ADVANCES IN E X P L O R A T I O N G E O P H Y S I C S 5
ANALYSIS OF G E O P H Y S I C A L P O T E N T I A L FIELDS A Digital Signal Processing Approach
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ADVANCES
IN
EXPLORATION
GEOPHYSICS
5
A N A L Y S I S OF GEOPHYSICAL P O T E N T I A L FIELDS A Digital Signal Processing Approach
P R A B H A K A R S. N A I D U Indian Institute of Science, Bangalore 560012, India AND
M.P. M A T H E W 2 Church Street, Geological Survey of India, Bangalore 560001, India
19981 ELSEVIER
Amsterdam
- Lausanne - New York - Oxford
- Shannon
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- Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerharstraat 25 P.O. Box 521, 1000 AM Amsterdam, The Netherlands
Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.
ISBN: 0-444-8280 I-X 9 1998 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or othewise, without the prior written permission of the publisher, Elsevier Science B.V. Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations .[br readers in the U.S.A. - This publication has been registered with the
Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or othewise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. (~ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands.
Preface Gravity and magnetic surveys are inexpensive and are routinely carried out over vast stretches of land and sea. Modem high technology flying combined with precision navigation and computerized data acquisition make it possible to acquire high quality digital magnetic data over vast tracks of the earth's surface. The availability of large quantity of high quality data affords us an opportunity to apply the tools of digital signal processing, a field which has been extensively used in seismic exploration but has not been used as extensively in gravity and magnetic exploration. The primary purpose of the present monograph is to introduce the subject of digital signal processing (dsp) and its application to the analysis of potential field data. The book covers such topics as digital filtering, parameter estimation, spectrum analysis inverse filtering as applied to potential field data. The style of writing is closer to that of a monograph for self-study and reference. However, the book can be used as a reference book in a course on gravity and magnetic exploration at graduate level. The book is particularly useful in any advanced level course on geophysical data processing. The thought of writing the monograph occurred while colloborating with Dr. D.C. Mishra of the National Geophysical Research Institute, Hyderbad, India. Dr. Mishra pursued practical applications of some of the ideas elaborated in this monograph. We are grateful to him for many lively discussions. We would also like to thank the then director of NGRI, Dr. V. K. Gaur for allowing us to use the library. We would like to thank the officers of the Airborne Mineral Survey and Exploration, in particular M/s A. G. B. Reddy, M. R. Nair and Dr. S. N. Anand, for their support. One of us (PSN) wishes to thank the Alexander von Humboldt Stiftung, Bonn for support in the form of an Apple Notebook computer on which the present monograph was prepared. Prabhakar S. Naidu M. P. Mathew
Bangalore, India January 1998
Dedication This book is dedicated to the memory of Professor P. K. Bhattacharya, formerly of the Indian Institute of Technology, Kharagpur, W. B., India
vii
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Chapter 1. G e o p h y s i c a l Potential Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.
1
1.2.
1.3.
P o t e n t i a l field s u r v e y s for m i n e r a l a n d h y d r o c a r b o n e x p l o r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.
Brief description of G & M surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2.
I n f o r m a t i o n c o n t e n t in potential fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Role o f digital signal p r o c e s s i n g (dsp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1.
D i g i t a l filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.2.
Parameter estimation ...................................................................
9
1.2.3.
I n v e r s e filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.41
Spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.5.
Image processing .......................................................................
11
1.2.6.
Some reservations ......................................................................
A c o m p a r i s o n w i t h s e i s m i c signal p r o c e s s i n g . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 12
1.4.
Prologue ............................................................................................
13
1.5.
Notation ........ ....................................................................................
16
1.5.1.
16
Conventions .............................................................................
References ............................................................
.....................
~. . . . . . . . . . . . . . . . .
Chapter 2. Potential Field Signals and M o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.
2.2.
2.3.
2.4.
17
19
P o t e n t i a l field in source free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.1.
Fourier transform .......................................................................
20
2.1.2.
P o t e n t i a l field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.3. 2.1.4.
P o i s s o n relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 24
2.1.5.
Singularities o f potential field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Potential field in source filled space
26 27
2.2.1.
G r a v i t y potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2.2.
M a g n e t i c potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2D source models
30
2.3.1.
L i n e source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3.2.
Cylinder with polygonal cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.3.
Dyke ......................................................................................
33
2.3.4.
Fault ......................................................................................
34
2.3.5.
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3D s o u r c e m o d e l s 2.4.1. P o t e n t i a l field in f r e q u e n c y d o m a i n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 36 36
2.4.2.
Variable density/magnetization model ..............................................
39
2.4.3.
U n i f o r m vertical p r i s m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
viii
2.5.
Contents
2.4.4.
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5.
Prism with p o l y g o n a l cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44 45
Stochastic models I: r a n d o m interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 5 . 1 . Stochastic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49 49
2.5.2.
R a n d o m interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3. 2.5.4.
Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prism model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56 57
2.5.5. L a y e r e d strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic model II: r a n d o m m e d i u m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58 62
2.6.1. 2.6.2.
Thin layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thick layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62 64
2.6.3. 2.6.4.
H a l f space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Undulating layer with r a n d o m density or magnetization . . . . . . . . . . . . . . . . . . . . . . . .
66 67
2.6.5. Relation between gravity and magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70 72
Chapter 3. Power Spectrum and its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.2.
Spectrum o f r a n d o m fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.2.1. 3.2.2. 3.2.3. 3.2.4.
R a n d o m functions (2D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autocorrelation and cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum and cross-spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial and angular spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76 76 78 79
3.2.5. Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6. Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete potential fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81 82 83
3.3.1. 3.3.2.
Sampling theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Folding o f spectrum and aliasing error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83 83
3.3.3. 3.3.4.
Generalized sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantization errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86 91
2.6.
3.3.
3.4.
3.5.
Estimation o f p o w e r spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Discrete Fourier transform (dft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2, Fast Fourier transform (FFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3. 2D discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. Properties o f dft coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92 92 97 99 101
3.4.5. 3.4.6. 3.4.7. 3.4.8.
Statistical properties o f dft coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation 2D spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation o f coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
05 106 109
3.4.9. Spectral w i n d o w s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depth estimation from radial spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 3.5.1. Single layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Fractal models o f susceptibility variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
111 114 114 116
3.5.3.
M a n y layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
3.5.4. 3.5.5.
Depth variation o f susceptibility/density: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface m o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 124
Contents
3.5.6. 3.5.7. 3.6.
3.7.
ix
Physical significance o f 'spectral' depths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation o f radial spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125 126
3.5.8. Effect o f quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A n g u l a r spectrum .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1. A n g u l a r spectrum o f u n i f o r m l y m a g n e t i z e d layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128 128 129
3.6.2. 3.6.3.
Estimation o f angular spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation o f a fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130 131
3.6.4. Application to real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 136
3.7.1. Stochastic m o d e l for the density and susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2. Isostatic c o m p e n s a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137 142 142
Chapter 4. Digital Filtering of Maps I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145
4.1.
T w o - d i m e n s i o n a l digital filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. L o w p a s s filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145 146
4.1.2. 4.1.3. 4.1.4.
Polygonal support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G i b b ' s oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design o f an finite 2D filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147 149 153
4.1.5.
Polygonal filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.1.6. T r a n s f o r m a t i o n o f 1D filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7. Elliptical pass band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I m p l e m e n t a t i o n o f digital filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Spatial and frequency d o m a i n approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158 162 166 166
4.2.2. 4.2.3.
Fast convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166 168
4.2.4. Additional refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering for signal e n h a n c e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
169 169
4.3.1.
170
L o w p a s s filtering for removal o f regional fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2. Directional filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital filters for analytical operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172 174 175
4.4.2. 4.4.3.
Derivative m a p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total field . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . .
176 181
4.4.4.
Continuation o f field for enhancing deep seated anomalies . . . . . . . . . . . . . . . . . . . . .
181
Reduction to pole and equator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Reduction to pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. L o w latitude effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3. Reduction to equator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183 183 185 186
4.5.4. 4.5.5.
188
Pseudogravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distortion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190 193
Reduction to a plane surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Least squares approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Iterative filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
194
R e m o v a l o f the terrain effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200
4.7.1.
202
Filters to r e m o v e terrain effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
x
Contents
4.7.2. 4.8.
Correlation filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .......................................
203
W i e n e r filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
210
4.8.1.
Basic theory ............................
.................................................
210
4.8.2.
E x t r a c t i o n o f p o t e n t i a l field s i g n a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211
4.8.3.
Signal distortion ........................................................................
213
4.8.4.
W i e n e r filter for r e d u c t i o n - t o - p o l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215
4.8.5.
W i e n e r filter for s e p a r a t i o n o f fields f r o m d i f f e r e n t levels . . . . . . . . . . . . . . . . . . . . . .
216
4.8.6.
M a t c h e d filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
References ...................................................................................................
219
C h a p t e r 5. Digital Filtering of M a p s II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
223
5.1.
223
5.2.
5.3.
5.4.
Inverse filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.
Irregular interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
223
5.1.2.
Density maps ............................................................................
227
5.1.3.
Susceptibility maps .....................................................................
228
5.1.4.
Undulating layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229
L e a s t s q u a r e s i n v e r s i o n (2D d i s t r i b u t i o n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233
5.2.1.
Discrete model ..........................................................................
233
5.2.2.
Least squares solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
5.2.3.
M e a s u r e m e n t error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
239
5.2.4.
Backus-Gilbert inversion .............................................................
239
5.2.5.
Resolution ...............................................................................
L e a s t s q u a r e s i n v e r s i o n (3D d i s t r i b u t i o n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
242 243
5.3.1.
Discrete model (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
243
5.3.2.
C o n s t r a i n t least s q u a r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245
5.3.3.
Linear programming ...................................................................
246
Texture analysis ..................................................................................
248
5.4.1.
Non-linear transformations ............................................................
248
5.4.2.
Textural spectrum ......................................................................
251
5.4.3.
Textural features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
253
References ...................................................................................................
254
C h a p t e r 6. P a r a m e t e r Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257
.1.
6.2.
Maximum likelihood (ML) estimation ........................................................
257
6.1.1.
Basic detection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
6.1.2.
Parameter estimation ...................................................................
260
6.1.3.
Cramer-Rao
261
6.1.4.
Properties of ML estimates ...........................................................
262
6.1.5.
M L estimation and Gaussian noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
262
bound ....................................................................
M L estimation source parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
264
6.2.1.
264
Point mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2.
Point mass - location parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269
6.2.3.
Point mass CR bounds .................................................................
273
6.2.4.
Dipole ....................................................................................
275
6.2,5.
Dipole CR bounds ......................................................................
277
Contents
6.3.
xi
6.2.6.
Vertical prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
279
6.2.7.
D a m p e d sinusoids C R b o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
285
L east squares inverse (non-linear) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. G a u s s - N e w t o n method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
288 289
6.3.2.
L e v e n b e r g - M a r q u a r d t modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
291
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
294
Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
295
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Chapter 1
Geophysical Potential Fields
1.1 Potential field surveys for mineral and hydrocarbon exploration
Among the potential fields (that is, those satisfying the Laplace equation), gravity and magnetic (including aeromagnetic) fields constitute the commonly measured geophysical information in mineral and oil exploration and also in deep crustal studies. Gravity and magnetic surveys, being inexpensive, are routinely carried out over vast stretches of land and sea. Magnetic surveys are often carried out in the air using modem high technology flying combined with precision navigation and also from remote satellite. Computerized data acquisition systems make it possible to acquire high quality digital magnetic data over vast tracts of the earth's surface The availability of large quantities of high quality data affords an opportunity to apply the tools of digital signal processing, a field which has been extensively used in seismic exploration but has not been as extensively used in gravity and magnetic (G&M) exploration. The primary purpose of the present book is to introduce the subject of digital signal processing (dsp) to geophysicists engaged in G&M exploration. For some time, geophysicists have used the concepts of filtering, spectrum, parameter estimation, etc. but often on an ad-hoc basis. They are not very familiar with dsp principles and how they can be applied to potential field maps. This book is aimed at filling this gap. With this in mind, we first describe the potential field signals and their analytic properties to be exploited later in signal processing. Several examples of application of dsp to potential field data are also included largely to encourage more geophysicists to adopt the dsp approach in G&M exploration.
1.1.1. Brief description of G&M surveys Gravity surveys are often land-based or ship-borne. Gravity surveying from the air is still not perfected enough to be useful in practice. The instrument used is called a gravimeter which measures relative gravity field in units of milli gal (1 gal = 1 dyne). The gravity survey is often carried out along available tracks on land or at sea. Along the tracks, the sampling is often very dense but the spacing between the tracks is wide and the tracks themselves are scattered in different
Geophysical Potential Fields directions. Consequently the network of gravity observation stations is far from the ideal square grid format required for dsp application. In Chapter 4, we look into methods of converting from an irregular network to grid format. While gravimeters are rugged and precise (better than 0.01 mgal), the uncertainties in station location, gravimeter drift, limitations of terrain correction and unstable platform (mostly in marine surveys) are some of the major limiting factors. It is generally believed that a digital gravity map may have an error as large as 0.1 mgal. The measurement error has an important bearing on the precision with which the model parameters can be estimated. Large gravity maps are prepared by combining many smaller maps. In this process, special care has to be exercised in tying all base stations to a common base value and avoiding discontinuity at the boundaries. Aerial magnetic surveys are routinely carried out by flying a magnetometer, such as a proton precision magnetometer, for measuring the total magnetic field in units of "y (1 - y - 10 -5 gauss) at a height which depends upon the purpose of the survey. For mineral and hydrocarbon exploration, the flight height is a few hundred meters but for deep crustal studies the flight height is a few kilometres. In mountainous terrain, since the flight paths closely follow the rough topographic surface, the plane of measurement is an undulating surface closely following the rugged mountainous terrain. The magnetic measurement on such a rough surface will have to be reduced to a grid format on a horizontal plane before attempting any dsp algorithm. The measurement accuracy of modem magnetometers is adequate (for example, caesium vapour magnetometer has better than _+0.1 "y or nanotola resolution) for precision surveys. Similarly the positional accuracy (_+10 m), thanks to the GPS (geographic position system) satellite, is adequate for precision surveys. Higher precision can be achieved but only at increased cost [7]. Perhaps the most important limiting factor is the errors caused by temporal variations of the earth's magnetic field and the drift of the instrument. The long period diurnal variations are normally corrected by subtracting the base station magnetometer readings. But the fast changing micro pulsations in the geomagnetic activity are far more difficult to handle. The errors can be as large as 10 "y. One possible solution to the problem of inadequate correction of diurnal and other fast temporal variations is to measure the horizontal or vertical derivative. This requires the use of two or more precision magnetometers mounted in the wing tips and tail of the aircraft [ 13]. As an additional benefit, we show in Chapter 2 that the line spacing can be doubled by use of a three magnetometer system. Many such sources of error will have to be taken into account in any high resolution survey. From the point of view of signal processing the presence of such errors, often termed as jitter, may lead to significant lowering of signal-to-noise ratio in the high frequency band. We explore this aspect in Chapter 5.
Potential field surveys for mineral and hydrocarbon exploration
(a) v
FliGht line
,
,, y
,,..._
y
Figure 1.1. Distribution of observation stations in (a) an aerial survey and (b) a ground survey. Notice in the aerial survey that there is almost uniform coverage but in the ground survey the coverage can be very uneven depending upon the availability of public roads. As a result there is a possibility of large interpolation error in a land survey. To obtain digital gravity or magnetic data in a grid format, it is invariably necessary to do interpolation and at times extrapolation (to fill in the gaps in the survey) from the observed data set which may consist o f almost parallel profiles or randomly scattered profiles. Along the profiles the observations are densely packed but the profiles themselves are relatively widely spaced as in most aerial surveys. In ground surveys, since the observations are made along public roads which criss cross the country, we have essentially a collection of randomly scattered profiles (see Fig. 1.1). The error in the reduction to grid format largely depends upon the average spacing o f the observations in relation to the Nyquist sampling interval, that is,
Geophysical Potential Fields
100 -~
NPT=2 NPT=4
-~
10 -1
NPT=6
.
10O0 ~
-----.....
10_3
10-4
~
2
3
4
Wavelength
5
6
Figure 1.2. Mean square interpolation error as a function of the wavelength. The mean sampling interval is one unit. Ten percent sampling error is assumed. NPT stands for the number of samples used in the Lagrange interpolation scheme [1].
the minimum sampling interval required to achieve an error free reconstruction of a band limited function (see Chapter 3 for more details on sampling). To give some idea of the magnitude of error due to interpolation, a study was carried out on unit amplitude sinusoids of different wavelengths ranging from 2 to 6 units. Each sinusoid was sampled at unit interval with a small sampling error of 10%. Such a model (known as a jitter model) corresponds to a profile taken at right angles to the flight line in an aerial survey. The Lagrange interpolation scheme was used to obtain interpolated values at unit interval [1]. The mean square error as a function of wavelength of the sinusoids is shown in Fig. 1.2. The minimum wavelength used was two in order to satisfy the Nyquist criterion. Fig. 1.2 shows that the interpolation error can be large unless the wavelength is greater than the minimum required by the Nyquist criterion. Unfortunately the geophysical potential fields are not generally band limited. The effective bandwidth, which contains, say, 99% of the power, itself depends upon the depth of the source. Hence the required sample spacing in a digital gravity or magnetic map will depend upon the expected depth of the sources. Let us consider a worst case situation. Let there be a thin layer where the density is varying rapidly, such as white noise. The gravity field observed at a height of h units above will possess a spectrum (power spectrum) given by o~ exp(-2hs) where ~ is a constant and s is spatial (radial) frequency (see Chapter 2). Let So be the radius of a circle within which 99% of spectral power is contained. We obtain the following simple equation connecting h and So:
Potential field surveys for mineral and hydrocarbon exploration
(1 § 2hso ) = 0.01 exp(2hs0) An approximate solution of the above equation gives hs0=3.345. From this result it may be stated that the Nyquist sampling interval is equal to the depth to the layer. But, if the sampling is irregular and interpolation needs to be carried out, we require a sampling interval about half the Nyquist sampling interval. Thus, a rough rule of thumb is that the sample spacing should be less than or equal to half the depth to the source. Using this rule for a given sampling interval the shallow anomalies originating from a depth less than the sampling interval would be affected most by interpolation error. The Nyquist sampling rate is relevant only when it is desired to reconstruct a function without error from the samples. Very often the aim may be just to estimate a finite number of unkno.wn parameters of a model. In this context the important question is whether the available data possess enough degrees of freedom, at least equal to the number of unknowns required to be estimated [2]. The degrees of freedom may be estimated from the rank of a suitably defined data matrix (see Chapter 5 for discussion on this topic).
1.1.2. Information content in potential fields The potential fields are caused by the variation in density and magnetization in the earth's crust. Since the potential field is observed over a plane close to the surface of earth and the field in the free space above is analytically related to the observed field and hence is not independent, it is not possible to estimate the entire three-dimensional distribution of density and/or magnetization from the two-dimensional observed field. However, if the density and/or magnetization variations are properly modelled, consistent with other geological information, it is possible to fit the model to the observed potential field. The model parameters are then observable. The models may be grouped into three types: (a) Excess density or magnetization confined to a well-defined geometrical object. All model parameters can be estimated even in the presence of noise, except those pertaining to the underside of the object. The presence of noise, including the measurement errors, can seriously limit the complexity of the model that can be fitted. Further, if we have a collection of objects, the problem of model identification becomes unstable in the sense that a small change in the observed field can produce a large change in the model. An isolated anomaly caused by a single object is perhaps the most effective candidate for application under this category. Shallow objects such as a dyke, intrusive bodies of cylindrical or tabular shape often encountered in mineral exploration are some of the examples belonging to this category.
Geophysical Potential Fields (b) The second category of models pertain to a geological entity, such as a basin, young sedimentary basin or ancient metamorphosed basin with many intrusive bodies. The sedimentary basins are of great interest on account of their hydrocarbon potential. The sedimentary rocks are generally non-magnetic hence contribute very little to the magnetic field. The observed magnetic field is probably entirely due to the basement on which sediments are resting. This is also largely true of the gravity field. Therefore, it is possible to estimate the basement configuration from the potential fields. This is a problem of inverse filtering or deconvolution filtering but its success is conditioned on certain idealization on the part of magnetization or density. Since the depth of a sedimentary basin is large (1-10 km), large changes in the basement may produce only a small change in the observed field. It is therefore not practical to expect fine details in the study of basin configuration based on the potential fields. Ancient basins present a different picture. The original sedimentary rocks are now metamorphosed, often structurally disturbed, and intruded with dykes and granitic rocks. Magnetic maps are of great help in the mapping of different rock units because of their characteristic magnetic expression. The intrusive rock units are easily localized. But the question of great significance is 'Can one go beyond the visual correlations?'. For example, what is the meaning of the magnetic expression in relation to the genesis of rock? Some of these questions cannot be answered today, but with continued effort it may be possible to answer them in the future. (c) With available resources and technology, it is now possible to cover the whole country or continent or even the entire earth with G&M surveys. Such maps would naturally cover many geological provinces or basins and therefore allow inter-basin studies of, in particular, structural features such as faults, basin boundaries, etc. Deep seated features pertaining to the crust-mantle boundary are seen as very low frequency signals in potential field maps. Such signals can be extracted only from large regional maps after careful low pass filtering.
1.2. Role of digital signal processing (dsp) When some useful information is hidden behind a mass of unwanted information, we often resort to information processing used in its broad sense or specifically to signal processing when the useful information is a waveform. We would like to extract the signal waveform or to measure some of its broad characteristics, such as its spectrum, position, or its amplitude. It is a common practice to model the useful signal as a stationary stochastic process for the simple reason that it represents a wide class of signals of considerable complexity. The tools of the signal processing remain same for all members of this class of signals. In
Role of digital signal processing (dsp)
TABLE 1.1 Summary of the information that can be obtained from the potential field maps (note that the size of map is most crucial to the depth at which the information is sought) Size of map
Geological targets amenable to investigation
Information from potential field maps
A few kilometres" depth of interest to about a kilometre
Shallow isolated objects such as dykes, sills, faults etc.; mainly for mineral exploration
Several tens of kilometres; depth of interest to about 10 km Several hundreds of kilometres; depth of interest to about a few tens of kilometres
Geological province, for example, a sedimentary basin for hydrocarbon exploration A collection of many geological provinces; crustal structure and upper mantle inhomogeneities
The objects may be modelled as regular geometrical bodies whose parameters are then estimated from potential fields Basement configuration, density and susceptibility maps Determination of faults and grabens; Moho boundary and upper mantle density variations
geophysical surveys, in particular in aeromagnetic and gravity surveys, from the measured field, which m a y be considered as a sum total contribution of all that lies beneath, it is often difficult to say much about any one specific target unless it is close to the surface and well isolated from the rest. Yet, considering a map as a large data set, it is possible to extract the broad features of the earth's crust. Naturally, it makes sense to model the aeromagnetic field as a stochastic field representing a class of signals of considerable complexity and to seek signal processing tools which are applicable in a variety of geological problems. The digital signal processing approach enables us to bring out the underlying model of the source, that is, the geological strata and their structure. Some of the tools of dsp such as digital filtering, spectrum estimation, inversion, etc., have found extensive applications in aeromagnetic and gravity map analysis. There are other emerging applications of dsp in the area of inverse filtering, three-dimensional visualization, etc. In the field of signal processing one often comes across words like signal and noise. Traditionally, noise often relates to the measurement errors including the noise from the electronic components in the instrument used in the survey. In the context of potential field analysis, we have two other types of noise, namely, earth noise and model errors. The earth noise is simply the contribution from the sources in which we have no interest. This often includes the field generated by near surface sources. Modelling of a signal source is the fundamental step in the analysis of potential fields. Evidently the complexity of the model used will
Geophysical Potential Fields
measurement noise
Observed Potential Field
Signal from ~lesired source
earth noise
model errors
Figure 1.3. A schematic of the potential field signal and noise. The earth noise is the most dominant component.
depend upon the quality of dat~ available computational power and the utility of such a model. Under practica('considerations, the preferred model is often a simple model, which will then leave a good part of the actual signal unaccounted. This model error is now clubbed with noise. A common property of the noise is the presence of significant power in the high frequency band; in the limiting case we have white noise which has equal power in the entire frequency band of observation. The measurement noise is more likely to be white than the other two types of noise. Finally, a schematic of the potential field signal and noise is shown in Fig. 1.3.
1.2.1. Digital filtering In plain language, digital filtering is simply weighted smoothening of the map data carried out for different purposes, for example, removing noise, enhancing certain components, such as a slowly varying field, or carrying out some mathematical operations, such as differentiation, reduction to pole, etc. In two dimensions, the weighting coefficients or the filter coefficients are a matrix of real numbers chosen in a specific manner in order to achieve the desired goal. The task of implementing the filtering operation, particularly over a large map data, becomes a serious computation intensive problem, requiring fast filtering algorithms. A low pass filter is often used to separate the potential field, which originates at near surface, from that at deep subsurface. Since the spectra of the fields are overlapping, a perfect separation is not possible. Use of sharp cut-off filters may only contribute to artifacts like sidelobes which may be mistaken for an anomaly. These are some of the issues of great concern to those involved in the map data analysis. Further, some of the filters, for example, downward continuation, are basically unstable. It is then necessary that such filters are preceded by a well designed low pass filter.
Role of digital signal processing (dsp) 1.2.2. Parameter estimation
The entire effort of signal processing must ultimately lead to reliable estimation of the unknown parameters of the model describing the signal source. There are two main issues connected with the question of parameter estimation, namely, the algorithm and quality of the estimate. The maximum likelihood algorithm is the one where the parameters are so selected that the probability of observing the signal is maximized. It involves maximization of a likelihood function, which is a ratio of the conditional probability density function given that the signal is present and the conditional probability density given that no signal is present, with respect to the unknown model parameters. The above optimization problem is by no means a trivial mathematical exercise. There are often simpler alternatives, but at the cost of the quality of the estimate, that is, the mean square error. Fortunately, it is possible to define lower bounds, known as Cramer-Rao bounds, on the mean square error. In some sense, these bounds indicate what best can be done from the given data. The signal-to-noise ratio (SNR) is often the limiting factor. When the number of unknown parameters becomes large, for example, in a problem of continuous variation of density or magnetization we may have a large number of parameters depending upon the number of blocks of uniform density or magnetization used for approximating the continuous variation. Then the parameter estimation problem assumes a larger dimension of complexity. Another possibility is where the shape of the body is highly irregular, although the density or magnetization is uniform. The maximum likelihood approach becomes impractical particularly when the unknown parameters occur in a non-linear form, for example, the boundary defining an object. In such a case, linearization in some small neighbourhood is first attempted. The problem is then reduced to solving a large system of linear equations, usually ill-conditioned. The singular value decomposition (SVD) approach is often used to compute an approximate inverse of the kernel matrix. Use of prior information derived from other sources will help to improve the inversion. The most important issues are resolution and information content of the observations [3]. 1.2.3. Inverse filtering
There are situations, although somewhat idealized, where the aeromagnetic or gravity map signal can be expressed as a convolution between an interface separating two mediums of different density or magnetization and the impulse response function. An example of such an idealized model is sedimentary strata overlying a crystalline basement. The interface may be an undulating ancient
Geophysical Potential Fields
10
topographic surface. We can think of a filter which is an inverse of the impulse response function and when operated on the map signal we get an estimate of the interface. The inverse filter is prone to instability due to amplification of noise and hence a careful preconditioning of the data through low pass filtering is mandatory.
1.2.4. Spectrum analysis Spectrum analysis is another basic tool in signal processing. It shows how the signal power is distributed as a function of spatial frequencies. For effective spectrum estimation we must exploit the underlying signal model whose parameters are then determined as a part of the spectrum estimation procedure. The importance of the spectrum stems largely from the fact that certain information is TABLE 1.2 A quick survey of how dsp can be used in the analysis of potential field maps DSP tool
What does it do?
Geophysical application
Discrete Fourier transform
Takes Fourier transform of the potential fields, 1D or 2D
Digital filtering
Removes noise, decomposes field into low and high frequency components and performs analytic operations like continuation, derivative, etc.
Inverse filtering
Removes convolutional effects
Spectrum estimation
Computes power as a function of spatial frequencies
Parameter estimation and inversion
Estimates the unknown parameters of signal model in presence of noise
Since convolution integral is reduced to simple multiplication, many relations in the potential field are greatly simplified Cleans up noisy observations, separates deep seated anomalies from the shallow ones. Filtering enhances the directional features like faults. Reduction-to-pole, Reduction to equator Pseudo gravity, and derivative maps Basement mapping, magnetization maps Gives a measure of characteristic magnetic expression of rock units. Depth to magnetized layers. Angular spectrum can detect linear features Enables us to estimate the depth and shape of complex signal models. G&M data may be used to improve upon the existing subsurface knowledge
Role of digital signal processing (dsp)
11
best obtained in the frequency domain rather than in the spatial domain. Take a simple example of an aeromagnetic field caused by a thin horizontal sheet of magnetic sources. The depth to the sheet is easily obtained only from the spectrum, that is, from its decay rate. In this case, the depth cannot be estimated using any other approach. The angular variation of the spectrum is intimately related to the structural elements of the rock strata such as average dip and strike. When we have two sets of map data, possibly of different types, for example, marine magnetic data and ocean floor topography, we define a quantity called the cross-spectrum, which basically gives us a measure of the common energy between the two types. From the cross-spectrum we can define a transfer function which characterizes the common link between the two types of data, if any.
1.2.5. Image processing Map data are like a picture. Instead Of grey levels we have real numbers representing the measured magnetic or gravity field. From a contour map we can easily prepare a grey level photograph for a prescribed angle of illumination and an angle of viewing. A large number of such photographs may be generated giving different views of the surface. The human eye is good at deciphering patterns and textures in a photograph. This ingenuity of the human eye may be effectively used for the visual processing of the map data, in particular for identifying long linear features such as faults or variations in the texture. Adding colour to aeromagnetic map representation further accentuates the effect of visualization of a map.
1.2.6. Some reservations The dsp will succeed only when there is a good understanding of the signals and the corrupting noise. A clever way of exploiting the underlying differences between the signal and noise structures may lead to a successful extraction of the signal from the observed map data. Unfortunately our understanding of the signal models and sources of noise is far from complete. Moreover, from what little we understand there is no clear difference between the signal and noise. For example, the noise generated by the near surface sources is spectrally similar to the signal generated by deeper sources. They differ only in the rate of decay. Thus, digital filtering may not be as effective as one would like, except perhaps when the noise is white noise mostly due to the measurement and gridding errors. Another important limitation is the assumption of stationarity or homogeneity of the potential field. This certainly is not true over a large area covering different geological provinces. The spectrum analysis of such maps is untenable. In inverse
12
Geophysical Potential Fields
filtering suggested for basement mapping, it is necessary to assume that the density or the magnetization of the underlying rocks is uniform, which indeed is very unlikely in the real world. Thus, the models assumed in dsp applications are only approximate to the real earth. This indeed is also true with all other natural sciences. Only through our continuous efforts to improve the models will progress be made. With these words of caution we invite the reader to explore what follows in the remaining chapters.
1.3. A comparison with seismic signal processing Application of digital signal processing to seismic signals is well known, In fact, some of the dsp tools, for example, linear prediction, high resolution spectrum analysis, principle of maximum entropy, were first used in seismic signal processing and later introduced to the signal processing community. A number of books dealing with seismic signal processing have already appeared [8-10]. In contrast, the use of dsp in potential field analysis has been slow, as evidenced by the absence of any book dealing with the use of dsp in potential field analysis. Often the practitioners of potential field methods in their interpretation are reluctant to go beyond a semi-quantitative approach. It is the basic nature of the potential fields that they represent the sum total effect of the earth that lies beneath the surface, that is, the observation plane. The seismic field is a propagating phenomenon. It takes finite time to travel from point A to point B. The travel time can be mapped into distance. This combined with the well known phenomenon of reflection, transmission and diffraction of propagating waves enables us to derive localized information of the earth beneath. To some extent this is also true of the electromagnetic waves but the speed with which these waves travel is just too high for making any accurate travel time measurements. In some sense the potential field may be considered as a field that propagates at infinite speed. In the Helmoltz equation, which governs the propagating field, if we let the speed to go to infinity, the equation reduces to the familiar Laplace equation governing the potential fields. While the seismic field is created externally under controlled conditions, the potential fields, being a native property of rocks, are present everywhere without any external control. The seismic field can be focussed onto a target of interest through a use of spatial arrays of sources and sensors. The seismic field possesses an extra dimension of temporal frequency which may be used to control both resolution and depth of investigation. For example, a high frequency seismic survey is often used for detailed subsurface mapping. Unfortunately the potential fields do not possess such advantage. Thus, the seismic field is best suited to derive a detailed image of the subsurface rocks
A comparison with seismic signal processing
13
and rock structures. Thousands of geophysicists around the world with large budgets and expensive computers are constantly striving to achieve this cherished goal. Turning to the potential fields, it may be argued that the global picture that a potential field map paints has also an important role to play in achieving the cherished goal of subsurface mapping or imaging. The 'global' nature of the potential field will enable us to obtain a 'global' view of the subsurface image. The details may be filled in later with a seismic survey, when and where required. This is a well accepted view among the geophysicists. Since the potential field maps are subjected to the same kind of noise and interference as in a seismogram, it is natural to employ the tools of signal processing to help us unravel the information buried in a potential field map. While in seismic signal processing the main aim is to identify and to measure the arrival times of reflections from surfaces separating different geological strata, in potential field signal processing one tries to fit a model (deterministic or stochastic) of excess mass or magnetization to the observed field. The dsp tools differ for the simple reason that in seismic signal processing we have to process a collection of time series but in potential field signal processing we have to process a 2D map data. The dsp tools used in potential field signal processing are closer to those used in picture processing; for example, vertical derivative processing in a potential field is akin to edge enhancement in picture processing. In contrast, the dsp tools used in seismic processing are closer to those used in radar and sonar.
