major attraction of the Basis Transformation is that a surface can be exactly transferred into the Fourier domain, have filtering operations applied and then be exactly transformed back into the space domain. When the same basis function is used for both the original and new surfaces, the convolution will often be only approximate. For example, interpolation with v(x ) = x gives a piece-wise linear curve with first derivative discontinuities at the nodes. Most convolution operations have some element of smoothing. Therefore, the sharp edges at each of the nodes will become rounded and no piece-wise linear curve defined on the same set of points will be

182

able to capture this behaviour. Similarly, the cubic spline fits a cubic polynomial between data points and application of an arbitrary filter is unlikely to preserve the cubic shape between nodes. 2. The method can be exact for the sinc function; The original interpolation with the sinc function leads to a surface that is band limited, with no frequency content above the Nyquist limits imposed by the nodal spacing in the ABF expansion. Any filtering operations will not affect this property and may even cause the Nyquist to reduce; e.g. a low pass filter. Therefore, the new surface will still be able to be modelled with the sinc function as a basis.

For the general case the transformation will only be approximate, therefore there is no point in looking for an exact solution. I require that the approximate solution, g˜(x ), of Equation (13) is close to the exact solution, g(x ), of Equation (9) and choose the least squares criterion as a measure of fit, J(j ) = °ℜ d g˜(x ) − g(x ) 2 dx

(14)

By Parseval’s theorem (e.g. Bracewell, 1986) the least squares error, or power, in the space and frequency domains are equal, 2 J(j ) = °ℜ d G˜ (u ) − G(u ) du

(15)

Substituting in the Fourier transforms of G˜ (u ) and G(u ) I find, J(j ) = °ℜ d F(u )

2

M j m exp(−2oiu $ x˜ m ) − H(u )L(u ) S m=1

2

(16)

du

M where L(u ) = S m=1 k m exp(−2oi u $ x˜ m ) is the Fourier transform of the original weights. At the least

squares solution ¹J(j )/¹j n = 0 for all n. By recognising that F 2 = F & F for the complex variable F and after applying some algebraic manipulation followed by differentiation with respect to j n , I find that ¹J(j ) = 0 = °ℜ d F(u ) 2 exp(2oiu $ x˜ n ) ¹j n

M j m exp(−2oiu $ x˜ m ) − H(u )L(u ) S m=1

du

(17)

Moving the two terms inside the square brackets to opposite sides of the equation and evaluating the inverse Fourier transforms leads to an N % N system of linear equations in the space domain, M M j m v˜(x˜ n − x˜ m ) = S m=1 k m y˜ (x˜ n − x˜ m ) S m=1

(18)

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The derivation is a simple reverse application of the Fourier transform result derived in the previous chapter. The new basis functions are given by v˜(x ) = °ℜ d F(u ) 2 exp(2oiu $ xˆ )du

(19)

y˜ (x ) = °ℜ d F(u ) 2 H(u ) exp(2oiu $ xˆ )du

(20)

and

Solution of Equation (18) requires inversion of a large matrix equation (which could of course be achieved using the methodology of Chapter 5) and the calculation two new basis functions. However, the new basis functions are at least as difficult to compute as the basis function, y(x ) required for the exact method described in the last section. The implications of Equation (18) are interesting as they show that the minimum norm solution does not involve matching the approximate and exact solutions at the nodes (i.e. with v(x ) and y(x ) in Equation 18) as one might expect. Instead the correspondence must occur between two basis functions derived from the originals by squaring of their Fourier transforms.

2 An exception is the sinc function, where F(u ) = F(u ), and hence

v˜(x ) = v(x ) and y˜ (x ) = y(x ).

To avoid the difficulties raised in the previous paragraph we return to Equation (17) and search for a solution that minimises the least squares error over a finite number of points. Having M free parameters to determine suggests that the error be minimised over M points, u k ; k = 1, ..., M, converting Equation (17) to

0=

¹J(j ) N = S k=1 F(u k ) 2 exp(2oiu k $ x˜ n ) ¹j n

M j m exp(−2oiu k $ x˜ m ) − H(u k )L(u k ) S m=1

(21)

The factor in square brackets is independent of n and can be made zero for all k because there are M coefficients j m and M frequency co-ordinates u k . This is achieved if M j m exp(−2oiu k $ x˜ m ) = H(u k )L(u k ) S m=1

When F(u k )

2

(22)

is finite for all u k this will cause Equation (21) to be zero (which may rule out basis

functions such as splines that have discontinuities at the origin). Note that, the solution gives zero

184

least squares error when calculated at the M frequency co-ordinates, u k . For the special case that the u k and x˜ m are regularly spaced, the j m can be obtained by an inverse DFT. For example in one

dimension

jm =

M/−1

S

k=0

Λ(u k ) exp 2oi km H(u k )L N

(23)

For the more general case of irregularly distributed nodes, x˜ m , solution of the Fourier inversion specified by Equation (22) is more difficult. Naive inversion would require O(M 3 ) operations which is unfeasible given the typical size of geophysical surveys.

Additionally, calculation of Λ(u k )

involves a DFT on scattered data that naively requires O(M 2 ) operations. Dutt and Rokhlin (1995) give a series of algorithms for the 1-D case that calculate both the inversion and the DFT for the Λ(u k ) to within an accuracy of eε in O[M log M + M log( 1ε )] operations. At the time of writing the

algorithms had not been generalised to the 2-D case.

Depending on the size of the problem (and the availability of a fast code for calculating the DFT of 2-D data), these computational issues may be one reason to prefer defining the ABF expansion on a regular grid, preferentially with less nodes than sample points. In that case, a degree of implicit smoothing occurs and both the forward and inverse Fourier transforms can be calculated rapidly, in O(M log M ) operations, using a standard FFT algorithm.

4 APPLICATION OF THE ABF METHODOLOGY TO A SIMPLE ONE-DIMENSIONAL EXAMPLE

In this section I apply the ABF methodology to a simple one-dimensional example. The intention is, firstly, to illustrate that the ABF approach is very general and can be used for many different types of filtering.

Secondly, I aim to identify potential problems with the methodology and use any

knowledge gained to aid real two-dimensional applications.

The one-dimensional example consists of 80 regularly spaced samples derived from two zero order Bessel functions, f(x ) = J o (15 x ) + 0.5J o (50 x ) in the range [-1, 1); see Figure 1a. The frequency cut-off is 20 cycles per sample which represents a situation of over sampling (Figure 1b). The function is real and symmetric which implies firstly, that its spectrum is real and secondly, that there will be minimal endpoint problems. Such an idealised example is considered initially as there will be few complications to distract from the major aspects of ABF filtration.

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The samples are first convolved with a Gaussian PSF which then has synthetic noise added. The noisy data is deconvolved, then examples are shown of low and high pass filtering, derivative filtering and finally, multiple filters calculated in a single step (derivative of the deconvolved response). The analysis is first performed with the sinc function and then with the cubic spline.

4.1 Convolution with the sinc function as a basis

The sinc function approach closely resembles the standard method for Fourier domain processing, especially when the data are collected with regular sampling. The ABF approach still has much to offer with regard to edge effects (Beatson and Newsam, 1998b) and interpolation of desired results to intermediate points. When the data spacing is regular the standard processing method is as follows: 1. Calculate the DFT of the data; 2. Multiply the result by the appropriate filter; 3. Calculate the inverse DFT to give the filtered response.

For a method based on the ABF methodology using Weight transformation the steps are 1. Calculate the weights which, for the sinc function with coincident nodes and data samples, are the same as the function values, M s(x ) = S m=1 f(x m ) sinc[2u max (x − x m )]

(24)

where, u max = 1/(2Dx ) is the Nyquist sampling cut-off; 2. Multiply the result by the appropriate filter; 3. Calculate the inverse DFT to give the new weights (by Equation 23),

The ABF and standard DFT methods give identical results as the weights and function values are identical and the transform of the sinc function is unity up to the Nyquist cut-off. The method using Basis Transformation has the same initial step as the Weight Transformation, then proceeds as follows: 2. Form a new basis function by inverse Fourier transforming a truncated version of the filter, y(x ) = ° −u max H(u ) exp(2oi ux )du u max

(25)

186

3. Calculate the filtered response at the sample points by evaluating the new ABF expansion (by Equation 13).

Convolution with a Gaussian transfer function To illustrate the differences in the two ABF approaches consider convolution with a Gaussian transfer function, 2 2 h(x ) = a exp −(ax ) u H(u ) = exp −(ou/a ) o

(26)

where a = 10 is a positive constant and the factor at the front of the first equation ensures that the transfer function integrates to unity (Figures 1c-d). The intermediate steps and the final result for both methods (only one method is displayed as they are virtually identical) are shown in Figure 1, which is organised with space domain curves down the left column and Fourier domain down the right column. Weight transformation follows a flow path from top left to right (DFT of data), then downwards to the bottom right (multiplication by the filter) and finally to the bottom left (inverse DFT). Basis transformation follows a flow path straight down the left hand column. The middle step is an explicit spatial convolution with the new basis function calculated using Equation (25). Because the Fourier transform of the sinc is unity up to the Nyquist (20 cycles per second) and the transfer function is effectively zero above this frequency, the new basis function and the transfer function are virtually identical (up to a scaling constant), i.e. y(x ) = Dxh(x ). Thus, for the sinc function, Weight transformation is nothing more than the Fourier domain path to convolution, while Basis transformation involves explicit convolution in the space domain. Note that; 1. One of the main reasons the DFT (in the form of the FFT) is so widely used, is that it considerably speeds up convolution operations. The Fourier method requires two applications of an FFT plus a single length N multiplication giving a total operation count of 2O(N log N ) + O(N ). This compares to the O(N 2 ) operations required for the explicit (and naive) calculation of the spatial convolution. For large N the Fourier technique is the preferred method, but note that with irregular sampling and fast methods for interpolation there can be good reasons for preferring the spatial method. 2. The exact relationship between the new basis function and the convolution operator, y(x ) = Dxh(x ), only applies when the operator is zero outside the Nyquist limits imposed by the

sampling. For example, if the sample spacing were 1/4 of its present value the Nyquist cut-off would be at 5 cycles per sample and the new basis function would be the inverse Fourier transform of a truncated version of the Gaussian (see Figure 1).

187 1.5

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Figure 1: Sinc interpolation applied to convolution with a Gaussian PSF. Graphs in the right hand column are the frequency domain versions of the left hand column; (a) and (b) Bessel function input to the convolution equation sampled at 80 equal increments between [-1 1); (c) and (d) The new basis function; (e) and (f) Convolved response.

188

Deconvolution with a Wiener filter A more common scenario in geophysics is that the convolved response (or noisy samples of it) is available and the underlying signal needs to be obtained by deconvolution. To illustrate this process, I added Gaussian noise (mean zero, standard deviation 0.05) to the convolved response obtained in the last section (Figure 2a). The addition of noise and the very rapid attenuation of any high frequency signal by the convolution, prevents reconstruction at the highest resolution. Therefore I implemented the Wiener filter of Chapter 4 with white noise and a constant power law for the signal,

W(u ) =

exp −(ou/a )

2

1 2 + r 2 exp (ou/a )

(27)

The noise cause the data to be asymmetric, which implies that the spectrum will contain both a real and imaginary component. Therefore, I plot the magnitude of the Fourier coefficients (Figure 2b). In the frequency domain the filter resembles the reciprocal of the transfer function for low frequencies, before beginning a very rapid decrease towards zero (Figure 2d). I calculated the space domain version of the filter using a FFT of its frequency domain representation to illustrate that a numerical calculation often suffices.

The new basis function needs to be known within the range [0, 2], which means that the maximum frequency spacing that can be used is 0.25 cycles per sample (using x max = 1/(2Du ); the spatial Nyquist). The number of samples (or equivalently the extent of the sampling in the frequency domain) then determines the grid spacing on which the function will be known. To prevent aliasing this has to be greater than 10 cycles per sample (the effective support of the Wiener filter in the frequency domain). To be known with the same sample spacing as the data the frequency cut-off has to be 20 cycles per second; as Dx = 1/(2u c ). I sampled a region several times this length to display the smooth nature of the new basis function (Figure 2c). The new basis is oscillatory and rapidly decreasing with distance.

The deconvolved response models the low frequency structure in the original, predicting the central peak height and width almost exactly (Figure 2). As expected the finer details of the original are not reconstructed accurately. The example illustrates the difficulty in juggling the signal-to-noise ratio with a Wiener filter even when we have a perfect model and substantial information on the characteristics of the noise. Note, that the situation is particularly unfavourable in this example because the Gaussian kernel severely attenuates any high frequency signal.

189 1

(a)

(b)

0.25

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Signal

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(c)

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Figure 2: Sinc interpolation applied to Wiener deconvolution with a Gaussian PSF. Graphs in the right hand column are the frequency domain versions of the left hand column; (a) Convolved response from Figure 1 with Gaussian noise added; (b) Magnitude of the spectrum of the noisy samples; (c) and (d) The new basis function; (e) and (f) Deconvolved response compared to the true response (light curve).

190

Low and high pass filtering For the sinc function, determination of an ideal low or high pass filter is very straightforward, as it only involves changing the scaling constants. A low pass filter with a cut-off of u c has a basis function given by uc sinc(2xu c ) w low (x ) = u max

(28)

where u max is the Nyquist limit. The corresponding high pass filter is w high (x ) = v(x ) − w low (x ), uc sinc(2xu c ) w high (x ) = sinc(2xu max ) − u max

1

(a)

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(b)

Noisy data Low pass filter

(29)

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Figure 3: Sinc high and low pass filters (cut-off of 5 cycles per sample) applied to the noisy convolved samples of Figure 2a. (a) Application of low pass filter; (b) Application of high pass filter; (c) Basis functions for high and low pass filtering.

1

191

Low and high pass filters of the noisy convolution samples with a frequency cut-off of 5 cycles per sample, along with the corresponding basis functions, are shown in Figure 3. Note, that the sharp cut-offs imposed by these filters can introduce oscillations in the space domain.

Practical

applications usually involve the use of some type of tapering such as a cosine roll-off (e.g. Craig, 1995; Blakely, 1995). 4

(b) 40

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Figure 4: Sinc low pass derivative filter applied to the noisy convolved samples of Figure 2a; (a) and (b) Basis function for a low pass derivative filter with a cut-off of 5 cycles; (c) and (d) Application of the derivative filter. Derivative filter In the frequency domain the derivative filter is H(u ) = 2oiu, and the new basis function is the inverse Fourier transform of this linear trend truncated at the Nyquist limit (or equivalently the derivative of the sinc function, see Figures 4a-b). A direct application of the derivative filter would result in accentuation of the high frequency noise. To determine the derivative of the low frequency part of

192

the spectrum it is a simple matter of zeroing the filter above the desired maximum frequency. These operations can most easily be achieved by directly differentiating the low pass filter given at Equation (28),

w(x ) =

¹w low (x ) uc [cos(2oxu c ) − sinc(2oxu c )] = xu max ¹x

(30)

Notice that this new basis function is odd, i.e. w(x ) = −w(−x ). The results of applying this low pass derivative operator to the noisy convolution samples are shown in Figures 4c-d.

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Figure 5: Sinc interpolation and the derivative of the deconvolved response; (a) and (b) New basis function; (c) and (d) Results of applying the filter to the noisy samples of Figure 2a. “Reference” is the deconvolved response amplified by a factor of 5.

193

Multiple filters The previous example showed that is possible to cascade several filtering operations and use a single basis function. We give a further example of that process here, with a one step calculation of the derivative of the deconvolved response directly from the data. The frequency domain version of the filter (Figure 5b) is H(u ) = 2oiuW(u ), which is imaginary and odd, where W(u ) is the Wiener filter given at Equation (27). The corresponding space domain representation was calculated using a DFT and is odd (Figure 5a). The result of applying the filter to the noisy convolution samples along with an expanded version of the original function are shown in Figures 5c-d.

4.2 Convolution of curves defined by a cubic spline

The previous examples showed that ABF filtration with the sinc function is very similar to standard grid based techniques. The cubic spline, on the other hand, emphasises the considerable differences in the approaches. Basis Transformation generalises the process of discrete spatial convolution to include an explicit cubic spline interpolation. Weight transformation as we will discuss in a moment, is difficult to apply when filtering a spline curve.

The are several important aspects of filtering a cubic spline that deserve specific mention; 1. The Fourier transform of a curve defined by a cubic spline is

S(u ) =

3 4 4(ou )

M k m exp(−2oi ux˜m ) S m=1

(31)

The spline approximates some underlying function f(x ) so that S(u )~F(u ). Therefore, if F(u ) contains appreciable signal at high frequencies the spectrum of the weights must contain approximately u 4 times as much signal (equivalently u 2 as much power) by Equation (31). The result is that any high frequency content in the original function is substantially amplified in the spectrum of the weights. Therefore, even to fit relatively smooth functions, such as the composite Bessel function of Figure 1a, gives cubic spline weights that are highly oscillatory (Figures 6a-b).

Such highly oscillatory

functions are difficult to approximate with Fourier transforms, creating difficulties in using Weight Transformation with a cubic spline. Additionally, the singularity invalidates equations (22 and 23) for the new weights. These issues will be discussed further in a section 5.3. 2. The fourth order singularity at the origin of a spline curve (Equation 31) is not numerically integrable. However, the singularity can be eliminated if a polynomial approximation, T(u ), to the

194 filter near the origin can be determined (see Appendix J). By subtracting it from H(u ) and adding it back in elsewhere there are two well behaved integrals to calculate:

y(x ) = 3 4 4o

°ℜ T(u )u −4 exp(2oi ux )du + °ℜ [H(u ) − T(u )]u −4 exp(2oi ux )du

(32)

In geophysics, we usually deal with real convolutions, which have even real and odd imaginary components in the frequency domain. Therefore, I assume that I can approximate H(u ) near the origin by T(u ) = a o + a 2 u 2 + i[a 1 u + a 3 u 3 ]

(33)

For real, even spatial convolutions, a 1 = a 3 = 0, while for real odd spatial convolutions a o = a 2 = 0. I show in Appendix B that the new basis function is then given by

3a 3a 3a y(x ) = a o x 3 + 1 x 2 sgn(x ) − 22 x − 33 sgn(x ) + y r (x ) 2o 2o 4o

(34)

where sgn(x ) is the sign function (+1 for x m 0, 0 for x = 0 and -1 for x < 0) and y r (x ) is a remainder component calculated by y r (x ) = 3 4 4o

°‘ [H(u ) − T(u )]u −4 exp(2oi ux )du

(35)

The leading order behaviour near the origin is [H(u ) − T(u )]~u 4 , thus cancelling the singularity so that the remainder component is finite and well behaved at the origin. To evaluate it near the origin may require additional terms in the polynomial filter up to u 6 . As long as [H(u ) − T(u )] decreases faster than u 4 , the remainder term can be calculated numerically without difficulty. The first four terms in Equation (34) are proportional to, respectively, the original spline and then its first, second and third derivatives. 3. With a cubic spline there is a linear polynomial, p(x) = a o + a 1 x which is transformed by filtering to q(x) = b o + b 1 x by the equation (see Appendix K),

bo b1

=

H(0 ) 0

1 ( ) 2oi H u 0

H(0 )

ao a1

(36)

195

In summary, application of filters based on the cubic spline reduces to a problem of determining polynomial approximations to the near origin behaviour of H(u ), the numerical calculation of the remainder and transformation of the polynomial terms. I now present the basis functions for each of the filtering operations considered in the last section, but do not show results as they are similar to the sinc application (except possibly near boundaries). 4

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Figure 6: The cubic spline applied to convolution of the Bessel function of Figure 1a; (a) and (b) The weights (at the 80 equally spaced dots) that exactly interpolate Figure 1a. (c) and (d) Remaining component of basis function after removal of the near origin constant and quadratic behaviour of the filter. Convolution with a Gaussian transfer function The filter is given by Equation (26) and can be expanded in even monomials with n 2n a 2n = (−1 ) (ou/a ) /n!. The new basis function is then

y(x ) = x 3 + 3 2 x + y r (x ) 2a

(37)

196

The remainder component is of relatively short duration in both the space and frequency domain (Figures 6c-d) and is small in magnitude with a maximum in the space domain at the origin of 6x10-4. As H(0 ) = 1 and H u (0 ) = 0 the polynomial terms are unchanged by the convolution.