1.4. Prologue The purpose of writing this book is to bring numerous tools of dsp to the geophysical community, in particular, to young men and women who are entering the geophysical profession. Also, we hope that the practising geophysicists involved in the aeromagnetic and gravity data analysis using the commercially available software packages will find this book useful in answering their questions on 'why and how?'. It is hoped such a background would enable the practising geophysicist to appreciate the prospects and limitations of the dsp in extracting useful information from the potential field maps. A background of college level mathematics, in particular, topics like Fourier transforms, elements of stochastic process, and linear algebra are expected of a reader. Although dsp is a branch of electrical engineering the reader need not be unduly worried over that fact, as dsp can be mastered by practically anyone with a good base in mathematics. However, this book does not aim at teaching dsp as there are many excellent texts on this and related topics, for example, the books by Proakis [4]
14
Geophysical Potential Fields
on dsp, by Naidu [5] on spectrum analysis and by Lim [6] on two-dimensional signals provide a state-of-the-art coverage in the respective areas. However, the present book does provide the essential background in digital signal processing that would be necessary to understand its applications in potential field analysis. In a sense the book is self-contained but a serious reader is strongly advised to refer to the above-mentioned or any other books of his/her choice. While application of modem digital signal processing in seismic applications is well covered in many recent texts, for example, see [8-10], there is no book that is devoted to potential fields. Some books, namely Bath's book on spectrum analysis [11] and Buttkus' book on spectrum analysis and filtering [ 12], devote one or two chapters to potential field applications. It is hoped that the present book will fill that gap. This book is divided into six chapters. Chapter 2 is devoted to characterizing the potential field signals in free space in Section 2.1 and potential field in the space filled with sources in Section 2.2. Idealized models such as the sphere, cylinder, dyke, and other two-dimensional objects are considered in Section 2.3 and a basic 3D model such as a vertical prism which would enable us to build more complex models is covered in Section 2.4. The potential field signals possess some interesting properties, namely, their singularities lie at the comers of the model. This fact has been exploited in parameter estimation in the frequency domain. From the idealized models we go on to stochastic models (Sections 2.5 and 2.6) where the density and magnetization are assumed to be random functions. They include some of the important practical problems such as an undulating basement, a thin sheet of magnetized strata, semi-infinite medium, etc. Stochastic models bring out the gross features of the source and hence they are ideal for modelling large data sets such as potential field maps covering one of more geological provinces. This chapter by its very nature turns out to be more mathematical than the rest of the book. However, since it provides the required theoretical background it must be carefully read, although not at the first attempt. Chapter 3 is devoted to the study of the spectrum, radial and angular spectra, cross-spectrum, and coherence of the two-dimensional stochastic process which is used as a model for potential fields. In Section 3.1 we introduce these quantities and later we look into their estimation from the data. The properties of the discrete data, such as sampling in two dimensions, aliasing, folding of the spectrum are covered in Section 3.2. The discrete Fourier transform, estimation of the spectrum and the role of windows in spectrum estimation are covered in Section 3.3. Application of the spectrum for depth estimation and that of the angular spectrum for average strike direction are also dealt with in Sections 3.4 and 3.5. Finally, the applicability of the fractal model to potential field is explored in Section 3.6.
Prologue
15
The topic of digital filtering of maps is covered in Chapters 4 and 5. The digital filtering approach is useful for implementation of a variety of processing procedures under a common framework. The first of these two chapters deals with some of the basic tools of filtering. In the first section, we introduce the basic concepts of 2D filters, design of pass filters, in particular low pass filters used for separation of regional and residual. The issues pertaining to implementation of 2D filters, in particular fast convolution, are discussed in Section 4.2. Extraction of signals generated by well defined objects in the presence of noise is an important problem as in many other areas of engineering. Wiener filters for this purpose are well known. We explore the topic of Wiener filters for extraction of a potential field signal in Section 4.3. Many analytical operations such as upward and downward continuation, differentiation, reduction to pole or equator, etc. can be implemented through digital filtering. This approach is described in Sections 4.4 and 4.5. The potential field surveys are often carried out, whether on the ground or in the air, over an uneven surface. However, the final digital potential field is required to be on a plane horizontal surface. We describe a fast iterative method for the reduction of potential field measured over an uneven surface to a horizontal surface. The potential field caused by terrain undulation acts as a major interference. A method of correlation filtering is suggested in Section 4.7 for the removal of the terrain effect. In Chapter 5 inverse filtering is suggested for the mapping of the basement surface, particularly where a thick sedimentary strata is lying over denser and/or magnetized basement rocks. Also included is preparation of familiar density and susceptibility maps as an inverse filtering problem. Linear inverse theory has been extensively used for estimation of sub-surface distribution of density and magnetization. It is true that the potential fields by themselves cannot provide a unique answer but they do provide additional information which would help us to improve upon the existing subsurface geological knowledge. In Sections 5.2 and 5.3 we briefly describe the theory of linear inversion. Finally in Section 5.4 we explore the topic of texture analysis of aeromagnetic maps as an aid in geological mapping. The parameters of an idealized model such as a horizontal cylinder with polygonal cross-section or a vertical prism can be estimated in the frequency domain in the presence of noise. The maximum likelihood estimation of these parameters is covered in Chapter 6. When the number of unknown parameters is large, maximization of the multi-dimensional likelihood function becomes impractical. Simpler alternatives such as non-linear least squares and damped non-linear least squares may be used. Throughout the book several numerical examples, some of them using real data, are included to highlight both the strengths and weaknesses of the dsp approach.
16
Geophysical Potential Fields
1.5. Notation
The following is a list of conventions, symbols and abbreviations used in this book:
1.5.1. Conventions (i) In free space, that is, above the earth's surface we take the z axis pointing upward but in the source space, that is, below the earth's surface we take the z axis pointing downward into the earth (see Fig. 2.7). This choice of dual convention appears to be consistent with the geophysical and signal processing literature. (ii) Lower case and upper case characters are used to represent a function and its Fourier transform. For example, f(x) and F(u) constitute a Fourier transform pair. (iii) Vectors and matrices are shown by bold characters. (iv) ( )v, matrix transpose; ( )n, Hermitian transpose; ( )-l, matrix inverse; ( )+, pseudo inverse. (v) A parameter and its estimate are distinguished by the presence of a hat on the estimate. For example, if 0 is a parameter, then its estimate is 0.
Symbols
f ,(x), .f2(x,y), f3(x,y,z) iT( ) ) ) bl~F,W S
p K
G ol,/3 and y m
=
(Ix, Iv, Iz)
functions of the coordinates x; x,y and x,y,z respectively gravity field total magnetic field gravity potential magnetic potential spatial frequencies along x, y, and z axes respectively radial frequency density susceptibility gravitational constant direction cosines magnetization vector and its components components of a polarizing vector spatial spectrum off(x,y) spatial cross-spectrum between f(x,y) and g (x,y)
Notation
cohj ( ,v) A(O) Anorm(0)
E{} Var{ } E = diag{X1, )~2, X3, ...} U,V ZXx, Ay
17 coherence between f (x,y) and g (x,y) radial spectrum off(x,y) angular spectrum normalized angular spectrum expected operation in probability theory. variance of the quantity inside the braces diagonal eigenvalue matrix eigenvector matrices sampling intervals
Abbreviations CR col diag dsp 1D 2D 3D fft dft idft MSE rn'ls
FIR IIR G&M SNR SVD
Cramer-Rao lower bounds column (matrix) diagonal (matrix) digital signal processing one dimension two dimensions three dimensions fast Fourier transform discrete Fourier transform inverse discrete Fourier transform mean square error root mean square finite impulse response infinite impulse response gravity and magnetic signal-to-noise ratio singular value decomposition
References [1]
P. S. Naidu, A statistical study of the interpolation of randomly spaced geophysical data, Geoexploration, 8, 61-70, 1970. [2] S. Twomey, Information content in remote sensing, Appl. Optics, 13, 942-945, 1974. [3] D.D. Jackson, Interpretation of inaccurate, insufficient and inconsistent data, Geophys. J. R. Astr. Soc., 28, 97-109, 1972. [4] J. G. Proakis and D. G. Manolakis, Digital Signal Processing, Theory, Algorithm, and Applications, Prentice Hall, New York, 1988. [5] P.S. Naidu, Modern Spectrum Analysis of Time Series, CRC Press, Boca Raton, FL, 1995. [6] J.S. Lim, Two Dimension Signal and Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1990.
18
[7] [8] [9] 10] [ 11] [12] [13]
Geophysical Potential Fields
W.E. Featherstone, The global positioning system and its use in geophysical exploration, Exploration Geophys., 26, 1-18, 1995. R.H. Stolt and A. K. Benson, Seismic Migration Theory and Practice, Geophysical Press, London, 1986. P.L. Stoffa (Ed.), Tau-p" a Plane Wave Approach to the Analysis of Seismic Data, Kluwer, Dordrecht, 1989. E.R. Kanasewich, Seismic Noise Attenuation. Handbook of Geophysical Exploration, Vol. 7, Pergamon Press, Oxford, 1990. M. Bath, Spectrum Analysis in Geophysics, Elsevier, Amsterdam, 1974. B. Buttkus, Spektralanalyse und Filtertheorie in der Angewandten Geophysik, SpringerVerlag, Berlin, 1991. R.E. Sheriff, Geophysical Methods, Prentice Hall, Englewood Cliffs, NJ, 1989.
19 Chapter 2
Potential Field Signals and Models
In this chapter we describe the properties of a potential field signal caused by different distributions of mass and magnetization; both deterministic and stochastic models are considered. We start with some basic properties of the potential field in the frequency or Fourier domain. The frequency domain approach is preferred for two reasons. Firstly, most dsp tools are easy to understand and easy to implement in the frequency domain, particularly when a large amount of data is required to be processed. It is with this view we have analyzed many signal models, both deterministic and stochastic, in the Fourier domain. Secondly, it turns out that the Fourier domain characterization of the potential field signals caused by a large variety of source models is easy and concise. Yet, with nature being immensely complex, we can only analyze a few highly simplified models. In Sections 2.1 and 2.2 we deal with the potential field in source-free space and source-filled space. The following four sections describe different types of models: the deterministic models; starting from a simple point source we go on to cover more complex models such as a vertical prism with polygonal cross-section in Sections 2.3 and 2.4 and the stochastic models in Sections 2.5 and 2.6. The concept of singularity of a potential field signal is introduced. It turns out that the position of singularity is closely related to some of the important features of the shape of the anomaly causing body; for example, at each comer of the body there is a singularity. Under stochastic models, first we consider a model of a random interface separating two otherwise homogeneous media and next, we consider a model where the density or the magnetization in a medium varies randomly. The main result that emerges from the stochastic analysis is an analytical relationship between the spectrum of the potential field and the spectrum of the source. This chapter provides the theoretical backbone for what follows in the rest of the book.
2.1. Potential field in source free space
The potential field Laplace equation,
~p(x,y,z) in free space, i.e. without any sources satisfies the
20
02q5 F-
Potential Field Signals and Models
02~5+ 02~5--b72
0
(2.1a)
When sources are present the potential field satisfies the so-called Poisson equation
~
02~5 02~5
OX------~-~
Z)
(2.1b)
OZ2
where O(x,y,z) stands for density or magnetization depending upon whether 4~ stand for gravity or magnetic potential, cb(x,y,z) on account of Eq. (2.1) has certain very useful properties particularly in source-free space such as the atmosphere where most measurements are made. Some of these properties are: (i) given the potential field over any plane we can compute the field at almost all points in the space by analytic continuation; (ii) the points where the field cannot be computed are the so-called singular points. A closed surface enclosing all such singular points also encloses the sources which give rise to the potential field. The singularities of the potential field are confined to the region filled with sources. These properties of the potential field are best brought out in the Fourier domain. We shall first introduce Fourier transform in two and three dimensions.
2.1.1. Fourier transform The Fourier transform in one dimension is found in most books on applied mathematics and mathematical physics. However, the Fourier transform in two or three dimensions, a straightforward extension of the Fourier transform in one dimension, possesses additional properties which are worth recalling [1]. In two dimensions the Fourier transform is given by
qS(x,y) e x p ( - j ( u x + vy)) dx dy
~I,(u, v) --
(2.2a)
oo
and its inverse is given by
r (x , y ) - - 4 7 2
9 (u, v)exp(/'(ux + vy)) du dv
(2.2b)
Potential field in source flee space
21
Eq. (2.2b) is also known as the Fourier integral representation of cb(x,y). The integral in Eq. (2.2a) exists only if
(2.3) oo
which condition is generally not satisfied in most geophysical situations except where we have an isolated anomaly. In practice it is found that there is a neverending succession of anomalies. Then the condition implied in Eq. (2.3) will not be satisfied. Later in this chapter we introduce the generalized Fourier transform. The Fourier transform of a real function in two dimensions possesses the following symmetry properties:
9 ( - ~ , ~) = ~ * ( u , - ~ )
9 (~, ~) = ~ , ( _ ~ , - ~ ) , 9 (o, o) -
~(x,y) dx dy
(2.4a) (2.4b)
oo
2.1.2. Potential field Let cb(x,y,z) be the potential field in free space satisfying Eq. (2.1a) and its Fourier integral representation be
~(x,y,z) -- ~
'ff§
~(u, v)H(u, v,z) exp(-j(ux + vy)) du dv
(2.5a)
oo
where H(u,v,z) is to be determined by requiring that it satisfies the Laplace Eq. (2.1a). This is possible if H(u,v,z) satisfies the following differential equation:
d2H dz 2
(u 2 --1- v2 )H -- 0
whose solution is given by
H(u,v,z) = exp(+sz) where s -
(2.5b)
4(u2-+ - v2). Further, because the potential field must vanish at
22
Potential Field Signals and Models
z
Free Space
/ Figure 2.1. Coordinate system used in the free space above the earth's surface. Note that +ve z points upward into the free space.
z ~ +oo (+ve z points upward, see Figure 2.1), we must select the negative sign in Eq. (2.5). Here ~(u,v) would refer to the Fourier transform of the potential field a t z - 0. Using Eq. (2.5b) in Eq. (2.5a) we obtain
l f f_
qS(x,y,z) -- ~ 2
~(u, v ) e x p ( - s z ) exp(/'(ux + vy)) du dv,
z _> 0
(2.6)
O(3
Eq. (2.6) is a useful integral representation of the potential field. It forms a basis for derivation of many commonly encountered results. It is important to remember that Eq. (2.6) is valid for potential fields, e.g. gravity and magnetic fields in free or homogeneous space. Letfx, fy, andf~ be three components of a potential field which is defined as the negative gradient of the potential. The Fourier transform of the components measured over a plane are related to each other through the following set of equations:
Fx(u, v,z) -- - j u ~ ( u , v ) e x p ( - s z ) Fv(u , v,z) - - j v ~ ( u , v ) e x p ( - s z ) F~(u, v,z) --
s~(u, v) exp(-sz)
(2.7)
Potential field in source free space
23
From this it is clear that any one component is adequate to describe the potential field in the free space. Indeed, all three components of the potential field are related through the following linear relation
v,z) -ju
--
v,z) -jv
=
Fz(u,v,z) s
= ~(u, v ) e x p ( - s z )
(2.8)
2.1.3. Poisson relation The gravitational potential and the magnetic potential are closely related if the sources generating them are exactly identical. Let O(x,y,z) be the potential generated by an element of mass with density 0 and volume dv and ~o(x,y,z) be the magnetic potential due to an elemental dipole located at the same spot with magnetic moment.
m--mxex
+ myey + mzez
where ex, ey,, ez, are unit vectors in the direction of x, y and z axes and mx, my, mz are dipole moments along these axes respectively. A dipole may be thought of as two equal and opposite charges placed an infinitesimally small distance apart. To evaluate the potential field due to a dipole it is enough to consider the potential due to each charge separately and then add the two contributions. Thus, the magnetic potential, ~o(x,y,z), due to an infinitesimal vertical dipole will be given by
~pz(X,y, z) -- mz O(x, y, z - 89~z) - O(x, y, z + 89zXz) Gp ~z
-mzO0 Gp Oz
~z--~ O
(2.9)
Similarly, we can calculate the potential field due to a horizontal dipole in the same manner. For a magnetic dipole oriented in an arbitrary direction, the magnetic potential is given by
~(x,y,z)
--l (
O O O) mx -~x + my~yy+ mz --~z qb(x , y , z )
(2.10)
Potential Field Signals and Models
24
Making use of Eq. (2.6) in Eq. (2.10) we can write the following relationship known as the Poisson relation in terms of the Fourier transforms of the potential on a plane:
O(u, v) - -G--pp-1(l'mxU +jmyv - mzS)~(u, v)
or
O(u,~)
~(u,~)
jmxu -t-jmyv- mzS
Gp
(2.11)
We later generalize the above relation for a more complex source model (see Section 2.6).
2.1.4. Hilbert transform Sometimes the potential field is independent of one axis, say the y axis. It is then a function of two coordinates, x and z. The integral representation of such a potential field may be derived in the same manner as Eq. (2.6).
lf+
~(u) exp(-lulz)exp(/ux) du
vS(x,z) - ~
(2.12)
O0
where l,(u) is the Fourier transform of 4~(x,0).There are certain interesting properties of a 1D potential field (z is held fixed). From Eq. (2.12) we can show that
02e(x,z) Ox2
lf+
2~
o~
--U2~)(bt) e x p ( - l u l z ) e x p ( / u x ) du
and
02 ea(x, z) cox&
lf+ lf+
2rr
oo
2w
O0
-ju]ul~(u) e x p ( - l u l z ) e x p ( / u x ) du
--jbt2sgn(u)~(u) e x p ( - l u l z ) e x p ( / u x ) du
(2.13)
Potential field in source free space
25
where sgn(u) stands for the sign of u. Note that the integrands in Eqs. (2.13) and (2.14) differ only in jsgn(u) function. This is the essential difference between a signal and its Hilbert transform [2, p. 71]. Hence --oZt~/OX 2 and -02~b/0x0z constitute a pair of Hilbert transforms. Thus, the horizontal derivatives of the horizontal and vertical components of gravity field are Hilbert transform pairs. To obtain the corresponding result for the magnetic field we must note the relationship between the gravity and magnetic potential fields (see Eq. (2.10)).
-1[ m x ~0x + mz -~z c~(x,z)
g~(x,z) -- ~ p
(2.15)
In the frequency domain Eq. (2.15) becomes
O(u z ) - - 1 [l'mxU- mzlUl]'~(u z) ' --G-tip '
(2.16)
Now, consider the Fourier transform of the horizontal and vertical components of the magnetic field,
FT { - O~(x'
} _ -1 [mxU2 + m~juZsgn(u)](~(u,z)
(2 17a)
} _ -1 ~.mxu2sgn(u) _ mzuZ]~(u ,z)
(2.17b)
9
and
FT { - O~(x'
Note that the right hand side of Eqs. (2.17a) and (2.17b) differ by jsgn(u) (Note: sgn(u) x sgn(u) = 1). Thus, we note that the horizontal and vertical components of the magnetic potential field constitute a Hilbert transform pair. A complex analytic signal is defined using the Hilbert transform pair as follows:
L(x)
0f~
0fz
(2.18)
where fx and fz are horizontal and vertical components of the gravity field and similarly for the magnetic field. The concept of a complex signal has been utilized in model parameter estimation [3,4,29].
Potential Field Signals and Models
26
For a magnetic field, the analytic signal is given by fc mag (X, Z) =
fxmag(X,z) --t-jfzmag(X,z)
where
fxmag(x,z)
--l /"cooo -~p l (mxU2 +jmzsgn(u)u 2) exp(l'UX)'~(u, z) du
and
fzmag(x,z) - - l f ~
l-~--(jmxsgn(u)u2 - mzu2) exp(l'UX)~(u,z) du
The envelope of the analytic signal is given by
gnv{fcmag(x,z)}- V/l~xmagl2
+gzma12
/m2x + m2z
: V ~ - - ~ ~ { [f_c~ U2(I)(u,Z)exp(/'ux)du]
+
2 +
u2sgn(u)
Note that, if m is the magnetization vector with components, m~ and mz, Iml 2 - m2x + mz; 2 thus the direction of the polarization vector does not affect the envelope of the analytic signal. This useful property of the analytic signal was first noted by Nabighian [3].
2.1.5. Singularities of potential field Every potential field must possess singularities somewhere in the entire space; otherwise, the potential field will be zero everywhere. The singularities are the only points where the potential is maximum (or minimum). Since the potential field cannot possess a singularity in the free space, all singularities must be confined to the interior or at the most on the surface of the source. It must be pointed out that the singularities are encountered only when the external potential field is analytically continued into the region occupied by the source. The analy-
27
Potential field in source free space
tically continued potential found inside the body is not equal to the true potential found inside the body. The importance of the singularities lies in the fact that these can be uniquely determined from the observed field. They are often closely related to the shape parameters of the source. Each singularity is described by three sets of parameters, namely, location, amplitude and order of singularity. A group of singularities associated with a source may be enclosed by a convex surface. Often it is the top most singular point on the convex surface which is of great interest, and it is also the one which can be determined relatively easily. Let h be the depth to the top most singular point, then there exists an upper bound on the Fourier transform of the signal due to the source [5], 9 (u, v) < ~ ( c ) e x p ( - [ h - cls),
c-+ 0
where s - V/(b/2 --[- 1;2). For a gravitational field, K(e) should be replaced by the total mass. Analytic continuation beyond the first singular point is not possible. However, there are methods [6] which would enable us to continue the field around the surface enclosing all singularities.
2.2. Potential field in source filled space
2.2.1. Gravity potential Consider a space filled with sources, mass of variable density or variable magnetization. To obtain the potential field in such a source filled space, we need to solve Eq. (2. l b), first for the gravitational potential.
+(x,y,z
- o) -
o
p(xo,yo,zo)
cx~
[(X -- XO):~ 7 ~ ~~00i 2 -Jr-Z21 1/2 dXOdyO dZO
(2.19)
where G stands for the gravitational constant. Eq. (2.19) holds good everywhere including at a point inside the space filled with matter [7]. To compute the potential at a point (x,y,O) we shall divide the space into two halves separated by z = 0 plane (see Figure 2.2) and write the potential as a sum of the contributions from two semi-infinite spaces. Note that the coordinate system used in the source filled space differs from that in the free space in the previous section. As shown in Figure 2.2 the +ve z axis points downward.
28
Potential Field Signals and Models
Figure 2.2. To compute the potential at a point (x,y,0) we divide the space into two halves separated by the z = 0 plane. Thus, (2.19) may be rewritten as follows: O(x,y,z
-- O) -- Olo(X,Y,Z -- O) + O u p ( X , y , z -- O)
where q51o(x, y, z -- O) -- G
dxo dyo dzo
oo .
Oop(x,y,z - o) - G
i
oo
{(~-x0) ~ + (y-y0)
~
~ + (z0)~]l/~-
p(xo,yo,zo)
+ ( y - Y ~ ~ + (z0)~],/~ dXo dyo dzo o~[(X-Xo) ~ (2.20)
Let us now replace the density function in Eq. (2.20) with its Fourier integral representation (Eq. 2.2b). We obtain the following result:
Potential field in source filled space
O(x ' , , z
29
)1
O)
~ { f + Oe ~ P,o(,, ~, w) dw -exp(/'(UXs + vy)) du dv ~~ f f + Oe -(s 7-jw-)
G /f+oe{f+oePup(U,V,W) +~
Oe
Oe
(s +jw)
dw} -1exp(/'(ux + vy)) du dv s
(2.21) where
f +OeoefOe p(xo,Yo,Zo) exp(--j(uxo
Plo(U, v, w) --
Pup(U, v, w) --
+ vyo + WZo)) dxo dyo dzo
f f+oe fO p(xo,Yo,Zo) exp('j(ux o + vyo + WZo)) dx o dy o dz o (2O O0
and we have used the known result [8]
fo ~ rJo(sr) dr
v/r2 + a2
exp(-sa)
s
2.2.2. Magnetic potential We now derive the magnetic potential inside a source filled space. The magnetic potential in the spatial domain is given by
~ ( x , y , z -- O) --
/ + o e J o e (x - xo)m x + (y - Yo)my --Zorn z
oe J-oe [ ( x - Xo)2 --]- 0 " - YO)2 4-Z2] 3/2 dXO dyO dZO (2.22)
where mx, my and m~ are functions of x0, Y0, and z0. As in the case of gravity potential, we divide the entire space into two halves and express the potential as a sum of the contributions from two semi-infinite half spaces. We replace mx, my and m~ with their respective Fourier integral representation (see Eq. (2.2b)). After some straightforward but tedious algebraic simplification (see Section 2.4 for some details), we obtain
30
Potential Field Signals and Models
-1 //+oo [foo Mlo(U,V,W) ~(x,y,~
- o) - a 7
~
1
x - exp(/'(ux +
s
oo
(~ - j w )
dw +
/OOMup(U,V,W) j ~
(~
7)~i
vy)) du dv
dw
(2.23)
where Mup(U ,
v, w) -juMUp(u, v, w) q-jvM;P(u, v, w) - sMUP(u, v, w)
Mlo(U , v, w) - " juMxlo (u,
v, w) -~--jvMl~ , v, w) - ,M'z o(u, ,,, w)
(2.24)
The Fourier transforms M up(u, v, w), ... and MJ~ v, w ) , . . , are defined as in Eq. (2.21). When the upper half space is filled with air of negligible density the second term in Eqs. (2.21) and (2.23) vanishes.
2.3. 2D s o u r c e m o d e l s
2.3.1. Line source Consider a line source (horizontal) with density p and cross-section dx0dz0. The line source is located at (x0, z0). The vertical component of gravity field at (x, z = 0) due to such a line source is given by
2Gpzo dxo dzo d S z ( x , z - o) -
(
x-xoi
+7ool
Note that the +ve z0 axis points downward towards the line source (see Figure 2.2). The Fourier transform of the above vertical component of the gravity field on z - 0 plane is given by
dFz(u) = 2rrGpexp(-lU[Zo) exp(-juxo) dxo dzo
(2.25)
Note that the potential field due to a line source has a singularity at ( x - x0, Z -- Zo).
2D source models
31
Figure 2.3. A horizontal cylinder with polygonal cross-section which is specified by the coordinates of the comers.
2.3.2. Cylinder with polygonal cross-section To evaluate the total field due to a cylinder with a polygonal cross-section we must integrate Eq. (2.25) over the cross-section of the cylinder (see Figure 2.3).
Fz(u) - 2~Gps f e•
e•
dxo dzo
where ~2 stands for the cross-section of the cylinder. Following Stoke's theorem [9] the integral over the polygonal cross-section may be reduced to a line integral taken along the perimeter of the polygon. This method was first used by Ford [10] to compute the filter coefficients whose frequency response is unity over a polygon in the frequency plane and zero outside the polygon.
F~(u) -- 2rcGp / e x p ( - l u [Zo) exp(-juxo) ds,
-ju
Fz(O) -- -27rGp / Zo ds
u 4= 0
(2.26a)
(2.26b)
where the contour integral is taken along perimeter c of the cross-section ~2. and ds stands for infinitesimal element on the perimeter. Let the comers of the polygon be at ~i, 2i, i = 1,2,...,M.
N:'
I
+
N>
II
-
II
a,
~
:r
_
-
I
,.--,,--,
,,,.--
"4.
I'g:,
+
,..._
I'...i
I
II
_
+
hi>
~
oo
"--.
I
I
t-t-
...,
~"
-.,
I
II
I
I
,.,-
I
~>
?" ,,,,.
,--.
I
Jr-
x _
L'b1~
II
'-"
~
9
i,,~o
=l
0',
,.Q
IxJ
.-....I
~
~
~
.-...I o"
~--
~
0
o
.
~
'7
-.-.-,I
I
'T
i,,- t
I,--Lo
=
0"r
("1)
0
i-,,o
i,,,,-,,
0 ("1)
2D source models
33
2.3.3. Dyke
To illustrate an application of Eq. (2.28), let us consider an example of a vertical dyke (Figure 2.4). There are four sides. From Eq. (2.28) we can write Fz(u) --
27rGp
ju
(Contributions from sides AB, BC, CD and DA)
(2.30)
Since ZA -- ZB = 0 and also ZD - Zc = 0, the contributions from side AB and DC are zero. The contribution from line BC is given by
Contribution from BC"
H
1-
exp(-]ulO)] exp(-lulh -jua) lulH
1 --exp(-lulh
I.I
-jua),
H ~ ec
(2.31)
and similarly from DA
Contribution from DA 9
H
.1- exp(+lulH)] exp(-lul(g + h) +jua) lulH 1 lul e x p ( - l u l h
+jua),
H ~ oo
(2.32)
Substituting Eqs. (2.31) and (2.32) into Eq. (2.30), we obtain .
.
.
.
" ......
A
__~
B
'
2a
"-~ . . . . .
:
A
:
:
h
~--
/
Vortical dike
density: p
D
C
Figure 2.4. A vertical dyke of uniform density. A method of computing the gravity field in the frequency domain is described in the text.
34
Potential Field Signals and Models
Fz(u) = j
2rcGPsgn(u){exp(-lul(S + h) +jua) + exp(-lulh -jua) u2
- e x p ( - l u l h +jua) -exp(-luL(H + h) -jua)}
(2.33a)
Fz(O) = 47rGpaH and for infinite dyke (taking a limiting form of Eq. (2.33a) H --+ oe), we have
Fz(u)
-
4rraG________~psgn(u)sin(ua) u
exp(-[u[h)
ua
(2.33b)
Fz(O) = 47ra GpH,
H--~ oc
2.3.4. Fault A fault in a layer of rock may be modelled as the effect of two semi-infinite planes vertically displaced as shown in Figure 2.5. First consider the gravity effect of the left half layer. We shall apply the result previously derived for vertical prism. We need to introduce two modifications, namely, shift the origin from the midpoint of AB to B and evaluate the contribution of side AD, which is now at infinity. The contribution of the side DA becomes zero as the thickness of the layer goes to zero when the top surface AB and the bottom surface DC meet at infinity,
Contribution of DA - H (1 - exp(JulH)) exp(lu](H + h) +jua)--+ O,
lu[H
H---~ 0
Thus, the effect of the semi-infinite layer is essentially due to side BC given by Eq. (2.31). After shifting the origin to point B we obtain the gravity effect of the left half layer as
Contribution of left half layer
(1 - e x p ( - l u l H )
luf
e x p ( - [u]h))
Similarly, we can evaluate the effect of the right half layer. The final result is given by
35
2D source models
F(u) =
27r Gp ]u
I
(1 - e x p ( - l u l H ) )
(1 -
exp(-lu] (h))
(2.33b)
I"1 xp(-I.IH))
exp(-lul(h + ~ h ) ) ]
(2.34)
2.3.5. Singularities Comparing Eq. (2.33) with Eq. (2.25) we observe that the field due to a rectangular cylinder may be looked upon as a field caused by four fictitious line sources placed at (+a, h) and (+a, h + H). Such fictitious line sources are presumed to have a frequency dependent radiation strength equal to +j(Gp/uZ)sgn(u). The position of the line source is naturally a point of singularity of the potential field. In the same manner the singularities for a fault may be found by examining the expression (Eq. 2.34). They are located at (0, h), (0, h + H), (0, h + Ah), and (0, h + Ah + H). The fictitious line sources at the points of singularity possess strength equal to +j(Gp/u2)sgn(u). The concept of singularity of a potential field has been widely discussed by several Russian scientists [11-14]. It is easy to extend the concept of singularity to any two-dimensional model with polygonal cross-section. In fact this approach was taken in Ref. [ 15] for the estimation of the comers of a polygon. We return to this topic in Chapter 5 where we deal with parameter estimation.