Deconvolution with a Wiener filter The filter is given by Equation (27) and can be expanded in even polynomials about the origin by use −1 of the binomial theorem (1 + z ) = 1 − z + z 2 − z 3 + ..., with z = b 2 u 2 + b 4 u 4 + ..., where the b n ’s are

obtained from a Taylor expansion of the numerator of Equation (27). After some tedious algebraic manipulation I find that the new basis function is given by

y(x ) =

1 x 3 − 3(1 − r 2 ) x + y (x ) r 2 1 + r2 2(1 + r 2 ) a 2

(38)

The remainder component requires the calculation of the u 4 and u 6 terms which can be obtained from the binomial formula with appropriate attention to the contributing components (e.g. u 4 term uses b 22 − b 4 ). The remainder component is shown in Figure 7 and can be simply calculated using a

numerical procedure based on the FFT. As H(0 ) = 1/(1 + r 2 ) and H u (0 ) = 0 the polynomial terms are reduced in magnitude by 1/(1 + r 2 ).

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1

-2 -20

-15

-10

-5

0 5 Frequency

10

15

20

Figure 7: The cubic spline applied to deconvolution with a Wiener filter and Gaussian PSF; (a) and (b) Remaining component of basis function after removal of the near origin constant and quadratic behaviour of the filter.

197

Application of the method to the noisy convolution samples of Figure 2a gives the dark curve shown at Figure 8. Near the endpoints the deconvolved response either under (at one end) or over predicts the equivalent response obtained from the sinc function deconvolution. At either end of the domain (neglecting the linear polynomial) there is an equal and opposite linear growth in the cubic spline (very similar to that illustrated previously at Figure 1 in the last chapter), which causes the observed contamination. To eliminate this effect, two extra nodes can be included with the additional degrees of freedom used to constrain the spline curve to a compact domain (light curve of Figure 8). Note that this procedure is similar to the trend removal which is sometimes used to minimise end effects in standard Fourier processing. 1.5

Deconvolved response

1

0.5

0

-0.5 Normal cubic spline Compact domain -1 -1

-0.5

0 Distance

0.5

1

Figure 8: Cubic spline, Wiener filter applied to the noisy convolution samples from Figure 2a. The dark curve is the ordinary cubic spline, while the lighter curve is after two additional nodes have been used to ensure that the spline has compact support. Low and high pass filtering For a low pass filter T(u ) = 1 and the remainder is the negative of the high pass filter y low (x ) = x 3 − y high (x ), where

y high (x ) = 6 4 4o

° uº

c

cos(2oux ) ¹u u4

(39)

The new basis function can be obtained by numerically evaluating this integral. The frequency and space domain versions of the high pass filter are shown in Figure 9. For a low pass filter H(0 ) = 1 and H u (0 ) = 0 leaving the polynomial terms unchanged, while a high pass filter has H(0 ) = 0 and H u (0 ) = 0 thus annihilating the polynomial.

198 -5

6

-5

x 10

1.4

x 10

1.2

Spectrum of high pass filter

Spatial convolution function

4

2

1

0.8 0.6

0

0.4

-2

0.2 -4 -2

-1.5

-1

-0.5

0 0.5 Distance

1

1.5

2

0 -20

-15

-10

-5

0 5 Frequency

10

15

20

Figure 9: Cubic spline and high pass filter with a cut-off of 5 cycles. Derivative filter For a derivative filter T(u ) = 2oiu and there is no remainder term unless some low pass filtering needs to be applied. In that case y(x ) = 3x 2 sgn(x ) + y r (x )

(40)

where the remainder relates to the low pass filtering and is obtained analogously to Equation (39) with u −3 and sin(2oux ) in place of u −4 and cos(2oux ), and an additional factor of 2o outside the integral (Figure 10).

Note, that the first term in Equation (40) could be obtained trivially by

differentiating the cubic spline basis. For the derivative filter, H(0 ) = 0 and H u (0 ) = 2oi so that b o = a 1 and b 1 = 0 which is what one would expect from differentiation.

Multiple filters The basis for differentiation of the deconvolved response can be obtained by multiplying the polynomial terms for the deconvolved response by 2oiu or, alternatively, differentiation of the basis already given for deconvolution. This new basis is

y(x ) =

3 x 2 sgn(x ) − 3(1 − r 2 ) sgn(x ) + ¹y r (x ) 2 1 + r2 ¹x 2(1 + r 2 ) a 2

(41)

199

where the derivative of the remainder can be obtained by inverse Fourier transforming 2oiu times the Fourier transform of the original remainder. The remainder component and its Fourier transform are shown in Figure 11. As H(0 ) = 0 and H u (0 ) = 2oi/(1 + r 2 ) the polynomial terms are transformed to b o = a 1 /(1 + r 2 ) and b 1 = 0. -3

2

-4

x 10

4 3

Imaginary component of spectrum

Spatial convolution function

1.5

x 10

1 0.5 0

-0.5

2 1 0

-1

-1

-2

-1.5

-3

-2 -1

-0.5

0 Distance

0.5

-4 -20

1

-15

-10

-5

0 5 Frequency

10

15

20

Figure 10: Negative of remainder component for cubic spline interpolation and a low pass derivative filter. -3

0.02

3

x 10

Remainder component (Imaginary axis)

0.015

Remainder component

0.01 0.005 0

-0.005

2 1 0

-1

-0.01

-2

-0.015 -0.02 -1

-0.5

0 Distance

0.5

1

-3 -20

-15

-10

-5

0 5 Frequency

10

15

20

Figure 11: Remainder component for the derivative of the deconvolved response for cubic spline interpolation.

200

Comments on Basis Transformation Expanding the near origin behaviour of a filter by a polynomial gives a series of terms for the new basis function which have algebraic decay at infinity (i.e. decay as O(u −n ) for some integer n). It is well known that with this decay the original function has a discontinuous (n − 1 )-th derivative (e.g. Bracewell, 1986). For example, the second term in Equation (37) for convolution with a Gaussian behaves as O(u −2 ) at infinity and its inverse transform is the linear spline, x , which has a discontinuous first derivative at the origin. Gaussian convolution is a smoothing process, causing a very rapid decrease of the transform with increasing frequency (much faster than any algebraic decay). This must imply that the derivative discontinuity in the linear spline is exactly cancelled by a discontinuity in the remainder term. This cancellation is evident in Figure 12a which shows the contributions to the convolved response from the linear spline and the remainder term. The linear spline curve is piece-wise linear with derivative discontinuities at the nodes. The remainder term has discontinuous derivatives at the nodes which are of opposite sign and presumably of equal magnitude. Notice that the contributions from both terms are highly oscillatory and partially cancel to leave a smooth curve. For the examples considered in this section, the delicate cancellation which seems to be going on, did not create a noticeable effect on any of the convolved curves (i.e. they all looked reasonable and very similar to the sinc examples). However, it would be prudent to give some indication of an alternative method which would avoid these cancelling discontinuities.

An alternative may be to expand the near-origin behaviour of the filter by a serious of terms of the form

u 1 + u2

n

or u n exp(−d 2 u 2 ). These have the advantage that the near origin behaviour is still

polynomial, but each term decays more rapidly, so that discontinuities will be restricted to higher order derivatives or even eliminated (for the exponential). To determine whether this procedure is feasible, and indeed, whether it confers any advantage, we will reconsider the Gaussian convolution example (Equation 37). This time we will expand the near origin behaviour of the filter as 2 2 T(u ) = 1 − o u2 exp(−d 2 u 2 ) a

(42)

I show in Appendix A2 that the new basis will be given by 2 2 y(x ) = x 3 + 3 2 x erf ox + d3/2 exp − o 2x 2a o d d

+ y r (x )

(43)

where y r (x ) is the remainder term obtained in the usual way (Equation 32), but with T(u ) as in Equation (42). When x is large relative to d (or when d d 0) the term in square brackets approaches x which shows that the exponential multiplier only effects the solution away from origin in the

201 Fourier domain. With d ! 0 the effect of the second term is to eliminate the derivative discontinuity that occurs in the linear spline. Letting d = o/a and calculating the contributions from the second and remainder terms, we see that the large oscillations and derivative discontinuities have been eliminated (Figure 12b). Clearly, the solution of Equation (43) has much better numerical properties than the original solution of Equation (37). Note, that for a practical application, we would usually interpolate only once, with a single basis function constructed from the individual components. Therefore, we would not usually see the individual components. The remaining issue with the method of Equation (43) is the selection of the parameter d. The closer d is to zero the more the solution will resemble that given with the linear spline. As d increases the

second term will become smoother and its contribution to the curve will diminish. In the limit that d d º the contribution will be zero, and the burden of solution will be on the residual component.

There is presumably some happy medium value where the solution will be optimal in some sense. However, the choice of the parameter takes us beyond the scope of the current work. From my own experience with the method, it appears a good choice is one where the discontinuity in the spline is eliminated, but at medium distances (say more than 10 nodes distant) the curve approaches the linear spline.

(b) 1.5

1.0

1.0

0.5

0.5

0.0

0.0

Signal

Signal

(a) 1.5

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

Second term Remainder term

Second term Remainder term -2.0

-2.0 -1.0

-0.5

0.0 Distance

0.5

1.0

-1.0

-0.5

0.0

0.5

Distance

Figure 12: Contribution of second and remainder terms to Gaussian convolution with the cubic spline. The left hand figure is for no damping, while the right hand 2 one is with exp − ox . a

1.0

202

5 EXTENSION TO TWO-DIMENSIONAL CONVOLUTION AND DECONVOLUTION

Weight transformation with the sinc function is similar to standard grid based methods. As for the 1-D case, Weight transformation using Equation (23) on splines and multiquadrics has met with little success. Specific reasons for this and a method to overcome the difficulties are given in section 5.3. For Basis transformation with the sinc function, the original basis is v(x ) = sinc(2xu max ) sinc(2yv max ), where u max and v max are the Nyquist cut-offs imposed by the nodal spacing, and the new basis is calculated by y(x, y ) = ° −u max u max

° −vv

max max

H(u, v ) exp(2oi [ux + vy ])dudv

(44)

As the integral extends only over a finite segment of the real plane, the calculation should present few difficulties and could be achieved by a method based on an FFT. Note, that any Gibb’s effects caused by the truncation of the filter are real, in the sense that they are imposed by the frequency cut-off in the sinc basis. The main consideration with interpolation and convolution using sinc functions is handling edge effects. In Chapter 5 some success was achieved by extending the distribution of the nodes in the sinc expansion beyond the limits of the survey boundary. However, the weights near the extended boundary are very large and highly oscillatory so that when this construction is deconvolved or convolved, noticeable artefacts extend a significant distance into the survey area. The sinc method has many similarities with a grid based approach and a rigorous method for eliminating edge effects has recently been developed (Beatson and Newsam, 1998b). Therefore, rather than trialing other ad hoc methods for eliminating boundary effects (e.g. synthetically extend the survey data with a cosine roll-off used to taper the values to zero at the new survey edges) we do not consider sinc convolution further in this thesis.

For bases such as the multiquadric and the thin-plate spline the issue, once again, is dealing with the singularity at the origin in the frequency domain. The transform of the multiquadric is of the form q −3/2 K 3/2 (2oqa ) which behaves as q −3 at the origin, while the thin-plate spline transform is

proportional to q −4 . In the last section we showed that the singular behaviour of the cubic spline could be removed with the aid of a near-origin polynomial expansion of the filter. Let us attempt to repeat this process here for the thin-plate spline, with a radially symmetric filter, H(q ). This has ÖH(0 ) = 0 and a near-origin polynomial expansion of T(q ) = H(0 ) + a 2 q 2 . I show in Appendix J3 that

the new basis function is then

203 a y(r ) = H(0 )r 2 log(r ) − 22 log(r ) + y r (r ) o

(45)

where y r (r ) is the remainder term calculated in the usual manner. The problem with this equation is that the log term introduces a singularity at each of the nodes that, in theory, is removed by the remainder term. In practice this cancellation of singularities is not possible as the remainder term has to be calculated numerically.

5.1 Thin-plate spline Basis transformation with radially symmetric filters

For the cubic spline, damping the near origin behaviour of the filter by an exponential eliminated problems with discontinuous derivatives. Repeating this process for the thin-plate spline with, T(q ) = H(0 ) + a 2 q 2 exp(−a 2 q 2 ), I show in Appendix J3 that the logarithmic term in Equation (45) is

replaced by

y 2 (r ) =

−a 2 o2

° 0or/a 1 − expz −z (

2)

dz

(46)

4

3

New basis function

2

1

0

-1

-2 New basis function Logarithm -3 0

500

1000

1500

2000

Distance (m)

Figure 13: Comparison of the basis function for exponential damping (a = 247.5) with log(r ). The integrand behaves as O(z ) at the origin, hence y 2 (r ) = 0. Calculation of the integral for an arbitrary r is straightforward. Note, that as z increases the exponential rapidly decays to zero so that the integrand approaches 1/z. The anti-derivative of 1/r is log(r ), which implies that y 2 (r ) behaves as log(r ) as r is increased (i.e. it resembles the result for no damping). This observation is evident in

Figure 13 where I compare y 2 (r ), using a = 247.5, with log(r ), appropriately scaled, and find that the

204

functions differ only at small arguments. This is a similar situation to the 1-D case where the damped basis function was smooth at the origin and approached x as x increased. Thus, we see that exponential damping only affects the new basis functions near the origin.

The method requires that a damping factor be selected and the remainder term inverse Fourier transformed. The damping factor is scale dependent, but for many filters we will presently see that there is a natural choice of a. To illustrate the method we consider deconvolution of the thorium band of the Jemalong survey using the transfer function, H(s ), developed in Chapter 3, assuming it is symmetric (i.e. we ignore aircraft movement). I use the simplest Wiener filter which is

W(s ) =

H(s ) H(s ) 2 + r 2

(47)

with H(0 ) = 1 and r = 0.25 (this simple filter was used as these calculations were completed prior to the development of the Wiener filter in Chapter 4).

A near-origin polynomial approximation,

W(0 ) + a 2 q 2 + a 4 q 4 + a 6 q 6 , to the filter was obtained by a least-squares fit to the Wiener filter in the

range [0, 0.002]. At the origin, W(0 ) = 0.941 and the best fit was obtained with a 2 = 7.79 % 10 4 , a 4 = −4.77 % 10 9 and a 6 = 3.51 % 10 14 (Figure 14).

The exponential damping modifies this

polynomial approximation by changing the q 2 term to q 2 exp(−a 2 q 2 ). For example, Figure 14 shows the original and exponentially damped approximations for a = 247.5. Note, that we are not too concerned at how the approximation fits away from the origin. The important consideration is that it eliminates the singular behaviour of the thin-plate spline at the origin.

2.0 Wiener filter No exponential Exponential

Wiener filter

1.5

1.0

0.5

0.0 0.000

0.005

0.010

0.015

0.020

Spatial frequency (m-1)

Figure 14: Wiener filter for deconvolution of thorium and near origin polynomial approximations with and without exponential damping (a = 247.5).

205

Turning now to the calculation of the remainder term, its Fourier transform is

Y r (q ) =

W(q ) − T(q ) W(q ) − W(0 ) − a 2 q 2 exp(−a 2 q 2 ) = 2o 3 q 4 2o 3 q 4

(48)

which near the origin behaves as Y r (q ) = 1 3 [(a 4 + a 2 a 2 ) + (a 6 + a 2 a 4 /2 )q 2 ] 2o

(49)

Figure 15a shows Y r (q ) for three values of a = 200, 247.5 and 295, where it is evident that the function increases to a maximum before decreasing rapidly towards zero. Numerical solution should present few difficulties. From Equation (49) we see that Y r (0 ) = a 4 + a 2 a 2 , which leads to a natural choice of a =

−a 4 a 2 if a 2 and a 4 have opposite sign, so that Y r (0 ) = 0. With this value of a (=247.5

in this case) the basis function for the remainder term has zero integral, which implies that the integrated effect of each weight on the deconvolved surface will be zero. Figure 15b shows the new basis function for this natural choice of a. It initially increases and reaches a maximum at a distance of 15 m, then rapidly decreases to a minimum at approximately 125 m. It then decays and is effectively zero for distances greater than 500 m.

(a)

(b)

4.0e+7

2000.0 α = 295 α = 247.5 α = 200

3.0e+7

Remainder term

1500.0 Remainder basis function

Remainder term

2.0e+7

1.0e+7

0.0e+0

-1.0e+7

1000.0

500.0

0.0 -2.0e+7

-3.0e+7

-500.0

0.000

0.005

0.0010

Spatial frequency (m-1)

0.015

0

100

200

300

400

Distance (m)

Figure 15: (a) Effect of the parameter a on the remainder component in the Fourier domain; (b) the space domain form of the remainder component for a = 247.5.

500

206 Regardless of the choice of a, the net effect of y 2 (r ) + y r (r ) should always be the same. Figure 16 shows y 2 (r ) and the summation of the two components for a = 247.5. The effect of the remainder term is concentrated around the origin, and will have little influence on the deconvolved surface at distances greater than 500 m.

5.0e+3 Second term Combined 0.0e+0

New basis function

-5.0e+3

-1.0e+4

-1.5e+4

-2.0e+4

-2.5e+4

-3.0e+4 0

250

500

750

1000

Distance (m)

Figure 16: The summation of the second and remainder components for Gaussian damping with a = 247.5. I applied the method to the GCV fit to the 3 % 3 km2 subset of the Jemalong survey considered previously in section 5, Chapter 6. Figure 17 shows the original and deconvolved surfaces where it is clear that the latter image has been sharpened without any obvious increase in noise. Specific comments on the improvement contributed by the deconvolution will be reserved to the next section. Here our interest is on the relative contributions of the y 2 (r ) and y r (r ) terms (Figure 18). The variation in the y 2 (r ) surface is significantly larger than the remainder term (-0.97 to 0.61 ppm eTh compared to -0.25 to 0.26 ppm eTh). However, the remainder term does contribute an important component to the sharpening effect of the Wiener filter.

In conclusion, we have successfully implemented an exact method for deconvolution of thin-plate spline surfaces with a radially symmetric Wiener filter.

The method uses a scale dependent

exponential damping, but a natural choice of the parameter suggests itself. Numerical solution of the remainder term presents little difficulty as it decays rapidly with frequency, and is well behaved throughout its domain. The first obvious extension of the method would be to the symmetric but non-radial case (as occurs for the true PSF for a radiometric survey).

However, rather than

attempting this approach, I consider an approximate method for the inverse Fourier transformation

207

(first suggested by Newsam, pers. comm.) that can be applied to many basis functions whose Fourier transforms decay rapidly to zero.

eTh (ppm)

Figure 17: Original and deconvolved surfaces for GCV thin-plate spline fit to Thorium with Gaussian damping(a = 247.5 ) applied.

eTh (ppm)

Figure 18: Interpolated surfaces for the second and remainder terms with Gaussian damping(a = 247.5 ).

208

5.2 An approximate method for thin-plate splines and multiquadrics The Fourier transform of an ABF surface may be written as S(u ) = F(u )L(u ) + P(u ), where L(u ) is the DFT of the weights and the other symbols have there usual meaning. The Fourier transform of the convolved surface is obtained by multiplying this equation by the filter, H(u ), to give, G(u ) = H(u )F(u )L(u ) + H(u )P(u )

(50)

Subtracting the value of the transfer function at zero and adding it back in elsewhere, I find G(u ) = [H(u ) − H(0 )]F(u )L(u ) + H(0 )F(u )L(u ) + H(u )P(u )

(51)

For a symmetric transfer function, H u (0 ) = H v (0 ) = 0, so that H(u )P(u ) = H(0 )P(u ) and I find G(u ) = [H(u ) − H(0 )]F(u )L(u ) + H(0 )S(u )

(52)

For the thin-plate spline, the last term in the equation is a scaled version of the original spline surface, while the first term is now finite at the origin. To see this, recall from Appendix B in Chapter 6, that near zero, the Fourier transform of a set of spline weights has leading order terms proportional to O(u 2 + v 2 ), as the constant and linear terms are eliminated by the conditions imposed by the

polynomial (i.e. S k n = S k n x n = S k n y n = 0). The transfer function, H(u ), is symmetric, implying that ÖH(0 ) = 0 and with the removal of the constant term it also behaves as O(u 2 + v 2 ) near the origin.

The combined leading order behaviour of the weights and transfer function is enough to counter the −2 O([u 2 + v 2 ] ) singularity of the spline at the origin.

The main computational burden of the approximate method, is the numerical calculation of the inverse Fourier transform of the first term in Equation (52), onto a regular grid in the space domain. This first requires that the Fourier transform of the weights is evaluated on a regular grid in the frequency domain. The highest frequency in this grid is determined by the cell size of the space domain image, while the grid spacing is determined by the spatial extent of the image. However, if we calculate the inverse FFT from this grid, we find that the correction term is very smooth and physically unrealistic. This occurs because an FFT without appropriate attention to edge effects creates a “St-Georges Cross” type contamination in the power spectrum (e.g. Figure 3, Chapter 6). The contamination is absent from our Fourier description as it was obtained by exact Fourier transformation of the thin-plate spline surface. To overcome this difficulty we need to evaluate the

209

Fourier transform on a slightly finer frequency grid. This, in effect, extends the limits of the space domain image and defines an implicit method for edge matching.

An advantage of this approximate method over a standard grid based method is that the initial Fourier transformation is exact. The rapid decay of the spline basis with frequency causes the Fourier transform to approach zero at the edges of the frequency grid. Thus, even though the inverse transformation is only approximate, there are at likely to be few concerns with edge mismatch. The principal disadvantage of the method is that the computational burden rests on the DFT of the weights which, as we have previously indicated, cannot currently be evaluated rapidly when the weights are irregularly distributed. Note, that with radiometric data we usually grid at a finer spacing than the sampling rate so that the computational effort will scale at worse than the square of the number of data points.

Three further advantages of the approximate method are that no polynomial approximations to the near origin behaviour are required, non-radially symmetric filters can be accommodated and it can be applied to several other basis functions without modification.

For example, the multiquadric

increases at O(q −3 ) near the origin, but is always selected to be orthogonal to constants so that the weights are O(q ). When this is combined with the O(q 2 ) behaviour of the filter (minus its value at the origin), the resultant is finite. Multiquadrics decay even more rapidly to zero than do splines so that edge mismatch will again not be a concern. Applications of the method to both thin-plate splines and multiquadrics are given in section 6.

5.3 Extending Weight Transformation to splines

Filtering of spline surfaces using Weight Transformation could theoretically be achieved using the exact formula of Equation (18). However, this requires the calculation of two new basis functions that are likely to be even more difficult to calculate than the one required for Basis Transformation. Attempts to use the simpler discretised formula of Equations (22) and (23), which leads to a relationship between easily calculated discrete Fourier transforms, met with little success. A possible explanation is that the discretisation is a poor approximation to the continuous version because the weighting factor, F(u ) 2 , is ignored. I outline (but do not implement) an alternative method that could be used for Weight Transformation when filtering with a real, symmetric transfer function.

The method proceeds in a similar manner to the approximate method considered in the last section. Returning to Equation (52) the new spline surface is given by a reverse application of the Fourier transform result derived last Chapter,

210

g˜(x ) = S m=1 (H(0 )k m + j m )v(x − x˜ m ) + H(0 )p(x ) M

(53)

The goal of Weight Transformation is to find a set of weights, j m , that approximate the product of the transfer function and the old set of weights as closely as possible. This involves minimising M J(j ) = S k=1 F(u k )

2

M j m exp(−2oiu k $ x˜ m ) − [H(u k ) − H(0 )]L(u k ) S m=1

2

(54)

for some set of points u k . The discussion immediately proceeding Equation (52) established that the last term in the above equation is O(u 4 + v 4 ) near the origin. This implies that the DFT of the new weights will also have this behaviour near the origin. From the near origin expansion of the Fourier transform derived in Appendix B in Chapter 6, this gives a set of side conditions that the weights must satisfy, M j j m x km y m = 0 for k, j = 0, ..., 3 and k + j [ 3 S m=1

(55)

Solution of Equation (54) subject to (55) is a constrained least squares problem, that should be solvable quickly by an iterative method if a fast DFT is available. The reason for this is that airborne geophysical data is distributed almost on a regular grid, which means that exp(2oiu k $ xˆ m ) should be a −1 reasonable approximation to [exp(−2oiu k $ xˆ m )] . Therefore, preconditioning the equations by the

inverse DFT will give mostly diagonal entries, hence a low condition number and a quick solution.

The technique described above is suitable for any symmetric transfer function, not just those that are radially symmetric. Additionally, with a few modifications it could be extended to other basis functions such as the multiquadric (for the same reasons discussed in the last paragraph of section 5). However, we will not attempt to implement the method here.

6 EFFECT OF ABF PARAMETERS ON THE DECONVOLUTION OF RADIOMETRIC DATA

The last section addressed the problem of how to convolve and deconvolve surfaces defined by an ABF expansion. In this section we consider how the results of the deconvolution of radiometric data are affected by the basis function, the placement of the nodes and whether the original surface exactly interpolates or smoothly fits the data. I again consider the 3 % 3 km2 subset of thorium data analysed in section 5 of Chapter 6 and section 5.2 of this chapter. Four different examples are considered: (i)

211 exact thin-plate spline; (ii) exact multiquadric (with c = 100); (iii) GCV thin-plate spline; and (iv) LSQR thin-plate spline. The surfaces are shown interpolated to a 25 m grid in the left hand columns of Figures 19 (exact fits) and 20 (smooth fits). The power spectrum of the weights multiplied by the transforms of the basis functions were previously shown at Figure 5, Chapter 6. The Wiener filter given at Equation (47) was used with r = 0.25 and a point-spread function that accounted for the movement of the aircraft during the integration time. The deconvolutions were achieved using the approximate method outlined in section 5.2, which required calculation of the Fourier transform of the weights on a regular grid. To minimise edge effects the spacing of the frequency grid was selected so that it mimicked a 4 % 4 km2 image in the space domain. After deconvolution, the extended parts of the grid were ignored and the image added to a scaled version (by W(0 ) = 0.941) of the original surface (Equation 52). The resulting deconvolved surfaces are shown interpolated to a 25 m grid in the right-hand columns of Figures 19 (exact fits) and 20 (smooth fits).

Exact interpolation: Effect of basis function When the two surfaces constrained to honour the original data are deconvolved, they give very similar results (Figure 19).

Both the original and deconvolved surfaces obtained with the

multiquadric basis are sharper than their thin-plate spline equivalents. This is to be expected as surfaces defined by splines minimise second derivatives. The differences between the two results can more clearly be seen in a difference image (left-hand side of Figure 21). As one would expect, the major differences between the two images are mostly between the aircraft flight lines, although note that there are obvious patterns that extend over multiple flight lines. Apart from these subtle differences there is little to separate the two images. While the deconvolution obviously sharpened the spatial detail it also appeared to magnify unwanted noise.

Smooth surfaces: Effect of smoothing method The results for the two smooth surfaces are similar (Figure 20), although the GCV thin-plate spline images appear to be slightly sharper for both the interpolation and deconvolution. This observation highlights one of the deficiencies of the LSQR method in that the smoothing occurs implicitly through the number and spacing of the nodes in comparison to the data. It is easy to under- or over-smooth at which point the entire process needs to be repeated if an improved result is required. This factor, coupled with the implied periodicity of the weights in the Fourier domain, have to be weighed against the (currently) significant computational savings that occur by distributing the nodes on a regular grid. Although, we must also bear in mind that the solution of the interpolation equations by LSQR is more expensive than GCV (Chapter 5).

212

eTh (ppm)

Figure 19: Deconvolution of thorium in a 3 by 3 km subsection of the Jemalong survey. Images on the left are the raw thorium data interpolated to a 25 m grid, while the images on the right are deconvolved versions. From top to bottom the surfaces are the exact thin-plate spline and the exact multiquadric. Comparison of smooth and exact surfaces The differences between the exactly and smoothly interpolating surfaces are much more significant than the differences caused by choice of basis function or smoothing method. Variations of ! 2 ppm eTh are not uncommon in a difference image of the exact and GCV spline surfaces (right-hand side of Figure 21), which are quite large considering the mean value of the deconvolved GCV surface is 10.7 ppm eTh. Many of the significant differences between the images are located over measurement points. These are noisy observations which diverge even further from the GCV spline surface when deconvolved. The parameter r 2 controls the action of the Wiener filter at different frequencies. For

213

a fixed value it will accentuate more noise in the exact surface than the GCV spline surface, as in the latter case noise has already been partially removed during the interpolation process. In fact, for the example considered here, σ r 2 needs to reduced by a factor of four to noticeably accentuate noise in the GCV spline surface.

eTh (ppm)

Figure 20: Deconvolution of thorium in a 3 by 3 km subsection of the Jemalong survey. Images on the left are the raw thorium data interpolated to a 25 m grid, while the images on the right are deconvolved versions. From top to bottom the surfaces are the GCV and LSQR thin-plate splines. Of the four methods considered here, the GCV spline is the best. Firstly, noise is removed prior to deconvolution which means it is less likely to be magnified by the Wiener filter. Secondly, the

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amount of smoothing is controlled by the characteristics of the data rather than implicitly through a prior estimate of the expected frequency cut-off as with LSQR.

eTh (ppm)

Figure 21: Difference between deconvolved surface for the exact thin-plate spline and (on left) exact multiquadric and (on right) GCV thin-plate spline deconvolution. The data locations are overlaid. 7 DISCUSSION

The ABF approach to convolution/deconvolution has some attractive theoretical and practical aspects. An ABF expansion defines an interpolant that is defined over all of space and, in contrast to grid based approaches, the entire surface is moved in and out of the Fourier domain, rather than just discrete samples. As discussed last chapter, this has multiple advantages with respect to issues of finite domain, edge mismatch and quadrature approximation.

With the Basis transformation method the convolution or deconvolution of an ABF surface can be calculated exactly and leads to a new ABF surface with a different basis function. The only error incurred in the whole process is in expressing an arbitrary function of signal as an ABF surface. The method has the advantage that the data or ABF weights never have to be moved into the Fourier domain. The entire effect of the transformation is encapsulated in the new basis function. For large geophysical surveys, rapid interpolation with the new basis function requires a fast method for ABFs. As many different basis functions are likely to be encountered, with some only known numerically, the adaptive, Beatson and Newsam (1998a) algorithm, would be required.

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For sinc functions the calculation of the new basis function was found to be straightforward. For splines and multiquadrics the principal issue was dealing with the singular behaviour at the origin in the Fourier domain. This singularity could be removed by a near origin polynomial approximation. However, to prevent this process from introducing a further singularity or discontinuous derivative, a Gaussian damping factor was applied. Fortunately, a natural choice for the damping factor suggested itself and the method was successfully applied to 2-D radially symmetric deconvolution with thin-plate splines. For non-radial symmetry, and indeed in any situation where the new basis function is difficult to calculate or unsuitable for a fast method, two alternative methods were considered.

The first method was termed Weight transformation, as it involved calculating a new set of weights while retaining the original basis function. While this method could only be approximate in most cases, it has several attractive features. The first is that it allows the reuse of any data structures already set up for the fast interpolation of the original surface. Secondly, it does not require any near-origin polynomial approximations or numerical calculation of a new basis.

The main

disadvantage of the method, is that it requires both a forward and inverse DFT, processes that are currently O(N 2 ) and O(N 3 ), respectively, when the nodes are irregularly distributed. This is one incentive for placing the nodes on a regular grid as both the forward and inverse transformations can be achieved with an FFT in O(N log N ) operations. Lastly, the singular behaviour of splines and multiquadrics are a problem and we have yet to successfully implement the method on them.

A second alternative method for singular basis functions, that could be used with any symmetric convolution kernel, was developed. The ABF surface was exactly transformed into the Fourier domain at which point the expression was split into two components. The first component was a scaled version of the original ABF surface and could be transformed exactly. The second component was calculated on a regular grid in the Fourier domain before being transformed via an inverse FFT. While this transformation of the remainder was only approximate, the rapid decay of the spline and multiquadric bases meant that there were no concerns with finite domain and edge mismatch. The main computational burden of the method lies with the Fourier transformation of the weights, which generally need to be calculated at more points in the frequency domain than there are weights.

In the last section of this chapter, we considered the effect of the basis function, the placement of the nodes and exact or smooth fitting on the deconvolution of radiometric data. The difference between an exact fit with the thin-plate spline or the multiquadric was found to be quite small, as was the difference between the choice of smoothing method (LSQR or GCV). The biggest differences were found between the exact and smooth thin-plate spline surfaces. The best approach was felt to be the GCV spline, as significant noise removal occurred before application of the Wiener filter.

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CHAPTER 8

DISCUSSION There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact. Mark Twain

1 INTRODUCTION

The aims of this thesis were:

1. To quantify the influence of the survey geometry (detector height, flight line spacing, background radiation etc.) on the spatial resolution and assay uncertainty of maps of potassium, uranium and thorium distribution.

2. To increase the spatial resolution and accuracy of geochemical maps by implementing improved methods for interpolation, Fourier transformation and deconvolution of radiometric data.

Addressing the first aim, required the development of a model of gamma-ray spectrometry, a method for calculating assay uncertainty and a criterion for determining spatial resolution based on the action of a Wiener filter. These methods were used successfully to determine the assay uncertainty and spatial resolution of the Jemalong survey. The analysis of the spatial resolution indicates that radiometric surveys are usually sampled adequately along-lines but under-sampled across-lines. These results are tentative, as the auto-correlation model used for the signal strictly applies only to the Jemalong survey.

Achieving the second aim required the development of a general framework for interpolation, Fourier transformation and deconvolution of irregularly sampled geophysical data. The Arbitrary Basis Function (ABF) approach was found to be very promising in this regard, with many of the technical details addressed successfully. The main difficulty with the method was the computational effort required by each of the processing steps including, (i) solving the interpolation equations; (ii) interpolation to a regular grid; (iii) Fourier transformation; and (iv) deconvolution onto a regular grid.

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2 SUMMARY OF THESIS

In Chapter 2, the fundamentals of airborne gamma-ray spectrometry were discussed and the requirements for its use as an aid to soil mapping were established. This resulted in six specific objectives being developed which it was felt would facilitate the application of gamma-ray spectrometry to soil mapping. I now discuss these objectives in the light of the results obtained in this thesis.

2.1 Modelling the gamma-ray signal

Objective 1 of the thesis was to: Develop a model of gamma-ray spectrometry that accounts for the shape of the detector and the movement of the aircraft during the integration time.

The results presented in Chapter 3, showed that a geometrically based model of primary unscattered radiation could be used to simulate gamma ray surveys. The model calculates the total count rate by considering the characteristics of a detector (the solid angle, thickness and attenuation coefficient) that influence the number of interactions within its volume. The total count rate is converted to a peak count rate by multiplying by the detector photo-fraction which, from Monte Carlo modelling, was shown to be approximately constant with incidence and azimuthal angles.

The aircraft

movement during the integration time can be simply incorporated by either numerically integrating the response, or by multiplication of its Fourier transform with the sinc function.

The geometrical detector model predicts a smaller detector footprint than a uniform detector model. For example, for the uniform model, the 90% contributing area for thorium at 60 m is 10.9 ha compared to 7.6 ha for the geometrical model. This reduced footprint has significant implications for the calculation of spatial resolution and the deconvolution of radiometric data. Additionally, the aircraft movement can exert a strong influence when the ratio of survey altitude to distance travelled during integration is low. 2.2 Spatial resolution, assay precision and Wiener filtering

The second objective of the thesis was to: Use error propagation to determine the expected assay uncertainty for a given survey.

The approach of Lovborg and Mose (1987) was generalised successfully to airborne surveys. The most significant source of error for potassium and thorium was found to be from the Poisson counting

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error in the detection of gamma-rays. For uranium (and for potassium when its concentration was low) the contamination of the windowed count rates from radon and scattered gamma-rays was also important. The Poisson error is determined by the number of gamma-rays detected which depend largely on the aircraft height, the isotope concentrations, the background contamination and the live-time.

Errors introduced by height correction were predominantly caused by incorrect

background removal and appeared to be significant only in areas with large deviations in topography and/or survey altitude.

With standard concentrations of potassium, uranium and thorium (2 %K, 2 ppm eU and 8 ppm eTh) and average radon (25 cps in Uranium window), the assay uncertainties for potassium, uranium and thorium at 60 m elevation are 7.8%, 49% and 16%, respectively. At 120 m these uncertainties are increased to 11%, 77% and 21% respectively. These numbers are a rough guide to the uncertainties expected with four-channel processed data. Multichannel techniques would cause a reduction in the relative errors with the most noticeable impact on uranium. For example, using NASVD by cluster Minty and McFadden (1998) found 2.4-, 3.4- and 2.5-fold reductions in uncertainty for potassium, uranium and thorium, respectively.

Objective 3 of the research was to: Develop a method for estimating the spatial resolution for potassium, uranium and thorium for a given survey.

The Wiener filter for deconvolution of radiometric data was used to estimate the spatial resolution by analysing its effect on signal and noise in the frequency domain. While the estimated frequency cut-offs are based on subjective criteria, they can at least be used to obtain a rough indication of the spatial resolution and its change with aircraft height, assay uncertainty etc. The spatial resolution was found to depend on the logarithm of assay uncertainty, which implies that mutiplicative improvements in the signal-to-noise ratio only additively change the resolution.

The principal difficulty with the approach was obtaining realistic estimates of the auto-correlations of the potassium, uranium and thorium signals. The method I used was to fit a power law to the Jemalong radiometric survey. Using this power law model indicates that radiometric surveys are usually sampled adequately along-lines but under-sampled across them.

Realistically, complete

sampling could probably be achieved only at low elevations, where the cost of surveys are already high, and the efforts to achieve maximum resolution are justified. However, before too much confidence was placed in the results, the auto-correlations of other, preferentially lower altitude, higher definition airborne surveys, would need to be calculated.