Figure 2.5. A layer of rocks is faulted along ff. The infinite layer is now reduced to two semiinfinite layers. We consider the gravity effect of the left half and right half semi-infinite layers.
36
Potential Field Signals and Models
2.4. 3 D s o u r c e m o d e l s
2.4.1. Potential field in frequency domain We describe two-dimensional signals generated by idealized models of excess mass or magnetization in the frequency domain. We start with an element of mass p dxo dyo dzo located at point (xo,Yo,Zo) (see Figure 2.6). The vertical component of a gravity field on the horizontal plane z = 0 is given by
fz(x,y)
Gpzo dxo dyo dzo [(X -- Xo)2 -I-- (y -- yo) 2 -1- Z2]3/2
(2.35)
where p is the mass density. The Fourier transform of the vertical component may be obtained as follows:
Fz(u, v) --
ff+
fz(x,y) exp(-j(ux + vy)) dx dy
oo
e x p ( - j ( u x + vy)) cc {(x - x0) 2 + (y - y 0 ) 2 + z2] 3/2
Gpzo dxo dyo dzo 7 "+~
dxdy (2.36)
Let r c o s 0 = x -x0, rsin0=y above equation reduces to
- Y0, u = s c o s ~
and v = s . sinc~. The
Figure 2.6. An elementary volume of density P and volume dv = dx0dy0dz0.
3D source models
37
Fz(u, v) -- 2rcGpzo dxo dyo dzo exp(-j(uxo + vyo))
f
~
rJo(,r)
[r 2 --1-Z213/2 dr
where we have used the integral
lf02
2re
e x p ( - j s r cos(O - a)) dO - Jo(sr)
The integral involving Jo(sr) can be evaluated using the results given in Ref. [8, p. 488]. The final result is
Fz(u, v) = 2rcGp dxo dyo dzo exp(-sZo) exp(-j(uxo + Vyo))
(2.37)
To obtain the corresponding result for the magnetic field we make use of Eq. (2.11) in Eq. (2.37) and derive the magnetic potential due to a dipole oriented in an arbitrary direction as given below
O(u,v)
-- -1
-~p (jmxu + jmyv - Smz)
Fz(,.,.)
(2.38a)
Next, substituting for Fz(u,v) from Eq. (2.37) into Eq. (2.38a), we obtain the Fourier transform of the magnetic potential
~(u, v) -- -2rC(l'mxU +jmyv - Smz)
exp(-sz0)
~xp(-j(xou + yo~)) dxo +o dZo (2.38b)
where m~, my, mz refer to the horizontal and vertical dipole moments, respectively. The magnetic field is measured along a given direction, for example, along the earth's magnetic field. For this we compute the gradient of ~p in the direction of the earth's magnetic field. Let a, /3, and y be direction cosines of the earth's magnetic field in the free space. The gradient operator in the direction of the field is given by 0
0
0)
Potential Field Signals and Models
38
In the frequency domain the gradient operator is equivalent to multiplication with a factor, (jau + j / 3 v - 7s). Let FT(U,V) represent the Fourier transform of the magnetic field in the direction of earth's field (remember that in an aeromagnetic survey the measured magnetic field is in the direction of the earth's field, that is, the total field is measured). Then
Fv(u , v) = 2rc(j'au +j/3v - 7S)(l'mxU + j m y v - Smz) •
exp(--SZo)
e x p ( - j ( x o u + yov)) dxo dyo dzo
(2.39)
To convert the Fourier transform of the gravity field due to a point mass to the total magnetic field due to a dipole of unit strength, the conversion filter is given by
1 (l'aU +jt3v - sT)(jlxu + j l y v - l~s) g ( u , v) -- Gp s It turns out that the same conversion trical body with uniform density and From Eq. (2.39) we note that for magnetic field may be approximated
(2.40)
filter is valid for any well defined geomemagnetization contrast. large s the Fourier transform of the total after ignoring the angular variation by
FT(u, v) => s exp(--SZo)exp(--j(uxo + vyo)) dxo dyo dzo,
s -+ oo
but for gravity, the large frequency behaviour is given by
Fz(u, v) ==~e x p ( - s z 0 ) e x p ( - j ( u x o 4- vyo)) dxo dyo dzo,
s -+ oc
Further, we note that FT(U,V) ~ 0, (u,v) ~ 0 but F~(0,0) ~ proportional to the total mass. This, in fact, is the basic difference between the gravity and magnetic field arising out of their fundamental difference in the monopole and bipole structures of the source. We have so far dealt with point or dipole sources. Now we generalize these relations to more complex source distributions.
3D source models
39
Figure 2.7. Variable subsurface density. The gravity field at the observation plane h units above the surface is obtained in frequency domain.
2.4.2. Variable density~magnetization model Let O(xo,Yo,Zo) represent density variation below a horizontal surface (ground surface) as shown in Figure 2.7. The Fourier transform of the gravity field may be obtained by integrating Eq. (2.37) over the volume containing excess mass.
Fzl.. ~, O) --
f !!!f !!!f p(Xo.Yo.Zo)~p(-.Zo/~xp(-j(UXo+ UYo))dXo dyo dZo volume
= 27rG
f0 ~P(u, V, Zo)exp(-sz0)
dzo
(2.41)
where oo
P(u, V, Zo) --
ff_+
oo P(x~176176 exp(-j(uxo + vyo)) dxo dyo
Further, express
P(u, ~,zo) - ~1
foo P(., ~, w)~xp(iwz0) dw
(2.42)
Potential Field Signals and Models
40
Substituting Eq. (2.42) in Eq. (2.41) and after evaluating the integral with respect to z0 and continuing the field to the observation plane, h units above the surface into free space, we obtain (see Figure 2.7).
F~(.. ~. h) - C f oo P(u. ~. w) exp(-sh) dw (s - j w ) (3O
(2.43)
The corresponding result for the magnetic field (total field) may be written as follows:
F~(.. v. h) -
(j~u + j ~ u - ~)[/UMx(U. ~. w)
(2O
+ jUMy(U, ~, w) - ~Mz(U, ~, w)]
exp(-hs)
(~ -jw)~ dw
(2.44)
where Mx(u,v,w), My(u,v,w) and Mz(u,v,w) stand for Fourier transform of the three-dimensional magnetic moment distribution. We can simplify Eq. (2.44) by assuming uniform direction of magnetization, namely, I = Ixex + Iyey + Izez
where ex, ey, and e~ are unit vectors in direction of x, y, and z coordinates respectively and similarly Ix, ly and Iz are components of the polarization vector in direction of x, y, and z coordinates respectively. Let K(x,y,z) stand for magnetic susceptibility variation; then
Mx(U. ~. w) = ~xK(.. ~. w) M.(u. ~. w) = I,K(u. ~. w)
Mz(U. ~. w) = IzK(.. ~. w) where K(u,v,w) is the Fourier transform of the susceptibility variation. Hence, [l'uMx + j v M y - sMz] -- K(u, v, w)(l'UI x + j v l y - slz) -- K(u, v, w)I. (]Uex + j v e v - Sez)
(2.45)
3D source models
41
Using Eq. (2.45) in Eq. (2.44), we obtain
FT(u, v, h) -- f ~ K(u, v, W) (l'C~u +j/3v - 7S)(l'UIx +jvly oo
slz) e x p ( - h s ) ( s - j w ) s dw (2.46)
As a consequence of non-uniqueness in potential field interpretation or principle of annihilation [32], an interesting interpretation of Eq. (2.41) follows. Eq.(2.41) may be looked upon as a Laplace transform,
Fz(u , v, h -- O) - 27rG f o ~ P(u, ~, zo) ~p(-,z0)
az0
Consider the possibility that
P(u, V, Zo) = P0(u, v)cos(pz0)exp(sz0) then the above integral is simply a delta function leading to Fz(u,v,h = 0) = 0 or fz(x,y,h = 0) = 0 for any p 4= 0, independent of the actual density distribution. In the case of magnetic field under a similar possibility we get fT(x,y,h = O)= O. Thus, the true distribution of density or magnetization can never be ascertained from the measured field in the free space as we can always append a source distribution of the type mentioned above without affecting the potential field. Thus we may express the three-dimensional density variation as a sum of two parts, namely, the observable and unobservable,
p(x0. y0. z0) = po~. (x0. y0. z0) + p..ob. (x0. y0. z0) The Fourier transform of Punobs(XO,Yo,Zo) is a separable function of the type, P1 (u, v)f(z0) exp(sz0) wheref(z0) is any periodic function with integer number of periods within the vertical extent of the source layer. The gravity field produced on the surface by such a distribution will be zero (also the magnetic field will be zero), independent off(z0).The implication of this observation is that in any 3D modelling, any amount of unobservable component may be appended to the model without affecting the observed field.
Potential Field Signals and Models
42
Figure 2.8. A rectangular prism with uniform density contrast O0 or susceptibility contrast K0.The prism is just below the surface. The observation plane is h units above the surface into free space. The potential field is computed directly in the frequency domain.
2.4.3. Uniform vertical prism As an example of a complex signal source let us consider a vertical prismatic body of uniform density or susceptibility variation. Let the prism have the dimensions 2a x 2b x 2c (see Figure 2.8). It is easy to show that
P(u, v, w) = 8abcpo sin c(au) sin c(bv) sin c(cw) exp(-jcw)
(2.47)
On substituting Eq. (2.47) in Eq. (2.43), we obtain the Fourier transform of the gravity field on the observation plane, h units above the surface (see Figure 2.8)
Fz(u, v, h) = 8Gabcpo sin c(au) sin c(bv) e x p ( - s h ) x
/
o~ sin c(cw)
oo s - j w
exp(-jcw) dw
(2.48)
The last integral in Eq. (2.48) can be evaluated by referring to tables of definite integrals( see, for example, Ref. [16, p. 170])
43
3D source models
c(cw) e x p ( - j c w )
o~ ( s - j w )
dw - rc (1 - e x p ( - 2 c s ) ) cs
Finally, we obtain Fz(u, v, h) - 87rGabpo sin c(au) sin c(bv) e x v ( - s h ) (1 - e x p ( - 2 c s ) )
(2.49)
s
This expression is derived in Ref. [ 17] and also by Bhattacharya [ 18] who used an approach of direct Fourier transformation of the spatial signal. Note that Fz(u,v,h) ~ 27rGoo2a2b2c as (u,v) --~ O. The corresponding expression for the magnetic signal (using the conversion filter (Eq. 2.40)) is given below: F r ( u , v, h) - 87rabt%(l'C~U + j f l v - sT) (l'uIx + j v l y - slz) x
sin c(au) sin c(bv) e x p s2 ( - s h ) (1 - e x p ( - 2 c s ) )
(2.50)
Notice that the Fourier transform of the total magnetic field is real at the poles or at the equator where it also vanishes along a radial line whose slope is equal to the negative of angle of declination (see Figure 2.9). In addition, if the inducing magnetic vector is horizontal, the Fourier transform vanishes along a radial line
Figure 2.9. The Fourier transform of the total magnetic field due to a uniformly magnetized prism in low latitudes vanishes along the radial lines. Here 6 stands for declination of the earth's magnetic field and 60 is declination of the inducing magnetic vector.
Potential Field Signals and Models
44
whose slope is equal to the negative of angle of declination of the inducing magnetic vector. Further, Fs(u,v) --+ 0 as (u,v) ~ O.
2.4.4. Singularities In order to show the location of singularities, we rewrite Eq. (2.49) as:
Fz(u , v,h) = -2rcGpo[exp(l'(au + bv)) - e x p ( / ' ( - a u + bv)) - e x p ( / ' ( a u - bv)) + exp(-j(au + bv))]
Fz(u,v,h)
=
exp(-sh)
(1 - exp(-2cs))
SI,t V
-2rrGpo [ exp(/'(au + bv)) e x p ( - s h ) suv
- exp(/'(-au + by))exp(-sh) - exp(/(au - by)) e x p ( - s h )
+ exp(-j(au + bv))exp(-sh) - e x p ( / ( a u + bv))exp(-s(h + 2c)) + e x p ( / ( - a u + bv))exp(-s(h + 2c)) + e x p ( / ( a u - bv))exp(-s(h + 2c))
-exp(-j(-au + bv))exp(-s(h + 2c))]
(2.51)
Note that each term inside the brackets in Eq. (2.51) corresponds to a field due to a point source (see Eq. (2.37)) located at (-a,-b,-h), (a,-b,h), (-a,b,h), (a,b,h), (-a,-b,h + 2c), (a,-b,h + 2c), (-a,b,h + 2c) and (a,b,h + 2c), respectively. Note that the z coordinate of the singular points is measured with respect to the observation plane. The resulting field is multiplied by a factor of 1/suv. The mass at the above location is either 0 o r - 0 according to the sign in front of each term in Eq. (2.51). The location of the point mass is a point of singularity, for the field in the close vicinity is bound to be very high, approaching infinity as the point is approached. Thus, the singularities of the gravity field due to a prism are located at all eight comers of the prism (see Figure 2.8). At each singularity
3D source models
45
point, a source with radiation pattern, 2~rG/suv, is said to be present. A similar interpretation is possible for the magnetic field due to a prism.
2.4.5. Prism with polygonal cross-section We now apply the method of computing the field in the frequency domain to a vertical cylinder with polygonal cross-section (Figure 2.10). The prism is just below the surface. The observation plane is h units above the prism. The density contrast is p0. The gravity field in the frequency domain is described by Eq. (2.42) in which the unknown function is P(u,v,w), the Fourier transform of the density distribution. In the present case,
p(xo,Yo,Zo) -- Po 0
when (x0,Y0) E P and 0 <_ z0 <_ 2c otherwise
where r is the area bound by the polygonal cross-section of the prism. Hence,
P(u, v, w) -- Po [ [ / " exp(-j(uxo + vyo + WZo)) dxo dyo dzo JJJ
pof f exp(-j(UXo+Yo)) dxodyof
exp(-jwzo) dzo
2poc sinc(cw)exp(-jcw) f /exp(-j(uxo + vyo)) dxo dyo (2.52)
Figure 2.10. A vertical prism with polygonal cross-section. The potential field is computed directly in the frequency domain.
Potential Field Signals and Models
46
The surface integral is expressed as a line integral as shown in Section 2.2.
f / exp(-j(uxo + vyo)) dxo dyo - - f exp(-j(uxO_jv+ Vyo)) dxo
f
exp( - j
--fyodxo
(UXo + vyo) ) dyo
whenu-v-0
when v 4= 0
when u 4= 0
(2.53)
When u 4 : 0 and v 4= 0 one may use any one of the first two line integrals in Eq. (2.53). The line integrals are evaluated in the same manner as in Section 2.2. We give the final result,
a(" exp( - j
j.
(UXo + vyo ) ) -jr
dxo
M-I
Z 1 (*; -jv*;-') exp(_j(u2i_l _k- V ~ i _ l ) ) ( 1 - exp(--cti))ozi exp(
when v 4= 0
- j (UXo + vyo ) )dyo -ju
M-I (Y* 1
)
exp(--j(u2i_l -+- v~i_l ) ) ( , 1 - exp(-cti) i
when u 4= 0
ju 1M_I
-
yodxo
2Z
(2i - ki_, )@i - Yi-1 )
when u -- v = 0
1
where oli =j[u(2i-2~_,) + v(~g-.Yi-I)]. Substituting Eq. (2.53)into Eq. (2.52) and using the result in Eq. (2.43), we can now write down the Fourier transform of the gravity field at the observation plane:
47
3D source models
m m l
84
Fz(u, v, h) = Gpo e x p ( - s h )
exp(-j(udci_l + v~i_, )) 1
jv
~o o sin c(wc) s -jw
•
dw
The last integral in the above equation is evaluated as in Eq. (2.48). We obtain finally,
=
27rGpo
• ~
exp(-sh)(1 - exp(-2cs))
(ki - xi-1) exp(-j(uki_l + vj;i_,)) 1
JV
(
1 - exp(-ai) OLi
)
(2.54a)
This is valid when v 4: 0. Similarly we can derive an expression valid for u 4 : 0 and when u = v - 0. This we will leave as an exercise for the reader. The aeromagnetic field (total field) of the polygonal prism, in particular, its Fourier transform may be obtained from Eq. (2.54a) by multiplying by the gravity-tomagnetic conversion filter (Eq. 2.40). We obtain
FT(U, v, h) -- 27rt%(j'au + ; f l v - sT)(jlxu +jlyv - Izs) exp(-Sh)s2 ( 1 - e x p ( - 2cs))
X~1 (Xi- Xi--1)exp(_j(uSci_l_+_vYi_l)) (1 1
JV
-- exp(--ozi))
(2.54b)
OLi
Eq. (2.54) is a useful frequency domain representation of the potential field signal. Since the Fourier transform of the potential field signal is directly available, there is no need to evaluate complex integrals often encountered when evaluating the Fourier transform integral (see, for example, Ref. [ 18]). From its frequency domain representation it is easy to go to the space domain by computing the inverse Fourier transform. As an example we show the gravity and magnetic field of a cube using the method of representing the potential field signal in the frequency domain and then computing its inverse Fourier transform. The model and its parameters and the computed profiles are shown in Figure 2.11.
48
Potential Field Signals and Models
0.06~-
W o.. f-
gravity field matrix size A9 - 16 x 16 B - 32 x 32 C-64 x64
~0.04 !,._ i. m C "0
I O.O: A
o
,',,\',r
5
10 distance
15
3
Figure 2.11. The vertical gravity field due to a cube (2 • 2 x 2 units 3) computed using the frequency domain approach described here. Open circles represent gravity field computed using the formula given in [19].
Further details may be found in Ref. [17]. As a note of caution it may be mentioned that unless the size of the signal matrix is large (several times the depth expressed in units of sampling interval) there may be serious truncation and aliasing errors [17]. In the above computations we have used different matrix sizes and compared the results with those obtained directly from the formula given in Ref. [19]. The frequency domain approach for the potential field signal has been applied to more complex 3D source models such as a polyhedron with triangular facets [20]. Indeed, a very complex source model may be decomposed into many simple prisms and later linearly combined to form the closest approximation to the actual model.
3D source models
49
2.5. Stochastic models I: r a n d o m interface
The central theme in the current section and in the following section is that the potential fields are caused by stochastic sources of two types, namely, a random interface separating two homogeneous media; for example, sedimentary rocks overlying a granitic basement and a horizontal layer of finite thickness wherein the density or magnetization is varying randomly. Our aim is to relate the stochastic character of the potential fields to that of the interface or the layer, in the hope that from the stochastic properties of the observed potential field we can determine some gross features of the source model, for example depth to the interface. The rationale behind stochastic modelling is that the natural variation of the density or magnetization, being the outcome of several independent factors, is likely to be extremely complex and is best described under a stochastic framework.
2.5.1. Stochastic field We first develop the necessary tools for stochastic characterization of the potential field caused by a random distribution of sources. The spectral representation of a stochastic field is a useful tool in deriving a compact form of useful results connecting the second order statistical structure of the observed field and the unknown source distribution. Let fix,y) be a homogeneous (stationary) stochastic field. It is possible to represent it in a form analogous to the Fourier integral representation of a finite energy function. Note that a stationary random function does not meet the condition of finite energy, hence the standard Fourier integral representation cannot be used. The spectral representation (also known as the Cramer representation) of a homogeneous random field is given by
f (x, y) --
l j'j"
dF(u, v) exp(j(ux + vy))
where
dF(u, v) = F(u + du, v + dv) - F(u, v),
(du, dv) --~ 0
and F(u,v) is the generalized Fourier transform offlx,y) (see for more details Ref. [21]). dF(u,v) possesses many interesting properties, some of which are listed below:
Potential Field Signals and Models
50
E ~ 21
dF(u,v)}
-- E{f(x,y)},
--0
i.e. mean
of f(x,y)
when u -- v - 0
when (u, v) 4= 0
E{dF(u, v) dF* (u', v') } -- 0
(2.55)
where the set of points centered at (u,v) and the set at (u',v') do not intersect (see Figure 2.12). When the two sets are completely overlapping we have
1 1 E ~ 1 dF(u,v) -~2dF*(u, v) } - --~g~2Sf(u, v) du dv
(2.56)
where S~u,v) is the spectrum (power spectrum) of the stochastic field f(x,y). The spectrum of a stochastic process is discussed further in Chapter 3. Next, we derive some of the basic properties of the random potential field in free space, similar to those in Section 2.1. Let ~p(x,y,z) be a random potential field in free space. Let us express the spectral representation of ~,(x,y,z) as follows:
99(x,y,z) -- ~
dO(u, v)H(u, v,z) exp(/'(ux
+
vy))
(2.57)
(2O
where H(u,v,z) is selected so that ~o(x,y,z) satisfies Laplace's Eq. (2.1) and dO(u, v) is a random function. Substituting Eq. (2.57) into Eq. (2.1) we obtain an equation governing H(u,v,z),
Figure 2.12. (a) Non-intersecting and (b) intersecting sets of points in the Fourier plane.
51
Stochastic models I: random interface
d2H(u, v,z) dz 2
-- (U 2 -t- v 2 ) H -- 0
The solution of the above differential equation is simply given by H(u, v, z) -- exp(4- V/b/2 Jr- 1/'2 Z)
(2.58)
Note that the +ve sign is not acceptable as it would lead to a non-decreasing potential as z --+ +oo. Substituting Eq. (2.58) in Eq. (2.57) we obtain a spectral representation of the potential field [22-25].
g~(x, y, z) -- ~
,jJ+
dO(u, v)exp(-sz)exp(/'(ux + vy))
(2.59)
oo
The spectral representation of three components of the potential field is easily obtained by differentiating Eq. (2.59) with respect to x, y and z. 0~ 0x
(2rr) 2
fy(x ' y, z) -
Oqp _ Oy --
1 2 f / o o ooJv dO(u, v)exp(-sz)exp(l'(UX + vy)) (27r)
fz (x, Y, z) -
Oqpoz-- ~,rr))2 1 f j l ~176 --`2 s dO(u, v)exp(-sz)exp(/'(ux -+- vy) )
fx(x,y,
=
l f f_ooJu dO(u, v)exp(-sz)exp(/(ux
+ vy))
(2.60)
co
The spectrum of the potential field and its components may be obtained from Eq. (2.59) and by using a relation between the spectrum and the differential of the generalized Fourier transform (see Eq. (2.56)). We obtain the following results:
S~(u, v, h) - S~oo(u, v)exp(-2sh) Sfx (u, v, h) - S~oo (hi, V)U 2 e x p ( - 2 s h ) Sly (hi, v, h) -- Sqpo (hi, V)v 2 e x p ( - 2 s h )
Sfz (u, v, h) - S~o (u, v)s 2 exp(-2sh)
52
Potential Field Signals and Models
Figure 2.13. Random interface separating two media of uniform density (magnetization). The gravity (magnetic) field may be related to the interface, considered here as a homogeneous random surface.
where S~0 (u, v) is equal to the spectrum of the potential field at z = 0 level and S~(u, v, h) is the spectrum of the potential at z = h level. It is interesting to note that given the spectrum of any one component we can deduce the spectrum of other components. The relationship is
U2
V2
S2
=S~(u,v,h)
(2.61)
We can compute the cross-spectrums between different components using Eq. (2.60). Without going into the details, the final result is
SLt) (u, v, h) - uvS~o (u, v) e x p ( - 2 s h ) SLIi (u, v, h) -- -jusS~o (u, v) e x p ( - 2 s h )
(2.62)
Sf,jz (u, v, h) - -jvsS~o (u, v) e x p ( - 2 s h ) From Eq. (2.62) we note that the vertical component is 90 ~ out of phase
53
Stochastic models I: random interface
(lagging) with respect to the two horizontal components, but the horizontal components are themselves in phase.
2.5.2. Random interface In Figure 2.13 we show a random interface separating two media. Let the potential field be observed on a plane h units above the interface. We assume that both media are otherwise uniform in density or magnetization. As far as the potential field in medium I is concerned, we can replace it with a vacuum (that is, density/magnetization = 0) and the lower medium with a medium whose density or magnetization is equal to the difference in density or in magnetization of medium I and medium II. Thus the problem is reduced to a semi-infinite medium whose upper surface is a random function. We first present a solution to a simpler problem of gravitational field caused by an interface. We model the interface as a homogeneous random function zXZ(xo,yo) so that the mean surface (shown by a dashed line) is a horizontal plane. Now it is easy to see that the observed field will consist of two components, namely, a constant part representing the field due to a semi-infinite medium with its upper surface as a horizontal surface, and a random component representing the field due to the random interface. The vertical gravity field due to a random interface on the observation plane may be obtained from Eq. (2.35) by integrating over the striped region (see Figure 2.13b) where the density is -Z~O below the mean surface and +Ao above it, where
Z~k/3-- (Pmedii --/3medi).
fz(x,y,h) - - f f +~~ fzXz
GAp(h + Zo)
[ ( x - x0) 2 + ( y - y0) 2 + (h + z0)2] 3/2
dxo dyo dzo (2.63)
where (xo,yo,zo) represents a point in the striped region. We assume that 2~Z(xo,yo) is a homogeneous (stationary) random function. The reference plane is the mean plane passing through the interface (dashed line in Figure 2.13). We hope to express Eq. (2.63) in the frequency domain in terms of the generalized Fourier transform of ZXZ(xo,Yo).We first evaluate the last integral. Let p = h + z0 and express the inner most integral in Eq. (2.63) as /x~
~o
GAp(h §
Zo)
[(X- X0) 2 + ( y - y0) 2 -Jr-(h -4--Zo)2]3/2 dZO
54
Potential Field Signals and Models
f
h+~
GApp
[ ( x - Xo)2 4- ( y - yo) 2 q_p213/2 dp GAp
GAp
[ ( X - XO)2 -+- ( y - - y o ) 2 -Jr- h2]l/2
[(X -- XO)2 -[- (y -- yo) 2 + (h + Az)2] 1/2 (2.64)
Consider the Taylor series expansion of
g(z) --
[(X -- XO) 2 -}- (y --yo) 2 Jr- Z2] 1/2
in the neighbourhood of z = h Z~z2 ZXz3 g(h + A z ) - g(h) + g ' ( h ) A z + g"(h)--aT, + g'"(h)--aT, + . . . z! --.3! where all derivatives of g(h) may be written in terms the 2D Fourier transform of g(h). Let
G(u, v,h) -
f f + ~ exp[--j(u(x- Xo) + v O, -- Yo))] d(x- Xo) d(y cx~ [(x - Xo) 2 Jr- (y - - y o ) 2 -f- h2] I/2
-Yo
) (2.65)
Differentiating with respect to h on both sides of Eq. (2.65), we obtain
0 O~ a(,,, ,,, h) .
.
f / + o o hexp[-j(u(x - Xo) + v(y -yo))]d( x . . [ ( x. - Xo) . 2 .q-- ( y. - yo) . 2 .q- h2]. 3/2
xo)d(y
Yo)
oo
= -2re e x p ( - s h ) Therefore,
G(u, v, h) - 2re
exp(-sh)
(2.66)
Stochastic models I: random interlace
55
All derivatives in the Taylor's series expansion may be expressed in terms of
G(u,v,h)
lff§ lj]+ g"(h) -- ~ g'(h) - ~-~
- exp(-sh) exp(l'(Ux + vy) ) du dv co
s exp(-sh)exp(/(ux § vy)) du dv oo
g'"(h)
: ff_+oo _s 2 exp(-sh) exp(/(ux + vy)) du dv
G
(2.67)
and so on. Using the Taylor's expansion of g(h) in Eq. (2.64) we obtain
fz(x,y,h) -
//+cooc { g'(h)Az
+
gttAz2
z~XZ3}
(h)--~-. + g'"(h)-f(-. + . . . GAp dxo dy o (2.68a)
When the interface is of infinite extent, we will have to use, in place of the ordinary Fourier transform, the generalized Fourier transform or spectral representation of a random function (see p. 49). Using Eq. (2.67) in Eq. (2.68a)the following result is obtained:
-s)"-' exp(-sh) exp(/(ux + vy)) fz(x,y,h) -- --1~ G A p f foo y~. dAZn(u , v ) ~ :xD n = l
n!
(2.68b) where
zXzn(x,y) -- ~ 1 f f_ ~ dAZn(u , v)exp(/'(ux + vy)) co When AZ(xo,Yo) << h it is sufficient to consider the first term in Eq. (2.68b) giving
fz(x,y,h) - --1~ G A p f f ~ dAZ(u, v)exp(-sh)exp(j(ux + vy)) (?o
(2.68c)
56
Potential Field Signals and Models
2.5.3. Magnetic field The magnetic field due to a random interface may be obtained either by repeating the mathematical procedure for a magnetic field or by making use of the relationship between the magnetic and gravity potential (Eq. 2.11) and the gradient operator in the frequency domain. The latter approach is simpler and hence we adopt it. First we need to obtain the gravity potential from Eq. (2.68). It is given by
c~(x, y, h ) =
GAP2rc/ f + ~ ~cx) gAIn(U,s v) (--s)n-ln! exp(-sh) exp(j(ux + vy)) cx) n--1
Now use Eq. (2.1 l) to obtain the magnetic potential. Then, multiply the magnetic potential with the gradient operator, namely (j'au + j [ 3 v - ys) where (c~,/3,y) are the direction cosines of the gradient direction. Thus, we obtain
f
(x,y,h) = - - - l ff+ 27r
~
r(u,v) s
dzXZ,(u, v) n=l
nV
exp(-sh) exp(j(ux + vy)) (2.69a)
where 1-'(u,v) = (jmxu + jmyv - mzS)(j'au + j [ 3 v - 3's) and (mx,my,mz) are components of magnetization. Note that mx = IAXK, my = lyZ~K and mz = IzAK, where AK = (KmedII- Kmedl).When IzSe(xo,yo)l << h,
1 ff+ fv(x,y,h)
2Ir J J _ ~
r(u v)
~ ' s
exp(-sh)dAz(u v) exp(j(ux + vy))
(2.69b)
Eqs. (2.68c) and (2.69b) may be rewritten to emphasize the filter concept. Let H(u,v) be filter transfer function and AZ(u,v) be the input. Then, the output of the filter is the gravity or the magnetic field (see Figure 2.14),
Fz(H , 1,',h) = HGrav(U , v, h)AZ(l,l, v) FT(H , V, h) -- HMag(U , v, h)/kZ(u, v)
Stochastic models I" random interface
57
Figure 2.14. The role of the interface is shown as an excitation function to a filter, the output is the observed potential field.
where (2.70a)
HGrav(U , 12, h) -- - 2 7 r G A p e x p ( - s h )
and HMag(U , V ,
271h)
--
- -
s
(jmxu + j m y v - mzS)(l"au + j / 3 v - 7 s ) e x p ( - s h )
(2.70b)
In Eqs. (2.68) and (2.69) a constant term representing the gravity or magnetic field due to half space is not included. 2.5.4. Prism model
There is an alternate method of computing the effect of an undulating interface separating two homogeneous media. The interface is replaced by a layer enclosing the interface as shown in Figure 2.15. Consider a small prism of height 2d (the thickness of the equivalent layer) and cross-section dx dy. The average density of the material inside the prism is equal to Observation Plane
Po
h o,ace
. Figure 2.15. An interface separating two media is replaced by a layer of slender prisms of constant height but variable density. The prisms are truncated from above by upper surface and from below by lower surface. The height of the prisms is equal to the vertical separation between the surfaces.
Potential Field Signals and Models
58
1 Since our interest is in the spatially varying potential field we shall ignore the constant term in the density expression. Referring to Eq. (2.49) we express the gravity field caused by a single prism located at (x;, y;) in the frequency domain,
z~Fz(u V, xi,Yi) -- 7FG(Do - Pl) exp(-sh) (1 - exp(-2ds)) ' sd
x Az(xi, Yi) dxi dyi exp(--j(ux i + v dyi) )
Finally, the total field due to all prisms, assumed to be finite, will be given by exp(-sh) sd ( 1 - e x p ( - 2ds) )
Fz(u , v) = 7rG(p 0 - Pl)
Z
A z ( x i , Y i ) dx i d y i e x p ( - j ( u x
i -+- v dyi) )
all prisms - 7rG(po - p,)
exp(-sh) sd (1 - e x p ( - 2 d s ) ) A Z ( u ,
v)
(2.71)
where AZ(u,v) is the discrete Fourier transform of 2~z(xi,y~). Similarly, we can write down an expression for the magnetic effect of an interface separating two media with finite susceptibility contrast but polarized by the same field.