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2.3 The ABF approach to interpolation

The forth and fifth objectives of this research were to:

Implement alternative methods for

interpolating the line data to a regular grid that can exactly honour (Objective 4) or smoothly fit (Objective 5) the original data, are independent of the cell size and can be applied to large data sets.

In Chapter 5, I presented a general approach for interpolation and smooth fitting of geophysical data based on Arbitrary Basis Functions. The framework accommodated many different interpolation methods including smoothing and tension splines, radial basis functions, kriging and sinc functions. The interpolated surfaces inherit certain characteristics from the basis function, including minimum variance for a kriging variogram, smoothness for a spline and a band limited Fourier transform from a sinc function. The requirement to exactly fit noisy radiometric data can be relaxed, with the compromise between data fidelity and smoothness estimated by generalised cross validation.

Naive solution of the resulting matrix equations is O(N 3 ) in computation and O(N 2 ) in memory, which has prevented application to problems with more than a few thousand points. Using iterative schemes based on the Lanczos approach, the computational complexity was decoupled into two factors O(N )O(N 2 ); the O(N ) relates to convergence and the O(N 2 ) to the matrix-vector product required at each iteration. To reach an acceptable solution none of the iterative schemes (with the exception of the LSQR spline example) required a significant number of iterations, relative to the problem size. Therefore the cost lies somewhere in the lower part of the range O(N 2 ) to O(N 3 ).

The implementation is still O(N 2 ) in memory and cO(N 2 ) in computation, where c is a relatively large positive factor which technically is dependent on the problem size. The computational and memory scaling of the present implementation makes application to large geophysical problems unfeasible, except possibly on super computers with substantial memory. However, the eventual objective is to reduce the computational burden down to a level that allows everyday application on desktop computers.

The first obvious extension of this work is to incorporate preconditioning to reduce the factor c related to the number of iterations, especially for LSQR fitting of spline surfaces. In this context, Beatson (pers. comm.) has conducted some numerical experiments on the thin-plate spline equations which indicate that a good solution can be obtained with very few iterations (often < 30). Furthermore, the method appears to be applicable to other basis functions.

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The second extension would be to use fast methods for the matrix-vector product. These are able to reduce the cost of a single matrix-vector multiplication from O(N 2 ) to O(N log N) or even O(N ). In addition, the computational structures needed to implement these methods result in memory requirements scaling at much better than O(N 2 ). Furthermore, many of the computational structures can be reused after they have been set-up. This means that they don’t need to be regenerated at each iteration or for different datasets defined on the same set of nodes. Existing fast methods for the thin-plate spline already exist in the form of fast-multipole methods (e.g. Greengard and Rohklin, 1987; Beatson and Newsam, 1992). However, a more flexible approach is the fast moment method algorithm described in one dimension by Beatson and Newsam (1998a). This algorithm is adaptive and could be applied to any of the basis function considered in Chapter 5. It is currently being extended to 2-D applications by Beatson (pers. comm.).

The ABF approach is flexible because the matrices interact only with the Lanczos solution algorithms through their action on vectors.

Therefore, implementation of the preconditioning and fast

matrix-vector product routines alluded to in the last couple of paragraphs, would require little modification of the existing software base. It requires only removal of the direct routines and insertion of the fast methods. 2.4 Fourier transforms, convolution and deconvolution of ABF surfaces

The sixth objective of this research was to: Implement a method for direct deconvolution of the line data using a model that accounts for detector shape and aircraft movement. Recognising that this objective was to be achieved by Fourier methods, required the development of methods for both forward and inverse Fourier transformation of ABF surfaces. In Chapter 6, I considered Fourier transformation of ABF surfaces and compared them to standard methods that use a Fast Fourier Transform (FFT) on an imaged version of the surface. The FFT based approach requires consideration of a number of difficult issues including edge mismatch, survey gaps, quadrature approximation and grid cell size. In contrast, the formal expression of the Fourier transform of an ABF surface has a particularly simple expression in the Fourier domain and can be calculated exactly on an arbitrary grid. Thus, the only error incurred in computing it is the approximation error between the original function and the ABF interpolant. No numerical quadrature errors are introduced and the issue of choosing appropriate quadrature points and weights is avoided. Additionally, when using splines and multiquadrics there are no concerns with opposite ends of the survey matching or with survey gaps as the ABF surface extends over the whole real plane. However, if the nodes are defined on a regular grid there is an implied periodicity that can create artefacts due to edge mismatch.

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Application of the grid and ABF based approaches to the Jemalong survey illustrated the benefits of the latter approach. The power spectra derived from the grid based method showed obvious artefacts from edge mismatch that were absent from the ABF calculation. Additionally, using generalised cross validation to fit a smooth surface to noisy radiometric data results in a Fourier transform with reduced noise contamination.

The principal difficulty with the ABF approach is the computational cost associated with the calculation of the Discrete Fourier Transform (DFT). If there are M nodes that need to be Fourier transformed onto N points in the Fourier domain the DFT requires O(MN) operations. Both M and N can be large for many geophysical surveys (M ~ 1,000,000 is not uncommon) which makes naive application unfeasible. Timely calculation requires a fast algorithm for the Fourier transformation that could be applied to scattered data. Currently, an algorithm does not exist for two dimensional applications.

However, this is likely to be only a temporary impediment as there are several

promising algorithms under development (e.g. Newsam, pers. comm.). These include generalisations of existing 1-D algorithms (e.g. those of Dutt and Rohklin, 1993; 1995) and an approach based on prolate spheroidal functions (Newsam, pers. comm.).

In Chapter 7, I considered the problem of inverse Fourier transformation of ABF surfaces after they had been linearly filtered. Basis transformation involved using the same set of weights with the effect of the convolution or deconvolution, encapsulated by a new basis function.

It has the

advantage that both the forward and inverse Fourier transformation are exact and the DFT of the data or weights never has to be calculated. For splines and multiquadrics, the major difficulty was accommodating the non-integrable singularity at the origin in the Fourier domain. A polynomial with Gaussian damping was successfully used to eliminate the singularity concerns for the special case of radially symmetric filters. However, further work is required to: (i) determine whether matrix-vector products with the new basis functions can be calculated rapidly by the fast algorithm of Beatson and Newsam (1998a); and (ii) assess whether the technique can be generalised to non-radial filters.

When the new basis cannot be calculated, or is unsuitable for a fast method, two approximate methods were developed. The first, Weight transformation, could be applied to well behaved bases such as sinc functions and Gaussians. Application to splines and multiquadrics was more difficult, and a method was proposed but not implemented. The second approximate method was intended for these singular basis functions and was used to successfully deconvolve spline and multiquadric surfaces with symmetric point spread functions. Its main disadvantage was the computational effort required to calculate the Fourier transform of the weights on a regular grid in the frequency domain.

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2.5 Integration of methods

I conclude this section by integrating each of the methods developed in this thesis and illustrate their application on a 3 % 3 km2 subsection of the thorium data for the Jemalong survey. The radiometric data were processed using the NASVD technique of Hovgaard (1997) and then interpolated to a 25 m grid using two different methods (Figure 1). The first surface fitting method was an exact thin-plate spline with the interpolation equations solved by conjugate gradients (Chapter 5). The second was a GCV thin-plate spline with the equations solved by the Fast GCV method (Chapter 5). The data were then exactly transformed into the Fourier domain (Chapter 6) and then deconvolved using the approximate method for deconvolution (Chapter 7). The Wiener filter used the geometrical detector model, including the 70 m movement during the integration time (Chapter 3) and the estimates of the signal and noise auto-correlations developed for the Jemalong survey (Chapter 4).

After deconvolution, the exact thin-plate spline surface has clearly been sharpened considerably, although the noise appears to have been magnified in the process (Figure 1). On the other hand, the GCV spline surface has been noticeably sharpened without any obvious increase in noise. Thus, the overall goal of the thesis has been reached; that is, the implementation of an improved and rigorous method for creating gamma-ray images from noisy, blurred and irregularly sampled data. The challenge for the future is implementation of fast methods for ABFs, which will render feasible the application of the technique to large radiometric surveys.

3 IMPLICATIONS AND DIRECTIONS FOR FUTURE RESEARCH

The processing of airborne gamma-ray data has improved markedly over the duration of this thesis. At the outset, the only processing technique widely available was the standard 4-channel method. Gamma-ray images were usually created by a local interpolation procedure, followed by application of the minimum curvature smoothing kernel. Methods for sharpening the images by deconvolution were known but not yet implemented. Since that time, there have been significant advances in multichannel processing techniques (e.g. Hovgaard, 1997) and the implementation of a deconvolution procedure (Craig et al., 1998). The combination of the two developments has resulted in sharpened images with less noise contamination (e.g. Taylor, 1998). This thesis has contributed to the advance in gamma-ray image creation by the implementation of improved methods for interpolation and deconvolution.

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eTh (ppm)

Figure 1: Exact (top) and GCV (bottom) thin-plate spline surfaces interpolated to a 25 m grid cell. Images on the left are the original surfaces, while those on the right have been deconvolved by the Wiener filter developed in chapter 4. 3.1 Soil mapping implications

The development of improved methods for creating images of potassium, uranium and thorium distribution has important implications for soil mapping. The better spatial resolution that results from deconvolution would have a follow-on effect to the mapping resolution of a soil survey. The improved interpolation methods, especially the GCV thin-plate spline, reduce the noise levels in a gamma-ray image and should result in better estimates of isotope concentration between flight-lines. For parametric soil survey, the ability to spatially extrapolate soil properties would be improved

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while for field survey design by classification of radiometric data, the number of pixels incorrectly assigned to a given class would be reduced.

The ABF methodology improves the quality of gamma-ray images, although the level of improvement in both noise reduction and spatial resolution is unclear. One possible method to quantify the improvement would be to map potassium, uranium and thorium distribution in detail at ground level and then compare the results of a standard and ABF approach to deconvolution. Indeed this approach was attempted during this research, by mounting a large crystal on a quad-bike and surveying a number of paddocks in the Jemalong area. However, there were problems with the detection system that prevented the survey from being used for its intended purpose. At present, we have to be content with the more rigorous basis of the ABF method, the greater control on the interpolation between flight-lines, the method’s ability to remove noise during the interpolation process and the capacity to exactly deconvolve an ABF surface.

3.2 Practical implementation

In this thesis, the ABF approach has been successfully applied to the deconvolution of a small subsection of the Jemalong radiometric survey. Applying the technique to larger surveys would require implementation of the improved computational methods discussed in sections 2.3 and 2.4 above. This would be the first logical extension to this work.

A further consideration for practical deconvolution is the estimation of the signal and noise auto-correlations. When applied to a new survey, the signal auto-correlation could be estimated by a power law fit to the radially averaged power spectrum (as illustrated in Chapter 4). Assuming white noise, the noise auto-correlation could be estimated by linear error propagation. While the results of the deconvolution are not overly sensitive to the noise estimation, the extremely large reductions in assay uncertainty contributed by the NASVD by cluster technique may be quite significant. Application of this multichannel method to other radiometric surveys would be required to assess the level of error reduction on a case by case basis.

The GCV thin-plate spline fit to radiometric data results in a considerable reduction in the contribution of noise to the interpolated surface. However, the Wiener filter derived in this thesis was developed based on noisy observations. This consideration may indicate that some means of tuning the signal-to noise ratio will be required for a practical deconvolution method. This approach may be as simple as the inclusion of a multiplicative scaling constant, as already implemented in the Craig et al. (1998) method.

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3.3 Application of the ABF methodology to other types of data

The ABF methodology developed in this thesis could be used to interpolate, Fourier transform and convolve/deconvolve any type of data, either regularly or irregularly sampled.

With good

preconditioners, and fast methods for matrix-vector products and discrete Fourier transforms, the ABF methodology would make an ideal basis for a software package for surface fitting and manipulation. Alternatively, it could be used to supplement the interpolation methods and Fourier processing available in existing commercial software packages.

From a software-engineering

viewpoint, the ABF methodology is ideally suited to an object-oriented approach that typifies best practice because many types of interpolation method can be implemented from within a small core of routines. The goal from an ease-of-maintenance viewpoint is to distil the essence of the algorithm from each method and cast the data handling and hierarchy of approaches into a natural order that shares as much commonality as possible.

Within geophysics the next logical extension of the method would be to magnetic surveys. There, efficient computational methods would be essential, as magnetic data are usually collected at ten times the along-line sampling rate of gamma-ray data. Many operations on magnetic data are accomplished by Fourier methods including, first and second vertical and horizontal derivatives, upward and downward continuation, reduction to the pole and depth to basement calculations. The thin-plate spline would be ideal for magnetic data, as the spline smoothness penalty arises naturally for potential fields as they satisfy Laplace’s equation (e.g. Blakely, 1995). Alternatively, the equivalent source technique of Cordell (1992) could be used for interpolation as it is also encompassed within the ABF framework.

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APPENDIX A: EFFECT OF DEPTH VARIATIONS IN CONCENTRATION AND ATTENUATION A1 Does soil density affect the radiometric signal?

Soil density affects the depth of material that contributes to the signal recorded in airborne radiometrics. Less dense soils have a greater depth of penetration than denser soils. It is often assumed that such density differences can be directly measured using radiometrics due to their effect on the attenuation coefficient. However, this is not the case as shown in the following analysis.

Assume that there is a constant isotope concentration per unit mass of n and that this quantity of isotope emits k gamma rays per second. According to Kogan et al. (1969) the gamma ray attenuation for dry soil material depends only on the density and not on the mineralogical composition. This implies that the attenuation per unit mass, c is also constant. With an arbitrary depth distribution of density, q(r e ), the isotope concentration per unit volume is np(r e ) and the linear attenuation coefficient is l e = cp(r e ) LetG(r e ) = ° q(t)dt, 0 [ t [ r e ; then ¹G/¹r e = q(r e ), and from Equation (2), Chapter 3, the emission from a vertical rod is given by cos h nk ° ¹r e ¹G/¹r e exp(−cG(r e ) ) = −n(k/c ) cos h ° ¹r e ¹[exp(−cG )]/¹r e

(A1)

with the integration limits assumed to extend from 0 to º. The integral is now elementary and equal to unity giving an emission of n(k/c ) cos h

(A2)

which is independent of density. Equation (A2) shows that density variations are not detectable in gamma ray spectrometry.

A2 Effect of depth variations in attenuation and isotope concentration

The two-dimensional approximation to the full, three- dimensional gamma ray transport problem derived at Equation (5), Chapter 3, effectively assumes that there are no depth variations in concentration or attenuation in the top 50 cm or so of soil. In this appendix, I present the point spread functions that would arise under different assumptions about the depth distributions.

237

In non-dimensional co-ordinates assume the detector is located at (0, 0, 1 ). The number of gamma rays, arising from material directly below a point (X, Y ), that are emitted from the surface in the direction of the detector are given by (Equation 2, Chapter 3):

° ¹Z n(Z ) exp − ° l e (t )¹t

, 0 [ Èt È [ Z/Å(X 2 + Y 2 + 1 ), º < Z < 0

(A3)

The t integral is along the straight line connecting the source to the detector and is truncated where this line intersects the Earth’s surface (Figure 1, Chapter 3).

Evaluating this integral for the

following four depth distributions, substituting back into Equation (A3) and combining with Equation (5), Chapter 3, I find the following point spread functions: Ÿ

A uniform model. With R 2 = X 2 + Y 2 + 1and w = arctan(Y/X ), the point spread function is then: P U (R, w) = D e (R, w ) exp(−G a R)/4oG e R 3

Ÿ

(A4)

A two- layer model with the second layer at a depth of b (in units of 1/l e ) and with a ratio of concentration to attenuation of a times the ratio in the upper layer. The two- layer profile can be used as a simple model of either moisture or isotope concentration variations within the soil. The PSF is: P S (R, w) = [1 + (a − 1 ) exp(−bR )]D e (R, w ) exp(−G a R)/4oG e R 3

Ÿ

(A5)

A linear change in isotope concentration per unit mass given by f(z) = (1 + jl e z), where j is the magnitude of the gradient change over a depth of 1/l e . The PSF is then P L (R, w) = [1 + j/R ]D e (R, w ) exp(−G a R)/4oG e R 3

Ÿ

(A6)

An exponential change in isotope concentration, N(Z) = N o 1 − f exp(−Z k/D ) , as Lovborg (1984) found to exist with radon. The factors in the equation above terms in the above equation are the radon emanation coefficient, f (amount of radon that escapes into the soil pores), the diffusion coefficient of radon in the soil, D, and the decay constant of radon, k = 2.1 % 10 −6 . The PSF is then: P E (R, w) = 1 + f R G e / R G e + k/D

D e (R, w ) exp(−G a R)/4oG e R 3

(A7)

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APPENDIX B: CALCULATION OF DETECTOR EFFICIENCY In this appendix I describe how to calculate the average total count efficiency (Equations 16 and 17 in Chapter 3) for a photon incident on a rectangular detector with dimension (a, b, c ). The photons are assumed to be incident at an angle, h, to the vertical unit vector (0, 0, 1 ) and with an angle w, to the horizontal unit vector (1, 0, 0 ) (Figure 5, Chapter 3). To simplify the computation the coordinate system is always rotated to ensure that both h and w lie between 0 and 90 o . These values of (h, w ) are related to the non-dimensional Cartesian co-ordinates, (X, Y, 1 ) and R 2 = X 2 + Y 2 + 1 by, X/R = sin h cos w, Y/R = sin h sin w, 1/R = cos h

(B1)

with each component greater than or equal to zero. Depending on the values of (h, w ) either one, two or three detector sides will have photons passing through. The efficiency must be calculated for each of the visible sides. Here I will give a general formula that can be used for each side by appropriately interchanging (X, Y, 1 ) and (a, b, c ).

The average efficiency for the base side is calculated using Equation (16), Chapter 3, −1 e Txy = 1 − (ab ) ° ° exp[−l d & r d (u  , v  , X, Y )] ¹u  ¹v  , 0 [ u  [ a, 0 [ v  [ b

(B2)

where r d (u  , v  , X, Y ) gives the path length of an incident photon, that hits the detector at position (u  , v  ). The easiest way to evaluate the integral in Equation (B2) is to split the region of integration

into the five parts shown in Figure 5, Chapter 3, and evaluate each component separately. If the contribution from region n is j n , then the average efficiency of the base side is given by −1 e Txy = 1 − (ab ) [j 1 + j 2 + j 3 + j 4 + j 5 ].

(B3)

The region boundaries can be calculated using simple geometric arguments and are located at u = min(cX, a ), v = min(cY, b ).

(B4)

239

The value of u and v are limited to maximums of a and b, respectively, otherwise the region of integration would extend outside the crystal. Within region 1 all photons have a common path length of cR, and integrating from u to a and v to b I find j 1 = (a − u )(b − v ) exp[−l d cR ].

(B5)

To calculate the contribution from region 2 note that the integral from 0 to v for all u  between u and a is given by g(v ) = ° ¹v  exp(−l d v  R/Y ) = (Y/Rl d )[1 − exp(−l d vR/Y )].

(B6)

Integrating this constant expression for u  between u and a I find, j 2 = ° ¹u  g(v ) = (u − a )(Y/Rl d )[1 − exp(−l d vR/Y )].

(B7)

If cY < b, then I can replace the exponential term in Equation (B7) by exp[−l d cR ], which is the same exponent that occurs in the Equation (B5) for region 1. If cY m b, then the exponential term becomes exp(−l d bR/Y ).

Region 3 is analogous to region 2, except the roles of u and v are interchanged and X and b is used in place of Y and a. I find, j 3 = (v − b )(X/Rl d )[1 − exp(−luR/X )].