FT(U, v) = 7r(,% -- '~I )
e x p ( - s h ) (1 - exp(-2ds)) s sd
• (l'Ua +jv/3 - sT)(l'ul x + j v l y - s l z ) A Z ( u , v)
(2.72)
where r0 and El are susceptibility constants in the upper and lower mediums, respectively. For small d, Eqs. (2.71) and (2.72) reduce to Eqs. (2.68c) and (2.69b). 2.5.5. L a y e r e d strata
Perfectly horizontally layered strata resting on a flat basement rock will not
59
Stochastic models I: random interface
Figure 2.16. Layered strata resting on an uneven basement rock. produce any potential field signal. However, when the layered structure is deformed or folded (see Figure 2.16) it produces an effect characteristic of the deformation. We relate such a signal to the nature of deformation. Consider a stack of layers, each of thickness Ah and of uniform density. All layers are parallel to each other. The interface separating two layers is modelled as a homogeneous random surface, 2tZ(xo,yo). The media below and above the stack of layers are of uniform density. The vertical field due to a single interface is given by Eq. (2.68c). Since the process is linear, the field due to N interfaces is the sum of fields due to N interfaces taken individually. For example, take the case of two interfaces. This may be looked upon as a superposition of two simpler models as shown in Figure 2.17. The gravity field due to each interface may be easily obtained by applying the formula (Eq. 2.68c). We obtain an expression in the frequency domain for the gravity field caused by the layer on the observation plane h units above the mean surface of the first layer,
dFz(u, v, h) --- -27rG[(pl - Po) + (P2 - P l ) e x p ( - s A h ) ] e x p ( - s h ) dAZ(u, v) (2.73) Continuing in the above manner we can derive an expression for the general case of N interfaces of ( N - 1) layers.
dFz(u, v,h) - -2~rG[(pl - P 0 ) + (P2 - P l ) e x p ( - s A h ) + (/93 /92)exp(-2sAh) -
-Jw"""-Jr-(jON_1
-
-
-
D N - 2 ) e x p ( - ( N - 2)sAh)] e x p ( - s h )
dAZ(u, v) (2.74)
Potential Field Signals and Models
60
P
h
P2
Ah
(a)
P3
h
(b)
h+ A h
(c)
P2
0
P3-P 2 Figure 2.17. (a) A single layer may be represented by two simpler models (b and c), each having one interface whose effect is computed as in Eq. (2.68c). Fields due to simpler models are linearly combined. Notice that the expression inside the square brackets may be expressed as the Laplace transform of a density gradient. To show this, let us consider the limiting form (Eq. 2.74)
(P3 -- P2) (p, - po) + (p~ - p , ) exp(-2sAh) exp(-sAh) + Ah Ah Ah
61
Stochastic models I" random interface
...+
,
f
Ah-+O
(PN-1
-
-
Ah
" dp(h) ~
dh
PN-2) e x p ( - - ( N - 2)sAh)] Ah
J
exp(-sh)dh
(2.75)
where H - ( N - 2)Ah is the total thickness of the layered strata. Evidently Eq. (2.75) is a Laplace transform of the vertical derivative of the density function for large H. Using the above result in Eq. (2.74) we obtain
dFz(u v, h) - -2rcG exp(-sh)dAZ(u v)/oo ~dp(h) exp(-sh) dh /
,
N
'
'
do
(2.76a)
dh
The integral in Eq. (2.76a) may be written in terms of the Laplace transform of the density log,
-
1 fj~+ao P(s)exp(sh) ds
which, on differentiating both sides with respect to h, we obtain
dp(h)
dh
1
ri/oo+aosP(s) exp(sh) ds 2 rrj d -joo+ao
or after taking the inverse Laplace transform on both sides we obtain
sP(s) - foo dp(h)dh_ e x p ( - s h ) d h Finally, it is possible to express (2.76a) in terms of the Laplace transform of the density log
dFz(u, v, h) - -2rcG exp(-sh) dAZ(u, v)sP(s) An analogous expression for the magnetic case is as follows:
(2.76b)
Potential Field Signals and Models
62
Input:
Gradient of depth dependent density/susceptibility
AZ(Xo,Yo)
Output: Potential field signal
Figure 2.18. The role of an interface separating two media is that of an excitation function of a filter. 27r dFT(U , v, h) - - - - (]'au + j / 3 v - 7 s ) ( j l x u + j l y v - Izs) e x p ( - s h ) d A Z ( u , v)
x
f
d
(h) _ exp(-sh) dh
dh
(2.77)
where K(h) is the susceptibility variation with depth and other symbols are as in Eq. (2.72). In terms of the Laplace transform of the susceptibility log, Eq. (2.77) may be expressed as 27r
d F r ( u , v) = - - - ( j a u + j ~ v
- 7 s ) ( j l x u + j l y v - l~s) e x p ( - s h ) d A Z ( u , v ) s K ( s )
(2.78) We have an interesting interpretation of Eqs. (2.76) and (2.78). The role of the interface is that of an excitation function to a filter whose transfer function is given by the Laplace transform of the vertical gradient of density or magnetization (see Figure 2.18). In the absence of interface undulations, that is, A z ( x , y ) = 0, the depth dependent density or magnetization does not produce any potential field signal. Such source models have been discussed by Serbulenko [26]. 2.6. Stochastic model II" random m e d i u m
In this section we consider a stochastic model of layered rocks where the physical parameters such as density or magnetization are varying randomly inside a thin layer, thick layer, and finally in semi-infinite medium. 2. 6.1. Thin layer
As a simple example of a random medium, we consider a thin horizontal layer (see Figure 2.19) of thickness dz0. Consider a small element of volume dxodyodzo.
Stochastic model II: random medium
63
ObservationPlane
z0
~.-,,y,,,i,,_v,,.`'~`:'.`:.'~:.`'.``..~`y.`.~`~`~'~y'`y~y~>5~y~.>y.<~'`y~y~y.`y.`y.`~`y~y~y~y~y~ dZ 0 Figure 2.19. A thin layer of random density or susceptibility variation at a depth z0. The gravity/ magnetic effect on the surface is given by Eqs. (2.81) and (2.79) respectively.
This acts as a small dipole with a dipole moment m(xo,yo,zo)dxodyodzo where m(xo,Yo,Zo) is a vector representing dipole density. Let mx(xo,Yo,Zo), my(xo,Yo,Zo) and mz(xo,Yo,Zo) represent the three components of magnetic moment density. The magnetic field (total component) due to an element of volume of magnetized layer may be derived from Eq. (2.39) where we replace the constants mx, my and mz by mx(xo,Yo,zo)dxodyodzo, my(xo,Yo,zo)dxodyodzo and m~(xo,Yo,zo)dxodyodzo respectively and integrate over the entire layer. Further we treat mx(xo,Yo,Zo), my(xo,Yo, z0) and m~(xo,yo,zo) as homogeneous random functions in (x0, y0), that is in the horizontal plane with a spectral representation; for example,
mx (Xo, Yo, Zo) -- ~
dM x (u, v, z o) exp (j"(ux0 4- vyo ) ) oo
Using the spectral representation of mx, my and mz we obtain the following result:
fr(x,y)
~
dM(u, V, Zo)(l'Cm +j/3v-'ys) exp(-szo) dz ~ exp(/(ux + vy)) oo
S
(2.79) where
dM(u, V, Zo) --ju dMx(u, V, Zo) + j r dMy(u, V, Zo) - s dMz(u, V, Zo)
(2.80a)
When the magnetization is due to induction alone we can express Eq. (2.80) in terms of the susceptibility variations and the inducing field,
64
Potential Field Signals and Models
dM(u, V, Zo) - )'uIx +jvly - slz] dK(u, V, Zo)
(2.80b)
where I = (Ix,Iy,Iz) is the inducing field vector and dK(u,v,zo) is the differential of the generalized Fourier transform of random susceptibility variations within the layer. Let us now consider the corresponding problem of gravity field. Let O(xo,Yo,Zo) represent density variation inside the layer. Consider a small element of volume dxodyodzo. The mass of the material inside the elementary cell is equal to o(xo,Yo,zo)dxodyodzo. The gravity field (vertical component) due to an elementary cell may be obtained from Eq. (2.37). Further, the field due to the entire layer is obtained by integrating over the layer and making use of the spectral representation of o(xo,Yo,Zo), considered here as a homogeneous random function in (xo,yo).
G dP(u, V, Zo) exp(-szo) dz o exp(/'(ux + vy))
fz(x,y) -- ~
(2.81)
oc
where dP(u,v,zo) is the differential of the generalized Fourier transform of a random function representing the density variations within the layer. It is interesting to observe that the magnetic field and the gravity field due to a thin layer given by Eqs. (2.79) and (2.81) respectively are similar in form. Indeed the two fields are related to each other provided there is a functional relationship between magnetization and density variations.
2.6.2. Thick layer From the thin layer model, we go on to a thick layer model. The field due to a thick layer may be obtained by integrating Eq. (2.81) with respect to z0 from the surface to the bottom of the layer (see Figure 2.20).
1 /f+~ fz(x,y) -- ~
Gexp(l'(UX + vy)) oo
I h+AhdP(u, V, Zo) exp(-szo)
dzo
(2.82)
Jh
and the total magnetic field due to a layer where the susceptibility varies randomly is obtained by integrating Eq. (2.79) with respect to z0. We obtain
1 /f+oo (l.ul x +jVIy - SIz)(l'aU +jflv - "ys) exp(/'(ux + vy)) f T (x, y, h ) -- -~-~ oo s
65
Stochastic model II: random medium
Figure 2.20. A thick layer of rocks with random density or susceptibility. Its gravity and magnetic effects are given by Eqs. (2.82) and (2.83) respectively.
x
f
(h+~Xh)
dK(u, V, Zo)exp(-sz0) dzo
(2.83)
where Ah is the thickness of the layer and h is the depth to the top of the layer (see Figure 2.20). Sometimes it is more convenient to treat a layer as made up of many tiny prisms with uniform but random density or magnetization (see Figure 2.21). Other types of randomly distributed models, for example, circular cylinders, dykes, have been considered in Refs. [30,31]. Let each tiny prism have a rectangular cross-section, 2a x 2b and height Ah. The gravity field due to a single prism is given by Eq. (2.49), which is now summed over the entire layer to obtain the gravity effect,
fz(x, y, h) - -2 / / + 0 0 71"
Gab sin c(au) sin c(bv) exp(-sh)
oo
S
x (1 - exp(-Ahs))P(u, v)exp(l'(UX + vy)) du dv
(2.84a)
where oo
P(u, v) -- y ~ m=-cxD
oo
~
,ore,,,exp(-j(2amu + 2bnv) )
n-cx~
and Pm,n is the density of the (m,n)th prism. We give without derivation an expression for the magnetic case where we assume that the susceptibility is random but the direction of magnetization is fixed.
Potential Field Signals and Models
66
Observation Plane
h
Figure 2.21. A thick layer of rocks is made up of many slender rectangular prisms with uniform but random density/magnetization.
fT(x,y,h)
--
2ff_
-
7r
absinc(au)sinc(bv)(jlxu +jlyv-Izs)(j'cm + j f l v - T s )
cx)
exp(-sh) S2
(1 - e x p ( - A h s ) ) K ( u , v) exp(j(ux + vy)) du dv
(2.84b)
where oo
oo
K(u, v) -- ~
~
m---oc
~,,,,,, exp(-j(2amu + 2bnv))
n-oo
2.6. 3. Half space We now obtain the potential fields due to a half space filled with rocks whose density or magnetization is randomly varying. For this purpose, in Eqs. (2.82) and (2.83) we take h = 0 and let Ah--+ oo. We need to evaluate the integral with respect to z0. In Eq. (2.82) we can express
lf+
dP(u, V, Zo) -- ~
dP(u, v, w)exp(/'WZo)
oo
and then evaluate the integral with respect to z0. We obtain
Stochastic model II: random medium
Loo
67
1/+oo
UP(u, V, Zo) e x p ( - s z 0 ) dzo - - ~
UP(u, v, w)
foo
e x p ( - ( s - j w ) z o ) dzo
oo
1 f ~ dP(u, v, w) 2re J_oo s - j w
(2.85)
Using Eq. (2.85) in Eq. (2.82) we obtain
fz(x, y) -- -~2
oo G exp(l(UX + vy) )
oo
(2.86a)
s -jw
and a similar expression for the magnetic field
f r ( x ' Y ) -- ~ x
1f
oo
v.U x
s
+j~v-Ts) exp(/'(ux + vy))
aX(u, v, w) oo
- s -jw
(2.86b)
Gudmundsson [27] appears to have been the first to obtain Eq. (2.86b) with a minor difference.
2.6. 4. Undulating layer with random density or magnetization We now combine the stochastic model of an interface with that for the threedimensional medium, in particular we consider a random layer bounded from above and below by random surfaces as shown in Figure 2.22. We assume the layer is thin and that the variation of density or magnetization is a function of x and y only. Above the layer there is material of constant density or magnetization (e.g. sea water) and below the layer the material is of uniform density or magnetization. The mean depth to the top of the layer is h l and that to the bottom of the layer is h2. Such a model has been used to represent a mid-oceanic crust [33]. The first step to solve the above problem is to express it as a linear combination of two problems, namely: (a) semi-infinite random medium is bounded from above by the top random surface. The density variation in the semi-infinite medium is given by
Potential Field Signals and Models
68
Oo + Ao(xo,Yo). (There is no vertical variation). (b) Semi-infinite random medium is bounded from above by the bottom random surface. The density variation in the semi-infinite medium is given by
~Xo(xo,yo). In problem (a) let the density variation be given by O0 + Ao(xo,Yo) and correspondingly the density variation in problem (b) is given by Ao(xo,Yo). The potential field due to a semi-infinite medium bounded from above by an undulating surface may be derived following the approach used in deriving Eq. (2.64) where the limits on the integral are now from h + Az to ~ (see Ref. [32] for details). The final result for model (a) is
fza(x,y) -
ff_
+~
G[p o +/kp(xo,Yo)]{g(hl) + g ' ( h l ) A Z -5 s
Az2
+ g"(hl)--~-. +
g'"(h
Az3
1)-~--.i+"" "} dxo dyo
(2.87a)
and for model (b)
fzb(X,y) -- G
ff_
+~ Ap(xo,Yo)){ g ( h 2 ) + g'(hz)AZ + O(3
Az 2 Az 3 + g " ( h 2 ) ~ + g"'(h2)7. + . . . }dxo dyo
(2.87b)
where g'(h), g"(h), g'"(h), ... are given in Eq. (2.67). Define
Figure 2.22. A layer of random density or magnetization bounded by two parallel random surfaces at average depths hi and h2.
Stochastic model II: random medium
69
qa (Xo, YO) = (Po + Ap(xo, YO))Az'~ (Xo, YO) b
n
q. (Xo. Yo) - Ap(xo. Yo ) z~kz1(Xo.Yo) Note the difference between our result (Eq. 2.87) and that given in Ref. [32]. This is due to the fact that a +ve z0 axis points downward. Using Eq. (2.67) in Eq. (2.87) and using the spectral representation of qa'b(x,y) we obtain for problem (a) 1
ff+oo
fa (x, y) - - 2---sG j
j
(;~
cc
1()
~ a ~ (u, v) .-s "_~exp(-Shl) n!
n--O
exp(/'(ux + vy))
and for problem (b)
f zb( X . y ) --
_Z 2 rc G
~ oc
d Qb~( u v) ( - s ) ~-----~lexp(-sh 2) exp(/'(ux + vy))
n--O
n!
~
where
lff+ lff§
(Po + Ap(xo.Yo))Az~(xo.Yo) -- - ~
dQa(u, v)exp(/'(UXo + vyo))
oo
Ap(xo.Yo)Az~(xo.Yo) -- ~ 2
dQb~(u, v)exp(/(UXo + vyo))
oo
By subtracting the field due to (b) from that of (a) we obtain the field due to the model shown in Figure 2.22.
f~(x,y) --
_ •2rcG 1
fj+ oo ~
Cx:)
n--0
~~" +oo oc n=0
dQa(u, v) ( - s ) ~ ~ exp(-sh 1) exp(/'(ux + vy)) n!
_s)n-' n!
exp(-sh:) exp(/(ux + vy)) (2.88a)
70
Potential Field Signals and Models
The corresponding expression for the magnetic case for the same model but with uniform magnetic polarization vector is as follows: 1
f v(x, y)
2re 1 -[- ~
oo
+o~ 1-'(u, v) ~ dQa( u v) (-s)"-' exp(-Shl) exp(/'(ux + vy)) oo s ~=0 ' n-----~ +~ 1-'(u, v) ~ cx3
S
dQ~(u, v) (-s)" 1 ~
n!
n=0
exp(-sh2)exp(/'(ux -Jr-vy)) (2.88b)
where
lff+
(~o + Aec(xo,Yo))Az'~(x,Y) ---~2
dQa(u, v)exp(/'(ux0 + vyo))
OO
Aec(xo,Yo)AZ'~(xo,Yo) -- ~
' ff_+
dQ~(u, v)exp(j(ux0 + vyo))
(2.88c)
oo
An interesting possibility arises when oo
dQa'b(u, v) (-s)"-' n! = 6(u)6(v)
(2.89)
n--O
Then, the left hand side of Eq. (2.88) vanishes for some distribution of density/ susceptibility and layer undulations; which leads to non-uniqueness in the estimation of density or magnetization variation within the layer [32,33]. This phenomenon is parallel to what we have observed in Section 2.4.2 in the context of three-dimensional distribution of density/magnetization.
2.6.5. Relation between gravity and magnetic fields We have seen in Section 2.1 that the gravity and magnetic fields due to simple objects are linearly related (see Eq. (2.11)). Here we show a similar relationship between gravity and magnetic fields caused by more complex models. Consider the potential fields produced by an undulating interface separating two media. From Eqs. (2.68c) and (2.69b) we can obtain the ratio of the gravity field to the magnetic field in the frequency domain as
71
Stochastic model II: random medium
dFz(u, v, h) dFv(u , v, h)
sGAp r(u, v)/kt~
(2.90)
For a thin layer we obtain from Eqs. (2.79) and (2.81) the analogous ratio
dFz(u , v, h) dFT(U , v, h)
sG dP(u, v, h) r(u, ~) dx(~, ~, h)
(2.91)
Finally, for semi-infinite medium we obtain from Eq. (2.86)
dF~(uv, h) dFr(u, v, h)
sG f,_%dP(u#,w) s --jw r(u, v) f_~ ~(ue,wl s-jw
(2.92)
From Eqs. (2.90)-(2.92) it may be observed that there exists a simple linear relationship between magnetic and gravity fields independent of the source model whenever the density and magnetization variations are linearly related. This includes the trivial case of constant density and magnetization. The transfer function of the filter which will map the magnetic field into a gravity field is given by
sGAp ,i~z(U, v) _ r(u, v)A~
(2.93a)
This, indeed, forms the basis of pseudogravity maps derived from aeromagnetic maps. The gravity field can also be transformed into a magnetic field using the inverse of Eq. (2.93a),
Hz~v
r(u,v)/Xk sGAp
(2.93b)
The magnetic (pseudo-magnetic) field thus obtained has been used for the estimation of direction of polarization in Ref. [28].
72
Potential Field Signals and Models
References [1] [2] [3]
[4] [5] [6] [7] [8] [9] [ 10] [ 11]
[ 12] [13] [14]
[15] [16] [17] [18] [19] [20] [21] [22]
H.J. Nussbaumer, Fast Fourier Transform and Convolution Algorithm, Springer-Verlag, Berlin, 1981. L.R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Prentice Hall, New Delhi, 1978. M.N. Nabighian, The analytic signal of two dimensional magnetic bodies with polygonal cross-section, its properties and use for automated anomaly interpretation, Geophysics, 37, 507-512, 1972. N.L. Mohan, N. Sunderarajan and V. S. Seshagiri Rao, Interpretation of two dimensional magnetic bodies using Hilbert transform, Geophysics, 47, 378-387, 1982. V.N. Strakhov, Smoothing of observed potential field, Parts I and II, Izv. Geophys. Ser., 897-904, 1964. I.P. Nedyalkov and P. Kh. Bymev, The analytic continuation of gravitational anomalies, Izv. Geophys. Ser., 566-572, 1963. W.D. MacMillan, The Theory of the Potential, Dover Publications, New York. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Maths Series, Vol. 55, National Bureau of Standards, 1964. H. Jeffrey, Mathematical Physics, Cambridge University Press, Cambridge, 1946. W.R. Ford, Application of Green's theorem in two dimensional filtering, Geophysics, 29, 693-713, 1967. V.N. Strakhov, The analytic continuation of two dimensional potential fields, with applications to the solution of the inverse problem of Magnetic and gravitational exploration, Bull (Izv) Acad. Sci. USSR, Geophys. Ser., 5, 1962. G. Ya. Golizdra, The distribution of the singular points of a gravitational field for one class of two dimensional bodies, Izv. Geophys. Ser., 11, 1030-1034, 1963. A.V. Tsirul'skii and M. I. Sirotin, On the analytic continuation of a logarithmic potential, Izv. Geophys. Ser., 1, 59-61, 1964. A.V. Tsirul'skii, The relation between the problem of continuing a logarithmic potential analytically and the problem of determining the boundary of an anomalous body, Izv. Geophys. Ser., 11, 1021 - 1023, 1964. D. O'Brian, "Compudepth"- a new method for depth to basement computation, 42ndMeeting Soc Exploration Geophys., Anaheim, CA, 1972. I.S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series, and Products, McGraw-Hill, New York, 1965. P. S. Naidu, Characterization of potential field signal in frequency domain, J. Assoc. Exploration Geophys. (India), 8, 1-16, 1987. B.K. Bhattacharya, Continuous spectrum of the total magnetic field anomalies due to a rectangular prismatic body, Geophysics, 31, 97, 1966. D. Nagy, The gravitational attraction of right rectangular prism, Geophysics, 31, 362-371, 1966. L.B. Pedersen, Wavenumber domain expressions for potential fields from arbitrary 2-,21/2 and 3-dimensional bodies, Geophysics, 43, 626-630, 1978. A. H. Yaglom, Introduction to Theory of Stationary Random Functions, Prentice Hall, Englewood Cliffs, NJ, 1962. P.S. Naidu, Spectrum of the potential field due to randomly distributed sources, Geophysics, 33, 337-345, 1968.
References
[23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
73
P.S. Naidu, Statistical properties of random potential field, Geophysics, 32, 88-98, 1970. P.S. Naidu, Stochastic models in gravity and magnetic interpretation, J. Assoc. Exploration Geophys. (India), 4, 1-11, 1983. V.N. Glazen, V. I. Pavlovskii and A. B. Rayevskii, Autocorrelation function of potential fields caused by a horizontal layer with random sources, Izv. Earth Phys., 14, 598-601, 1978. M.G. Serbulenko, Construction of some exact linear operators for separation of potential fields, (inRussian), Geol. Geophys., Acad. Sci. USSR, 12, 96, 1963. G. Gudmundsson, Spectral Analysis of magnetic surveys, Geophys. J. R.. Soc., 13, 325337,1967. M. Muniruzzaman and R. J. Banks, Basement magnetization estimates by wavenumber domain analysis of magnetic and gravity maps, Geophys. J., 97, 102-117, 1989. V . L . S . Bhimasankaram, R. Nagendra and S. V. Seshagiri Rao, Interpretation of gravity anomalies due to infinite dykes using Fourier transform, Geophysics, 42, 693-713, 1977. L.B. Pedersen, A statistical analysis of potential fields using a vertical circular cylinder and a dyke, Geophysics, 43, 943-953, 1978. A. Spector and F. S. Grant, Statistical models for interpreting aeromagnetic data, Geophysics, 35, 293-302, 1970. R.L. Parker and S. P. Huestis, The inversion of magnetic anomalies in the presence of topography, J. Geophys. Res., 79, 1587-1593, 1974.
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75 Chapter 3
Power Spectrum and its Applications
3.1. Introduction
The spectrum of a homogeneous random field is a most useful property; indeed if the random field is Gaussian, it completely characterises the random field. This chapter is devoted to the study of the spectrum of potential fields. Sections 3.2 and 3.3 introduce spectrum and other functions such as radial spectrum, angular spectrum, cross-spectrum, coherence, transfer function, etc. and the role of sampling and quantization on the spectrum. In Section 3.3 computational aspects are covered. The basic tool for the spectrum estimation is 2D discrete Fourier transform. A brief discussion on bias and variance is included. The applications of spectrum are covered in the next three sections. The most interesting application of the spectrum is in the estimation of depth variation of magnetization, in particular, 'spectral depths' from the radial spectrum. The angular spectrum is useful for obtaining directional information such as on polarization, faults and other structural features. Finally, the last section is devoted to the study of coherence between gravity and magnetic fields and between gravity and topography for isostatic compensation.
3.2. Spectrum of random fields
Most geophysical fields, in particular gravity and magnetic fields, are caused by an ensemble of sources distributed in some complex manner, which may best be described in a stochastic framework. Then, the resulting potential field measured on the surface is characterized by a random function of two spatial coordinates. The stochastic model is particularly useful in the study of a large map as a whole, but not in bits and pieces. The statistical approach was first advocated by Horton et al. [1 ] in 1964 and later by Mundt [2] in 1968. The theory of random fields may be found in many books on stochastic processes, for example, the book by Yaglom is strongly recomended [3]. The spectral theory of stochastic processes is well documented (see, for example, a recent text [4]). However, the spectral theory of two-dimensional random fields is not dealt with in most books.
76
Power Spectrum and its Applications
For this reason we briefly review the spectral theory of 2D random fields emphasizing those points where the spectral theory of 2D fields differs from that of time series [4].
3.2.1. Random functions (2D) When a continuous random variable becomes a function of two spatial coordinates a random function in two dimensions is realized. Conversely, a random function observed at a fixed point in a plane is a random variable. The more familiar stochastic process is a random function where the independent variable is time. In potential field characterization we take the view point that the potential field, being the result of randomly distributed large number of sources, is best modeled as a random function. In this process we do not look at the microscopic variations but rather at the macroscopic or broad picture of the source distribution. A random function may be characterized by means of a few quantities particularly when it is a homogeneous function which is also known as the stationary function for a stochastic process.
3.2.2. Autocorrelation and cross-correlation A homogeneous random field measured at two points, namely, at (xl~Yl) and may be related to each other through a covariance function, defined as follows: (Xz,y2)
Cf(Z~c, /ky) - E{ ( f (xl,Yl ) - mf ) ( f (x2,Y2) -
mf ) }
(3.1)
where A~c = x 2 - x 1 and zXy = Y2 - Y l . Here we have shown that the covariance function is a function of the difference in the spatial coordinates. This, indeed, is a special case of great importance. Such a random field is known as a homogeneous (also called stationary) random field. The mean and variance of a homogeneous random field are always constant. The covariance function has a number of important properties. They are
1.
Cf(z2xx,Ay) -- CU(-,Xx, - A y )
2
Cf(Zkv, A y ) <
9
9
.
~
foo oo
Cf(O O ) - 0 .2 ~
.1"
(symmetry) (decreasing function)
Cf(x 2 - x l ; y 2 - yl)~(xl,Yl)qS(x2,Y2) dx 1 dx 2 dy 1 dy 2 ~ 0 oo
(positive definite)
Spectrum of random fields
4.
Cf(zSJc, Ay) --~ 0,
77
Iz~l,
[Ayl---> oo
When property (4) is valid the random field is said to be ergodic. This enables us to replace the expected operation by an arithmetic averaging operation. Mean, variance and covariance functions are then defined using arithmetic averaging: 1
N
[Zf -- ff ~'~f(xi,Yi) i--1
1
N
6~ - - N - 1 ~
[f(xi'Yi)
--
fir]2
i--1
1
N
V(xi -4- Ax, yi 4- Ay) - [z]V(xi,Yi) -- [Z],
C f ( ~ , Ay) - N - 1 ~
N ~ oo
i--1
(3.2) When we have two different types of random fields, the relationship between them is expressed in terms of the cross-correlation function defined below: Cfg(X2 --xl,Y 2 --Y,) -- E { ( f ( x l , Y l ) - #f)(g(x2,Y2)- lZg)}
(3.3)
where ~f and ~g are means off(x,y) and g(x,y) fields, respectively. For an ergodi random field, the expected operation in Eq. (3.3) can be replaced by arithmetic averaging: 1
Cfg(Ax, Ay) - - N -
N
(f (xi'Yi) - lZf)(g(xi + z~"Yi + Ay) - fig),
1~
N ---~oo
i=1
(3.4) where/2f and fig are computed via arithmetic averaging, that is 1
N
[zf -- ~ ~ f n=l
1
(x,,, y,,),
N
fig -- ~ ~-~g(x,,, y,,), n=l
N---~ oo
Power Spectrum and its Applications
78
The cross-correlation function possesses a different set of symmetry relations, namely -
_/Xy)
Two random fields are said to be uncorrelated when Cfg(ZSoc,Ay) = 0 for all ~c and Ay. This can happen whenever two fields being correlated do not possess a common generating mechanism. A random field is said to be uncorrelated when its autocorrelation function is a delta function: (3.5)
Such a random field, often known as a white noise process, is one whose field measured at two points, however close they may be but not overlapping, is uncorrelated. This is merely a mathematical concept as no real physical field can be purely white. The term 'coloured noise' is often used to refer to any noise that is not white. For some processes, the autocorrelation function may become close to zero beyond certain lag, that is Q.(zSoc, Ay) ~ 0
for ~xx > z2xx0 and Ay > Ay 0
The random field is said to possess a correlation distance of v/(~xx20 + Ay20) (for white noise the correlation distance is zero). A good example of white noise is pseudo-random numbers generated on a computer. They are a uniformly distributed and uncorrelated sequence. A uniformly distributed random sequence can be mapped into a Gaussian random sequence using Box-Mueller transformation [5].
3.2.3. Spectrum and cross-spectrum We next introduce the power spectrum or simply the spectrum of a random field as a Fourier transform of its autocorrelation function,
Sf(u) -
f oo o0 Cf(z~)exp(-juz2xx)
dzSJc
(3.6)
79
Spectrum of random fields and the cross-spectrum as a Fourier transform of a cross-correlation function
Sfg(u) --
f
oo
o0 Cfg(ZSJc)exp(-juAx) dzXx
(3.7)
The spectrum of a two-dimensional stochastic field is similarly defined as the Fourier transform of an autocorrelation function. The spectrum is a positive real function with the following symmetry properties:
sz(.. v) - s j ( - . . - v ) s y ( - . , v) - s z ( . . - v )
(3.8a)
The cross-spectrum between two random fields has similar properties
Sh(u. v) - s ; f ( - . . - ~ ) (3.8b)
s h ( - u , v) - s ; ~ ( . . - ~ )
The property of equality of a spectrum (complex conjugate equality of a crossspectrum) is illustrated in Figure 3.1. Further,
var { f ( ~ . y ) } -- ~
Sj(.. v) d . av oo
+oo
jff_
sj(o, o) -
~ 0(~,
Ay) d ~
day
(3.9)
3.2.4. Radial and angular spectrum A two-dimensional spectrum may be expressed in a condensed form as two one-dimensional spectra, viz. a radial spectrum and an angular spectrum. The radial spectrum is defined as
lfo'
Rf(s) -- ~
Sf(s cos(O), s sin(O)) dO
(3.10)
80
Power Spectrum and its Applications
Figure 3.1. The symmetry property of the spectrum and cross-spectrum is illustrated above. The opposite quadrants are symmetric (conjgate symmetric for cross-spectrum) Indeed, it is enough if the spectrum or cross-spectrum is specified in the upper or lower half frequency plane. where s - ~/r(u2 -[-- 1,'2) and 0 -- tan -l (v/u) are polar coordinates and the angular spectrum is defined as
1 /'~0+~"Sj.[ s cos(0),s sin(0)] ds A(O) - -~S a~o
(3.11)
where z~s is the radial frequency band starting from So to So + A~s, over which the averaging is carried out. In order to free the angular spectrum of any radial variation, a normalization with respect to the radial spectrum is applied. Thus, the angular spectrum is expected to bring out the angular variations, if any, of the 2D spectrum of the potential fields. We may normalize the spectrum with respect to the radial spectrum (Eq. (3.10)) and then define the angular spectrum 1 11 '0+~x'~Sf[s cos(0),s sin(0)] ds ANorm(0) -- ~ss R/.(s)
(3 12)
The numerical computation of the radial and angular spectra requires averaging of the 2D spectrum over concentric rings and wedges as shown in Figure 3.2. Because of the averaging the radial and angular spectra tend to be smooth
Spectrum of random fields
81
Figure 3.2. A template for computation of the radial spectrum (upper) and the angular spectrum (lower) functions. The presence of peaks in the angular spectrum gives an indication of linear features in the map. The radial spectrum is a measure of the rate of decay of the spectral power which may represent a deep seated phenomenon such as a deep basement.
3.2.5. Coherence While the spectrum provides information on how the power is distributed as a function of frequency, the cross-spectrum provides information on the common power that may exist between two fields. In fact we define a quantity called the coherence function which is a measure of the relative common power,
Cohfx (w) _
Sfg(W)
(3.13)
The coherence function is a complex function as the cross-spectrum is a complex function, although the spectrum is a real function. The magnitude of a coherence function lies between 0 and 1. It represents the fraction of power that can be
82
Power Spectrum and its Applications
predicted in field f given the field g, or vice versa [4]. Two fields are said to be non-coherent when the coherence function between the two is zero at least over the frequency band where the signal power is significant. Similarly two fields are coherent when the coherence function magnitude is 1 or in practice close to 1, for example, any two components of the magnetic field are coherent. The presence of noise will decrease the coherence magnitude even though the two fields are theoretically perfectly coherent. In fact in such a case the coherence magnitude becomes a measure of the signal-to-noise ratio (SNR).
3.2.6. Transfer function Often a physical process can be modelled in terms of a linear system with one or more inputs and one or more outputs. In the context of potential field we have noted in Chapter 2 (p. 56) that the potential field of a layered medium with undulating interfaces can be modelled as a linear system consisting of a layered medium which is 'driven' by an interface (a plane separating the layers is now an undulating surface). The output is the potential field. The input and output are related through a two-dimensional convolution integral,
fout (X, y ) _
ff
oo
h ( x t ,y t ) f n ( X - x
,,y-
y t) dx t dy t
(3.14)
oO
where h(x,y) is known as the impulse response function of the linear system. Taking the Fourier transform on both sides of Eq. (3.14), we obtain Fout(U , v) ~- S ( u , v)Fin(U , V)
(3.15a)
or
Fou,(.,
H(u,
-
>Tini;
(3.15b)
where H(u,v) is the Fourier transform of the impulse response function or transfer function of the linear system. When the input and output to a linear system are stationary random functions, the transfer function of a linear system can be expressed in terms of the cross-spectrum and the spectrum [4]
H(u,
- S*nou, (_U:
(3.16)
Spectrum of random fields
83
3.3. Discrete potential fields The potential fields are naturally continuous functions of space coordinates. They have to be sampled and digitized for the purpose of processing on a digital computer. This brings us to the question of sampling and quantization.
3.3.1. Sampling theorem A homogeneous random field f(x) is sampled at x = n~Jc, n = 0, +1, +2,... where Ax is the sampling interval. The adequacy of sampling is often judged by whether by using the samples we can retrieve the original function with as small an error as possible. Consider the following linear combination: 0(3
~C(x) _ Z f -~
(n~c) sin a ( x - nAx) a(x - nAx)
(3.17)
where c~ = 7r/zfloc. A perfect retrieval is achieved when the mean square error is identically zero, that is, ELf(x ) -)?(x)l 2 - 0 . Indeed this can be achieved under some special conditions [4], namely (i) the spectrum of the field is band limited and (ii) the sampling rate (= 1/Ax) is at least twice the highest frequency present. The spectrum of the field is said to be band limited when s x ( . ) - o,
!.1 _> .o
Let ~,0 be the smallest wavelength corresponding to the highest frequency u0 (u0 = 2~/~,0). Condition (ii) implies that Zk,c-- ~0/2. A random field with a band limited spectrum is important from the theoretical point of view, although it is doubtful if we ever find a truly band limited process in practice.
3.3.2. Folding of spectrum and aliasing error For any arbitrary sampling interval the spectrum of the reconstructed function, j~(x), is related to the spectrum of the original function through the following relation: oO
Sj.(u) -- Z n~--O0
Sf(u + 2na)
(3.18)
84
Power Spectrum and its Applications
1.5
E
. . . . I
~
'i -o ,i o
I
I:
I
,i,! ,i
1.0
0
I
-o E:
I I ! I
JE I-"
~
I
ei
::3
,i
I
,
I
,i ,i ,i ,i
Q.
co
0.5
I I I I I
,i ,i
I I
0.0
~
0
200
400
.
600
. . . .
~ .....
~
800
1000
Frequency (Hz)
Figure 3.3. Folding of a 1D spectrum. The spectrum of a time series is obtained by successively folding the spectrum of the continuous process as shown above. The first fold is at half the sampling frequency. It is interesting to picture the mathematical operation involved in Eq. (3.18). Draw a spectral density function on a graph sheet. Mark points on the frequency axis, o~ angular frequency apart. Fold the frequency axis at the points marked on it. The width of the folded sheet will be 2~. There will be an infinite number of stripes stacked one over the other. Now, transfer all segments of the folded graph onto the base stripe and sum up all segments. The resulting quantity is what appears on the left-hand side of Eq. (3.18). The procedure is demonstrated in Figure 3.3 for an exponentially decaying spectrum. Naturally, we call Si.(u) the folded spectrum. In general, Si.(u ) ~ Sf(u) except when Sf(u)= 0 for lul > a. The error introduced by sampling, that is, (Sj~(u)- Sf(u)) is often called an aliasing error. A random field satisfying the property S/(u) = 0 for mum> a is called a band limited function. Such a function when sampled with a samp|ing interval less than or equal to half the smallest wavelength would result in an error free reconstruction, that is to say with no loss of information provided an infinite number of samples are available for interpolation. A random field in two dimensions can be sampled in more than one way, for example, we may use a square or rectangular grid, polar grid, or hexagonal grid. Such a choice of sampling patterns is not available for a one-dimensional field.
Discrete potential fields
85
Figure 3.4. Spectral folding in 2D. The spectrum of discrete map data is obtained by folding the spectrum of a continuous map data along the dashed lines, which are spaced at half the sampling frequency. As a result, an interesting situation may arise. A long linear feature in the spectral domain is broken, as shown by the thick dashed line, into three linear features. However, from practical considerations for implementation of dsp algorithms, a square or rectangular sampling grid is preferred. Also, in this case m a n y results of one-dimensional sampling can be extended to two-dimensional sampling. For example, the folding of spectrum in two dimensions is given by
Sic(u, v) - ~ Z Sf(u -+-2mUnyq, v --I-2nVnyq) m
(3.19)
17
where Uny q -- 7f/~f and 1,'nyq = " I f / ~ y . In Figure 3.4 we demonstrate folding of a spectrum associated with two-dimensional sampling. On a graph sheet the frequency axes are now marked U,yq and v,,yq angular frequency apart on the u and v axis respectively. The axes are folded on the points marked. There will be infinite number of rectangles stacked one over the other. Now, transfer the spectrum from each rectangle onto the base rectangle and sum up all transferred spectra. The
86
Power Spectrum and its Applications
procedure is demonstrated for a line spectrum, which may be caused by a set of linear features in a map. The folded spectrum shows many lines which may be mistaken as different sets of linear features.
3.3.3. Generalized sampling Let f(x,y) be a common input to P linear systems whose outputs are measured and sampled. Let gk(x,y), k = 1,2,...,P be the outputs of P linear systems. Sometimes, it is easy to measure and sample these P outputs, gk(x,y), k = 1,2,...,P rather than the input, which can then be reconstructed using the samples of the ouput. In this process, because of the multiple views of the input function, it is possible to reduce the minimum sampling rate required for error free reconstruction of f(x,y) [29]. For example, in aerial surveys one can increase the flight spacing if two transverse derivatives are measured. Let us represent the output of the kth system in the frequency domain +-
o0 dF(u, v)Hk(u, v)exp(j(ux + vy)) dx dy,
gk (x, y) -- -~2
k = 1,2,...,P (3.20)
where Hk(u,v) is the transfer function of the kth system. We assume that the spectrum of the input process is band limited to [+_B0,+_B0]. Let us sample the output function at intervals zXx and Ay which are related to the maximum frequency as ~ ~
~yII
/j I
71"
Bo
where ~, is a constant yet to be determined (1, = 1 for Nyquist rate sampling of
f (x,y)). Let ak(x - mz~c, y - nay), k = 1,2,...,P be a set of interpolating functions defined as
j'(x, y)
-- ~ k
~ m
~ ga (mAx, nAy)ak (x -- max, y - n a y )
(3.21)
n
We wish to estimate the interpolating functions and the sampling intervals such that the mean square error between f ( x , y ) and f(x,y) is minimum. The mean
Discrete potential fields
87
square error is given by
E { f ( x , y ) - j ' ( x , y ) [ 2 } -- Cf(O, 0) + @(0, 0) - 2Cf2(0.0) where
Cf(O, O) - E{f2(x,y)} @(0, O) -- ~
... ~
m
Cg~g,((m - m')zXx, (n - n ' ) A y ) a k ( x - mZkv, y - nay) l
x a l ( x - m ' z ~ , y - n'Ay) and
Clio(O, O) - ~
~
m
~ n
Cf gk (x - mzhx,y - nAy)ak(x -- mZXx,y - nay) k
Further, we have
% gl((m - m') ZXv, (n - n') Ay) 1 4rr 2 ,!
J-Bo Sf(u, v)Hk(u , v)H[ (u, v) exp[/(u(m - m')z2xx + v(n - n')Ay)] du dv
Cfgk ( x - ms
y - nay)
l[[+Bo
47r2 s J-8o Sf(u, v)Hs (u, v)exp[l'(u(x- mZkr) + v ( y - nay))] du dv
(3.22)
Define,
Ak(u,x; v,y) -- Z m
ak(x - mzSx,y -- nAy)exp(-j(umZkx + vnAy))
~ tl
Using the above quantities we can rewrite the mean square difference as follows:
Power Spectrum and its Applications
88
E{f(x,y) -j~(x,y) l2}
1 JJ-Bo [[ +~~Sf(u, v) [ 1 + -- 47r2
[~
Ak(u,x; v,y)Hk(u , v) 12
- 2 expO'(ux + vy) ) ~--~Ak(u,x; v, y)H~ (u,
du dv
For the mean square error to become zero, that is, for perfect reconstruction the quantity inside the square brackets must be zero (since Sf(u,v) > 0)). For this to be true we must have
~--~ H~, (u, v)Ak(u,x; v,y) - exp(-j(ux + vy))
(3.23)
k
B0). Note that Ak(u,x; v,y) is periodic in u and v with a for all (x,y) and (lul,lvl period o f [ + Bo/v, + Bo/v]. Let us divide the frequency range [ + B0, + B0] into v2 equal divisions (square tiles as shown in Figure 3.5), 2Bo -Bo+--,-Boto
-Boto
-Bo+
2Bo /
//
2Bo -Bo to - B o + - - /./, - B o -Bo+
iBo //
to - B o +
(tile#l)
L/
+
2Bo //
(i + 1)Bo //
to - B o +
,-Bo
+ jBo //
4Bo /
(tile #2)
//
to - B o +
(]" + 1)Bo } //
(tile #(i,j)) Since Ak(u,x; v,y) repeats over these tiles, Eq. (3.23) gives rise to a system equations,
H~(~, ~)Ak(fi,x; ~,y) -- 1 k
+ k
2Bo t/
, ~)Ak(~,x; ~,y) -- exp - j
.2B0 ) --u X
of v2
Discrete potential fields
~H;(~,~ + 2B0 )Ak(fi, k
89
x; ~,y) -- exp
2B0 '~ -j---~-y)
P
2B0 k
, ~ + (u - 1) ~~-~)Ak(h, x; ~,y)
lJ
exp [ - j ((u - 1) 2B~ ~ + (~-
1 ) -2B~ - y - y ) "~]
(3.24)
where ~ and ~ are now limited to the first tile (see Figure 3.5). A factor exp(l'(~x + ~y)) has been absorbed into Ak(~t,x;~,,y). There are v2 equations in Eq. (3.24) which may be solved for Ak(ft, x;~,y), k = 1,2,...,P. For a unique solution we must have v = x/(P). Thus, we have a gain of v/(P) in terms of reduced sampling rate. In the context of a precision aeromagnetic survey the
Bo
-
Bo
B o
~
-
U
B o
Figure 3.5. The signal band is divided into p2 tiles each of size (2Bo/u)
x (2Bo/u).
Here ~,= 4.
Power Spectrum and its Applications
90
major limitation comes from the diurnal and rapid variations of the earth's field (see Chapter 1 for discussion). This problem may be overcome by measuring the spatial derivatives instead of the field itself. If the derivatives are measured across the flight direction (transverse derivatives) we can at the same time reduce the sampling rate and thus increase the line spacing. If we were to measure two horizontal derivatives using, say, three magnetometers, it is possible to double the line spacing. The required interpolation filters are derived in Example 3.1.
Example 3.1 Let us assume that two horizontal derivatives (first and second derivatives) across the direction of flight are measured with the help of three magnetometers (two in the wing tips and one in the tail). This specifies Hk(u) in Eq. (3.20)
gk(x,y) -- ~
dF(u, v)Hi,(u) exp(j'(ux + vy)) dx dy,
k-l,2
0(3
as
H, (u) - ju,
1-12(u)
-
-u
2
where we have assumed that the flight direction is along y. The sampling intervals in the x and y directions are ~ : = 7r/Bo and Ay = 7r/Bo. Now the interpolation filter Ak(u,x; v,y) will be periodic over rectangular tiles as shown in Figure 3.6. Eq. (3.24) reduces to a system of 2 x 2 equations,
H~((~t)A,(~t,x; v,y) + H~(h)A2(h,x; v,y) - 1 H~(h + Bo)A,(ft,x; v,y) + H~(h + Bo)Az(~t,x; v,y) - exp(-jBox ) where -B0 <_ u _< 0 and -B0 _< v _< B0 and an additional factor, exp(/'~x), is absorbed in the interpolation filters. The solution of above system of equations is quite straightforward"
Al(h,x;v,y) -
A2(ft, x; v,Y)
_(~ + B0)2 + ~2 exp(-jBox)
j Bo ( + Bo )
(~t § Bo) - ~t exp(-jBox) hBo ({t + Bo )
Discrete potential fields
91
V
-%
%
]:B0 Fig.3.6. The signal band is divided into two rectangular tiles each of size B0 • 2B0. It may be observed that the transfer function of the interpolation filters are singular at ~ = 0 and ~ = -B0.
3.3.4. Quantization errors Along with sampling we must account for quantization. Since numbers in computer memory are represented by a finite number of bits there will always be some error in the representation. Also, the sensors used for field measurements and the associated electronic circuitry impose a finite precision on the measured data, for example, in a typical aerial survey the magnetic field can be measured within an error of +0.53, [6]. The number of bits required to represent a measured quantity is dictated by the measurement precision and the dynamic range such that the error due to finite word length representation is matched to the measurement precision. The error due to finite word length is often known as the quantization error. Let fq(X) be the quantized field and e(x) be the quantization error. Then,
fq(X) - f (x) + r where e(x) is the quantization error, which is often modelled as uniformly dis-
Power Spectrum and its Applications
92
tributed and uncorrelated random noise. As an illustration of the quantization effect, an analog sinusoid, its quantized version and the resulting error are shown in Figure 3.7a. The spectrum of the quantized signal is given by (u)
-
sy( ) +
(3.25)
where cr2~is the variance of the quantization noise. From Eq. (3.25) it is clear that the quantization noise power can submerge the weak signal peaks if the quantization is too course. Let the quantization step be A, then, assuming e is uniformly 2 is equal to A2/12. distributed in the range +A/2 the quantization noise power cr~ The effect of quantization on a simple waveform such as a sinusoid is demonstarted in Example 3.2.
Example 3.2 A pure sinusoidal signal, sin(0.1t), is sampled and then quantized to eight levels(each level of size A = 0.125). Here t stands for time. The continuous waveform and quantized waveform are shown in Figure 3.7a and the difference between the two is shown in Figure 3.7b. Notice the noise like appearance of the difference waveform. For this reason the difference waveform is modelled as white noise particularly when the quantization is fine.
3.4. Estimation of power spectrum
The modem methods of spectrum estimation are closely linked with the underlying model of the time series [4]. Many of the models used in the analysis of time series have not found usefulness in the geophysical potential fields. Only the very basic model where the potential field is expressed as a linear combination of randomly distributed sources (equivalent to the moving average (MA) model of time series) has been used. The spectrum estimation approach is therefore based on classical methods, such as those of Blackman and Tukey and Welch [4]. We shall emphasize the Welch method which uses the fast Fourier transform algorithm to compute the discrete Fourier transform of large map data.
3.4.1. Discrete Fourier transform (dft) The Fourier transform of a two-dimensional function is defined as follows:
93
Estimation of power spectrum
(a)
1.0
i i i i i i iiiiiiiiiiiiiiii i il k
0.5 o
~ f,.,~
<
0.0
-0.5
-1.0
16
32
"/
/
48
64
Time
(b)
0.10 ......
i.. A...IP..
0.05 o
ijlv' i,"Y7 ! !iiiiiri!i}!i!_
0.00 "<
-0.05
-0.10
....... - ' i "
i
i
|
9 ,,, ....
,,, 9
0
16
32
48
64
Time Figure 3.7. A sinusoid sin(0.1t) is quantized to eight levels (each level of size A - 0.125). The quantized and original waveforms are shown in (a) and the difference (error) is shown in (b).
F(u, v) -
ff
+oo
oo f (x,y) exp(-j(ux + vy)) dx dy
A n a l o g o u s l y the discrete Fourier transform (dft) is defined as follows: M-1N-1
Fd(k , l) - ~
Zf
m--O n = 0
(m, n) exp{-jZTr[(mk/M) + (nl/N)]}
(3.26)
Power Spectrum and its Applications
94
0_
and
0_< 1 < N - 1
(3.27)
The discrete Fourier transform Fa(k,l) is quite analogous to the continuous Fourier transform F(u,v). The analogy is better brought out if we identify the indices k and I with the frequencies u and v respectively. Indeed the relationship is given by
u -
27rk M
and
v
27rl N
For this reason the indices k and l are often called frequency numbers. It may appear that Eq. (3.27) is a discrete version of Eq. (3.26). This, however, is not necessary as the discrete Fourier transform can stand on its own. For example it has its own inverse transform defined as follows"
f (m n) - 1 M-I N-I ' MN ~ ~--~Fa(k,l)exp{+j27r[(mk/M)+ (nl/N)]} k=0
0_<m_<M-
1
/=0
and
0_
1
(3.28)
When the data length is large, ideally infinite, Eq. (3.27) may be written as oo
Fa(u, v) -- ~ --00
oo
~--~f (m, n) exp[-j(um + vn)]
(3.29)
--0(3
and the corresponding inverse transform may be written as l f ( m , n ) --
Fa(u, v) exp[/'(um + vn)] du dv
(3.30)
Let us call Fd(u,v) as ideal dft obtained from an infinite data matrix. We now introduce a windowed dft, that is, a dft of a data matrix which has been tapered by a suitable function called a window function. Let fw(m,n) -- f (m, n)w(m, n) where w(m,n) is the window function having the following property:
Estimation of power spectrum
95
w(m,n) - 0 for m ~ 0 , . . . , M -
1
n~O,...,N-1 Let us now demonstrate the basic role of a window. For this we compute the dfl of a windowed data matrix. M-1N-1
Fd(k,l) - ~
~-~fw(m,n) exp{-j27r[(mk/M) + (nl/N)]}
m=0 n=0
= ~ --00
~-~f(m, n)w(m, n) exp{-j27r[(mk/M) + (nl/N)]} --00
47r2
71-
=Fw(k,/),
-- u, ~N -- v
Fd(u , v)W 0_
and
05
)
dudv
I_
(3.31)
where W(u,v) is the dfl of the window function and Fd(k,I) is the windowed dfl. From Eq. (3.31) it is clear that the windowed dfl is a convolution between the ideal dfl of infinite data matrix and the dfl of the window function. When there is no deliberate taper applied to the data, the window function is simply unity, that is
w(m, n) -- 1 for m E 0, . . . , M n E 0,...,N-
1 1
and zero outside. We call this a default window. The dft of the default window is given by w(u, v) -- sinc ( 89
sinc ( 89
exp [ - j ( 89
+ 89
where
sinc(x) - sin(x) x It is important to understand the role of a window function. Firstly, note that the windowed dft does not have a one to one relationship with the ideal dft. Because
Power Spectrum and its Applications
96
of the convolution operation the sharp peaks in the ideal dft, if any, are smudged. Further there is some amount of power leakage resulting in the appearance of spurious peaks. This effect is demonstrated in Figure 3.8. The spectrum of a pure sinusoid is a delta function as shown in Figure 3.8a. We take just 16 samples from this sinusoid and compute its spectrum and the result is as shown in Figure 3.8b.
0.5
,.
, .............. , , , , , , .
.
.
.
"
o
0.25
.
.
,
.
.
.
.
,
.
.
.
.
~
.
.
0.1
0.2
0.3
0.4
9
9
,
. . . .
.
9! . . . . . . . . . .
.........
"
. , , . , . , ,
0 . . . . . . - . i . . . . . . . . . . . . . . . . .
.
'
, . , , .
iiiiiiiiiiiiii!iiiiiiii!ii!iiii
o
. 5
.
~
0 . . . . . . . . . - o . . .
0
.
:
.
..........
I
.
!..........
0.5
fl
0.25
0 f~
0.1
0.2
0.3
0.4
0.5
Frequency in Hz Figure 3.8. The effect of a default window (rectangular window) on the spectrum of a unit magnitude sinusoid. (a) Spectrum of infinite length sinusoid and (b) spectrum of 16 samples of sinusoid (2 samples/period).
Estimation of power spectrum
97
The sharp peak corresponding to the sinusoid has now become a broad peak. Further, we notice many minor peaks created by the leakage of power through the side lobes of the dft of the window. Both smudging and creation of spurious peaks can b e controlled by appropriate design of the window function. The default window has the least smudging effect but the worst spurious peak phenomenon. Design of good windows is beyond the scope of this work. The readers are encouraged to refer to current books and papers, for example [4,7,8].
3.4.2. Fast Fourier transform (FFT) The Fourier transform of discrete data turns out to be one of the basic steps in a data processing exercise. Several fast algorithms are available for the evaluation of the discrete Fourier transform (dft). There are different variations of the now famous fast Fourier transform (FFT) algorithm, first invented by Runge and Konig [9] and rediscovered by Tukey and Cooley [10]. We briefly describe the basic algorithm emphasizing how the increase in the speed of computation is achieved. Evaluation of dft Eq. (3.27) in the normal course would require N 2 complex operations (one complex operation consists of one addition and one multiplication). We now demonstrate that by a simple trick of splitting the data into two halves the number of operations can be reduced to a little over N2/2. We split the given sequence into two equal length sequences as follows: ~(n,) =f(2nl) j z ( n l ) - - f ( 2 n l + 1),
nl-
0, 1 , . . . , I ( N - 1 )
j](nl) contains all even indexed data points and j~(rtl) contains all odd indexed data points. Now consider the dft off(n) in terms of the dft ofjq(nl) and j~(nl).
F(k)-~f(n)exp[-j(~)nk] n--O N/2-1
f(2n)exp[-j(~)2nk] +
1
[ f ( Z n + 1)exp - j
n--O
(2n
+ 1)k]
Power Spectrum and its Applications
98
=
~
jq(n)exp-j
N/2 nk
n=0
+ ~=ofZ(n)exp[-J(~2)nk]exp --Fl(k) +
Fz(k)exp[-j(~-~)k]
- F~(k) + Fz (k)exp [-j (~-~) k]
[
for k = O, 1 , . . . , N / 2
for k =
- 1
N/2,N/2 + 1,...,N- 1 (3.32)
The above equation shows how to combine the dft of two half sequences into the dft of the full sequence. This is often known as a doubling algorithm which is illustrated in Figure 3.9 where the signal flow diagram is often referred to as a butterfly. Let us now count the number of complex operations required to evaluate the dft using Eq. (3.32). The two half sequences would require N2/4 complex operations each; in addition we would require N operations to evaluate Eq. (3.32). Thus we now need N2/2 + N operations compared to N 2 operations using straightforward evaluation of Eq. (3.27). Nearly a 50% reduction could be achieved by a simple trick of splitting the full sequence into two half sequences. It is clear how
F(k) k=O... N/Z-I
F1(k) ~
_.
I~
~1
F(k)
Figure 3.9. Doubling algorithm where the Fourier transform of two half sequences are combined into one sequence. The multiplicative factor is known as the twiddle factor, Wk - exp [-j (-~)kI .
Estimation of power spectrum
99
one can exploit the above property in a repeated manner to significantly reduce the number of operations. It turns out that a sequence of N points requires just about N logzN operations (N is equal to a power of 2). The fast Fourier transform algorithm basically consists of several interconnected butterflies as shown in Figure 3.9. 3.4.3. 2D discrete Fourier transform
The discrete Fourier transform of a two-dimensional discrete data can be obtained by repeated application of dft first along rows and then columns. Each row is replaced by its dft and then each column is replaced by dft. The number of operations required would be NZlog2N while the direct evaluation would require N 4 operations. It may be noted that this popular approach requires matrix transposition, an operation that requires a fair amount of computation if the data are residing on a sequential storage device such as disc or magnetic tape. However, efficient methods do exist to carry out the matrix transposition operation [11 ]. The matrix transposition can be completely avoided if the dft operation is carried out along both directions. In addition, there is a further reduction in the computational load. We briefly outline the so-called direct method, also known as the vector radix method. This can be significant in reducing the heavy computational load in Fourier transforming large geophysical maps. Let us introduce the following transformations in Eq. (3.27): m--2ml+m0,
n--2nl+n0
and k -- Ml k~ + ko,
1--Nlll+10
where M~-
M z ,
m0--0,1,
N~
N -~
n0---=0,1
ml--0,1,...,Ml-1,
nl--0,1,...,Nl-1
k0 : 0 , 1 , . . . , M 1 - 1 ,
10=0,1,...,N~-I
kl = 0 , 1 ,
11 = 0 , 1
We obtain a set of equation analogous to Eq. (3.32), the doubling algorithm in 1D discrete Fourier transformation.
~,.=~ o
c~ =
II
~mL
.J
-j
c~
9 =
9 =
<
9
I
9
.J
.j
.j
.j
.j
,w.
.j
.J
.j
.J
.j
=
....
.j
.j
I
.J
II
<
,m.
.J
.J
.j
.J
~
Jr
.J
+
~s
L ~ j
i
|
i
I
i
I
I
!
!
.j
.j
,m,
t
II
II ,m,
il
.J
I
!
I
I
+
~s
.j
~ j
!
I
|
+
I
|
I
|
I
!
| I
~
.j
!
+ I
~ J
|
"J
~ 1 I~ I
I
I
|
~1o
~
+
II
II
1
II
II
.J
II
.j
II
.j
I
~ J
+
.J
+
.j
+
I
|
Jr
tl
II
.J
I
|
o
~,~o
9
Estimation of power spectrum
101
F((O, ko), (1, 1 o))
!::2
, (0, 1 O) F((1, ko), (1, 1 o))
F~
, k o), (0, lo))
Figure 3.10. Signal flow diagram of the doubling algorithm (Vector butterfly) in direct 2D-dft.
combines dft of four quarter matrices into a dft of full data matrix. The signal flow diagram, known as a vector butterfly, is shown in Figure 3.10. Note that, for the sake of clarity, only one half of the butterfly is shown. By repeated application of the doubling algorithm we can obtain 2D-dft recursively, starting with an elementary matrix of 1 x 1 whose dft is simply itself. The doubling algorithm requires three complex operations (i.e. three complex additions and three multiplications). There are log N recursive stages and in each stage the doubling algorithm is executed N 2 times; thus, we have 3N 2 log N complex operations. But in the usual row-column approach we have as pointed out earlier N 2 log N complex operations. It would therefore seem that the direct method would be slower than the row-column method. However, this is not true if we remove the redundant operations in the doubling algorithm. Note that out of 12 complex multiplications (in all four cases) only three multiplications are different and the rest are simple repetitions. As a result the count of complex multiplications reduces to 3/4 N 2 log N. Thus, the direct method becomes faster than the row-column method by about 25%. The drawback of the direct method is that the entire data matrix must reside inside the computer memory. The direct method was first suggested by Rivard [ 12]. A modem treatment of the discrete Fourier transform and the FFT algorithm may be found in Ref. [13]
3.4.4. Properties of dft coefficients The 2D-dft coefficients possess several useful properties which we quote without proof. These are straightforward extensions of 1-D dft coefficients [4].
Power Spectrum and its Applications
102
1.
F(k,I) = F(k + M, I + N)
2.
F(-k,-l)
3.
F ( - k , l) = F ( M - k, l)
4.
F(k,-1)=F(k,N-1)
= F(M-
k,N-
(periodicity) 1)
(symmetry)
(3.34a)
For real data, further s y m m e t r i e s in the dft coefficients exist, n a m e l y
5.
F(k, l) = F* ( - k , -1)
6.
F ( - k , l) = F* (k, - l )
7.
F(k, -1) = F* ( - k , l)
( s y m m e t r y for real data)
(3.34b)
As a result of the above s y m m e t r y properties, the 2-D dft coefficients o f a square data matrix are arranged in a m a n n e r illustrated in Figure 3.11. The zero frequency dft coefficient is located at k = 1 = 0. The dft coefficients for negative frequency n u m b e r s are located in II, III, IV quadrants as shown. A M x N data matrix will give rise to M x N dft coefficients. For the purpose
Figure 3.11. (a) Fourier transform of an infinite data matrix is periodic. The principal period is a square with its centre of coordinates at the centre of the square. (b) In the discrete Fourier transform of a square data matrix (which is replicated over the entire plane) the four quadrants of the principle period in (a) are rearranged as shown. This arrangement is often known as dft symmetry.
Estimation of power spectrum
103
of graphical illustration a small size coefficient matrix is not appropriate as it does not give a smooth appearance. Often it is necessary to interpolate between the coefficients. This is best achieved by padding a large number of zeros to the data matrix before computing its dft. The output contains as m a n y coefficients as the size of the input matrix which has been expanded by padding zeros. We show here that the resulting extra dft coefficients are the result of interpolation. Let a data matrix f(m,n), m = 0 , 1 , . . . , M - 1; n = 0 , 1 , . . . , N - 1 be expanded by padding zeros to matrix of size p M • p N where p is an integer. Some of the possible ways of padding zeros are shown in Figure 3.12. We consider the first type as it leads to a simpler mathematical expression. The dft of the expanded matrix is given by
Figure 3.12. For the purpose of interpolation, the data matrix is expanded by padding zeros. Some of the possible methods of padding zeros are shown above. In (d) the data are interlaced with zeros. This method is useful when the data are required to be interpolated.
Power Spectrum and its Applications
104
Fp(k,l) - ~
~-~fp(m,n) exp -j
m--0 n=0
-
-
km +
In
M-1N-1 ~ ~-~f (m, n) exp { -j [(,P--~)km+ 2(a/~---N) In1}
(3.35a)
m=0n=0
where
fp(m, n) -- f (m, n) 0.0
when0<m<M-1
and0
whenM<_m<_pM-landN<_n<_pN-1
Letk --pkl + ko and l --pll + lo where kl - 0, 1 , . . . , M - 1, k0 = 0,1,...,p - 1, ll 0, 1, ...,N - 1 and l0 - 0, 1, ...,p - 1. Making the above substitutions into Eq. (3.35a), we obtain -
-
M-1N-I
Fp(k, l) -- ~ ~-~f(m, n)exp
m=0 n=0
-~
{ -jZTr [(~ + _~) (~ k0 m +
~ {f(m ' n)exPI-J27r(k~ lon) k,pM+pNJl}exp
- m=O ,=o
4-
I~N)n ] }
I_j27r(k__~ +
l--~)l
which may be simplified to
Fp(k, 1) - F(kl , lI) x
sin [Tv(~ + kl) ] sin [Tr(-~ + l,)] sin[~ (-~ + kl)] sin[~ (-~ + l,)]
x exp{-j27r[(-~+kl)M-12
(3.35b)
Example 3.3 In this example we illustrate the interpolation scheme implied by Eq. (3.35b). Let the data matrix be expanded three times by padding zeros, that is, P = 3. The dft coeffecients of the data (before padding) are given on a grid shown by large
Estimation of power spectrum
105
(ki+I,]I+I) ~ W
JJL
A ,,
1
IzO
(k
~r"'
~i
F
2
;
Io
~"
,,
l
a h, ' I ,
I r
, 1 9
,i
z
2
,
i
i
i
,,,L
1,1 .~
Figure 3.13. Interpolation scheme as per Eq. (3.35b). The dft coeffcients of the data matrix are given by filled circles while interpolation is shown by empty circles. The data matrix has been expanded three times (P = 3) by padding zeros.
filled circles. By interpolation the dft coefficients are now computed on a finer grid with spacing of one-third of the original grid at the points shown by small empty circles.
3.4.5. Statistical properties of dfi coefficients When the observed potential field is modelled as a random field the dft coefficients also become random variables. They possess some important statistical properties, valid for large data size [4].