(B8)

Additionally if cX < a the exponential term is exp(−l d cR ) , otherwise it is exp(−l d aR/X ). For region 4 I proceed in the same manner as Equations (B6) and (B7), except this time the factor v  in g(v  ) depends on u  and is given by v  = u  Y/X. Substituting into Equation (B7) and integrating between 0 and u I find j 4 = (Y/Rl d ) ° ¹u  [1 − exp(−l d u  R/X )] = (Y/Rl d ) u − (X/Rl d )[1 − exp(−l d uR/X )] .

(B9)

Similar analysis for region 5 with the roles of u and v and of X and Y interchanged gives j 5 = (X/Rl d ) u − (Y/Rl d )[1 − exp(−l d Y/R )] .

(B10)

240

Equations (B2) to (B10) enable the calculation of the average efficiency for the base side (i.e. the side perpendicular to the vector (0, 0, 1 ). To calculate the average efficiency for the vertical sides of the detector interchange the roles of the unit vector and detector dimensions. Specifically, for the side perpendicular to (1, 0, 0 ) interchange 1/X for X and c for a, while for the side perpendicular to (0, 1, 0 ) interchange 1/Y for Y and c for b.

To calculate the average efficiency for the entire detector I weight the average efficiencies of each side with the solid angle of that side to find, e T = (e Txy 1/c + e Txz Y/b + e Tyz X/a )/(1/c + Y/a + X/a ).

(B11)

To better illustrate the form that the efficiency equations take, consider how the efficiency varies along the line Y = 0. Then, for the base and one vertical side, the contributions from j 2 , j 4 and j 5 are all zero. For the other vertical side all five components are zero. Assuming cX < a, j 1 = 0 for the remaining vertical side and inserting Equations (B5) and (B8) into Equation (B2) and then into (B11) I find e T = 1 − 2X/R + [l d (a − cX ) − 2X/R ] exp(−l d cR ) /[l d (cX + a )]

(B12)

With cX m a the contribution from j 1 is zero for the base side and I find e T = 1 − 2X/R + [l d (cX − a ) − 2X/R ] exp(−l d aR/X ) /[l d (cX + a )]

(B13)

241

APPENDIX C: CALCULATION OF LIVE-TIME The calculation of the live-time requires an estimate of the total number of gamma rays of all energies incident on a detector. One way to achieve this goal is to construct a series of component spectra, and obtain an estimated spectrum by a linear summation. This requires:

1. Pure potassium, uranium and thorium spectra.

These were obtained by Minty (1996) by

simulating the effects of altitude by varying thickness of wood. A principal component analysis was then conducted on the data, with only two components for each element being required to fit the variation in shape with height. These components can be used to construct simulated spectra for pure K, U and Th sources of arbitrary concentration, with a linear sum giving the composite spectra.

2. An aircraft background spectrum obtained during calibration. 3. A cosmic spectrum normalised to 1 cps in the cosmic channel. The spectrum for any cosmic count rate can then be determined by multiplication. I assume throughout this thesis, a typical cosmic count of 120 cps.

4. A radon spectrum obtained by calibration and normalised to 1 cps of radon in the uranium channel. The radon spectrum for an arbitrary level of radon can be obtained by multiplication.

Once these spectra have been obtained a good estimate of a real spectrum can be obtained within the energy range 0.41-2.81 MeV which corresponds to the total count window in four channel processing (e.g. IAEA, 1991). For example, Figure C1 plots the observed and modelled spectra at 60 m over a calibration range at Albury, Australia, which has 1.36 %K, 2.6 ppm eU and 15.9 ppm eTh and 16 cps of radon in the uranium window. The two spectra show excellent agreement down to 0.41 MeV. Below this level the theoretical spectrum tapers off, while the real spectrum shows a marked increase in count rate. This is due to a buildup in the sky-shine component, which are high energy gamma rays that have been Compton scattered through large angles. The sky-shine component is absent from the modelled spectra as they were obtained at ground level, with the effect of height simulated by wood.

To calibrate the dead time estimation I used calibration data collected by AGSO at 30 m increments from 60 m to 240 m over the Albury calibration range. For each height, I first obtained a composite potassium, uranium, thorium, cosmic and aircraft background spectrum. I then calculated a best fitting theoretical spectra within the range 0.41 to 2.81 MeV by allowing a linear gain in the composite spectra and an arbitrary radon concentration. The calculated radon concentrations were consistently around 16.5 cps with a range of 15.8-17.8 cps and achieved an excellent fit between

242

actual and modelled spectra (Figure C1). I then calculated the total count, T, within this range and regressed a straight line curve to the observed dead time (Figure 1 in Chapter 4). The calculated and observed dead times (in milliseconds) show excellent agreement with the model t d = 29.9 + 0.024351 & T

(C1)

Notice that the GR-820’s constant dead time of 10 ms has been replaced by 29.9 ms in order to compensate for the unmodelled sky shine component.

The estimation of the dead time for an arbitrary height, K, U, Th, cosmic and radon concentrations is straightforward to obtain. We only need to store the total count for a series of heights for unit concentrations of each component. The total count is then simply obtained by multiplying each component by it’s concentration and then summing the results.

80 Observed spectrum Modelled (16 γ/s radon)

70

Counts per second

60 50 40 30 20 10 0 0.4

0.8

1.2

1.6

2.0

2.4

2.8

Energy (MeV)

Figure C1: Observed and modelled spectra at 60 m elevation over the Albury calibration range.

243

APPENDIX D: CALCULATION OF THE ERROR IN THE AIRCRAFT AND COSMIC BACKGROUNDS The first processing step after energy calibration is usually the removal of aircraft and cosmic background, which requires a good aircraft background spectrum and a cosmic spectrum normalised to a certain number of counts in the cosmic channel. These are obtained by suitable calibration flights as described in, for example, Grasty and Minty (1995). Briefly, the aircraft background is constant while the count rates due to cosmic radiation increase exponentially with height above mean sea level in all spectral windows. Therefore, by flying a series of flights (minimum of two) at different altitudes over the sea (so there is no terrestrial radiation) and with an onshore breeze so there is little radon, the two components can be separated. The requirement for minimal radon is not essential as it will result in contamination in the shape of a radon spectrum, that can easily be removed as part of the radon background (Grasty and Minty, 1995).

The background calibration for the AGSO aircraft was conducted at 500 m increments from 1,000 m to 4,500 m over the sea with an onshore breeze blowing. Denote the count rate in channel i, at height h, by N i (h ) and the count rate in the cosmic channel by C(h ). Then the aircraft background, a i and

cosmic count, c i , in channel i per unit count in the cosmic channel are related by the equation N i (h ) = a i + c i C(h )

(D1)

If there are L heights, h l , the cosmic and aircraft backgrounds are obtained as the least squares solution of L

min: S l=1

(N i (h l ) − a i + c i C(h l )) 2 r 2 (N i (h l ))

(D2)

where r 2 (N i (h l )) = N i (h l )/t l , t l is the total live-time of each transect and we have assumed that there is negligible error in the cosmic counts. This last assumption is valid as each flight typically has a 10 minute measurement time giving observed cosmic counts in the range of 150,000 to 1,000,000 with a relative error of 0.1-0.2%.

Appropriate minimisation of Equation (D2) gives the least squares

solution, as well as estimates of the uncertainties in the aircraft and cosmic backgrounds, r(a i ) and r(c i ) (e.g. Press et al., 1992). By repeating the least squares procedure for all 256 channels

a spectrum of errors can be obtained (Figure D1). The spectrum of standard deviations show

244

prominent peaks related to potassium, uranium and thorium radiation from the aircraft, as well as a peak at 0.511 MeV due to pair production.

The aircraft and cosmic background uncertainties for the potassium, uranium and thorium windows can be obtained by summing the variances of the channels with each window (Table 3 in Chapter 4). For example, let K specify all the channels within the potassium window, then the variance of the aircraft background in the potassium channel, r 2 (a K ) is given by r 2 (a K ) = S r 2 (a i )

(D3)

icK

Note, that the summation for the extreme left and right channels may include only part of these channels, if the potassium energy window falls somewhere within the channel boundary.

0.06 Aircraft Cosmic (120 γ/s)

0.51 MeV Pair Production

Calculated standard deviation (γ/s)

0.05

0.04

1.46 MeV 40 K 2.62 MeV 208 Tl

0.03 1.76 MeV 214 Bi

0.02

0.01

0.00 0.4

0.8

1.2

1.6

2.0

2.4

2.8

Energy (MeV)

Figure D1: Standard deviations of the error in the calculated aircraft and cosmic backgrounds.

245

APPENDIX E: CALCULATION OF THE RADON BACKGROUND UNCERTAINTY The uncertainty in the radon background was determined by a Monte Carlo procedure developed by Minty (1996). The construction of a modelled spectrum required the same component spectra used for the calculation of live-time and as input, the ground concentrations of potassium, uranium and thorium, the aircraft height and the radon concentration.

Let the number of counts in the modelled spectrum in channel i be M i . Once the modelled spectrum has been generated the following process is repeated many times:

1. For each channel i, generate a random number, e i , from a Poisson process with mean M i . This then becomes the count rate of the noisy spectrum in channel i.

2. Remove the aircraft and cosmic background from the noisy spectra. 3. Fit a straight line (or exponential) curve to the Compton continuum and then calculate the counts in the low energy 214Bi window, L ob .

4. Calculate the counts in the potassium, K, uranium, U ob , and thorium, Th, windows. 5. Use Equation (6), Chapter 2, to calculate the radon counts in the U channel. I repeated the above process 3,000 times and then calculated the standard deviation of the radon correction in the U channel. The expected radon correction errors in the potassium and thorium windows were then determined by linearly scaling this standard deviation by 0.79 and 0.1, respectively. These are the ratio of the counts in the K and Th channels, to the counts in the U channel for a pure radon spectrum.

Note, that calculating the radon error in this way, effectively assumes that the radon calibration constants and the stripping ratios have negligible error. Obtaining meaningful estimates of these constants would be extremely difficult as the radon calibration process is complicated. Note further, that the Monte Carlo procedure could, in principal, be used to obtain estimates of the errors in the whole process. However, we have avoided adopting this approach as it tends to obscure the intuitive understanding that can be obtained using the more direct approach based on explicit error propagation.

I applied the Monte Carlo procedure to a wide range of potassium (0.5-4% K), uranium (0.5-4 ppm eU) and thorium (2-16 ppm eTh) concentrations, heights of 30, 60, 90, 120 and 150 m, and radon

246

counts of 10, 25 and 40 cps in the uranium window. In the majority of cases the radon standard deviation ranged from 1.5-3.5 cps with a mean value of 2.35 cps. The lowest standard deviation was 1.03 cps and corresponded to the case of lowest possible count rates; i.e. 0.5 %K, 0.5 ppm eU, 2 ppm eTh, a height of 150 m and a radon concentration of 10 cps in the uranium window. The highest absolute errors were found for large potassium concentrations and low radon conditions, which appeared to be counter to our intuition. The maximum standard deviation observed was 9.6 cps at 30 m elevation with 4 %K, 0.5 ppm eU, 2 ppm eTh and 10 cps of radon.

To clarify the cause of these large errors, Figure E1 shows the theoretical spectra obtained from two radon concentrations of 10 and 40 cps in the U window, at a height of 90 m, with ground concentrations of 4% K, 2 ppm eU and 8 ppm eTh (typical earth model with double the potassium concentration). For the low radon case the low energy 214Bi peak at 0.609 MeV only slightly exceeds the Compton Continuum, whereas in the high radon case the peak is quite distinct. When errors are added to the theoretical spectrum, uncertainties in the determination of the linear trend of the Compton Continuum for the low radon case have a larger effect than for the high radon case. This is reflected in the much larger standard deviation of 6.0 cps for low radon, compared to 2.2 cps for high radon for this example.

To avoid having to do a Monte Carlo simulation every time I want to calculate the radon error I formed a lookup table of radon standard deviations. The determination of the radon error for arbitrary conditions is obtained by linear extrapolation between table entries 70 Low radon High radon 60

0.609 MeV 214 Bi 1.46 MeV 40 K

Counts per second

50

40

30

2.62 MeV 208 Tl

20 1.76 MeV 214 Bi

10

0 0.4

0.8

1.2

1.6

2.0

2.4

2.8

Energy (MeV)

Figure E1: Theoretical noise free spectra for low (10 γ/s in U channel) and high (40 γ/s in U Channel) radon conditions at an altitude of 90 m for a high potassium concentration of 4% with 2 ppm eU and 8 ppm eTh.

247

APPENDIX F: EFFECT OF HEIGHT CORRECTION ON ASSAY PRECISION Equations (9) and (10) in Chapter 4, only give the error after height correction by a specified amount. In some cases we are interested to know what impact the height correction has over the survey as a whole. This requires some estimate of the range of heights that the pilot flies over the survey area, something almost impossible to predict before the survey.

To illustrate the impact of height

correction I conducted a Monte Carlo error analysis of the height correction for a number of simulated surveys at a nominal altitude, h 0 , of 100 m. These were constructed by drawing the fluctuations in nominal altitude from a Gaussian distribution with a specified standard deviation, r(h f ). While the Gaussian distribution may not be completely realistic (real surveys probably have

bigger tails and may be skewed to positive heights), it can be used to get some indication of the impact of height correction. Uncertainty in effective height was also accommodated by drawing errors from a second Gaussian distribution with standard deviation r(Dh e ). The survey was assumed to have been flown over uniformly radioactive ground with standard concentrations, f j , of potassium, uranium and thorium (2% K, 2 ppm eU and 8 ppm eTh) under conditions of both low (10 cps in U window) and high radon (40 cps in U window).

The Monte Carlo procedure involved repeating the following process 20,000 times:

1. Draw a current deviation, h f , and error in effective height, Dh e , from the appropriate Gaussian distributions;

2. Estimate the observed count rate at the an altitude of h f ; i.e. m j = s j (h f ) f j + b j , where s j (h f ) is the sensitivity at the height h f ; mj

3. Add Poisson noise, Dm j , with variance t (t is the live-time, Equation 6, Chapter 4) to the measured counts;

4. Obtain the standard deviation, r(r j ), of the radon background estimation error at height h f , and use it to generate a Gaussian perturbation to the background, Dr j ;

5. Background correct and the strip the measured counts, i.e. n = A(m + Dm − b − Dr ), where A is a matrix with entries given by Equation (11), Chapter 2.

6. Height correct the stripped count rates, n j (h 0 ) = n j (h f ) exp(l j Dh ) where Dh = h f − h o + Dh e and l j are the relevant height attenuation coefficients.

The relative error was then obtained by calculating the standard deviation in the stripped count rates and dividing by the mean value. Note, that when fluctuations in nominal altitude and error in

248

effective height are both assumed zero, the Monte Carlo procedure is equivalent (except it ignores the small errors in cosmic and aircraft background) to the linear propagation analysis developed in sections 2.2 and 2.3 of Chapter 4. Under the above conditions, and low radon, the Monte Carlo analysis gave relative errors of 9.7%, 58.4% and 18.4% for K, U and Th, compared to 9.7%, 57.9% and 18.6% for the linear error propagation analysis. Thus, both procedures give very similar results. Figure F1 shows the results of the Monte Carlo analysis for fluctuations in nominal altitude of r(h f ) = 0, 10, 20 and 30 m, and errors in effective height of r(Dh e ) = 0, 2.5 and 5 m. For potassium (Figure F1a), increases in both r(Dh e ) and r(h f ) cause increases in the relative error. However, both effects are quite small with the largest fluctuation, r(h f ) = 30 m, causing less than a 0.5% increase in relative error. Errors in effective height are usually of the order of 2%, so the small differences in relative error (< 0.2%) caused by r(Dh e ) = 2.5 m, show that they contribute little to the potassium relative error. Much larger errors in effective height (r(Dh e ) = 5 m) are required to cause a significant effect (1% increase in relative error).

For uranium, the deviations from the nominal altitude has a much larger effect on the relative error (Figure F1b), even when the height correction is precise. There are two reasons for this effect. The first is the larger value of the height attenuation coefficient for uranium over that of potassium or thorium (Table 2). This means that for a given height deviation, the correction for uranium is, relatively, the largest (Figure F2). The second cause is from errors in the calculation of the radon background which is approximately constant with height. Any uncorrected radon background gets amplified by the height correction process (see Minty, 1996). This effect can be seen in the larger influence of increasing r(h f ) on the relative error for high radon (7 % increase from r(h f ) = 0 to 30 m) over the low radon case (< 3% increase). Error in the calculation of effective height also has an impact but it is generally quite small (< 1%). For thorium, both r(Dh e ) and r(h f ) have little impact on the relative error (Figure F1c). For example, the difference in relative error between a survey flown at constant altitude, with zero error in effective height, compared to r(Dh e ) = 5 m and r(h f ) = 30 m, is only just over 0.5%. This small variation is due to the low background levels in the thorium window and the small value of the height attenuation coefficient (Table 2 in Chapter 4 and Figure F2).

To summarise, except for large deviations in altitude, which may occur when surveying over rugged topography, the height correction will have little impact on the relative error. Its effect will be most pronounced for uranium under conditions of high radon background.

249 0.120

0.85

(a)

0 m error 2.5 m error 5 m error 0 m error 2.5 m error 5 m error

(b) 0.80

0.115

Assay precision

0.105

Low radon

High radon

0.70

0.65 0.100 0.60

0.095

0.55 0

10

20

30

Variation about nominal altitude (m) 0.200

0

10

20

Variation about nominal altitude (m)

(c)

Assay precision

0.195

0.190

0.185

0.180 0

10

20

30

Variation about nominal altitude (m)

Figure F1: Effect of deviations from nominal altitude for potassium (a), uranium (b) and thorium (c). 40 Potassium Uranium Thorium

30

20 Height correction (%)

Assay precision

0.75 0.110

{ {

10

0

-10

-20

-30 -30

-20

-10

0

10

20

30

Deviation from survey altitude (m)

Figure F2: Size of height correction for potassium, uranium and thorium

30

250

APPENDIX G: THE LANCZOS METHOD In this appendix I show how the Lanczos procedure can be used to efficiently solve the matrix equations that arise in exact interpolation, smoothing by generalised cross validation and smooth fitting by least squares.

Given a starting vector, r, the Lanczos procedure applied to the matrix, C, can be written as (e.g. Golub and Van Loan, 1996), b o = Èr È; r = r/b o ; u = 0; k = 0 while b k ! 0 if k ! 0 then u = −b k r; r = v k /b k v k+1 = u + Cr k = k + 1; a k = r T v k ; v k = v k − a k r; b k = Èv k È end

The Lanczos procedure generates an orthonormal sequence of vectors, v k , such that the matrix V k = (v 1 , ..., v k ) satisfies V Tk CV k = T k , where T k is tridiagonal with diagonal entries (a 1 , ..., a k ), and

with entries (b 1 , ..., b k−1 ), above and below the main diagonal. Note, that one generally does not want to keep all the v k as only the most recent iterate is required to update the tridiagonalisation. Further, the matrix only interacts with the procedure through its action on vectors, which means fast methods for the matrix-vector product can be used in place of direct evaluation.

Numerical round off

generally prevents an exact b k = 0, and usually some other termination criteria is employed. When used as an iterative method for solving a matrix equation the termination criteria usually relates to some measure of accuracy of the iterated solution.