(3.36a) where ~Skk,-- 0 for k ~ k ~ and = 1 for k - k ~ and likewise ~5//,. The right-hand side of Eq. (3.36) is, by definition, the spectrum of random field. Hence, we have
Sf(k, 1) -M1 E{IF(k,I) i }
(3.36b)
Power Spectrum and its Applications
106
Eq. (3.36b) is useful for practical computation of spectrum of a random field using the dft coefficients.
3.4.6. Estimation 2D spectrum We now use the statistical properties of the dft coefficients of a homogeneous random process, in particular Eq. (3.36). The spectrum of a single random process (square data matrix, N x N) may be estimated
Sf(k, l) - ~1 E { IF(k, l) 12},
N ~ oo
(3.37)
Similarly where there are two stochastic processes, Ji(m,n) and j~(m,n) the crossspectrum is related to the dft coefficients of each process,
1 SAit (k, l) - - ~ E {F, (k, l)F~ (k, l)},
N --~ oo
(3.38)
There are two possible approaches to implementation of the expected operation in Eqs. (3.37) and (3.38). In the first method the expected operation is replaced by neighbourhood averaging. Specifically, we form an estimate of the spectrum ( or cross-spectrum) from
Sf(k, l) - po--1~m
,~~ IF(k + m, l +
n)l 2
(3.39)
where f~ represents some neighbourhood of a point k,l, say, a polygonal neighbourhood with P0 number of discrete points (see Figure 3.14). In selecting the neighbourhood it is essential to keep in mind the assumption that the spectrum is constant over the neighbourhood. Further, we assume that the digital map is sufficiently large so that the dft coefficients become uncorrelated. The map should be suitably windowed so that the power leakage through the sidelobes is minimized. This is important when large peaks are present in the spectrum. In many situations there is a slowly varying background whose effect is specially felt at the low frequency end of the spectrum in the form of a large magnitude peak. A considerable amount of power may leak into the neighbourhood of low frequency through the side lobes of the window function. Much of
Estimation of power spectrum
107
nl mml iii!ii I l l \ J~ mn mml %
Figure 3.14. The neighbourhood of a point (k,/) over which the averaging, as indicated in Eq. (3.39), is carried out.
this can be avoided by removing the slowly varying component by means of low pass filtering or by subtracting a least squares polynomial. In the second approach the digital map is divided into several submaps (these may be partially overlapping; see Figure 3.15). An estimate of the spectrum is now obtained as follows: 1
P0
sj(k, t) - -~ ~ IFp(k, t)
]2
(3.40)
p=l
where p0 is the number of submaps into which the entire digital map is divided. We further assume that each submap is uncorrelated with all other submaps. If the size of a submap is such that it is greater than the correlation distance, it is likely that a submap is uncorrelated with all other submaps including those in the immediate neighbourhood. Each submap is multiplied by a window function basically for the same reason as in the first method. One consequence of windowing is to reduce the weightage of otherwise perfectly good data which happens to lie along the edges of the submap. Thus windowing in fact throws away a portion of the good data. This drawback is partially overcome by overlapping the adjacent submaps to the extent of 50%. To improve the low frequency estimate the mean is subtracted from each submap.
108
Power Spectrum and its Applications
Figure 3.15. Overlapping of submaps for spectrum estimation. A large aeromagnetic map is first divided into many basic units (1 x 1 submaps). For spectrum estimation four or more unit submaps are combined into a larger submap with reuse. In this figure, two 2 x 2 submaps are formed with 50% overlap. The above discussion pertaining to the spectrum is equally valid for the crossspectrum between two digital maps. When the maps refer to two physically different phenomena it is possible that the scale factor of one may be quite different from that of the second. In such a case it is recommended that the two maps are suitably scaled so that the mean power in each is roughly the same. The discrete spectrum is estimated at frequency numbers, (k,l=O, 1,...,N/v/(Po)). We assume that we have a square map of X/v/(Po) x X/v/(Po). The frequency numbers are easily converted into frequencies (cycles per unit sample interval) by multiplying with a factor equal to v/(Po)/N. Note that the frequency numbers from 0 to N/Zv/P o refer to positive frequencies (0 to 1/Alx) and frequency number from (N/Zv/P o + 1 to (N/v/P o - 1 refer to negative frequencies (0 t o - 1 / ~ l x ) . From the estimated spectrum (or cross-spectrum) if you want to compute the autocorrelation function
Estimation of power spectrum
109
Figure 3.16. Zero padding around each submap is required to avoid the aliasing effect in the autocorrelation function computed using the dfl algorithm.
(or cross correlation function) by inverse Fourier transform we get the cyclic autocorrelation function which is different from the normal autocorrelation function. It carries an aliasing error on account of periodic repetitions of the autocorrelation function just as in the Fourier transform of a sampled signal. To overcome this, each submap is bordered by a blank or null map of width equal to half the width of the submap (see Figure 3.16). 3.4. 7. Bias and variance
In practice, the spectrum of a potential field is computed from the available finite map. Consequently, the size of a submap as well as the number of such submaps will be limited. The resulting estimate will suffer from two drawbacks, namely, the estimate becomes biased and its statistical variability increases. The bias arises on account of finite size of submaps. Recall that it was recommended that each submap should be multiplied by a window function. We now show the basis for such a recommendation. Using Eq. (3.31) in Eq. (3.40) and taking the expected value, we get
110
Power Spectrum and its Applications
ss( , z/- E{ s(k, t)}
1/f+
~ Sf(u, v ) [ W ( k - u, 1 - v)[ 2 du dv
47r2
(3.41)
From Eq. (3.41) it is clear that the average of the estimated spectrum is not equal to the true spectrum, Sx(u,v), except when W(u, v) is a delta function. For finite sized submaps W(u, v) will never be a delta .function. In general, for any chosen window function we have a main central lobe surrounded by many side lobes. Let us form a simplified mental image of the convolution operation performed in Eq. (3.41). The spectral estimate at frequency numbers (k,/) consists of two parts, namely, a contribution from the central lobe and a contribution from the side lobes. The first part may be approximated as an average of the true spectrum over a frequency cell of size 27r/N x 27r/N. The second part consists of all the power that has leaked through the various side lobes. The second part can cause serious problems, particularly when there is a small spectral peak in the neighbourhood of a large peak. It would be difficult to resolve the two peaks and to estimate the power in the smaller peak. The loss of resolution could also be due to the averaging effect of the central lobe. Thus, the bias in the spectrum estimate is closely related to the window function. Naturally, by suitable choice of the window function the bias can be minimized but can never be eliminated. The uniform window (or equal weighting window) is the worst type of window in so far as its sidelobe structure is concerned; it has, however, the narrowest possible main lobe. The art of selecting an optimum window is beyond the scope of the present work (see Ref. [4] for some details on the optimum window). The bias characteristics of the cross-spectrum estimate are quite similar to those of spectrum estimate. We now consider the variance calculation of the spectral estimate. Without proof we give the following results taken from [14]. var Sj.(k, l) - E{(Su(k,/) - E{Sf(k, l)}) 2} 1
-- P0
32 (k f
'
2 S2(k,l ) -
Po
N when k, l 4= 0 or 2v/p0
l)
otherwise
f
and for the cross-spectrum
t
1
(3.42)
111
Estimation of power spectrum
var Sfur2 (k, l) - E{lSy~ (k, 1)} -
E{Sj~(k,/)}12 }
1
- Po SA (k, 1)SA (k, l)
N
when k, l r 0 or 2 x / ~ + 1
1
= Po [Sf~ (k, l) + Sf~ (k, l)SA (k,/)]
otherwise
(3.43)
From Eqs. (3.42) and (3.43) we note that the variance of the spectral estimate can be reduced by increasing the number of submaps.
3.4.8. Estimation of coherence The coherence function is estimated by
c6hff2 (k, 1) -
Sff2 (k' l)
(3.44)
The statistical properties of the coherence estimate are highly involved. We note some of the simplified and approximate results due to Hinich and Clay [ 15] valid for large P0 and high coherence.
var
Ic0hH I
1
(1 - Ic0hAf212)2
-1 var 0 ~ ~ (1 - Ic0hf~ 1-2)
where 0 is the phase of the complex coherence estimate - tan- 1 I m c6hff2 ] Re c6hfr i Note that when the actual coherence magnitude is large the variance tends to be quite small even when the number of submaps is small. In practical problems, where the amount of available data is small, we can at most say whether the
112
Power Spectrum and its Applications
coherence is high (significant) or low (insignificant), but can rarely measure moderate coherence. Let P0 = 10 and the true coherence magnitude be 0.9; then the computed coherence lies between 0.88 and 0.95 with 90% confidence. When the true coherence magnitude is 0.4 the computed coherence magnitude lies between 0.16 and 0.7 with 90% confidence. Thus the variability is less at higher coherence values, which makes it easy to asce/'tain the occurrence of higher coherence. It is important to remember that for p0 = 1 the coherence estimate is always 1.0 whether the time series are correlated or not. Further details on the variability of coherence may be found in Ref. [4].
3.4.9. Spectral windows In Section 3.2 the spectrum of a random function was defined in terms of its autocorrelation function or its generalized Fourier transform. Both definitions assume that the random function is available over an infinite plane. In practice, however, the potential field data are available over a finite area, either because no measurements were made outside the area of investigation or the data are found to be homogeneous only over a finite area. Such a situation may be modelled as a product of a homogeneous random process and a function, known as a window, which has a finite value (<1.0) in the region where the data are given and zero where there are no data. The homogeneous random process used in the model coincides with the observed data and with its homogeneous extrapolation outside. However, since it is multiplied by zero, it is not necessary to carry out the actual extrapolation. We model the finite data as follows:
(3.45)
fo(m, n) -- f (m, n)wo(m, n)
where j~(m,n) equal to the observed field over a finite area and wo(m,n) be a discrete window function
wo(m,n) <_ l.O,
O <_m < _ M - 1 ,
wo(m, n ) - 0 . 0
otherwise
O <_n < N - 1
where we have assumed for simplicity that the area of investigation is a rectangle of size M • N. We use the spectral representation of a random potential field f(m,n) (see Section 2) and the Fourier representation of the window function in Eq. (3.45)
Estimation of power spectrum
f (m, n) -- ~
lff+ ljj+
113
dF(u, v)exp[+j(um + vn)]
71"
Wo(u, v)exp[+j(um + vn)] du dv
wo(m, n) - - ~ 2
7r
fo(m, n) -- f (m, n)wo(m, n)
lff+
Fo(u , v)exp[+j(um + vn)] du dv
47r2
7"f
where Fo(U, v) -- ~
dF(u', v')Wo(u - u', v - v')
The dft coefficients ofy~(m,n), 0 _< m < M -
Fo ( k, l) - -~gw2
dX ( u' , v' ) Wo
1 and 0 _< n _< N -
27rl u, N
v
~)
1 are given by (3.46)
7r
0
<M-
1 and0 < / < N -
1
Using the properties of the dft coefficients (see Section 3.4.5) we obtain an expression for the spectrum of a finite random potential field
sjo(k,z )
1
- E ~--AIFo(k,Z)
12}
lff_+ Sf (u',
471_2
71"
v')
W0 ~
u' ,--~ 27rl - v ,
du' dv'
(3.47)
Eq. (3.47) gives us a relationship between the spectrum determined from finite 2D data which is observed and the spectrum of infinite 2D data which is not observed. Imo(u,v)l 2 is the spectrum of the window function. Apart from the smoothing effect produced by the convolution operation and leakage of power, which is well documented [4], the angular variation of the spectrum of the potential field may be disturbed unless the window is isotropic.
114
Power Spectrum and its Applications
An ideal window for spectral analysis of a potential field must possess the following characteristics. (a) The spectrum of the window must be as close to a delta function as possible. Then, from Eq. (3.47) we note that
(b) The leakage of power is minimum, which is possible by controlling the height of sidelobes. (c) The spectrum of the window must be isotropic or close to being isotropic. In 1D spectrum analysis there is considerable interest in the design of spectral windows satisfying (a) and (b); for example, the optimum windows defined in terms of the discrete spheroidal wave sequences (see Ref. [8] for discussion on this topic). A window for 2D spectrum analysis may be created using a 1D window simply by spinning it around the centre point, but such a window will be on a polar raster instead of being on a rectangular grid.
3.5. Depth estimation from radial spectrum The depth to a source (magnetic layer) is a piece of information of great value in geological/geophysical interpretation of subsurface structure. Much of the literature on potential field interpretation is directed at estimation of the depth of a source, often modelled as a well defined geometrical object. We pursue this line of study in Chapter 6 as a problem in parameter estimation within the framework of digital signal processing. The parameter estimation approach soon becomes unmanageable when we are faced with more than two or more closely spaced sources. In practice such a situation is not uncommon; for example, a layer of magnetized rocks (basic rocks) where the magnetization is rapidly varying. Radial spectrum offers a unique possibility to estimate the depth to the magnetic layer. There are two types of models, namely, a randomly polarized magnetic layer and a random interface separating two homogeneous media for which depth estimation using the radial spectrum is feasible.
3.5.1. Single layer model First we consider a magnetized layer of finite thickness which is uniformly magnetized. However, the susceptibility variation is purely random. The spectrum of the total component may be derived from Eq. (2.83) and is given by [16,17]
115
Depth estimation from radial spectrum
Sr(u v, h) - ](j'ulx +jvly - SIz)(j'uc~ +jv/~ - sT)]2exp(-2sh) S2
x f+ I(1 -exp[-(s--jw)mh]l 2 oo
S 2 __1_W 2
SAt~ (bl, V, W) d w
(3.48a)
where SzxK(u,v,w) is the spectrum of the susceptibility variation in the rock layer. For a thin layer, Eq. (3.4 8a) reduces to
ST(u, v h) - I(j'UIx -+-jvly -- SIz)(j'uct -+-/v~ -- sT) l2exp!-2sh) S2
f
>( mh2
+oo
(3.48b)
SAt~ (hi, V, W) d w o(3
and for semi-infinite half space it reduces to
St(u,
v,
h)
-
[(julx +jvly
X
f+
-
slz)(l'UC~ +jv/3
-
s~/) [2exp(-2sh) s2
+ s~,~ (u, u, w)
oo
-~_-W2
(3.48c)
dw
In Eq. (3.48a) if we assume that S ~ ( u , v, w) = Szx~(u, v), that is, the susceptibility variation in the vertical direction is uncorrelated or white and noting that the resulting integral may be evaluated as +oo l1 _ exp[-(s - j w ) A h l l 2 oo
S 2 n t- W 2 7l-
& ( . . ~.h) - y IO.Ix + / .
dw - ~- [1 - exp(-2sAh)] s
- ~ z ) ( j u . + / v ~ - ~7)1 ~
x [exp(-2sh) - e x p ( - 2 s ( h + Ah))]Szx~(u, v)
(3.48d)
This corresponds to the sandwich model proposed by Jacobsen [24] which is sketched in Figure 3.17. Note that in Eq. (3.48) the dominant factor is exp(-2sh) which decays expo-
116
Power Spectrum and its Applications
Figure 3.17. A schematic representation of the Sandwich model [24]. The susceptibility in each layer is a stochastic function and is uncorrelated with all other layers
nentially. The rate of decay is proportional to the depth to the magnetized layer. From Eq. (3.48b) we can easily determine the radial spectrum Rpole(S ) --
12s2exp(_2sh)R/xh:(s)
(3.48e)
where
RA~(S)--
/kh 2 2rr
f f
ooSa~(scos(O),ssin(O), w) dw dO oO
is the radial spectrum of the susceptibility variations. For simplicity we have assumed only vertical polarization or the field has been reduced to pole.
3.5.2. Fractal models o f susceptibility variations Consider two different extreme cases which lead to relatively simple results. In the first case, we consider a thin layer whose spectrum of the susceptibility variations is a function of s only, that is, 2
S~x~(u v) - cr/x~ exp(-2ls) ,
-7-
where l is a positive constant depending on the rock type and O2A~is the variance
Depth estimation from radial spectrum
117
of susceptibility variations. The 1/s 2 term is introduced to account for the observed fractal nature of the magnetization [18,19]. Using the above model of magnetization in Eq. (3.48b), we obtain
Rpole (s) - crex~I 2 ~2 e x p ( - 2 s ( h + l))
(3.49a)
Interestingly the radial spectrum is a pure exponentially decreasing function. Next, we consider a semi-infinite medium where the 3D susceptibility variation has a spectrum of the form
f~ exp(-2/s), = 0
Iwl _< otherwise
where/3 and l i> 0 are unknown constants, depending upon the rock type. Using the above 3D stochastic model in Eq. (3.48c), we obtain
Rp~
(3.49b)
For 1/3 >> 1 and finite sl we approximate tan -1 (/3/s) ~ 7r/2. Then, Eqs. (3.49a) and (3.49b) become identical. The assumption that 1/3 >> 1 implies that the medium is more random in the vertical direction than in the horizontal direction. The radial spectrum now has 1/s dependance. There is some evidence suggesting that the natural magnetization possesses a fractal character, hence its spectrum may be expressed in the form f~ -
(S2 _+_W2) v/2 where v ~ 3 [ 19] and ~2 is a constant. Using the above model of magnetization, Eq. (3.48a) can be evaluated as follows"
St(u, v, h) - ](julx +jvly - slz)(juc~ +jv/3 - sT) ]2 e x p ( - 2 s h ) s2
Power Spectrum and its Applications
118
w2 (u/2)+l
+7) ](juI~ +jvly
- sI~) (juc~ + j v / 3
-
s~) l2exp(-2sh) S4
x s~_i-
~ (1 + x2) ~+1
f ~ v ~ r (~2-~) I(j'ulx
+jvly - slz)(j'ua +jv/3 - s~,) [2exp(-zsh)
r (~ -~- 1)
S u+3
(3.49c) Notice the presence of 1/s ~-1 which becomes dominant in the neighbourhood of s ~ 0. In support of the above model we give an example of the radial spectrum 27-
t E
22-
r,/') ~ ,,,,~
o
..]
17""
IO
20
30
Radial Frequency (rad/km) Figure 3.18. Radial specrum of the aeromagnetic field from the area where the German deep drilling project is located. Estimated value of the exponent, ",/= p - 1 = 2.07.
Depth estimation from radial spectrum
119
Figure 3.19. Vertical profile of rock susceptibilities obtained from drill cores of the German deep drilling project [19]. Missing values are due to incomplete recovery of the drill cores. of an aeromagnetic field from the area where the German deep drilling project is located [19]. Analysis of the rock susceptibilitiues obtained from the drilling core (see Figure 3.19), however, does not wholly agree with the spectrum model used in deriving Eq. (3.49c); but is perhaps closer to the band limited white noise model used in deriving Eq. (3.49b).
Power Spectrum and its Applications
120 For a gravity field the radial spectrum is pole reduced magnetic field with the only (3.48e) is absent. Hence the radial spectrum exponentially decreasing when the density totally uncorrelated, that is, white noise. model for density variation,
s p(u,v,w) -
(S 2 +
essentially the same as that for the difference that the s 2 term in Eq. of the gravity field becomes purely variation in all three direction is However, if we assume a fractal
W2)u/2
the radial spectrum of the gravity field will be given by
(w)
Rz(s) - exp(-2sh) f s~,+2
(~,/2)+1 d s
~176 + ~ )
= ~ v~r("+----t) exp(-2sh)
1)
(3.50)
As of now there is no practical evidence to suggest that v > 0.
3.5.3. Many layers Analysis of a single sheet of magnetization can be extended to a more complex model consisting of two or more layers or interfaces (Figure 3.20). Assuming the susceptibility variations in different layers are independent we write down the expression for total radial spectrum as follows: p Rpole(S) __ ~ - ~ O'iI 2 z2 exp(-2s(hi i=1
+ li) )
(3.51)
The radial spectrum in Eq. (3.51) is a sum o f p real exponential functions. A plot of the radial spectrum on a semi-logarithmic graph sheet shows some interesting features, namely, for small p there exist spectral windows in which linear segments of the radial spectrum can be seen and from the slope of the segments we can estimate the depth to different layers. For this to be possible, the magnetic layers must be well separated with significantly differing intensity of magnetization, that is, cr2 in Eq. (3.51). Figure 3.21 shows two examples of two and three layers whose radial spectra seem to exhibit linear segments with a slope close to the assumed depths. The intensity of magnetization of the deeper layer is assumed to be ten times stronger than that of the layer above it.
Depth estimation from radial spectrum
121
Figure 3.20. A model of three thin horizontal magnetic layers. The magnetization in each layer is independent and its spectrum follows from Eq. (3.48b).
3.5.4. Depth variation o f susceptibility~density:
When the number of layers is too large both the above graphical method and fitting an autoregressive model become impractical. We give an analytic method based on an inverse Laplace transform to estimate the variance of the susceptibility/density as a function of depth. Consider a stack of horizontal layers of equal thickness Ah << ~kmir, that is, much smaller than the smallest wavelength of the susceptibility variations (see Figure 3.17). Further, we assume that the spectrum of the susceptibility variation in the ith layer is given by (6h)2 f o o 27r S ~x~( u , v, w) dw oo
S2
exp(-21s)~Sh
(3.52)
where crzA~(i~Sh) i~ the variance of susceptibility in the ith layer. The susceptibility in the ith layer is uncorrelated with the susceptibility variation in any other layer. Using Eq. (3.52) in Eq. (3.48b) we obtain an expression for a thick stack of vertically polarized layers as shown in Figure 3.22.
Power Spectrum and its Applications
122
(a) 1~176 !, lO I
I
-.~.~
. . . . . . . . . .
.1
"'- "%
4,, 4 .01
9
1
2
"
Frequency
(radians)
(b) 1000 ]
lOO E
:3
~-
10
El.
"O
%
.01
0
1
%
4
....
2
3
Frequency (Radians)
Figure 3.21. Radial spectrum due to (a) two thin layers and (b) three thin layers. The dashed lines represent the radial spectrum due to individual layers at depths as indicated. O(3
Sfpo,~(s, h ) - 27rI2exp(-2s(h + l)) ~
cr2~;(i6h)exp(-2s6h i)6h
(3.53a)
i=0 or 0(3
crz~(i6h)exp(-2s6h i)6h i=0
Sfoo,~(s, h) 27rI~exp(-Zs(h + l))
(3.53b)
Depth estimation from radial spectrum
123
Figure 3.22. A thick rock strata is modelled as a stack of many thin layers. In each layer the magnetization is a function of two horizontal coordinates but the variance is a function of depth. In all other respects the strata are homogeneous.
The right-hand side is completely known. The left-hand side may be treated as a discrete Laplace transform of crzx~(z 2 ) and s as a complex frequency. By computing the inverse Laplace transform of the right-hand side, that is, evaluating the following integral on the imaginary axis 2
1
crzx,~ ( z ) - - ~ 2
f=~ IZ e x ppo,e (/co, h) )exp(/aJz) da~ ( - 2j co ( h + l)
(3.54)
Note that
Sfpole (J'0.), h) -- Sfp~ (S, h) 2 as the power at radial frequency s is equally distributed between +j~0 and -j~0. In practical applications, conversion of a real variable, s, into a complex variable j w is achieved by expressing the radial spectrum as a sum of exponential functions. This approach was first suggested by Klushin in 1959 [20] who must be given credit for the use of the radial spectrum of a potential field for depth estimation.We illustrate the inversion procedure through an example (Example 3.4).
124
Power Spectrum and its Applications
Example 3.4 Consider the sandwich model of a thick magnetic layer consisting of a large number of thin uncorrelated layers (Figure 3.17) but with lateral variation of susceptibility as in Eq. (3.52). The spectrum of the total magnetic field was derived earlier Eq. (3.48d) where we now assume in accordance with the model in Eq. (3.52) that 2
S A n ( u ~ v) ~ OAn s2
For simplicity we consider only the field reduced to pole. The spectrum of the field is given by
Sfpole(g, h )
-
-
7r ]lol2[exp(-2sh ) - e x p ( - 2 s ( h
+ Ah))]cr2A~
s
On substituting in Eq. (3.54) we obtain
2f 1
~2A,~(Z) _ cr/x~ 27r
~)-~
[1 -exp(-2j~Ah)]exp(l'~z) a~
2 The integral is easily shown to be the where ~2/x~(z) is an estimate of crzx,~. difference of two step functions, f~(z) - f~(z - 2z~h) where ~(z) is a step function. Hence
dz/x~(z) - crzA~[ft(z)- f ~ ( z - 2Ah)]
3.5.5. Interface model An interface separating two homogeneous media will give rise to potential field anomaly. In Chapter 2 we derived the gravity and magnetic field produced by an interface, modelled as a homogeneous random surface, separating two homogeneous media, for example, a column of sedimentary rocks overlying an ancient basement consisting of heavier granitic rocks. Let us consider the above model for the purpose of showing the possibility of estimating the depth to the basement. The spectrum of the gravity field observed on the surface, that is, on
Depth estimation from radial spectrum
125
the top of the sedimentary column may be obtained from Eq. (2.68c) when the basement is deep compared to the undulations. We give the final expression
Sf~ (u, v) - Ap2G2Szxz(U, v)exp(-2hs) The radial spectrum is given by
Rye(S) - Ap2G2R~z(S)exp(-2hs)
(3.55a)
Assuming that
S~z(U,V) - Cr~zexp(-2/s),
l>_ 0
the radial spectrum (Eq. 3.55a) reduces to
Rfz (s) - A p 2 G2crLexp(-2s(h + 1))
(3.55b)
It must be noted that the depth estimates given by Eq. (3.50) or Eq. (3.55b) will be an overestimate of the true depth depending upon the unknown constant l. A convenient method of estimating the depth to the layer or interface is to plot the radial spectrum on semi-logarithmic paper with the amplitude on the log axis and the frequency on the linear axis.
3.5.6. Physical significance of "spectral' depths In a physical model where there are distinct thin layers with 'white' magnetization, the spectral depths are actually the physical depths. When the magnetization is non-white, the spectral depth is always greater than the physical depth. When the layer is thick, it is difficult estimate the bottom of the layer. To show this let us go back to Eq. (3.48d) which we shall express in a slightly different form, S3
Sv(u, v, h) --- exp(--2sh)SA~(u, v) -- exp(-2s(h + Ah))SA~(u, v) (3.56)
126
Power Spectrum and its Applications
where ro(U , v) - (l'ulx + j v l y - slz)(l'ua + j v ~ - sT). For simplicity we assume that SA~(u, v) = constant. Eq. (3.56) may be treated as a difference between two exponential functions having different exponents. Such a spectrum, when plotted on semi-log paper, will not show two linear segments. For example, let h = Ah = 1; the spectral plot for this is shown in Figure 3.22a. In the low frequency region where we normally look for the deeper layer there is a deep null; however, the linear segment in the high frequency region is due to top of the layer. The depth to the bottom may be estimated by least squares fitting. We have fitted a difference of two exponential functions, e x p ( - a ~ s ) - e x p ( - a z s ) , to the radial spectrum in the range 0 - 1 . 0 radial frequency. The estimated values of a l and a2 for different background noise level are shown in Table 3.1. It, thus, appears that the spectral depth which is estimated from the slope of the linear segment or by least squares fitting, refers to a surface of sharp change in the susceptibility or the variance of the susceptibility. 3.5.7. Estimation o f radial spectrum
In Section 3.2 we introduced the radial spectrum as a radially averaged 2D spectrum of the potential field. It is more practical to estimate the radial spectrum -1
i
i
0.5
1
1
i
1
1.5
2
2.5
'
i
-2
_8'
-4 -5
-6 -7
-8 -9 -10
0
Radial Frequency
3
3.5
Fig. 3.22. (a) A plot of the right-hand side of Eq. (3.56). A layer of unit thickness at a depth of one unit and 'white' susceptibility variations (SaK(u,v) = 1) are assumed.
Depth estimation from radial spectrum
127
TABLE 3.1 Least squares estimates of depth to the top and bottom of a magnetic layer Actual values
Estimated noise std = 1
Estimated noise std - 2
al = 2.0 a2 = 4.0
1.998 3.960
2.004 4.005
The background noise is of zero mean and standard deviation equal to one and two. directly f r o m the F o u r i e r t r a n s f o r m o f the p o t e n t i a l field data. L e t f(m,n), 0 _< m, n _< N 1, be a s q u a r e data m a t r i x a n d F(k,1), 0 _< k, l _< N 1 be its F o u r i e r t r a n s f o r m . T h e radial s p e c t r u m is e s t i m a t e d b y a v e r a g i n g the m a g n i t u d e s q u a r e o f F(k,1) o v e r a set o f c o n c e n t r i c rings w i t h i n c r e a s i n g radius (see F i g u r e 3.23)
1
~
R(i)
Ni k,t c ith ring
where
Ni
ig(k,l)l 2
~
i-1
~
2~
'''1
S 2
1
stands for the n u m b e r o f p o i n t s in the ith ring. N o t e that w e h a v e
Figure 3.23. The magnitude square of 2D dft coefficients are averaged over a series of rings with incremental radius. Note that the central (zero frequency) dft coefficient is blocked as it may contain uncorrected power from the regional component.
128
Power Spectrum and its Applications
excluded the 0th dft coefficient as it may contain power from uncorrected or partially corrected regional components.
3.5.8. Effect of quantization We have shown earlier that the effect of quantization is to introduce white noise to the signal. The white noise will appear as a fixed component in the radial spectrum. To show this we have conducted a numerical experiment where the magnetic field due to a thin magnetic layer (unit thickness) at a depth of eight units was quantized by a step size of 10 3'. The sheet is made of unit cubes (22 500 prisms) with random susceptibility but polarized uniformly (inclination 15 ~ and declination 37~ The magnetic field is in the r a n g e - 8 6 t o - 1 3'. Because of the quantization error power, much of the high frequency radial spectrum is submerged under quantization noise (see Figure 3.24). Thus, the frequency window available for the depth estimation is drastically reduced.
3.6. Angular spectrum
A two-dimensional spectrum may be expressed in a condensed form as two one-dimensional spectra, viz. the radial spectrum and the angular spectrum, defined in Eqs. (3.10) and (3.11), respectively. In order to free the angular 2O i i
E
!
10
!
:
."
: ............................ ~.
I
i
9
i
i
"
!
:
Q
~ ,,,,,i
o -10 0
16
Frequency
32
48 Numbers
Figure 3.24. The radial spectrum of an infinite precision magnetic field due to a horizontal sheet (thick line) and quantized magnetic (step size 10 7), marked as Q.
Angular spectrum
129
spectrum of any radial variation, a normalization with respect to the radial spectrum may be applied. Thus, the angular spectrum is expected to bring out the angular variations, if any, of the 2D spectrum of the potential fields. The magnetic polarization vector imposes a strong directional variation of 2D spectrum. Thus, a study of angular spectrum of total magnetic field (i.e. aeromagnetic field) is very useful. 3.6.1. Angular spectrum o f uniformly magnetized layer Consider a horizontal infinite layer where the magnetic susceptibility is an arbitrary function (random function) of x and y coordinates but the direction of polarization is the same everywhere, that is, uniformly magnetized. The spectrum of the magnetic field (total field) caused by a layer of random magnetization may be obtained following the approach shown in [15,16]. We give the final result,
SfT (u, v) -- IV(u, v)I2S~ (b/, V)
exp(-2hs)
S2
(3.57a)
where &(u,v) is the spectrum of the susceptibility variation or the source spectrum and I'(u,v) is defined as P(u, v) = (lUC~ +jv/3 - s f ) ( l u I x +jvI~ - sI~)
(3.57b)
where (c~,/3 and 3') are direction cosines of the earth's field and I~, Iy and Iz are the three components of the inducing magnetic field. For the sake of simplicity we assume that the random magnetization is uncorrelated, making its spectrum a constant and the direction of the inducing field is same as the current earth's magnetic field. Hence, Ix = Ioc~, Iy = lo/3, and Iz = I0~/; Eq. (3.57b) reduces to
It(H, V)I2 - I2[(S~)2 + (b/O~+ V/~)2]2 __ I2S4 [,-)/2+ (Og2 +/~2)C0S2(0- 00)] 2
(3.58)
where 00 is the declination of the earth's field. Substituting Eq. (3.58) in Eq. (3.56), the spectrum of the total field is given by SfT(u , v) - exp(-2hs)12s4[~/2 + ( 2 +/32)cos2(0_ 00)]2
(3.59)
130
Power Spectrum and its Applications
From Eq. (3.59) the angular spectrum may be expressed as Anorm(0) _ Q[,),2 4-(o~2 4-/32)cos2(0- 00)]2
(3.60)
where ~2 is a constant. It may be noted that the angular spectrum is maximum in the direction of the polarization vector, as expected. This was actually verified by computing the angular spectrum of the total field observed on a plane two units above the horizontal layer (the total field is shown in Figure 3.25). The angles of declination and inclination of the polarizing vector are 37 ~ east and 15~ respectively. The computed angular spectrum and what is predicted in Ref. [6] are compared in Figure 3.26. Note that the theoretical results were derived under the assumption of white magnetization. Apart from the main peak in the computed angular spectrum corresponding to the direction of polarization, there are several sidelobes which are caused by square top prisms used in the modelling of the magnetic layer. The 2D source spectrum, S~(u,v) due to square top vertical prisms is like a star-shaped jelly fish which, when modulated with a F(u,v) function defined in Ref. [6], gives rise to a complex structure of sidelobes including those at 0 ~ and 90 ~ shown in Figure 3.26. The sidelobes will disappear if we remove the geometrical pattern imposed by regular prisms and the angular spectrum will approach a smooth function predicted by theory.
3.6.2. Estimation of angular spectrum The numerical computation of the radial and angular spectra requires averaging of the 2D spectrum over concentric rings and wedges as shown in Figure 3.2. Because of this averaging, the radial and angular spectra tend to be smooth functions. However, in the low frequency band the number of points in each angular wedge is highly variable causing increased variability in the estimated angular spectrum. This was overcome by converting the 2D spectrum from the rectangular grid to a polar raster using bilinear interpolation [21 ]. The resulting angular spectrum turns out to be smoother. An example of a smooth low frequency angular spectrum is shown in Figure 3.27. The presence of a peak in the angular spectrum is an indication of the presence of a linear feature in the map but not always. Some peaks in the angular spectrum may be attributed to the source spectrum; for example, the peaks at 0 ~ and 90 ~ in Figure 3.26 are caused by the spectrum of square top prisms. From Eq. (2.84b) the source spectrum of a horizontal sheet of unit cubes with random susceptibility is obtained by combining all terms pertaining to the source shape and susceptibility,
Angular spectrum
131
Figure 3.25. Total magentic field due to a randomly magnetized horizontal layer. S~ (u, v) - cr2~sinc 2 (u) sinc 2 (v)(1 - exp(-s))2 A plot of above function is shown in Figure 3.28. The angular spectrum will show two small peaks along the u and v axes, that is, along 0 ~ and 90 ~
3.6. 3. Orientation o f a fault We demonstrate the feasibility of estimating the direction of a fault from the angular spectrum. For this, consider the example of the faulted magnetized layer shown in Figure 3.29a. Only the deeper layer has a fault oriented at an angle of 45 ~ (measured clockwise with respect to the North). The magnetic field due to this model is shown in Figure 3.29b. The magnetization is uniform, in the direc-
Power Spectrum and its Applications
132 1.0 0.8
e
"-'
0.6 0.4
<=
iiiiiiii1115 iiiiiiiill i L2I J
0.2
0.0
0
18
36
54
72
90 108 126 144 162 180
Angle (deg) Figure 3.26. Angular spectrum of a magnetic field (total) due to a horizontal sheet made up of vertical prisms whose susceptibility is a uniformly distributed random variable in the range 0.00050.05, but the sheet is uniformly polarized (inclination 15~ and declination 37~ east). Theoretically the predicted angular spectrum is shown by a thick curve. tion, declination = 5 ~ and inclination = 15 ~ throughout the model (this corresponds to the earth's magnetic field vector at Bangalore, India). The presence of a fault in the lower layer is felt on the magnetic contours. Even an experienced interpreter would find it almost impossible detect the fault from the magnetic map. In order to reveal the contribution of the faulted layer from below, we have plotted in Figure 3.30 the magnetic field due to the faulted layer alone. Notice the field contours running along the fault. This effect of the fault is responsible for the peak in the angular spectrum. We have computed the radial and the angular spectra of the magnetic field matrix of size 128 x 128. The spectrum, that is, magnitude square of the Fourier transform was computed first. Next, to compute the angular spectrum we averaged the two-dimensional spectrum over the angular sectors shown in Figure 3.2. The averaging was carried over two frequency bands, namely, the low frequency band ( 1 - 1 0 frequency numbers) and the high frequency band (1 1-35 frequency numbers). Note that the Nyquist frequency is at frequency number 64. The angular spectra is shown in Figure 3.31 It is clearly seen that the presence of a fault at the correct position is shown by a peak in the low frequency angular spectrum but the corresponding peak in the high frequency band is absent. Other peaks in the angular spectrum, in particular the one at 90 ~, may be explained on the basis of direction of polarization, a tapered 2D window function used in the computation of the spectrum and the geometrical shape of the mag-
133
Angular spectrum
1.0-
E
0.8
(a)
L........ J
o.,. i
N
0.2
<
i X
,./J
..................................................
~
0.0 0
18
36
54
72
90
108 126 144 162 180
1.2-
(b)
1.0
!
0.6
.......
0.4
~ <
.........
......
k
......
I. .........
l_ .......
.~t ..........
=. . . . . . .
=. ..........
!
, 0.2
r"
"
0.0 0
18
36
54
72
90
108 126 144
162
180
Angle (deg) Figure 3.27. A comparison of angular spectra computed by averaging (a) over radial lines after conversion to a polar grid (Figure 3.26 is reproduced for convenience) and (b) over wedges as shown in Figure 3.2.
netized bodies. If the fault direction coincides with one of the horizontal edges of the rectangular prism, or the direction of polarization or is parallel to the edge of the data matrix, the angular spectra caused by the fault and the shape of the magnetized bodies will be similar. Then it is not possible to detect the fault. The effect of polarization may be removed by reducing the magnetic field to the pole and the effect of the tapered rectangular window may be isolated by rotating the map and repeating the processing. 3.6. 4. Application to real data
The aeromagnetic map over the eastern part of the south Indian shield [22] was used for the present study. The map, shown in Figure 3.32, was digitized at 1 km interval. This resulted in a data matrix of 192 x 192.
Power Spectrum and its Applications
134
!
O ~
o
.......
.
.
.
.
.
....
o
.....
~
~
-2
-3
~
,
,C, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i . . . . . . . .
-3
0 u
-2
-1
, . . . . .
1
2
3
Figure 3.28. Averaged 2D spectrum of unit cubes with random susceptibility.
We computed both the radial and angular spectra. The results are shown in Figures 3.33 and 3.34. The radial spectrum seem to support the possibility of three linear segments as marked on the graph. From the slopes of the segments at these points it may be inferred that the depths to the corresponding magnetic layers are 8.0, 3.0 and 1.1 km. The flight height in this survey was 1.5 km (above mean sea level). The depths to the magnetic layers with respect to the ground surface are 6.8 and 1.8 km and the third layer is probably due to surface rocks (surface is assumed to be 300 m above mean sea level). For the purpose of computing the angular spectrum the frequency band was divided into three sub-bands: a low frequency band 2-20 (frequency numbers); mid-frequency band 10-40; and a high frequency band 5080. The angular spectra for all three sub-bands are shown in Figure 3.34. The angular spectrum in the low frequency band has a prominent peak at 138 ~ which corresponds to an east northeast west southwest fault suggested in Ref. [22; fault B4 in Figure 4]. The angular spectrum in the mid-frequency band also shows a peak corresponding to this fault but with considerably reduced magnitude. The high frequency angular spectrum, however, does not show these peaks. Instead there are other peaks which may be explained as follows. The peaks at 0 ~ and 90 ~ in all three bands are caused by the tapered rectangular window used in
Angular spectrum
13 5
Figure 3.29. (a) A model of a faulted layer underlying a normal layer and (b) total magnetic field.
spectrum estimation, the large peak at 175 ~ in the low and mid-frequency bands corresponds to the direction of polarization, in the direction of the Earth's mag-
136
Power Spectrum and its Applications
Figure 3.30. Magnetic field due to the faulted layer alone. netic field at this latitude (declination -3 ~ and inclination 2.5 ~ and finally the peak at 162 ~ is probably due to a set of anomalies trending in east northeast direction in the upper part of the aeromagnetic map shown in Figure 3.32.
3.7. Coherence analysis The gravity and magnetic fields due to rock formation are likely to be correlated particularly when the density and susceptibility variations are connected through a common influencing factor. Indeed we have shown in Chapter 2 (see Eq. (2.90)) how to map a gravity field into a magnetic field when the density and the susceptibility are constant within a body and zero outside. When the density and susceptibility are variable and the body is a thin sheet or a semi-infinite
137
Coherence analysis
5.00e+6
-
4.oo~+6 u
L
3.00e+6
~ Z.OOe+6
=
I .OOe+6
<
.
....
.
~
-~
(a)
~9
.
.
.
.....\...
.
L............................i........ /.,k
~
l i/
/
K/
.................................................. .
.
.
.
.
.
.
~._....) \ /
9
O.OOe+O
le+5 8e+4
0
! i i i i,~,^....
-
~1
6e+4 .
.
.
.
.
.
.
.
.
.
r l
.
4e+4 <
2e+4 Oe+O-
0
18
36
54
72
90 108 126 144 162 180
Angle (deg)
Figure 3.31. Angular spectrum showing the presence of a fault in the lower layer. (a) Angular spectrum in the low frequency band (1-10 frequency numbers); (b) angular spectrum in the high frequency band (11-35 frequency numbers).
medium the resulting gravity and magnetic field are related as in Eqs. (2.91) and (2.92), respectively.
3.7.1. Stochastic model for the density and susceptibility Let us consider a stochastic model for the density and susceptibility variations, in particular, consider a thin sheet model and assume that P(u, v, h) - H(u, v)K(u, v, h) + N(u, v)
(3.61)
where H(u,v) is the transfer function and N(u,v) is that part of the density varia-
Power Spectrum and its Applications
138
Figure 3.32. Aeromagnetic map over the eastern part of the south Indian Shield. tions which are not related to the susceptibility variations. Using the above model in Eq. (2.91) we can compute the coherence between the resulting gravity field and the total magnetic field. 1 dF~(u, E ~ 1 dFz(u, v,h)-~5~2
_ A2 F*(u, v ) E { [ H ( u
sG
'
v,h)}
v)dK(u, v,h) + N(u v)]dK*(u, v,h)} '
_ A2 F* (u. v) H(u. v) 1 ~G ~ - ; S ~ ( . . u) d~ d~ where we assumed that
N(u,v) is uncorrelated with K(u,v,h). X is an arbitrary
139
Coherence analysis
10 7
9 9
~
9
10 6
i 10 5
E
i
9 9
9 ~
$
4
"
i
9
10 4
o
r/3
9
N ...............i_................ !
103 102
3 10 1 10 0
L
0
,
20
40
60
Rad Frequency
80
100
No.
Figure 3.33. The radial spectrum of south Indian shield data. Three linear segments are marked. The depth estimated from the slopes are 8.0, 3.0 and 1.1 km, respectively. For the purpose of estimating the angular spectrum, the frequency band was divided into three subbands: low frequency, 2-20; mid-frequency, 10-40; and high frequency, 50-80 frequency numbers. The Nyquist frequency is at frequency number 96.
constant and S~ (u, v) is the spectrum of the susceptibility variations. Similarly we can obtain
,
1 E -4~2dFz(u, v, h ) Tj~ dF; (u, v, h ) - ~2 IH(u, v) I2T ~1S ~ ( . , v )
d~a~
} __~_A2 1
and
{1
E ~ 2 dFr(u, v, h) ~
,}
dF~r(U,v, h)
-~2 SN(bt, V) dlg dv
140
Power Spectrum and its Applications
2 1
_ a~lr(", ~) T-~ & (u, ~) du du t sG 4e+5 -
(a) 3e+5
2e+5
OIO
/._~\,,j--Iv__. ,~
~~
.~ i ......
le+5
Oe+O 3e+4
(b)
2e+4
...........
U
le+4
t
Oe+OI
.
~ -, ._ ] L ,.,J
,o3{ ~o] 50
E
GO
[ 1!:I1
.........................................................................................
........lli......] .....Ij..........I......1 .......1..........i........111
20 10
0
18
36
54
72 Angle
90 108 126 144 162 180 (deg)
Figure 3.34. Angular spectrum over three different frequency bands. From the low frequency angular spectrum we infer that there is a major fault oriented at 48 ~ (a) Low frequency band; (b)mid-frequency band and (c) high frequency band.
Coherence analysis
-
"
141
T
-
(a) Transfer function \ -22
"r
,,
- 3 2
o'.5
O
if
'
..
,.o
WAVENUIdBER "'"
t
,15
km ~
:-'"
o,i~ ~j~,~,
'r ij ....
i
"'"
|
(b) Coherence (mag)
O5
o,
!
0
0.5
I
'
l .......
.
I
1.0
t.5
r'
i
(c) Coherence (phase)
,00r g
t
!I
o
-100~.~
o
oI~-"
,.o
',.'5
Figure 3.35. Coherence study of the gravity and topography across the Kane fracture zone [28]" (a) transfer function; (b) coherence magnitude; (c) coherence phase.
142
Power Spectrum and its Applications
where SN(U,V) is spectrum of the residual component in the density variations. Following the definition of coherence (see Eq. (3.13)) we obtain
cohffr(u,v)
{1
E ~ dFz(u, v, h) ~
~ E {~21
}
dFz(u, v, h) ~1 dFz (u, v, h)}E{4@ dFr(u,v , h)-~--~ dF~r(u, v,h)}
*(u,vlH(u,u) sG
~/(I
dF~.(u, v, h)
Ir(.,v)12
(3.62)
H(u, v)l 2+ s~(.,u)) I .G
Icoh j (u, v) l - 1 whenever SN(U, v)/S,~(u, v) -- 0; that is, the magnetic and gravity field are completely correlated whenever the susceptibility variations are linearly related to the density variations. The model given in Eq. (3.61) was first proposed in Ref. [25] but its demonstration on real data is not very convincing. Significant correlation between the Earth's gravitational field and the nondipole part of the geomagnetic field was reported in Ref. [26]. 3.7.2. Isostatic compensation A topographic surface not only produces its own gravity and magnetic fields, which are naturally correlated, but also influences the density variations deep inside the crust, a phenomenon known as isostatic compensation [27,28]. The gravity and topography are modelled as in Eq. (3.61). An example of coherence between the gravity and bathymetry profiles perpendicular to the Kane fracture zone of the mid-Atlantic ridge is shown in Figure 3.35, taken from [28].
References [ 1] [2] [3] [4] [5]
C.W. Horton, W. B. Hempkin and A. A. J. Hoffman, A statistical analysis of some aeromagnetic maps for northwestern Canadian shield, Geophysics, 29, 582-601, 1964. Von Wolfgang Mundt, Anwendung statischer verfahren in der magnetik und gravimetrie zur Tiefenerkunden, Pure Appl. Geophys. (Pageoph.), 69, 143-157, 1968. A.M. Yaglom, An Introduction to Theory of Stationary Random Functions, Prentice Hall, Englewood Cliffs, NJ, 1962. P.S. Naidu, Modern Spectrum Analysis of Time Series, CRC Press, Boca Raton, FL, 1996. S.M. Kay, Modern Spectrum Analysis, Prentice Hall, Englewood Cliffs, NJ, 1989.
References
[6] [7] [8] [9] [ 10] [ 11] [ 12] [13] [14] [15]
[16] [17] [18] [19] [20] [21 ] [22] [23] [24] [25] [26] [27] [28]
[29]
143
R.E. Sheriff, Geophysical Methods, Prentice Hall, Englewood Cliffs, NJ, 1989. F.J. Harris, On the use of windows for harmonic analysis with discrete Fourier transform, Proc. IEEE, 66 (1), 51-83, 1978. D.J. Thompson, Spectrum estimation and harmonic analysis, Proc. IEEE, 70 (9), 10551096, 1982. C. Runge and H. Konig, Die Grundlehren der mathematischen Wissenschaften, Vorlesungen uber Numerisches Rechnen, Vol. 11, Springer-Verlag, Berlin, 1924. W. Tukey and J. W. Cooley, An algorithm for machine calculation of complex Fourier series, Math. Comput., 19, 297-301, 1965. P.S. Naidu, FFT of externally stored data, IEEE Trans., ASSP-26, 473, 1970. G. Rivard, Direct fast Fourier transform of bivariate functions, IEEE Trans., ASSP-25, 250252, 1977. H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithm, Springer-Verlag, Berlin, 1981. P. S. Naidu, Estimation of spectrum and cross-spectrum of aeromagnetic field using fast digital Fourier transform (FDFT) techniques, Geophys. Prosp., 17, 344-361, 1969. M. J. Hinich and C. S. Clay, The application of the discrete Fourier transform in the estimation of power spectra, coherence, bispectra of geophysical data, Rev. Geophys., 6, 347-363, 1968. P.S. Naidu, Spectrum of potential fields due to randomly distributed sources, Geophysics, 33, 337-345, 1968. P.S. Naidu, Stochastic modelling in gravity and magnetic interpretation, J. Assoc. Exploration Geophys., 4, 1-11, 1983. M. Pilkington and J. P. Todoeschuck, Fractal magnetization of continental crust, Geophys Res. Lett., 20, 627-630, 1993. S. Maus and V. P. Dimri, Potential field power spectrum inversion for scaling geology, J. Geophys. Res., 100, B7, 12605-12616, 1995. I.G. Klushin, Investigation of the depth distribution of sources of gravitational and magnetic anomalies, Bull (Izv.) Acad. Sci. USSR Geophys. Ser., 9, 18-23, 1959. A.C. Kak, Tomographic Imaging with diffracting and nondiffracting sources, in: S. Haykin (Ed.), Array Signal Processing, Prentice Hall, Englewood Cliffs, NJ, pp. 351-428, 1985. A.G.B. Reddy, M. P. Mathew, B. Singh and P. S. Naidu, Aeromagnetic evidence of crustal structure in the granitic terrain of Tamilnadu-Kerala, J. Geol. Soc. India, 32, 368-381, 1988 P.S. Naidu and M. P. Mathew, Angular spectrum for detection of a fault, J. Geophys., 18, 189-194, 1997. B.H. Jacobsen, A case for upward continuation as a standard separation filter for potential field maps, Geophysics, 52, 1138-1148, 1987. M. Muniruzzaman and R. J. Banks, Basement magnetization estimates by wavenumber domain analysis of magnetic and gravity maps, Geophys. J., 97, 103-117, 1989. R. Hide and S. R. C. Malin, Novel correlation between global features of the earth's gravitational and magnetic fields, Nature, 225 (5233), 605-609, 1970. L.M. Dorman and B. T. R. Lewis, Experimental isostacy, 1. Theory of the determination of the earth's isostatic response to a concentrated load, J. Geophys. Res., 75, 3357-3365, 1970. K.E. Louden and D. W. Forsyth, Crustal structure and isostatic compensation near the Kane fracture zone from topography and gravity measurements - I Spectral analysis approach, Geophys. J. R. Astr. Soc., 68, 725-750, 1982. A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1977.
This Page Intentionally Left Blank
145 Chapter 4
Digital Filtering of Maps I
Digital filtering represents a major component of digital signal processing (dsp). Digital filters, both their design and implementation, are discussed in Sections 4.1 and 4.2, respectively. As a role, digital filters are used for signal enhancement, that is, to remove unwanted noise and enhance the desired signal. This topic is covered in Section 4.3. In a geophysical context, digital filters are routinely used to carry out certain analytic operations on the signal, such as, upward and downward continuation, reduction-to-pole, etc. These topics are dealt with in Sections 4.4 and 4.5. The next two sections deal with specialized topics. In Section 4.6 we develop an iterative filtering approach to reduce the potential field measured on an undulating surface to a plane surface and in Section 4.7 we describe a method of predictive cancellation of the effect of the terrain. The method is called correlation filtering. So far we have not been concerned with the nature of the background noise. In the presence of noise with known spectral characteristics, Wiener filters are designed to perform optimally. We consider a few simple examples of Wiener filters in Section 4.8.
4.1. Two-dimensional digital filters Since potential field data are generally in the form of a map, we require a twodimensional (2D) filter. The following section briefly discusses elements of twodimensional filters. Like a one-dimensional filter, a two-dimensional filter is a mathematical operation on a two-dimensional signal in order to modify its Fourier transform. We can express a 2D filter operation as p
g(m, n) -- ~
q
~
hm, nf (m - m', n - n')
(4.1a)
m=-p n=-q
On computing the dft of both sides of Eq. (4.1a) we obtain
G(k, l) - H(k, 1)F(k, l)
(4.1b)
Digital Filtering of Maps I
146 where
H(k, 1)- ~
~
hz,neXp - j m
m=-p n=-q
2p + 1
+n
\2q + 1
is the dft of 2D filter coefficients hm,, m = 0,+l,+2,...,+p; n = 0,+l,+2,+q. Since the filter is of finite size, that is (p,q) < o% it is said to be a finite impulse response (FIR) filter. When the filter size becomes infinite, it is said to be an infinite impulse response (IIR) filter. A special class of IIR filters consists of filters where 0 < p < oo and-oo < q < o o (half plane filter) a n d 0 < p < o o and 0 < q < oo (quadrant filter) (see Ref. [1] for more details on half plane and quadrant filters). The half plane or quadrant filter is appropriate where at least one of the independent variables is time and the concept of causality is meaningful, for example, in the study of wave fields. For our purpose, FIR filters are more appropriate and they are simple to design and easy to implement.
4.1.1. Lowpassfilters In many geophysical problems, it is required to separate the observed field into low frequency and high frequency components; for example, the classic example is the separation of regional and residual components representing the contributions of deep seated sources and shallow sources, respectively. A lowpass filter is often recommended for such separation. Another application of lowpass filtering lies in the suppression of white noise often caused by measurement errors, quantization, interpolation, finite precision numerical operations, etc. The potential field signal, in particular the gravity field, has most of its energy in the low frequency region. Naturally, a lowpass filter will help to preserve the signal energy. In 2D a lowpass filter can take different shapes, some of which are shown in Figure 4.1. The frequency response of an ideal lowpass filter is given by A(u,v)-I ---- 0
when(u,v) Ec otherwise
where c stands for a closed curve or polygon. The filter coefficients, given the frequency response function, are easily obtained, at least in principle, by evaluating the inverse Fourier transform of the frequency response function. We show
Two-dimensional digital filters
147
Figure 4.1. Different shapes of pass filters of some interest in digital filtering of maps.
how to compute the filter coefficients of a filter with polygonal pass region or support.
4.1.2. Polygonal support Consider a support region bounded by a polygon with N sides. For real filter coefficients the support region must bear diagonal symmetry. The filter coefficients may be obtained from l am'n ~
27r2
a-~-
A(u, v)cos(um + vn) du dv
where A(u,v) = 1 inside the polygon and 0 outside. By applying Green's theorem we can reduce the surface integral to a line integral taken along the boundary of the polygon [2].
148
Digital Filtering of Maps I
27r2n
am, n - -
1
27rZm
27r2
sin(um + vn) du,
sin(um+vn) dv, u dv
--
27r2
n r 0 mT~O
v du,
m - n -- 0
(4.2)
Let us represent the comers of the polygon by their coordinates, for example, each line segment of the polygon in parametric form,
(Uo, VO),(bll,V1),(btZ,V2),...,(Up,Vp).Next we represent
U - - Up_ 1 -~ t(blp - -
v-
Up_l) (4.3)
Vp_ l + t ( Vp -- Vp_ l )
where t continuously varies from 0 to 1. Using the mapping equations as in Eq. (4.3) we convert the integrals in Eq. (4.2) to be evaluated along the boundary of the polygon into integrals over the parameter t. For this note that, sin(um + vn) - sin(#p + rpt) where iZp -- (mUp_, + nVp_,)
and
~-p - (mUp + nVp) - (mUp_, + nVp_, )
du = (Up + Up_,) dt
and
dv-
(Vp - Vp_l ) dt
We can now evaluate Eq. (4.2). The final result is
)~P
am,n - - Z
p 27r27-p
sin(2#P; rP)sinrP
-2-
where )~P -
1(Up n
Up-l),
m--O
(4.4)
Two-dimensional digital filters
149
1 ----(Vp
m
-- V p _ I ) ,
n -- O
When m and n are not equal to 0 we can take either one of two choices. Finally, for m - n - 0 we have the following result: 1 a00
4,rr2 ~
(UpVp-1 -- V p U p _ l ) p
Example 4.1 Consider a fan shaped filter as shown in Figure 4.2a, as an example of a polygonal filter. The computed filter coefficients are shown in Table 4.1 and the filter response function of the computed coefficients is shown in Figure 4.2b. Finally, the mean square error in the computed response as a function of filter length is shown in Figure 4.2c.
4.1.3. Gibb's oscillations The presence of sharp discontinuity in the response function can pose a serious problem while approximating it with a finite order filter. To show the effect of a sharp discontinuity, let us consider a filter whose frequency response is given by a box car function having unit response in the range +w0 and zero outside. The filter coefficients are obtained by taking its Fourier transform which in this case is simply a sinc function sin(o:0r) 7i"7-
TABLE 4.1 8 x 8 Filter coefficients with a fan-shaped transfer function (see Figure 4.2) 0.0000 0.0003 -0.0101 0.0056 0.0000 -0.0362 0.0169 0.0236
0.0003 -0.0075 0.0121 -0.0212 -0.0056 -0.0135 0.0169 0.0225
-0.0101 0.0121 -0.0338 0.0304 -0.0507 -0.0169 0.1013 0.0169
0.0056 -0.0212 0.0304 -0.0675 -0.0507 0.2026 -0.0169 -0.0135
-0.0000 -0.0056 -0.0507 -0.0507 0.2500 -0.0507 -0.0507 -0.0056
0.0056 -0.0212 0.0304 -0.0675 -0.0507 0.2026 -0.0169 -0.0135
-0.0101 0.0121 -0.0338 0.0304 -0.0507 -0.0169 0.1013 0.0169
0.0003 -0.0075 0.0121 -0.0212 -0.0056 -0.0135 0.0169 0.0225
150
Digital Filtering of Maps I
11"
-'IT
0
Tr ~
-
/
U ---,--
-
"IT
-
b
/ 0
U --,-,-
TI"
15
10 C
,O I__ I_ tl.I
5
8
.....
~
I
32
..........
6'~
M
Figure 4.2. A fan-shaped filter, its approximation with a finite length digital filter (8 x 8) and mean square error as a function of filter length.
151
Two-dimensional digital tilters
1.2 1.0 0.8
~
-'9
0.6
0.4
0.2
0.0
I
0
64
-
!
128
192
256
Angular Frequency Figure 4.3. The phenomenon of Gibb's oscillations is caused by a sharp discontinuity in the filter transfer function. By increasing the size of the filter, the amplitude of the oscillations cannot be reduced. Transfer function of filters of different lengths: (a) 8, (b) 12, (c) 16.
The filter coefficients thus obtained are truncated to different lengths and then the response of each is obtained by computing the inverse Fourier transform of each sequence, padded with a sufficient number of zeros. The result of such an experiment is shown in Figure 4.3. We observe that the computed frequency response oscillates about the true response function. These oscillations are known as Gibb's oscillations and they arise whenever a discontinuity is present in the response function. The amplitude of the oscillations is practically independent of the filter length; however, the frequency of oscillations increases with increasing filter length. The amplitude of the oscillations can be suppressed only by replacing the discontinuity by a smoothly varying function.
Example 4.2 In the above illustration if we replace the step discontinuity by a smooth transition, the Gibbs oscillations are drastically reduced (see Figure 4.4). The effect of Gibb's oscillations is to introduce spurious variation in the output. This is best illustrated by taking an example of a one-dimensional signal, a
Digital Filtering of Maps I
152
,2] 1.00.8.~
0.60.4
\
0.2 r _
0.0 . 0
.5g
L~,
. . . . . . .
r~
Angular frequency Figure 4.4. The magnitude of the Gibb's oscillations has been reduced by providing a smooth transition from the pass band to the stop band.
band limited white noise process. A sample of the signal along with its spectrum are shown in Figure 4.5a,b. It is desired to carry out lowpass filtering with a pass band extending from 80 to 170 (frequency numbers). In the first approach, the signal is filtered using a filter designed following a simple minded approach. In this, the filter response is set to one in the pass band and zero in the stop band. An inverse discrete Fourier transform of the filter response thus defined is computed. Significant filter coefficients (in the present case the central 21 coefficients) are retained and the rest are set to zero. The frequency response of the truncated filter coefficients is shown in Figure 4.5c. In the second approach, an optimum lowpass filter which has a smooth transition between the pass band and the stop band is designed. The filter transfer function of the optimum filter is compared in Figure 4.5c with that of the simple approach. Notice the presence of large Gibb's oscillations both in the pass band and in the stop band. This results in a large difference in the filter outputs as shown in Figure 4.5d. Thus, the simple approach for the design of a lowpass digital filter will lead to serious errors, particularly when there is large power in the stop band. This simple idea of replacing a discontinuity present in a filter response function by a smooth function is extensively used in the design of lowpass filters. For example, a lowpass filter in two dimensions (2D) may be designed by surrounding the pass region by a transition region of specified width followed by a stop band as shown in Figure 4.6. The frequency response is unity in the pass region and zero in the stop region. In the transition band, the response varies smoothly from one to zero. The
153
Two-dimensional digital filters
(a)
4ooo
(b)
101e-
"! L
|
....
10 81 20o0
o
'17,
10 6 1
~
"-1 ~"
1041
I
2 i :7-:7:-i:::::~::: I !2 10 0'.ZIZ212L. :
10
<
~
"
i
,..... 10-2~] - .. ~ 10 -4 1 -I......i--J ,..kRIiJl
-2000
10 -6 ~ r " ~ - r 9 ! -4O00
32
(c)
1.2
64
96
128
160
Time Index
192
224
256
....
......
1.0
...1
64
96
128
-i
160 192
224 2 5 6
FrequencyNumbers 6oo
(
400
~ptimum
0.8
| 112
32
2O0
0.6
,-.
0.4
0 -200
0.2
-400
0.0 -O.2
/
64
0
128
192
Frequency Number
256
-600 0
9
! 40
9
i 80
!
120
-
i
160
9
!
200
Time Index
Figure 4.5. A numerical experiment was performed to show the difference in the output of two lowpass filters. The first filter is based on a simple idea of choosing the filter response as one in the pass region and zero outside. In the second approach there is a transition band separating the pass band and the stop band. The signal and its spectrum are shown in (a) and (b), respectively. The filter transfer functions are shown in (c) and the differences in the outputs are shown in (d). response in the transition region is chosen to m i n i m i z e the oscillations or ripples. This is an optimization problem o f fundamental importance in the design o f a digital filter.
4.1.4. Design of a finite 2D filter Let am,n, m = 0 , 1 , . . . , M 0 - 1, n - 0 , 1 , . . . , N 0 - 1, where the filter is o f size M0 x No, be the filter coefficients and ak,t, k - 0 , 1 , . . . , M 0 - 1, l = 0 , 1 , . . . , N 0 - 1 be the corresponding dft coefficients. N e x t w e define the continuous Fourier transform o f the filter coefficients,
Mo-1 No-1
f4(u, v) -- ~ m=0
~ n=0
am,. exp[-j(um + vn)]
(4.5)
Digital Filtering of Maps I
154
Figure 4.6. A pass band is surrounded by a transition band consisting of three rings with decreasing response values. After the transition band we have the stop band.
where u and v continuous frequencies. Substitute for am,n in terms of its dft coefficients. We obtain a relationship between the continuous Fourier transform and the dft coefficients of the filter.
Mo- l No- I J4(U, V) -- MoNol ~ ~-~ Ak,t k=0 ,=0
1 -- exp(-juMo)
1 - exp(-jvXo)
1 - e x p ~ "('-2\~k M0 u)]l-exp[
(4.6)
./'/'2~' - \ N 0 v)]
Naturally, we would like ] ( u , v) to be as close as possible to the desired frequency response which we denoted by A(u,v). We must select the coefficients Ak,l so that the difference between ] (u, v) and A(u,v) is minimized in some sense, for example, under the Chebychev norm,
I
A(u,v)
1
M~
N~
MoNo ~= ~/=0 Ak,l 1
1 - exp(-juM0)
1 - exp(-jvN0)
<~ (4.7)
Two-dimensional digital filters
155
for -re _< u, v _< re. 6 is a prescribed small number. Note that from Eq. (4.6) it is easy to see that 2rrk 2rrl) A u - - M0 , v - - N00J - A , , , The optimization in Eq. (4.7) is further constrained by requiring 2rrk 2rcl) A U-Mo,V-Nooj-Ak,1-1,
k, l c p a s s b a n d
and ( A u-
2rrk 2r~l'~ M 0 ' V - - N0J - - A k ' t - - 0 '
k, l c s t o p b a n d
The remaining dft coefficients lying in the transition band are unspecified. The reduced minimization problem has been solved using a linear programming approach, the simplex algorithm [3]. An example of the design of a lowpass circular filter with three rings of transition band is given in Ref. [4]. The design parameters are as follows: pass band, radius = 3~r/12.5; stop band, space lying between the outer most ring of radius = 6~r/12.5 and 27r x 27r square; filter size, 25 x 25; stop band attenuation, better than 67 dB. The response in the stop band is good with a maximum ripple magnitude of -67dB but the response in the pass band is not satisfactory. Better response can be achieved if we allow all dft coefficients to be unspecified and then minimize Eq. (4.7) over a fine grid [5]. The price we have to pay for this improved performance is the increased computer time even for a moderately sized 2D filter. A further simplification in the design of 2D filters may be introduced, that is, in each ring the filter response is held constant. Thus, if there are three rings, we will have to optimally determine just three unknowns. The cost of optimization can be substantially reduced at the cost of slight reduction in the stop band attenuation. We give a few examples of filters using this simplified approach.