G1 The conjugate gradients algorithm

Equation (4) of Chapter 5 for interpolation by ABF, must first be converted into a positive definite form by QR- factorisation before the conjugate gradients algorithm can be applied. If Q 2 is the relevant part of the QR-factoristion, Equation (4) reduces to the positive definite form Cl = z, where C = Q T2 AQ 2 , z = Q 2 f and k = Q T2 l..

251

Although most derivations of the conjugate gradients method are based on geometrical arguments it is possible to derive it from within the Lanczos framework. Conjugate gradients solves the matrix system Cl = z by iteratively minimising the functional h(k ) = 1/2 l T Cl − l T z

(G1)

Since Öh(l ) = Cl − z it follows that l = C −1 z is the unique minimisor of h. Given an initial guess, l o , and the sequence of Lanczos vectors obtained with a starting vector of r o = z − Cl o , the objective is to chose y k such that l k = l o + V k y k minimises Equation (G1). Substituting this expression into Equation (G1) and differentiating with respect to y k , I find that h is minimised if y k is chosen as the solution of (V Tk CV k )y k = V Tk (z − Cl o ) = V Tk r o

(G2)

The advantages of choosing y k from within the subspace spanned by V k are apparent from this equation. The factor in brackets on the left hand side is just the tridiagonal matrix T k formed during the Lanczos procedure. Further, V k , is an orthonormal matrix whose first column is v 1 = r o /Èr o È, which reduces the right hand side to b 1 e 1 , where e 1 = (1, 0, ...0 ) and b 1 was obtained from the first Lanczos iteration. The next iterate, y k , is then the solution of Tk yk = b1 e1

(G3)

which is a tridiagonal, positive definite matrix system. This can be solved efficiently using the Cholesky factorisation of T k which, for k + 1, can be updated using only the most recent iterates (e.g. Golub and Van Loan, 1996).

G2 Selection of the smoothing parameter by generalised cross validation

Sidje and Williams (1997) have developed a fast Lanczos based algorithm for choosing the smoothing parameter by generalised cross validation. The method can be applied after the usual QR-factorisation of the polynomial matrix has been used to transform the ABF matrix,C = Q T2 AQ 2 −2 −1 2 2 and data z = Q 2 f. Define the scalars h(m ) = z T (C + mI ) z/Èf È and g(m ) = u T (A + mI ) u/Èu È , then the

GCV functional can be approximated by 2 2 2 G(m ) l (N − K )Èf È h(v )/ (Èu È g(v) )

(G4)

252 where u is a vector with random entries drawn with equal probability from −1, 1 and N − K is the number of data points minus the number of polynomial coefficients. The objective of the algorithm is to find the m that minimises Equation (G4). When the Lanczos procedure is applied to C with the starting vector f, h(m ) can be approximated by −2

h k (v ) = e T1 (T k + mI ) e 1

(G5)

where e 1 = (1, 0, ..., 0 ) is a length k unit vector. Using u as the starting point of a second Lanczos procedure applied to C gives an approximation to g(m ), −1

g k (v ) = e T1 (S k + mI ) e 1

(G6)

where S k is the tridiagonal matrix formed after k iterations of this second Lanczos recurrence. The key observation of the Sidje and Williams algorithm is that only the upper left entries of the matrix inverses in both Equations (G5) and (G6) are required. These can be calculated without reference to the Lanczos basis vectors and only involve short term recurrences between entries in the tridiagonal matrices. Further, the Lanczos decomposition is independent of m and hence, once accurate estimates of h and g have been obtained for one m, they can be rapidly calculated for any other m. The recalculation does not generally require additional matrix vector products unless the approximations at the new m are not accurate enough. In that case either one or both Lanczos procedures resume until sufficient accuracy at the new m has been attained. Finally, once the optimal m has been found (usually by a simple 1-D search), the optimal solution k opt , can be obtained without additional matrix-vector products.

G3 Smoothing by least-squares

When there are less defining nodes than observations a least squares problem arises that involves the rectangular, M % N matrix A, the M % K matrix P and the N % K matrix P˜, minÈf − Ak − Pa È subject to: P˜ T k = 0 2

k,a

(G7)

Construct QR-factorisations of P = QR and P˜ = Q˜ R˜ in the usual way. The constraints on the weights imply that k = Q˜ 2 l for some l c ‘ N−K . As the matrix [Q 1 Q 2 ] is orthonormal T

253 Èf − Ak − Pa È 2 = È[Q 1 Q 2 ] T (f − Ak − Pa )È 2

(G8)

The equation can be separated into two components ÈQ T1 f − Q T1 AQ 2 l − Q T1 Pa È 2 + È Q T2 f − Q T2 AQ˜ 2 l − Q T2 Pa È

2

(G9)

By observing that Q T2 P = 0 and letting C = Q T2 AQ 2 and z = Q 2 f, the second term reduces to Èz − Cl È 2

(G10)

which is a least-squares problem involving the rectangular (M − K ) % (N − K ) matrix C. The weights can then be found by k = Q˜ 2 l and the first term in Equation (G9) made zero by setting Ra = Q T1 f − Q T1 Ak = Q T1 (f − Ak )

(G11)

where I have used Q 1 P = R. This is analogous to Equation (8) given previously in Chapter 5 for the polynomial coefficients and can be simply solved as R is an upper-triangular K % K matrix (typically K [ 3).

Observing that C T r ls = 0 for the residual r ls at the least squares solution, shows that solving Equation (G10) is equivalent to solving the augmented matrix system

I C CT 0

r l

=

z 0

(G12)

Paige and Saunders (1982) exploited the structure in this matrix to develop an efficient iterative method which they called LSQR. When the Lanczos algorithm is applied to the above matrix system with a starting vector f, one finds that the I and 0 parts of the matrix are unchanged and after k iterations the A and A T matrices are reduced to R k and R Tk . The matrix R k is (k + 1 ) % k and lower bidiagonal with diagonal elements (a 1 , ..., a k ) and entries below the main diagonal of (b 2 , ..., b k+1 ). The Lanczos procedure applied to the whole matrix can be summarised as follows

254 b 1 = Èz È; u 1 = f/b 1 ; v 1 = C T u 1 ; a 1 = Èv 1 È; v 1 = v 1 /a 1 for k = 1, 2, ... u k+1 = Cv k − a k u k ; b k+1 = Èu k+1 È; u k+1 = u k+1 /b k+1 v k+1 = C T u k+1 − b k+1 v k ; a k+1 = Èv k+1 È; v k+1 = v k+1 /a k end

Notice, one disadvantage of this bidiagonalisation is the need to compute the action of C T on a vector. By defining k k = V k y k ; r k = z − Ck k ; and t k+1 = b 1 e 1 − b k y k , it is possible to show (see Paige and Saunders, 1982) that r k = U k+1 t k+1 , where V k and U k are the matrices defined by the bidiagonalisation procedure above. As U k+1 is orthonormal and the residual, r k , needs to be as small as possible, this suggests that y k is chosen to minimise t k+1 . This leads naturally to the least squares problem minÈb 1 e 1 − R k y k È

(G13)

This is an upper bidiagonal least squares problem which should be compared to the tridiagonal system that is solved in conjugate gradients. The solution of Equation (G13) can be efficiently obtained by using a QR factorisation that only requires the most recent iterates for update. The bidiagonlisation is terminated when k k is deemed an acceptable solution. The LSQR algorithm, as implemented by Paige and Saunders, has a number of flexible termination criteria.

255

APPENDIX H: THE FOURIER TRANSFORM OF AN ABF EXPANSION The summation component of an ABF expansion, M s(x ) = S m=1 k m v(x − x m )

(H1)

where I have dropped the usual tilde over the x m , is a discrete convolution between the weights and the transfer function. In the Fourier domain, the summation reduces to S(u ) = F(u ) S m=1 k m exp(−2oi u $ x m ) M

(H2)

To prove this, take the continuous Fourier transform of equation (H1). Then, S(u ) = °ℜ‘d

M k m v(x − x m ) exp(−2oi u $ x m )¹x S m=1

(H3)

where d is the dimension of interest. Let y m = x − x m then S(u ) = S m=1 k m ° ℜd v(y m ) exp[−2oi u $ (y m + x m )]¹y m M

(H4)

I can now remove the x m component of the exponential outside the integral, at which point the integral becomes independent of m. After rearranging terms I find, S(u ) =

°ℜ

d

v(y ) exp(−2oi u $ y )¹y

M k m exp[−2oi u $ x m ] S m=1

(H5)

The factor in square brackets outside the summation is the Fourier Transform of the basis function from which the desired result follows.

256

APPENDIX I: AN APPROXIMATION TO THE FOURIER TRANSFORM ABOUT THE ORIGIN The Fourier transform of the summation part of an ABF expansion is given in Equation (H2). In this appendix I develop an approximation to S f (u ) when u is small. To simplify the calculation the angular frequency

ω = 2πu will be used.

About zero, a Taylor series approximation to the

exponential function can be constructed,

Σt=0 (−i ) x t /t! exp(−ix ) j S T

t

(I1)

where T is the number of terms required in the approximation. Substituting this expression into Equation (H2) I find π ) Σ m=1 Σ t=0 λk m (−i ) (z $ x m ) /t! F(z/2o Sf(z ) j Φ M

T

t

t

(I2)

In 1-D, (z $ x m ) = (zx m ) and after exchanging the order of the summation and rearranging, Equation t

t

(I2) reduces to T M t π ) Σ t=0 (−i ) z t /t! Σ m=1 λk m x tm F(z/2o Sf (z ) j Φ

(I3)

Taking all terms up to third order I find, S f (z ) j Φ F(z/2oπ)[Σ kλm − izΣ λk m x m − (z 2 /2 ) Σ λm x 2m + i(z 3 /6 ) Σ λ k m x 3m ]

where the summation is inferred to extend from m = 1, ..., M.

(I4)

For cubic spline interpolation,

Φ(z ) i 1/z 4 , which means that the Fourier transform will only be bounded if each of the four

summations above equates to zero.

In two-dimensions, t r t−r (z $ x m ) t = (z x x m + z y y m ) t = Σ r=0 t!/[(t − r )!r! ](z x x m ) (z y y m )

which reduces Equation (I2) to

(I5)

257 ) Σ t=0 (−i ) t Σ r=0 z rx z t−r Sf (z ) j Φ(z/2πo k m x rm y t−r y /[(t − r )!r! ] Σ m=1 λ m T

t

M

Taking all terms up to third order I find, S f (z ) j Φ(z/2π )[Σ λm − iz x Σ λm x m − iz y Σ λm y m − (z 2x /2 ) Σ λm x 2m − z 2y /2 (z x z y ) Σ λm x m y m + i(z 3x /6 ) Σ λm x 3m + i z 3y /6 i(z 2x z y /2 ) Σ λ k m x 2m y m + i z x z 2y /2

Σ λm y 2m −

Σ λm y 3m +

Σ λk m x m y 2m ]

(I7)

where the summations are again inferred to extend from m = 1, ..., M. Near the origin the transform of 4 the thin-plate spline behaves as Φ F(z ) i 1/Èz È = 1/ z 4x + 2z 2x z 2y + z 4y .

Therefore, the Fourier

transform of a thin-plate spline interpolant will only be bounded if all 10 summations above equate to zero.

258

APPENDIX J: CALCULATION OF A FILTER BASIS FOR CUBIC AND THIN-PLATE SPLINES In the exact method for ABF filtering outlined in Chapter 7, a new basis function needs to be calculated by Equation (10), y(x ) = °ℜd [F(u )H(u )] exp(2oi u $ x )du

(J1)

When the original basis function v(x ) is not absolutely integrable Equation (J1) can have a −4 singularity at the origin. For example the thin plate and cubic splines both behave as Èu È at the

origin, and the nature of the singularity is then determined by the behaviour of the filter near the origin. A high pass filter is zero about the origin and hence there is no singularity. In contrast, a low pass filter is unity near the origin and the singularity is then forth order. In this appendix I show that we can calculate the new basis function if we can express the near origin behaviour of the filter as a polynomial.

One method for calculating Equation (J1) is to transform the filter into the real domain and use the convolution equation, y(x ) = °ℜd v(x − y )h(y )dy

(J2)

This may be a good method of solution if the resulting convolution integrals can be calculated analytically. Note also, that if integrals against polynomials can be calculated that a new moment expansion may be obtained from the original (assuming the Beatson and Newsam (1998a) algorithm was being used for the matrix-vector product).

J.1 1-D application using the cubic spline

Assume that in the space domain the filter is real and absolutely integrable and that its near origin behaviour in the Fourier domain can be approximated by a polynomial. As the filter is real, it will be conjugate symmetric, i.e. the imaginary part of the transform is odd and the real part is even. This then implies that near the origin the filter can be expanded by a polynomial which is even in the real component, and odd in the imaginary component,

259 T(u ) = a o + a 2 u 2 + i[a 1 u + a 3 u 3 ]

(J3)

For real symmetric filters only the two real coefficients are required, while for real anti-symmetric filters the two imaginary components are required.

Now, I want to extract this polynomial behaviour from H(u) so that the leading term near zero will be u4 which when multiplied by F(u ) , will be well behaved. Adding and subtracting the polynomial from Equation (J1) I find y(x ) = °ℜ d [H(u ) − T(u )]F(u ) exp(2oi ux )du + °ℜ d T(u )F(u ) exp(2oi ux )du

(J4)

The first integral is the remainder after the polynomial behaviour near the origin has been subtracted and generally needs to be calculated numerically. Note that, the integral is finite at the origin and, as long as H(u) does not grow faster than O(u4), will approach zero as u increases. This allows the integral to be accurately approximated using some type of quadrature scheme based on the fast Fourier transform (e.g. Briggs and Henson, 1995). The second integral consists of a series of polynomial terms whose inverse Fourier transforms can be calculated analytically as I now show. For the cubic spline, F(u ) = c o u −4 , where c o = 3/(4o 4 ), which converts the second integral to

°ℜ

c o a o c o a 2 ic o a 1 ic o a 3 + 2 + + u exp(2oi ux )du u4 u u3

(J5)

The first term is trivially shown to be equal to a o x 3 , while each of the other terms can be found in a table of transforms of generalised functions (e.g. Gelfand and Shilov, 1964; Jones, 1966). However, it is more instructive and quite simple to derive the results using the transform of the cubic spline as a starting point. It is well known (e.g. Bracewell, 1986) that the Fourier transform of the derivative of a function is 2oiu times the Fourier transform of the function. Therefore, differentiating x

3

and

multiplying its transform by 2oiu I find i ¸[x 2 sgn(x )] = 2oiu 3 4 = 3 3 4(ou ) 2(ou )

(J6)

where ¸[$ ] denotes Fourier transformation and sgn(x) is the sign function which is positive one for x m 0 and negative one for x < 0. Repeating the process once leads too

260 i ¸[ x ] = 2oiu = −1 2 2(ou ) 3 2(ou ) 2

(J7)

and then repeating it again implies that

¸[sgn(x )] = 2oiu

−1 = −i 2 ou 2(ou )

(J8)

Combining Equations (J5) to (J8) and substituting into Equation (J4) the new basis function is 3a 3a 3a y(x ) = a o x 3 + 1 x 2 sgn(x ) − 22 x − 33 sgn(x ) + y r (x ) 2o 2o 4o

(J9)

where y r (x ) is the contribution from the remainder integral; i.e. the first term in Equation (A4). Note that [H(u ) − T(u )]u −4 can be calculated directly when u is away from the origin, but a polynomial approximation needs to be used near the origin. Continuing the polynomial approximation given at Equation (J3) up to terms of order u6, the near origin behaviour is given by [H(u ) − T(u )]u −4 = a 4 + a 6 u 2 + ia 5 u

(J10)

J.2 A cubic spline basis without cancelling discontinuities

In this section of the appendix, I consider the problem of calculating

y 2 (x ) =

for an arbitrary scaling constant, d ! 0.

3a 2 4o 4

°ℜ exp −d u2 (

2 2)

u

exp(2oi ux )du

(J11)

The calculation occurs when attempting to eliminate

discontinuous derivatives in components of the new basis function, by approximating the near origin behaviour of a real symmetric filter by a o + a 2 u 2 exp(−d 2 u 2 ). In the Fourier domain the equation is

Y 2 (u ) =

3a 2 exp(−d 2 u 2 ) 4o 4 u2

(J12)

The Fourier transform of the right-hand side is not listed in any of the standard tabulations of Fourier transforms (e.g. Oberhettinger, 1990) so I derive it here. Multiply both sides of (J12) by −4o 2 u 2 ,

−4o 2 u 2 Y 2 (u ) =

−3a 2 exp(−d 2 u 2 ) o2

(J13)

261

The Fourier transform of the left-hand side is

d2 y2 , while the right-hand side is a Gaussian (e.g. dx 2

Bracewell, 1986). Equation (J13) can then be converted to an ordinary differential equation 2 2 d 2 y 2 −3a 2 o = exp − o 2x 2 2 dx o d d

(J14)

A double differentiation reveals that

y 2 (x ) =

2 2 −3a 2 x erf ox + d3/2 exp − o 2x 2o 2 o d d

+ c1 x + co

(J15)

where the constant and linear terms occur because they do not effect the differential equation. However, c o = c 1 = 0, otherwise the Fourier transform of y 2 (x ) would contain delta functions or derivatives of delta functions. Notice, that in the limit as d d 0, y 2 (x ) d c x , where c is the constant outside the square brackets in Equation (J15). That is, we return the solution without the exponential damping.

J.3 2-D application using the thin-plate spline

At first sight, it would appear that a method for 2-D application using the thin-plate spline could be developed by generalising the 1-D method for the cubic spline. However, this turns out to be more difficult than one might expect.

For a radially symmetric transfer function, ÄH(0 ) = 0 and a

near-origin polynomial approximation is T(q ) = H(0 ) + a 2 q 2

(J16)

where q 2 = u 2 + v 2 is a radial frequency co-ordinate. Multiplying T(q ) by the Fourier transform of the thin-plate spline, F(q ) = c o q −4 , where c o = 1 3 , I find, 2o c H(0 ) c a F(q )T(q ) = Y 1 (q ) + Y 2 (q ) = o 4 + o 2 2 q q

(J17)

The inverse Fourier transform of Equation (J17) is H(0 )r 2 log(r ) −

a2 log(r ) o2

(J18)

262

and can be obtained by reference to a table of transforms of generalised functions (e.g. Gelfand and Shilov, 1964; Jones, 1966), or by the 2-D version of the Fourier derivative theorem after direct evaluation of the Laplacian of the thin-plate spline.

The problem with Equation (J18) is the

logarithmic singularity at the origin, which must be cancelled by the remainder component. However, the remainder component is calculated numerically, making this cancellation unfeasible.

The singularity occurs because the second term in Equation (J17) has algebraic decay at infinity. To avoid the singularity the a 2 q 2 term needs to be damped, as was done in the one-dimensional case (Appendix J2), by an exponential. The exponential damping converts Equation (J17) into Y 1 (q ) + Y 2 (q ) = c 0 H(0 )q −4 + c o a 2 q −2 exp(−a 2 q 2 )

(J19)

The first term returns a scaled version of the thin-plate spline, while the inverse Fourier transform of the second term, Y 2 (q ), is not listed in any of the standard references (e.g. Oberhettinger, 1990), so I derive it here. Subtract the thin-plate spline term and then multiply by −4o 2 q 2 to find

−4o 2 s 2 Y 2 (q ) =

−2a 2 2 2 o exp(−a q )

(J20)

The Fourier transform of the right-hand- side is the Laplacian of the new basis function, Ö 2 y 2 (r ), while the left-hand-side is a scaled version of the exponential (e.g. Bracewell, 1986), 2 2 ¹y 2 −2a 2 Ö 2 y 2 (r ) = 1r ¹ r = exp − o r2 a2 a ¹r ¹r

(J21)

Multiplying both sides of the equation by r and integrating from 0 to r I find

r

¹y 2 −2a 2 = a2 ¹r

° 0r z exp

− o z2 a

2 2

dz

(J22)

The integral is elementary and leads to

r

2 2 ¹y 2 −a 2 = 2 1 − exp − o r2 a o ¹r

(J23)

Dividing by r, integrating once more and using the change of variable z = or/a, I find y 2 (r ) =

−a 2 o2

° 0or/a 1 − expz −z (

2)

dz

(J24)

263

APPENDIX K: CONVOLUTION OF POLYNOMIALS The action of the given filtering operation on the polynomial terms can be derived either in the space domain through the convolution equation or in the frequency domain using generalised transforms. In this appendix I give the frequency domain derivation.

The known supplementary polynomial, p(x ) in the ABF approximation to f(x ) is related to the unknown polynomial, q(x ) in the ABF approximation to g(x ) by the convolution equation, Q(u ) = H(u )P(u )

(K1)

from which it follows that q(x ) = °ℜd H(u )P(u ) exp(2oiu $ x )du

(K2)

By reference to a suitable table of Fourier transforms of generalised functions (e.g. Gelfand and Shilov, 1964), the transform of a polynomial is given by

P(u ) = p

1 ¹ d(u ) = p 1 ¹ , ..., 1 ¹ d(u ) 2oi ¹u 2oi ¹u 1 2oi ¹u d

(K3)

where d(u ) is the Dirac delta function. For example, if the polynomial is linear in two-dimensions, i.e., p(x ) = a o + a 1 x + a 2 y, then

P(u ) = a o d(u ) +

a 1 ¹d(u ) a ¹d(u ) + 2 2oi ¹u 2oi ¹v

(K4)

A property of the Dirac delta function is that for a sufficiently well behaved function, F(u ),

°ℜ

d

¹d(u ) ( ) ¹F(0 ) F u du = ¹u ¹u

(K5)

Substituting this expression into Equations (K2) and (K3), I find

q(x ) = p

1 ¹ [H(u ) exp(2oiu $ x )] 0 2oi ¹u

(K6)

264 where the zero subscript implies that the expression is calculated at u = 0. For the linear 2-D polynomial this implies that

q(x ) = a o H(0 ) +

a 1 ¹H(u ) exp(2oiu $ x ) ¹u 2oi

+ 0

a 2 ¹H(u ) exp(2oiu $ x ) ¹v 2oi

(K7) 0

Each term in this expression can be calculated by using the product rule for differentiation, e.g. 1 ¹ [H(u ) exp(2oiu $ x )] 2oi ¹u

0

=

H u (0 ) + H(0 )x 2oi

(K8)

where H u (0 ) = ¹H(0 )/¹u. In all cases the analysis leads to an upper triangular matrix relationship between coefficients (as long as the relevant derivatives exist and are finite). The linear polynomial in 2-D gives the system

b0 b1 b2

=

H(0 ) 0 0

H u (0 ) H v (0 ) 2oi 2oi H(0 ) 0 0 H(0 )

ao a1 a2

(K9)

Consider a very simple example of a derivative filter along the x-axis. Then H(u ) = 2oiu, which implies H(0 ) = H v (0 ) = 0, while H u (0 ) = 2oi which shows that q(x ) = a 1 . That is, the filter extracted the linear component of the polynomial in the x-direction as expected.

SUPPLEMENT 1 VALIDATION OF SOIL MAPPING WITHIN THE JEMALONG-WYLDES PLAINS The material presented in this Supplement is based on a unpublished report by Billings and Turner (1997) to the Steering Committee of the Jemalong-Wyldes Plains Land and Water Management Plan. Where appropriate, sections of the report have been reproduced, with modifications, in section 5 of Chapter 2.

INTRODUCTION The Jemalong-Wyldes Plains are an area of extensive alluvial deposition within the Lachlan Catchment, NSW, Australia. Irrigation, with water diverted from the Lachlan River, has been conducted in the area for almost 50 years. However, in the early nineties it was discovered that there was a rising ground water mound threatening nearby Lake Cowal with salinisation. The lake is ephemeral, an important bird habitat and covers parts of several dryland farms. In response to this threat it was decided to develop a Land and Water Management Plan for the whole area. Good quality information on soils is central to the success of the plan, as soil characteristics largely determine the amount of water that enters the water table. Existing soils information for the area was limited to a geomorphologic map (Kelley, 1971) and a 1:250,000 soil landscape map under development (King, 1998). The flatness of the terrain and lack of obvious geological patterns made application of existing techniques difficult. It was therefore decided to commission a soil survey (Gourlay et al., 1996) based on interpretation of airborne radiometric data that had recently been flown over the area by AGSO. However, after the survey was completed there were serious questions raised regarding the validity and accuracy of the soil property maps (McGowan, 1996). Partly, this concern was due to the survey being the first detailed application of radiometrics over a large area and partly to questionable methodology. The radiometric measurement detects gamma rays emitted by naturally occurring isotopes in potassium and the decay chains of uranium and thorium. The concentration and distribution of these isotopes within about the surface 45 cm of the regolith determine the intensity of detectable gamma rays. The signal is dominated by the top 5 cm of soil which, depending on the soil density, can contribute over half of the total signal (e.g. Duval et al., 1971). In an aerial survey the gamma rays are recorded by a scintillation detector located in the fuselage of the aircraft. The aircraft height and the spacing between the flight transects determine the spatial and spectral resolution of the derived geochemical maps. The source material and subsequent weathering history of the material comprising the soil determine the isotope concentration. For example, soils derived from granites typically have much higher concentrations of isotopes than those derived from sandstones, and residual soils will generally have different proportions of the isotopes than depositional soils. These are all important determinants of soil properties but there is not necessarily a direct relationship between soils types or properties and the radiometric signal. Gourlay et al. (1996) derived their soil map by first classifying radiometric data acquired at 100 m elevation with a 400 m flight line spacing by the Australian Geological Survey Organisation in 1993. Gourlay et al. then sampled each of their 22 radiometric classes at eight different locations (total 176 sites) and then used simple averaging to determine the soil properties in each class. From this soil map, several maps of soil properties were produced (pH, texture & salinity) by assuming that every occurrence of the class had the mean value of the soil property under consideration. This technique of soil mapping used by Gourlay et al. (1996) assumes that areas with similar radiometric signature have similar properties. However, from Gourlay et al. (1996, p 29) “… a given signal can arise for different reasons. There is rarely a unique relationship between the gamma ray spectral values and soil properties.” They go

2

on to state (Gourlay et al. 1996, p 29) that “This limitation can be largely circumvented if the major blocks of parent material can be defined …”. However, Gourlay et al. make no attempt to explicitly delineate areas with similar parent material. This would be a difficult task in the predominantly alluvial environment of the Jemalong-Wyldes Plains as the parent material may have been derived from several distal sources. These include alluvial deposition of material from the Cowra Formation (Anderson et al., 1993) and aeolian deposition of predominantly sandy material around Lake Cowal (Bowler, 1983). The differences in soil composition could then relate purely to sorting of the one source material or to both sorting and differences in source material. McKenzie & Austin (1993) found that stratigraphic stratification is essential for successful soil mapping in alluvial systems. This requires a good geomorphic model and a preliminary field survey with a different approach to general soil mapping. This difficulty in defining the parent material may seriously degrade the performance of the Gourlay et al. (1996) technique for soil mapping within an alluvial environment. LOCATION AND DESCRIPTION OF STUDY AREA It was decided to test the map in an area of approximately 120 km2 felt to be representative of the soils expected in the irrigated parts of the Jemalong-Wyldes Plains. Its location was motivated by a detailed radiometric survey flown by AGSO over an area covered by an AIRSAR (airborne synthetic aperture radar) image acquired in 1993. The site was some 50 km east of Forbes along the Lachlan Valley Way, near the small settlement of Warroo (Figure 1). Figure 1: Location of the study area within the Jemalong-Wyldes Plains (adapted from Gourlay et al., 1996). The area covered by the entire map is approximately 56 km East-West and 73 km North-South.

3

Regional bedrock geology The Jemalong-Wyldes Plains lies within the Lachlan Fold Belt in the western part of the Bogan Gate Synclinorial Zone (Anderson et al., 1993). The irrigation area is bounded to the West by the Manna and Derbeys Ranges which are composed of rocks deposited during the Upper Ordovician (430 Million years ago) and Lower Devonian (390 Million years ago). These western ranges are subdivided into the lower Ootha group dominantly composed of marine lithologies, including phyllite, schist, micaceous and silty sandstone with some Andesite volcanics; and the Upper Derriwong Beds of boulder conglomerate, sandstone and breccia. To the East of the irrigation area lie the Jemalong and Corradgery Ranges which are composed of younger rocks deposited during the Upper Devonian (354-370 Million years ago). The rocks belong to the Nangar Subgroup which are a subdivision of the Hervey Group. They consist dominantly of the Weddin Sandstone which includes sandstone, conglomerate and siltstone. After deposition the rocks were gently deformed into the very open folds which roughly parallel the ridges. Unconsilidated alluvial fill During the Eocene and middle Miocene period the Lachlan River carved out a deep valley system. It was subsequently infilled by alluvial deposition of material that today is around 100 m thick. Bore hole data summarised in Anderson et al. (1993) reveal that alluvium consists of: the Cowra formation deposited in the Pleistocene (last 1.8 million years) and composed of clay and silt with lenses of sand and gravel. The mode of deposition appears to be by a low energy meandering river system in a semiarid environment. the Lachlan formation deposited between the late Miocene to Pliocene (1.8 to 10.5 million years ago) starting at about 80 m and comprising well sorted sands and gravels interbedded with silt and clay. It is confined to the deeper parts of the channel and its composition indicates deposition by a much larger and higher energy river system than the current Lachlan River. remnants of a third formation deposited during the Miocene (10.5 to 23.5 million years ago) consisting of well sorted quartz gravel terminated by a layer of red clay. Several well defined prior stream formations are present as elevated ridges above the plains and consist mainly as sandy sediments. They are recent remnants of Lachlan River tributaries with uncertain ages but are probably several thousand years old (Anderson et al., 1993). Topography Relief within the Jemalong-Wyldes Plains is minimal with a range of 205 to 225 m above mean sea level, and slopes of less than 30. However, changes in relief of less than a meter are significant in terms of hydrology, soil formation and plant distribution. The low relief prevents accurate elevation information being obtained, and hence limits the use of existing soil mapping methodology based on topographic analysis. Climate The climate of the Jemalong-Wyldes Plains is of arid-Mediterranean type with dry winter and spring months and wetter summer months. The area is characterised by a winter and/or spring drought (Bish et al., 1991), with average annual rainfall between 444 mm (as recorded at Condobolin) and 525 mm (as recorded in Forbes). Heavy isolated summer storms are a feature of the weather pattern and occasionally result in severe flooding. At Forbes the average annual maximum and minimum temperatures are 23.80C and 100C, respectively. Highest temperatures occur in January with a mean min-max range of 17.4-32.70C, while lowest temperatures occur in July with a mean min-max range of 2.8-14.20C.

METHODS The analysis is based on soil samples collected at 61 sites within a 10 x 12 km area within the Warroo area (Figures 1 and 2). This corresponded with two highly detailed radiometric surveys (100 m line spacing 60 m elevation and 200 m line spacing and 90 m elevation) that were flown by the Australian Geological Survey Organisation in December 1995. The area covered by the survey is representative of the soil variation within the wider area of the Jemalong-Wyldes Plains. The sampling strategy was designed by reference to a classification of the potassium,

4

thorium and total-count bands of the 100 m line spacing survey (Figure 6 in Chapter 2). Uranium was not used as there was little coherent signal when it was processed by standard 4-channel methods. Note, that we did not use Gourlay et al.’s (1996) classification to select our field sampling sites. This was due to • the quality of Gourlay et al.’s baseline radiometric data; • the method Gourlay et al. used to derive the classified image; • the patchiness of Gourlay et al.’s classification, as it contains many isolated classes (ie only a few pixels in extent).

Figure 2: Classification of radiometric data, with soil survey sites overlaid.

5

Classification of radiometric data In the corrected four channel data the total count band ranged from 1469 to 2589 γ/s, Potassium from 148 to 326 γ/s and Thorium from 40 to 112 γ/s. If these raw bands were used in the classification the result would be dominated by total count because of its much larger dynamic range. Therefore, before being used in the classification the bands had to be scaled in some way. One technique that has been used for example by Gourlay et al., (1995) is to scale each band to lie between 0 and 255 (a byte image). However, I used an alternative technique which attempts to take account of the relative noise levels in the data, by Z-scoring the data. The number of gamma rays in each energy window recorded by a detector approximately follows a Poisson distribution (e.g. Kogan et al., 1969). This implies that the variance is equal to the mean, and hence if there are N counts that the standard deviation is N . However, this standard deviation only expresses the standard deviation of the observed count rates, and not the final count rates after background and stripping corrections are applied. Lovborg and Mose (1987) found expressions for the standard deviations of the stripped and background corrected count rates for portable spectrometers. One can use the same equations to estimate the average error expected in an airborne survey by using the mean net count rates in each window, the mean background corrections and the stripping coefficients of the spectrometer (a more sophisticated procedure was developed in the thesis; see Chapter 4). Using mean unstripped count rates of 331 γ/s for K, 85.6 γ/s for Th and 2069 γ/s for total count, and the stripping and background constants in Table 2, in Chapter 4, the expected standard deviations for potassium, thorium and total count were 20.2 γ/s , 9.5 γ/s and 45.5 γ/s respectively (for total count I simply used 2069 ). The mean was subtracted from each band and the result scaled by the inverse of the standard deviation to ensure that one units distance in each band had the same level of uncertainty. That is, if the mean stripped count rate is and the standard deviation is then a count rate of N scales to (N − )/ 1. This scaling ensures that each band contributes to the classification in proportion to its relative certainty. The data was classified by a K-means classifier (Tou and Gonzales, 1974) that is implemented in the TNT-MIPS Image Processing and GIS software package (MicroImages Inc., 1995). No spatial considerations (as, for example, advocated in the classification methodology of Harrison and Jupp, 1990) were used in the classification as the field survey was intended to investigate the relationship between radiometric signal and soil properties. Several different target numbers of classes were trialed before it was decided to settle on 15 classes (Figure 2). The final number of classes is always a subjective decision and 15 was chosen as it adequately represented the range of variation in the area (i.e. no class standard deviations were overly large). Note, that at no stage I assume that the classes define a “pseudo soil map” (as assumed by Gourlay et al., 1996), they merely provide a basis for a stratified random sampling strategy. Within each class a number of replicates proportional to the relative area of the class (with a minimum of two for any class) were sampled. The largest number of samples was 15 for class 2, and the smallest was 2 for classes 6 to 15 (see Table 1). The original target number of sites was 60, but we ended up sampling one extra class to make a total of 61. The sample sites within each class were selected randomly within the survey area with three constraints. The first was that there was at least one sample of each class in the area covered by the detailed survey (if a class representative was present). The second was that the cluster (all the pixels connected to the sampled pixel that have the same class number) had to have at least two flight lines crossing it. The third was that the sample had to be at least one flight lines distance away from the boundary. These last two constraints ensured that the cluster was not just an artifact of the interpolation algorithm. In some cases the planned locations were inaccessible. Where possible, the sample was relocated elsewhere within the same cluster, or within the next closest cluster with the same class value. The locations of the sample sites are shown in Figure 2. Soil sampling The location of each site was determined at random within each class although note that there was a slight bias to sampling within the area covered by a highly detailed ground radiometric survey (Eastern edge of study area, Figure 2). We navigated to each site using a differential GPS with a positional accuracy of ± 5 m. At the site an auger hole of 10 cm diameter was excavated to a depth of 80 cm (auguring was only ceased before the required depth when physical barriers were encountered i.e., rock or impenetrable clay). The soil removed was partitioned into horizons

6

and described using McDonald and Isbell (1990). The analyses used soil properties rather than classified soil types (e.g. Stace et al., 1968; Northcote, 1979; Isbell 1996) which can mask the variation between profiles. An 800g soil sample for each horizon was collected in a 100 µm thick resealable polyethylene bag. Three bulk density samples located within 1.5 m of the augur hole were collected for each site. In order to reduce the error caused by variation of soil surface condition, the top 2 cm of surface material was removed before the bulk density sample was taken. The bulk density sample was sealed in an oven-bag and stored in a shaded box to prevent moisture loss before the sample was weighed at the end of each field day.

Table 1: Percentage area occupied by each radiometric class (from classification of the detailed 100 m line spacing survey) and the corresponding number of sites within each class.

Class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

% Area 10.62 20.86 18.09 11.64 6.46 3.57 2.91 4.16 3.30 3.70 2.73 3.78 4.25 2.63 1.31

# Sites 6 13 11 7 4 2 2 2 2 2 2 2 2 2 2

Table 2: Soil texture and it’s equivalent texture class number for the purpose of statistical analysis.

Soil Texture Sand Loamy sand Clayey sand Sandy loam Light sandy clay loam Loam Silt loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Light clay Light medium clay Medium clay Heavy clay

Texture class # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

7

Soil morphological properties (field texture, structure, consistency, fabric, plasticity and gravel content) were determined according to McDonald and Isbell (1990) and bulk density and moisture content calculated as per Rayment and Higginson (1992). The soil textures were assigned a numerical value for the purpose of statistical analysis, with sand = 1 through to a heavy clay = 16 (Table 2). The texture value represents an approximate clay content. The chemical properties (pH and EC1:5) were conducted on 1:5 soil:water extract and measured using Eutech Cybernetics hand-held meters. The pH and EC1:5 measurements were obtained according to Rayment and Higginson (1992). For each of our 61 sites we determined the class number given in the Gourlay et al. (1996) 22 class soil map. We found that we had at least one soil site within classes 1 to 11, class 17 and class 20 (Table 3). Classes 12 to 16, 18 to 19 and 21 to 22 were not sampled. The maximum number of samples of any class was 15 for class 1. The predicted soil properties were then found by reference to the average soil properties within each class. This corresponds to the approach used by Gourlay et al. (1996) to produce maps of pH, salinity and texture.

Table 3: Number of occurrences of our soil sites within each Gourlay et al. (1996) class. Class # # Sites

1 15

2 8

3 3

4 9

5 3

6 1

7 4

8 8

9 5

10 1

11 2

17 1

20 1

For the 61 sites we then conducted a statistical analysis to test the predictions of Gourlay et al. (1996). Two analyses were performed for each soil property under consideration using the S-PLUS package (Statistical Sciences, 1995). The first was a direct regression of the Gourlay et al. predicted property against our measured property. This tests the reliability of the Gourlay et al. prediction. Over the extent of the whole map, the assumption of a one to one relationship between radiometric class and soil properties may result in unreliable predictions. However, within a smaller area the map may have successfully delineated local soil variation. The second analysis was designed to test this possibility. It was an analysis of variance and investigated whether the Gourlay et al. (1996) classes could be used to group our measured soil properties in a consistent manner. Because our survey covers a relatively small proportion of the total area, this would show whether the Gourlay et al. classes are able to delineate local soil variation. At every site visited, Gourlay et al. (1996) took samples from the “A1”, “A2”, “B1” & “B2” horizons. These horizon definitions were chosen to simplify the statistical analysis and do not appear to be based on any accepted definitions. We used the horizon definitions as described by McDonald and Isbell (1990) and found that the number of horizons present varied from site to site. For example site 12 was a uniform profile of medium clay texture from 0 to 80 cm with no horizonation detectable in the field, while at site 34 their were four horizons with clear boundaries. The soil profile was made up of an A1, bleached A2, a yellowish red light medium clay B2 and a brown light medium clay B3. Many of the soil profiles did not contain B1 horizons while others displayed A11, A12 and A3 horizons. Due to the difficulty in determining which of our horizons corresponded with each Gourlay et al. defined horizon we only assess the predicted soil properties for the A1 and B2 horizons. Our horizons had the following characteristics (McDonald & Isbell, 1990): • A horizons: were the darkest in colour, displayed organic accumulation and lower proportions of clay and/or sesquioxides than underlying horizons, • A1 horizon: contained the most organic matter and darkest colour (dark brown to black), • B horizons: have one or more of the following characteristics: concentration of silicate clay, iron, aluminium, organic material; a structure and/or consistence and a higher chroma and/or redder hue then the A horizon above or any horizon immediately below, • B2 horizon: demonstrated illuvial processes (resulting in predominate clay content), contained the maximum colour chroma and exhibited strongest pedological organisation. The Gourlay et al. surface horizon was considered to correspond to our A1 horizon and we assumed the Gourlay et al. B2 was the actual B2 for the soil profile.

8

The EC1:5 value (from the 1:5 soil:water extract) was multiplied by a texture dependent factor (Taylor, 1993). This converts the EC1:5 reading to what would be measured from a saturated soil paste (Slavich and Petterson 1993), with the resultant quantity called ECe. The ECe values were assigned a salinity rating according to Taylor’s (1993) salinity classes. We used a numerical scale with 0 non saline (< 2000 µS/cm), 1 slightly saline (< 5000 µS/cm), 2 moderately saline (< 8000 µS/cm) and 3 very saline (> 8000 µS/cm). The EC1:5 value can be used when conducting salt balance studies but is not a good indicator of salinity effects on plant growth (Slavich and Petterson 1993), due mainly to the fact that EC1:5 measures the salinity of the soil (kg salts/kg soil), while the ECe is a measure of soil water salinity (kg salts/litre of soil water).

RESULTS The measured and predicted texture and pH for the A1 and B2 horizons and salinity in the B2 are shown in table 4. The easting and northing of each site along with the ERIC assigned radiometric class, are also shown. In Figure 3 we plot the measured values of each soil property against the Gourlay et al. (1996) predictions. Note that the lower number of visible points in the texture and salinity plots represent duplicate values. It is quite clear that the correlation between measured and predicted values for all five properties is extremely poor. In addition, for both horizons the Gourlay et al. (1996) predictions for texture and pH have a much lower range (refer to Table 5). For example, Gourlay et al. predicts a pH range of only 6.4 (mildly acidic) to 7.4 (mildly basic) for the A1 horizon, whereas we measured a range of 5.2 (acidic) to 9.3 (highly basic). Gourlay et al.’s (1996) samples were collected predominantly along road verges while our samples were collected within paddocks. Chartres et al. (1990) found that the soil pH of roadsides and paddocks can differ considerably which may partially explain the discrepancy. For the regression analysis up to two outliers were excluded to prevent bias. For example for the texture of the A1, site 5 was excluded as the predicted texture class (= 4, sandy loam) was five texture classes less than the next lowest texture class (=9, clay loam). When we regressed the predicted soil properties against the measured properties, the maximum R2 for any property was less than 0.015: Texture A1: pH A1: Texture B2: pH B2: Salinity B2:

R2 = 0.014 R2 = 0.0001 R2 = 0.0005 R2 = 0.008 R2 = 0.005.

A perfect fit between properties would have R2=1, a very good fit would have say R2> 0.8, and a poor fit would have R2<0.5. Gourlay et al.’s predictions thus have an extremely poor fit to the measured soil properties. For the texture of the A1 and B2 we analysed how often Gourlay et al. (1996) predicted the correct texture class. We found that for the A1 the prediction was correct at 20% (12/61) of sites compared to 38% (23/61) for the B2. The higher success rate for the texture of the B2 probably reflects the fact that 79% of Gourlay et al.’s predictions and 50% of our measurements had a light medium clay texture (texture class = 14). This appears to be the dominant B horizon texture encountered in the Jemalong-Wyldes Plains (King, 1997). The Gourlay et al. radiometric classes cannot be used to extrapolate texture, pH and salinity within the Jemalong-Wyldes Plains. This could be due to many (or all) of the Gourlay et al. soil classes being composed of one or more soil types. Therefore, the average soil properties within a class may not relate very well to the actual soil properties measured in that class. Consider a simple example. From Gourlay et al.’s analysis, class 1 has a B2 texture class of 10 (e.g. silty clay loam). Assume that class 1 is actually composed of two different soils, one with a texture class of 5 (light sandy clay loam) and the other with a texture class of 15 (medium clay). During the sampling, assume that four occurrences of each texture class were sampled. Then, each occurrence of class 1 would be assigned a texture of 10 in the B2 horizon. However, in this simple example all occurrences of class 1 have B2 textures of either 5 or 15, and not 10. Even though Gourlay et al. (1996) maps are unable to reliably predict soil properties over the whole survey area they may possibly be used to discriminate local differences in soils.

9

Table 4: Texture and pH of the A1 and B2 horizons plus salinity in the B2 for the 61 sites. The Gourlay et al. (1996) radiometric class of each site, along with their predicted texture and pH and salinity are also shown. Gourlay et al. (1996) predictions

Measurements Site #

Texture

pH

Texture

pH

Salinity

Class

Texture

pH

Texture

pH

Salinity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

15 12 8 8 9 12 8 8 10 8 8 15 15 13 7 7 7 9 13 12 11 12 8 12 15 10 11 12 13 7 12 12 12 13 12 12 11 9 9 10 12 13 10 12 13 12 12 7 10 7 13 6 6 9 7 6 9 7 7 13 10

8.2 8.5 8.1 5.6 5.6 6.6 5.9 6.9 6.3 5.4 7 6.8 9.3 6.3 6.3 5.4 6.7 7 6.9 7.2 6.9 7.1 5.5 6.3 8.2 6.7 6.8 6.9 6.6 6.7 8.7 7.8 6.2 6.6 5.2 6.3 6.7 7.1 5.8 6.8 6.8 6.7 6.2 7 5.8 5.8 5.6 5.8 6.4 5.6 6.5 6.1 5.4 5.8 6.1 7.1 6.8 5.6 6.7 6.8 6.4

16 9 13 11 3 12 11 11 15 11 11 15 15 15 7 15 14 14 15 15 11 15 15 15 16 11 15 15 14 13 15 15 15 16 13 15 15 15 15 13 15 15 15 15 15 13 12 13 15 5 15 13 13 15 16 13 15 15 12 16 15

9.4 9.3 9.4 7.5 8.7 9.2 8.5 8.5 8.7 8.3 8.9 8.1 8.4 7 8.2 8.4 9.5 9.3 8.8 9.3 7.7 9.3 7.2 9.4 9.2 7.8 9.3 8.5 9 9.6 7.9 8.1 7.8 9.1 7.3 8.4 7.6 7.8 7.6 8.2 8.1 9 8.5 8 7.1 7.4 7.1 7.5 8.2 7.5 7.7 8.6 7.8 7.8 8.9 8.9 8.8 8.7 7.3 8.7 7.9

0 0 3 0 0 0 2 0 1 0 0 0 1 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 3 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 11 3 11 20 1 3 7 5 17 1 1 1 1 4 9 2 1 2 1 10 1 4 1 2 4 4 4 8 9 4 3 8 5 4 1 1 9 4 7 1 1 2 1 8 1 2 2 2 8 5 7 8 9 7 8 8 4 9 8 6

10 12 9 12 4 9 9 13 13 9 9 9 9 9 12 9 10 9 10 9 12 9 12 9 10 12 12 12 12 9 12 9 12 13 12 9 9 9 12 13 9 9 10 9 12 9 10 10 10 12 13 13 12 9 13 12 12 12 9 12 12

6.7 7.4 6.4 7.4 6.5 6.5 6.4 6.8 6.7 6.8 6.5 6.5 6.5 6.5 6.7 6.7 6.7 6.5 6.7 6.5 6.4 6.5 6.7 6.5 6.7 6.7 6.7 6.7 6.4 6.7 6.7 6.4 6.4 6.7 6.7 6.5 6.5 6.7 6.7 6.8 6.5 6.5 6.7 6.5 6.4 6.5 6.7 6.7 6.7 6.4 6.7 6.8 6.4 6.7 6.8 6.4 6.4 6.7 6.7 6.4 6.6

15 15 14 15 12 15 14 15 15 15 15 15 15 15 14 15 15 15 15 15 15 15 14 15 15 14 14 14 15 15 14 14 15 15 14 15 15 15 14 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 14 15 15 15

6.7 8.4 7.4 8.4 6.9 7.5 7.4 7.3 7.1 8.1 7.5 7.5 7.5 7.5 7.2 7 6.7 7.5 6.7 7.5 7.7 7.5 7.2 7.5 6.7 7.2 7.2 7.2 6.9 7 7.2 7.4 6.9 7.1 7.2 7.5 7.5 7 7.2 7.3 7.5 7.5 6.7 7.5 6.9 7.5 6.7 6.7 6.7 6.9 7.1 7.3 6.9 7 7.3 6.9 6.9 7.2 7 6.9 6.9

2 2 2 2 0 1 2 1 0 2 1 1 1 1 1 0 2 1 2 1 0 1 1 1 2 1 1 1 0 0 1 2 0 0 1 1 1 0 1 1 1 1 2 1 0 1 2 2 2 0 0 1 0 0 1 0 0 1 0 0 0

10

Pedicted Texture A1

13 12 11 10 9 8 6

7

8

9

10

11

12

13

14

15

8.5

9

9.5

Measured Texture A1

Predicted texture A1

7.4 7.2 7 6.8 6.6 6.4 5

5.5

6

6.5

7

7.5

8

Measured pH A1

Predicted Texture B2

15

14

13

12 3

4

5

6

7

8

9

10

11

12

13

14

15

16

Measured Texture B2 8.5

Predicted pH B2

8

7.5

7

6.5 7

7.5

8

8.5 Measured pH B2

9

9.5

10

11

Predicted Salinity Class

2

1

0 0

0.5

1

1.5

2

2.5

3

Measured Salinity Class

Figure 2a-e: Measured soil properties versus the Gourlay et al. (1996) predicted soil property for the JemalongWyldes Plains.

Table 5: Measured and predicted range of texture and pH in the A1 and B2 horizons. Property

Measured Minimum Maximum 6 15 5.2 9.3 3 16 7 9.6

Texture A1 pH A1 Texture B2 pH B2

Predicted Minimum 4 6.4 12 6.7

Maximum 13 7.4 15 8.4

To assess this possibility we ran an ANOVA on Gourlay et al.’s radiometric classes (as factors) against our measured soil properties. As our survey area represents a small proportion of the total survey area, this analysis would show whether radiometric classes have relatively consistent soil properties in a local area. We found there was no significant difference between the Gourlay et al. radiometric classes for the pH of the A1 and B2 horizons and the texture and salinity of the B2 horizon. The texture of the A1 was the only soil property with a statistically significant (at the 5% significance level) relationship with Gourlay et al. class (Table 6). However, the variation within each class was relatively large with an adjusted R2 = 0.16. Therefore, even for the texture of the A1 the local soil discrimination of the Gourlay et al. classes is relatively poor.

Table 6: Predicted texture for Gourlay et al.’s (1996) radiometric classes and the measured texture class mean. ERIC’s class # ERIC’s texture A1 Measured texture A1 ERIC’s texture B2 Measured texture B2

1

2

3

4

5

6

7

8

9

10

11

17

20

9

10

9

12

13

12

13

12

9

12

12

9

4

12

11.1

9.3

9.8

12

10

7.8

9.9

7.8

11

10

8

9

15

15

14

14

15

15

15

15

15

15

15

15

12

14.3

14.5

13

13.4

15.3

15

13.3

13.3

14

11

10

11

3

12

DISCUSSION The results demonstrate that the Gourlay et al. soil map of Jemalong-Wyldes Plains is unable to reliably predict texture, pH or salinity. There are several factors that contribute to the poor predictive ability: • Location of sample sites: The majority of soil sites sampled were located on road verges. This enabled rapid sampling of the soils because the sites were readily accessible and there was no need to request permission to sample on private property. However, as Chartres et al. (1990) indicates the roadsides are not necessarily representative of the total soil variation within the survey area. • Measurement of the soil profile: As mentioned in the Methods section, at every site Gourlay et al. identified what they termed the A1, A2, B1 & B2 horizons of the soil. This approach facilitates statistical analysis as there is then an observation of each horizon at every site. However, we found that many of the soil profiles did not contain B1 horizons while others displayed A11, A12 and A3 horizons. The Gourlay et al. site descriptions appear to be oversimplifications of the true relationships found between horizons within the soil. This also implies that in many instances the property listed for a particular horizon may actually represent the property in a different horizon. • Calculation of average soil properties of each classes: Gourlay et al. (1996) used a linear scale to represent both textures and colour, a process that was questioned in the review of the original consultancy report (McGowen, 1996). We gave an illustrative example in the results section of a class that was composed of two distinct soils with texture class 5 and 15 in the B2. Using Gourlay et al.’s linear scale for texture, the average texture assigned to that class would lie somewhere between 5 and 15 (dependent on the number of samples of each soil type). This is clearly an inadequate procedure for calculating the average texture class of a given soil class. • Assumption of a one-to-one relationship between radiometric class and soil properties: The field sampling strategy and the subsequent calculation of class averages assumes that each radiometric class corresponds to a distinct set of soil properties. However, even Gourlay et al. (1996, p 29) concede that the same radiometric signal can arise for a number of different reasons but they make no attempt to overcome that difficulty. A possible solution would be to delineate areas of different parent materials and then assume that within each parent material type the properties were identical. • Use of a 50 m grid spacing: The radiometric map used by Gourlay et al. was derived from a 400 m line spacing geophysical survey. Gourlay et al. gridded these data to an 80 m pixel size, then resampled to create a 50 m grid spacing. Therefore, between each survey line there are 8 values that need to be estimated. Standard practice (e.g. Fitzgerald, 1996) is to produce at most five sample values between each flight line (80 m grid spacing) as it is not possible to constrain a larger number of extra points. Five sample points may even be too many and one should only extrapolate the data to four sample points (100 m) or less. The Gourlay et al. grid may contain significant effects related to the gridding algorithm and the resampling routine that are not well constrained by the data. In addition, the grid contains four times the number of data values (and hence is four times as complicated) compared to a grid with a spacing of 100 m. • The large footprint of a single measurement: A radiometric measurement is not a point measurement and represents a complex average of a relatively large area. At 100 m elevation the footprint that contributes 90% of the signal is approximately a 220 m radius circle centered directly beneath the aircraft (Chapter 3). Therefore, the contributing area of a single measurement is approximately 15 ha, and the resultant radiometric images are very blurred representations of the true signal at ground level. Any soils maps derived from this data will then not be able to resolve coarse paddock scale soil variation (at the 1-5 ha scale) and the location of any soil boundaries will be uncertain. Processing technology to partially remove this blurring is only now becoming available (Craig et al., 1997). Gourlay et al. (1996) had no control over the flying height of the radiometric data as it was a pre-existing survey flown by AGSO. They emphasises (and we concur) that a general requirement for soil mapping at a broad paddock scale using airborne radiometrics would require a lower survey height of the order of 60 m (~7 ha contributing area) to a maximum of around 80 m (~11 ha contributing area).

13

REFERENCES Anderson, J., Gates, G. & Mount, T. J., Hydrology of the Jemalong and Wyldes plains irrigation district: NSW Department of Water Resources (1993). Bish, S. and Gates, G., Groundwater reconnaissance Survey: Forbes-Condobolin-Lake Carelligo: NSW Dept. Water Resour., TS 91.033 (1993). Bowler, J. M., Lunettes as indices of Hydrologic Change: A review of Australian Evidence: Proc. R. Soc. Victoria. Vol. 95, No. 3, 147-68 (1983). Chartres, C. J., Cumming, R. W., Beattie, J. A., Bowman, G. M., & Wood, J. T., Acidification of Soils on a Transect from Plains to Slopes, South-Western New South Wales: Aust. J. Soil Res. 28, 539-48, (1990). Craig, M., Dickson, B. and Rodrigues, S., Correcting gamma ray data for flying height: Submitted to Geophysics (1997). Duval, J. S., Cook, B. and Adams, J. A. S., Circle of investigation of an airborne gamma ray spectrometer: J. Geophys. Res., 76, 8466-8470 (1971). Gourlay, R. C., Sparks, T., Williams, B (ERIC) and Wood, J., Soil Descriptions for the Jemalong/Wyldes Plains Using Airborne Radiometric Data: Report to NSW Department of Agriculture, Orange (1996). Fitzgerald, D., INTREPID image processing and visualisation tools reference manual volume 2: Desmond Fitzgerald & Associates Pty Ltd, Melbourne (1996). Harrison, B. A. and Jupp, D. L. B., Introduction to Image Processing: Part 2 of the MicroBrian Resource Manual: CSIRO Publications, Melbourne (1990). Isbell, R. F., The Australian Soil Classification: CSIRO Publishing, Sydney (1996). King, D. P., Soil landscapes of the Forbes 1:250,000 map sheet: Department of Land & Water Conservation, Sydney. In preparation (1997). Lovborg, L. and Mose, E., Counting statistics in radioelement assaying with a portable spectrometer, Geophys., 52, 555-563 (1987). McDonald, R. C. & Isbell, R. F., Soil Profile, in McDonald, R. C., Isbell, R. F., Speight, J. G., Walker, J. and Hopkins, M. S., Eds. Australian Soil and Land Survey Field Handbook: Inkarta Press, Melbourne (1990). McGowen, I., Comments on Draft Report - “Soil Descriptions for the Jemalong/Wyldes Plains Using Airborne Radiometric Data” by ERIC (1996). McKenzie, N. J. & Austin, M. P., A quantitative Australian approach to medium and small scale surveys based on soil stratigraphy and environmental correlation: Geoderma, 57: 329-55, (1993). MicroImages Inc., TNT-MIPS map and image processing system, Lincoln, NE, USA (1995). Northcote, K. H., A Factual Key for the recognition of Australian Soils: Rellium Technical Pub. NSW (1979). Rayment, G. E. and Higginson, F. R., Australian Laboratory Handbook of Soil and Water Chemical Methods: Inkata Press, Melbourne (1992). Slavich, P. G. and Petterson, G. H., Estimating the Electrical Conductivity of Saturated Paste Extracts from 1:5 Soil:water Suspensions and Texture: Aust. J. Soil Res., 31, 73-81 (1993). Stace, H. C. T., Hubble, G. D., Brewer, R., Northcote, K. H., Sleeman, J. R., Mulcahy, M. J. and Hallsworth, E.G., A Handbook of Australian Soils: Rellium Technical Pub., Glenside, SA (1968). Statistical Sciences., S-PLUS guide to statistical and mathematical analysis, Version 3.3: Seattle, StatSci, a division of MathSoft Inc. (1995). Taylor, S., Dryland Salinity: Introductory Extension Notes: 2ed, NSW Department of Conservation and Land Management (1993). Tou, J.T. and Gonzales, R. C., Pattern recognition principles, Addison-Wesley, London (1974).