Example 4.3 An example of 2D lowpass digital filter design is given in Figure 4.7. The pass band is a square of size 7r/3 x 7r/3 out of the total Fourier space of 2~r x 27r. We have assumed unit sample interval. The pass band is surrounded by three rings each of variable width, starting from 0 to 7r/16. To each ring a constant less than one is assigned; the ring closest to the pass band has the largest constant and the ring furthest from the pass band has the smallest constant. The filter size is fixed
156
Digital Filtering of Maps I
Figure 4.7. Lowpass filter surrounded by three transition rings. In each ring the filter transfer function is constant and is to be estimated in order to minimize the error in the pass band and in the stop band. at 21 • 21. The mean square error between the response of the filter and the ideal filter in the pass band and stop band is minimized with respect to the three unknown constants assigned to the transition band rings. Some sample results are shown in Table 4.2. This approach of designing a digital filter has been used to obtain a circular lowpass filter. A digital filter of size 21 x 21 was designed. The transfer function of the filter is shown in Figure 4.8. The filter coefficients (first quadrant only) are listed in Table 4.3. The number of rings in the transition zone was fixed at three. However, the peak ripple height is found to be exponentially decreasing with the increasing number of rings.
4.1.5. Polygonal filter Next we consider a bandpass filter whose pass band is bounded by a polygon of N sides or N comers as shown in Figure 4.9. Note that for filter coefficients to be real, the polygonal pass band must be symmetric in the opposite quadrants. Inside the polygon the filter response is unity. The pass band is surrounded by one or more polygonal rings. The filter response inside each ring is equal to a number
T w o - d i m e n s i o n a l digital filters
157
TABLE 4.2 The effect of ring width on the ripple magnitude: lowpass filter with a square pass band, 7r/3 x 7r/3 Ring width
Minimum error
Coefficients c l, c2, c3
Max ripple in pass band
Max ripple (magnitude) in stop band
0 2 4
44.78 4.56 0.619
0 0.7508, 0.4184, 0.1617 0.7864, 0.3947, 0.0841
1.182 1.159 1.032
0.107 0.051 0.011
oo 0 (
'10
Figure 4.8. The transfer function of (21 x 21) lowpass (circular pass band) digital filter. The contour interval is 0.2. The innermost contour has a value of 1.0 and the contours in the stop band are of value zero.
Digital Filtering of Maps I
158
TABLE 4.3 Filter coefficients (lst quadrant only) for lowpass filter whose response is shown in Figure 4.8 1119 1661 506 -2985 -7193 -8047 -1113 15009 36412 54895 62207
796 1043 372 -141 -546 667 1382 -173 -776 -900 -507 199 -1126 -1233 -750 -2667 -2929 -2038 -1038 63 -4593 -6479 -2187 -52 1211 -4090 -6917 -908 1294 1902 854 -514 2021 2193 1294 10501 1 3 7 8 6 6163 2021 -908 22484 3 2 5 0 3 10501 854 -4090 32503 48560 1 3 7 8 6 -514 -6917 36412 5 4 8 9 5 15009 -1113 -8047
-647 -543 27 846 1420 1211 -52 -2187 -4593 -6479 -7192
-395 49 650 1027 846 63 -1038 -2038 -2667 -2929 -2985
51 532 817 650 27 -750 -1233 -1126 -507 199 506
423 659 532 49 -543 -900 -776 -173 667 1382 1661
526 423 51 -395 -647 -546 9 -141 372 796 1043 1119
Multiply each coefficient by a factor of l0 -6. Circular pass band of radius equal to 5/16. Three transition rings each of width 5/64 and transition coefficients c~ = 0.921667, c2 = 0.48366, c3 = 0.06391 were used. The maximum ripple in the pass band is equal to 0.01056 and the minimum ripple in the reject band is -0.0073 (-42.33 dB). lying b e t w e e n 0 and 1. The rings act as a transition band, which is necessary for the reduction of ripples in the pass ana/or reject bands.
Example 4.4 As an e x a m p l e consider a bandpass filter with hexagonal pass band. The filter response in the first ring is 0.82, in the second ring it is 0.45 and in the last ring it is 0.17. The peak ripple in the pass band is found to be - 2 0 dB. The filter response in the transition zone in the above e x a m p l e is selected intuitively. By optimally selecting the three coefficients in the transition band the error m a y be reduced as shown in Figure 4 . 9 and Table 4.4. An alternate but suboptimal approach based on the m a p p i n g of a one-dimensional optimal filter into 2D filter has been suggested by McClellan [6].
4.1.6. Transformation of 1D filters Consider the following m a p p i n g of the frequency axis of a 1D filter onto the frequency plane of a 2D filter: cos(~) -- A cos(u) + B cos(v) + C cos(u) cos(v) + D
(4.8)
where A, B, C, and D are suitable constants. A point on a 1D filter frequency axis (w axis) is m a p p e d onto a contour in 2D filter plane (u,v plane). The origin of the 1D frequency axis is m a p p e d to the origin o f 2D frequency plane provided the constants satisfy an additional requirement (for a low pass filter):
T w o - d i m e n s i o n a l digital filters
159
(a)
i
L.
(b)
Figure 4.9. (a) Polygonal bandpass filter. It is surrounded by a three ring transition zone. The magnitude of the filter transfer function in each ring is determined in order to minimize the approximation error in the pass band and stop band. (b) The transfer function of the resulting digital filter (21 x 21). The contour interval is 0.2. The results are shown in Table 4.4. The innermost contour has a value of 1.0 and the contours in the stop band have a value of zero.
Digital Filtering of Maps I
160
A+B+C+D--
1.0
The shape of the contour is controlled by constants A, B, C, and D. The magnitude of the filter response on the contour is constant and is equal to the 1D filter response at the frequency point which was mapped into the contour. Nearly circular contours are obtained when A = B = C = - D = 0.5 and fan-shaped contours are obtained when A = - B = - 0 . 5 and C = D = 0.0 [7]. Several other types of transformations are listed in Ref. [8]. Any optimum 1D filter (odd length) with linear phase can be mapped into a 2D filter. The filter coefficients of a 2D filter are obtained in terms of 1D filter coefficients. Let a_N, a_N+l, ..., a_l, ao, al, ..., aN_l, a N be zero phase FIR filter coefficients. The filter response may be expressed as a polynomial in cos(o), N
A(~)-
N
~
ft. c o s ( n ~ ) - ~
n=0
b. cos"(~)
(4.9)
n=0
where b0, bl, b2, . . . , bN may be related to ~0, al, a2, --', aN 2 a l , . . . , {l u - - 2aN). N
(a0 -- ao, ~o --
N
A(~) - ~
h, Tn(r) - Z
n--0
b,r"
(4.10)
n--0
The Chebyshev polynomial can be expressed as a polynomial in its argument [9],
T~(r) - ~
n Fm Cm
m=O
which may be used in Eq. (4.10). By equating the coefficients of r m we obtain the following result (r = cos(w))" TABLE 4.4 The effect of transition ring width on the ripple magnitude: bandpass filter with a pass band as shown in Figure 4.9 Ring width
Minimum error
cl, c2, c3
Coefficients
Max ripple in pass band
Max ripple (magnitude) in stop band
0 2 4
89.60 5.20 0.452
0 0.8102, 0.4322, 0.1697 0.8659, 0.4168, 0.0671
1.219 1.079 1.027
0.137 0.060 0.019
Two-dimensional digital filters
161
N
n----0
N n=l
N
bi -
~-~cn~n n=i
and N~
b N -- eNaN
The coefficients c~ are listed in Ref. [9, p. 795]. Now using the transformation Eq. (4.8) in Eq. (4.10), we obtain N
A(u, v) -- ~ b,,[A cos(u) + B cos(v) + C cos(u) cos(v) + D] ~
(4.1 la)
n--0
which may be expressed in the standard form of a 2D filter response function N
N
A (u, v) - ~ ~ am,,, cos(mu) cos(nv)
(4. l i b )
m=0 n=0
am, n can be expressed in terms of bn or in terms of an, 1D filter coefficients. In practice it is not necessary to explicitly compute a~,, but use Eq. (4.1 l a) directly for filter implementation. Let F(u,v) be the Fourier transform of the input and G(u,v) be the Fourier transform of the output. These two are related through the following equation:
where
G(u,v) -A(u,v)F(u,v) N
-- ~ b,,(A cos(u) + B cos(v) + C cos(u) cos(v) + D)nF(u, v) n--0
(4.12)
Digital Filtering of Maps I
162 Define the following quantities:
Fo(u, v) -- F(u, v) F 1(u, v) - (A cos(u) + B cos(v) + C cos(u) cos(v) + D)Fo(u , v) Fz(u, v) -- (A cos(u) + B cos(v) + C cos(u) cos(v) + D)FI (u, v)
Fn(u , v) - (A cos(u) + B cos(v) + C cos(u) cos(v) + D)Fn_ 1(u, v)
(4.13)
Using the quantities defined in Eq. (4.12), we obtain the filter output N
G(u, v) - ~
b,F,(u, v)
(4.14)
n----0
Each term in Eq. (4.13) may be looked upon as a filter equation where the filter response is given by
H(u, v) -- A cos(u) + B cos(v) + C cos(u) cos(v) + D
(4.15)
and the corresponding filter is a nine point spatial filter with filter coefficients given as B
ho,o - D,
ho, l -- ho,_l = ~-,
A hi, 0 -- h
1,o
-2-
C
hi,1 -- hi,-1 -- h-l,l -- h-l,-1 = ~-
Example 4.5 An example of the transformation required for converting a 1D filter into a 2D lowpass filter with a circular base is given. In Figure 4.10 the spatial filter and its response function are shown. Notice that while the inner contours are close to circular shape the outer contours considerably deviate from the circular shape.
4.1.7. Elliptical pass band A 2D filter with elliptical pass band is of interest in directional filtering. The
Two-dimensional digital filters
163 /1; _
I
1 __ __ L ' 8 _
I _ _
1 L-~_
I
_
I 12 4
I---
I -1
7 I---
I
iiii
i
I
1 L-8_
I
1 L'~_
I
-/1;
_L,_F2_L_ 17 4 I"@ I
-/1;
(a)
(b)
Figure 4.10. (a) Nine point moving average filter and (b) its frequency response.
contour in the frequency plane to which the pass band edge of a 1D filter should map is described by
4-
-- 1
(4.16)
where u0 is a major axis and v0 is a minor axis (u0 > v0). In the mapping relation, Eq. (4.16), the constants A, B, C, and D will have to be determined in the least squares sense under the constraint that A + B + C + D = 1. This ensures that the pass band edge of the 1D filter is to be mapped as close as possible to the elliptic contour given by Eq. (4.16), that is, we minimize
cos(~o)-
cos(.) + B cos vo
+ c cos, , cos/v0 ,
1
700
Digital Filtering of Maps I
164
In the above analysis we have taken the long axis of the elliptic pass band contour along the u-axis. It will be of interest to tilt the long axis by an angle, say, 0. We can express the coordinates of a point on the elliptic contour in parametric form, u -- u0 cos(cp- 0),
v = v0 s i n ( w - 0)
The least squares estimates of A, B, C, and D are obtained by minimizing the following expression where ~o is a parameter: Icos(~o) - [A cos(uo c o s ( w - 0 ) ) + B cos(vo s i n ( w - 0)) + C cos(u0 cos(qo- 0)) cos(v0 sin(qD- 0 ) ) + D]lZ[min
Example 4.6 Given below is an example of the transformation required for converting a 1D filter to a 2D lowpass filter with elliptical contours. The pass band edge of the 1D filter and the elliptic contour parameters are as follows: a~0 = 0.257r,
u0 = 0.57r,
v0 -- 0.257r
The least squares estimate of the coefficients are A -= 2.58426,
B - - 0.16411,
C = 0.83529,
D - - -2.58366
The 2D lowpass filter can be expressed in terms of repeated application of the basic spatial filter given by Eq. (4.13). In Figure 4.12 we illustrate a scheme for implementation of a lowpass 2D filter. A potential field map to be lowpass filtered is repeatedly passed through a simple moving average filter (nine points for a circular pass region, four points for a fan-shaped pass region, etc.). The outputs are weighted and averaged to finally yield the lowpass filtered output. The basic spatial filter, H(u,v), depends on the shape of the pass region and the weighting coefficients depend on the selected 1D lowpass filter. The transformation approach is faster than the FFT approach for a moderate size of 1D filter, for example, to lowpass filter a map of size 256 x 256 points the transformation approach is faster than the FFT approach for a 1D filter of length less than 32 [10].
0 =
i,.~o
~.,,.
0
r
r ,.Q = C'I)
0
r
:z
~.,.Lo
!
f
,,
4
L .
"
t~
b~
Io
-~o-
I
k,
--"
OlD
I'"
I
I~
I "b.,
1
~ t~ I--"
i i.,.,
I
t 0
I
I
0 0c) co
t
I
i,,.,
tor
i,J
0
0 0 00
0 co 0o
0
--1'o--
t
I
I
I
o,,,
t~
c~ b,--L.
9
i.,,,,, 9
9
,-.]
Digital Filtering of Maps I
166
I bI
b2 .
b~ , ~
. . . . . . .
~
G(u,v) Figure 4.12. A scheme for implementation of a lowpass 2D filter. A simple nine point moving average filter is repeatedly used along with optimum 1D filter coefficients. 4.2. Implementation
of digital
filters
4.2.1. Spatial and frequency domain approaches Digital filtering may be carried out either in the space domain using the convolution operation given in Eq. (4.1 a) or in the frequency domain using a discrete Fourier transform of the map and the filter response function as in Eq. (4. lb). The choice is largely governed by the computational effort required to implement the two approaches. The spatial domain approach requires (2p + 1)2N2 real arithmetic operations (one operation = one multiplication plus one addition) where we have for simplicity assumed that the digital filter is (2p + 1) • (2p + 1) and the data matrix is N x N. In the frequency domain approach the filtering operation is carried out via the product of discrete Fourier transforms followed by inverse discrete Fourier transform of the product. The fast Fourier transform algorithm, which is now widely known, may be used to perform forward and inverse Fourier transformation required for convolution implementation.
4.2.2. Fast convolution The convolution operation in Eq. (4.1a) is implemented in the frequency domain where the discrete Fourier transforms of the data matrix and filter matrix are first computed and then the inverse discrete Fourier transform of the product is evaluated. At first glance this appears to be fairly straightforward; however practical difficulties arise largely on account of the implicit periodicity of the data matrix assumed in the definition of its discrete Fourier transform. Naturally, the convolution of two such periodic sequences shall beperiodic. There occurs what is known as the aliasing error unless the data matrices are properly bordered with zeros before convolution. This is an important practical consideration which calls
167
Implementation of digital filters
for clear understanding. Consider a convolution of two periodic sequences, obtained by periodic extension of the data and filter matrices. +oo
f'(m, n) -- f(m, n) 9 Z
+oo
~
~5(m- aM, n -/3N)
OZ - - - - O0 3 - - - - 0 0
+oo
a' (m, n) -- a(m, n) 9 ~
+oo
~
~5(m- ozM,n -/3N)
Ct -'- - - oo 3 " - - - - 0 0
The convolution of the extended sequence is given by oo
g'(m,n) -- ~
oo
Z f (m - p , n - q)d(p,q)
p=-oo q=-oo oo
= ~
oo
+oo
~f(m-p,n-q),
p=-oo q=-oo
+oo
* Z
~
+oo
~
5(m-p-o~M,n-q-/3N)a(p,q)
o~=-~ 3=-00
-k-oo
~
(5(p-a'M,q-/3'N)
Olt--.---O0 31~--00
(30
(4.17) -- Z Z ~ ~ g(m -(o~ - o/)M,n -(/3-/3')N) c~' # 3'=-oo where g(m,n) is the convolution output of finite non-periodic sequences. From Eq. (4.17) it may be observed that g'(m,n)is the periodically extended version of g(m,n). If g(m,n) is space limited, to M • N size, there will be no error caused by the neighbouring repetitions; then, it is said to be free from the aliasing error. When two data matrices of size (M • N) and (P • Q) are convolved the size of the resulting output convolution matrix will be ( M + P ) • ( N + Q). Now consider bordering each matrix with zeros to make it of size (M + P) • (N + Q). and convolving them. The resulting output matrix, now of size 2(M + P) • 2(N + Q), will consist of a central (M + P) • (N + Q) non-zero matrix surrounded by zero columns and rows. Following Eq. (4.17) such an output matrix is periodically extended with a period (M + P) and (N + Q). Because of the presence of zeros bordering the desired convolution matrix the overlap is avoided, hence, there is no aliasing error. The fast Fourier transform (FFT) algorithm is used to compute the 2D discrete Fourier transform.
168
Digital Filtering of Maps I
A complex 2D data matrix with the real part set to the data matrix and the imaginary part set to the filter matrix, which are properly bordered with zeros, is formed. The number of complex operations (one complex multiplication and one complex addition) for forward and inverse dft is equal to 7(M + P)(N + Q)log[(M + P)(N + Q)] where 3, is a constant (between 1 and 2 depending upon the optimization of the algorithm and programming [ 11 ]) In addition, we need 0.5(M + P)(N + Q) complex operations for multiplying the Fourier transforms of the data matrix and the filter matrix. In all, the required number of complex operations is given by Complex operations = (M + P)(N + Q)[3' log(M + P)(N + Q) + 0.5] or taking into account only the multiplications, one complex operation will be equal to four real multiplications; hence the fast convolution will require Real multiplications -- 4(M + P)(N + Q)[3'log(M + P)(N + Q) + 0.5]
4.2.3. Relative speed To compare the relative speed we define a quantity, the ratio of the number of arithmetic operations, Number of real operations required in fast convolution F(p,N) - N u m b e r of real operations required in ordinary convolution Enumerating the real multiplications in the definition of relative speed we obtain
V(p, N) -
4(N + P)212 7 log(N + p) + 0.5] (NP) 2
,~ 4(2"~ log N + 0.5) '~ p2 ,
(4.18)
N >>P
From Eq. (4.18) it is clear that the fast convolution approach becomes favourable only when p2 > 8",/, log N. For N = 256 and 3, = 1 it turns out that P > 8. Thus, fast convolution is computationally attractive for large map data and for moderate filter lengths, a typical requirement in any potential data analysis task.
Implementation of digital filters
169
4.2.4. Additional refinements Let the first matrix be the data matrix and the second matrix be the filter matrix; then (M,N)>> (P,Q). The filter matrix will therefore be very sparse. This helps to delete some of those operations which involve zeros. Such an algorithm known as a pruned dft algorithm has been suggested in Ref. [12]. Further optimization in the 2D FFT algorithm is possible through the use of 2D butterflies [13], mixed radix, avoiding the bit reversal operation, etc. As noted above, (M,N) >> (P,Q) and it might become difficult to store the data matrix in the computer memory (RAM), particularly when we are using a personal computer (PC). We briefly point out an algorithm known as 'overlap and save' [14] where it is sufficient to store a small part of the data matrix in the computer memory and the rest on the hard disk. The data matrix is partitioned into a number of submatrices such that at least one submatrix can be stored in the computer memory. The adjacent submatrices are overlapped by an amount equal to the dimension of the filter matrix. Each partitioned data matrix is convolved with the filter matrix which is now bordered with zeros to make it of size equal to that of the submatrix. A mosaic of convolved submatrices with overlap equal to the filter matrix dimension is created. The overlapped portion is summed and this results in a matrix equal to what we would have obtained if the entire data matrix had resided inside the computer memory [ 15,16]. On modem computers, including personal computers or workstations, memory limitation is rarely encountered for most routine processing jobs. Therefore, the above 'overlap and save' approach may not be required. But in another context where one may have to use spatially varying filters, as in the reduction to pole over a large map over which the earth's magnetic field, both magnitude and direction, may have substantially changed [17], 'overlap and save' is perhaps the right approach.
4.3. Filtering for signal enhancement In the context of geophysical map analysis the purpose of digital filtering is (a) to enhance the signal with the help of lowpass filtering for removal of noise or extremely low frequency power attributed to deep seated sources, (b) to condition the data before carrying out numerical differentiation or analytic continuation or reduction to pole/equator and (c) to shape the spectrum in order to enhance certain features such as lineations caused by geological faulting. We cover the applications (a) and (c) here in this section and application (b) in subsequent sections where we deal with digital filtering for implementation of analytic operations and reduction to pole/equator.
170
Digital Filtering of Maps I
1 4 84
i
1!
u
,,L']
4
,L
4
1 J I
"
4
Figure 4.13. A digital filter to remove a linear or quadratic component. The average of the field at four neighbouring points, three units away from the central point (P = q - 3), is subtracted from the field at the central point.
4.3.1. Lowpass filtering for removal of regional fields The potential field caused by sources which lie deep inside the earth is generally gently varying at least over the extent of the map under consideration. Such a field is often known as a regional field. The gentle variations may be accounted for by means of a low order polynomial, often a first or second order, although there is no clear agreement on what should be the order of the polynomial. What is left behind after taking out the linear or quadratic variation is called here a residual field which is presumed to be caused by sources closer to the surface. The regional field must be removed before we use the dsp techniques, in particular before computing the spectrum of the potential field. Otherwise the regional field will produce a large peak at very low frequency and some of this energy may leak into the adjacent frequency band on account of the window function used in the computation of the spectrum. The leaked energy may appear in the form of artefacts. Consider a simple 2D filter where four neighbours are averaged and the average is then subtracted from the field at the central point. 1
fres(m, n) -- f (m, n) -- -~ ~(m + p, n) + f (m -- p, n) + f (m, n + q) + f (m, n -- q)] (4.19)
171
Filtering for signal enhancement
The filter coefficients are placed at the grid points as shown in Figure 4.13. This filter template is moved over the entire map. It is easy to verify that when the digital map consists of a sum of linear components the output of the above filter is zero. When the digital map is a sum of quadratic components, the output of the filter turns out to be a constant which is easily removed by subtracting the mean from the filtered map. Thus, the 2D filter given in Figure 4.13 readily removes the linear or quadratic components without requiring the computation of the coefficients of the polynomial. The transfer function of the filter used in Eq. (4.19) is given by
H(u, v) - 1 -
1 (cos(pu) -~
+ cos(qv))]
(4.20)
From the plot of the transfer function (see Figure 4.14) we observe that apart from the removal of energy at very low frequency there is also significant attenuation of energy at low and mid-frequencies. To avoid this signal attenuation, it is necessary to look for a lowpass filter with an extremely narrow pass band with circular or polygonal support region depending upon the nature of the regional field. We have already explored the topic of lowpass filter design in Section 4.2 where we noted that a sharp transition from pass band to stop band is not possible and a transition zone is necessary. A least squares orthogonal polynomial approximation of a two-dimensional potential field can be described in terms of a low pass filter but one which is spatially variant. Its performance is found to be inferior to a standard low pass filter [39].
0
1
i
-1
/
-1 -2
( -3
.2
0 p=q=l
2
-3 -2
0 p=q=2
2
Figure 4.14. Transfer characteristics of a filter (4.20) for removing the linear or quadratic component. The filter characteristics in the region outside the low frequency band are far from satisfactory.
172
Digital Filtering of Maps I
4.3.2. Directional filtering In directional filtering the pass band consists of a wedge pointing along a selected angle. Outside the wedge the response is zero. Directional filters have been used in seismic exploration where they are known as fan filters [18], in picture processing for enhancement of linear features and for coding and transmission [19]. In potential field analysis, directional filtering has been used for enhancement of features such as dykes, fault zones, strike direction, etc. [20]. To design a directional filter we follow the method of digital filter design described in Section 4.1. The pass band is surrounded by three radial transition bands (see Figure 4.15) where the filter response is optimally determined so that the ripples in the pass band and stop band are minimized.
Example 4.7 A direction filter of size 21 • 21 pointing in the direction 135 ~ having a wedge angle of 40 ~ and a transition zone of 25 ~ was developed. The filter response in the transition zone is as follows:
Stopband
Stopband
Figure 4.15. Design of a directional filter. Notice the presence of a stop band at low frequency. The purpose is to suppress the strong power usually present in the low frequency region.
173
Filtering for signal enhancement
..____;
Figure 4.16. The response of the directional filter (21 x21). The contour interval is 0.2. The innermost contour has a value of 1.0 and the contours in the stop band are of zero value. Cl - 0 . 8 1 1 0 1 2 ,
c2 -- 0 . 3 2 6 9 9 6 ,
c3 -- 0 . 0 7 1 8 6 5
T h e m i n i m u m m e a n square e r r o r in the a p p r o x i m a t i o n w a s 8.87/4230. T h e r e s p o n s e o f the r e s u l t i n g filter is s h o w n in F i g u r e 4.16.
Example 4.8 To d e m o n s t r a t e the utility o f a d i r e c t i o n a l filter w e h a v e a p p l i e d the filter to
174
Digital Filtering of Maps I
Figure 4.17. A model of faulted magnetized rock layers. The top layer has a north-south fault and the lower layer has a northeast-southwest fault. enhance the effect of a fault. For this purpose we have simulated a complex geological situation consisting of magnetized rock layers. The upper layer has a north-south fault and the lower layer has a northwest-southeast fault. The magnetized layer was made of randomly magnetized rectangular blocks. The horizontal dimension of each block is one unit but the vertical dimension is equal to the layer thickness as shown in Figure 4.17. The computed field (reduced to pole) is shown in Figure 4.18a. The magnetic field thus generated is subjected to directional filtering. Figure 4.18b is the result of filtering when the pass band wedge is pointing southeast. The presence of a fault is indicated by high valued contours over the fault. It may be seen that there are many other closed contours elsewhere parallel to the fault but of lower magnitude.
4.4. Digital filters for analytical operations Certain linear mathematical operations such as differentiation, anaiytical continuation, rotation of a polarization vector (commonly known as reduction to pole) may be expressed in terms of a linear filtering operation. The filter response function for each analytical operation can be easily derived as shown in this section. Once the filter response is available it is straightforward to implement it as a digital filter. It turns out that the filter response for some analytical operations (e.g., derivative) is a non-decreasing function. As a result the background white noise gets amplified and the SNR in the filtered map comes down. It is therefore a common practice to filter out the background noise using a lowpass filter prior to filtering. A digital filter for most analytical operations must be accompanied by a suitable lowpass filter (see Figure 4.19).
Digital filters for analytical operations
175
4.4.1. Analytic continuation The potential field measured over a plane can be continued above or below the observation plane, but only in a source-free space. To obtain the filter response let us use the spectral representation of a potential field derived in Chapter 3. The field at height h (h is positive when the plane continuation is above the plane of observation and negative when it is below) with respect to the plane of observation is given by
c/)(x,y,h) - (2zr)2
1 ff+ ~
d~o(U, v) e x p ( - s h ) exp(/'(ux + vy))
(4.21)
where 4~0 is the potential field on the plane of observation and s = x/(b/2 nt- V2). From Eq. (4.21) we conclude that a filter for continuation of the potential field has a response given by
A(u, v) = exp(-sh)
(4.22)
For h > 0 the filter response for upward continuation is an exponentially decreasing function but for h < 0 for downward continuation it is an increasing function. Such a filter would be unstable when applied to noisy data, because the filter would amplify the high frequency noise. In the example shown below we demonstrate the effect of even a small amount of noise on downward continuation.
Example 4.9 A vertical prism is magnetized in the direction inclination 10 ~ and declination 3 ~ and its total field at a height of 10 units is shown in Figure 4.20a. To this field, random noise (variance =1/12) was added and the resulting noisy signal is shown in Figure 4.20b. The noise free and the noisy magnetic fields were then continued seven units downward, that is, three units above the prism. The results are shown in Figure 4.20c,d respectively. It is not surprising that the downward continuation of the noisy signal is totally useless. If, however, the downward continuation filter is preceded by a lowpass filter, the situation can be saved as shown in Figure 4.20e. The downward continued field is very close to the noise free case (Figure 4.20c). The lowpass filter should be so designed that as much of the signal energy as possible is preserved.
Digital Filtering of Maps I
176 4.4.2. Derivative maps
It is often desired to compute a derivative of the potential field, for example, a commonly used derivative is the vertical derivative. From Eq. (4.21) the vertical derivative is given by
fz (x, Y)
_
Oc~(x,y,h) _ 1 ff+~r Oz z=O--(2rc)2 J J-,~ da~o(U , v)s exp(j'(ux + vy))
(a) Figure 4.18a.
(4.23)
Digital filters for analytical operations
177
The filter response for vertical derivative c o m p u t a t i o n is given by
A(u, v ) -
V/u 2 + v 2
(4.24)
Similarly, for second vertical derivative c o m p u t a t i o n the response of the desired filter is given by A 2 (u, v) - u 2 + v 2
(b) Figure 4.18. Computer generated magnetic field due to faulted magnetized rock layers shown in Figure 4.17. (a) Magnetic field; (b) filter output when the directional filter is pointing southeast. The positive contours are coded as continuous curves and the negative contours as dashed curves.
178
Digital Filtering of Maps I
f(m
Low pass ! filter , ,.
I
Operations
Figure 4.19. A digital filter for analytical operation must be accompanied by a lowpass filter. The radius of the pass band can be ascertained from the radial spectrum of the signal.
The derivative filter, like the downward continuation filter is unstable as it is an increasing function. The energy in the high frequency region, which largely consists of measurement and quantization noise, and the contribution from the near surface magnetic sources, will be greatly increased. It is therefore mandatory that the data are lowpass filtered before subjecting them to derivative analysis. It may be pointed out that the so-called 'horizontal derivative' defined as
(oq kOxJ + \oyJ is a non-linear operation that cannot be described in terms of a linear filter operation. It is, however, claimed to define a sharp contact between two rock types having different magnetization [21]. This claim, however, does not appear to be true in a simple numerical simulation which was not done in Ref. [21 ]. We show in Example 4.10 the relative performance of the second vertical derivative and horizontal derivative. The idea of a 'horizontal derivative' may be extended to a total derivative defined as
[vrl-
N
+
+ N
where T is the total magnetic field. This has been termed an analytical signal with the property that the field is independent of the direction of polarization as in the case of a true analytic signal in a 2D potential field (see p. 26) [43]. This claim, however, was later shown to be wrong [44,45].
Figure 4.20. The effect of noise on downward continuation and how this can be overcome through a simple lowpass filtering is shown above. The outline of the prism is shown by a rectangle.
179
Digital filters for analytical operations
.
(a) -
. . . . . .
(d)
-
(r
.
.
.
, "~
7
........
Ce}
Digital Filtering of Maps I
180
. , ~ " ~'~
(-b)
Ca)
000
c22~
o
0
(c) Figure 4.21. Comparison of second derivative and 'horizontal derivative' maps.
Example 4.10 We have two vertically polarized vertical prisms (shown by thick outline rectangles). The total magnetic field, horizontal derivative (defined above) and second vertical derivative at a height of five units above the prism are shown in Figure 4.2 l a-c, respectively. The second vertical derivative was also computed through filtering of the noise added magnetic field (not shown). The result shown in (d), as expected, is close to that shown in (c). Clearly, the second derivative appears to define the outline of the prisms more closely than what the horizontal derivative is capable of.
Digital filters for analytical operations
181
4.4.3. Total field In aeromagnetic surveys we measure a component of the magnetic signal in the direction of the earth's field while in ground surveys we measure the vertical component of magnetic signal. Let c~, /3 and "?, be the direction cosines of the earth's field. The gradient operator in the direction of the earth's field is given by 0
0 +
+
0 Oz
Using the gradient operator on both sides of Eq. (4.21) we can express the total field as follows:
T(x,y)
lff§
~ d6Oo(U, v)(-juc~ - j v ~ + sT) exp(/(ux + vy))
(27r)2
(4.25)
Comparing Eqs. (4.23) and (4.25) for the transformation from vertical to total field, the filter response is
Afzr(U , v)
-juc~ -jv/3 + s~y
(4.26)
s
and for transformation from total to vertical the filter response is
s
AvL (u, v) -- -juct -jv/3 + s~/
(4.27)
Likewise, it is possible to devise a filter to transform the gradient of the potential field evaluated in any one direction to any other specified direction. Indeed one can prepare component maps showing the magnitude of a component in any prescribed direction. In a horizontal component map, it is observed that a linear feature perpendicular to the direction is emphasized; thus component maps may be used to highlight linear features [42].
4.4.4. Continuation of field for enhancing deep seated anomalies In an effort to enhance the basement anomalies which are often masked by
182
Digital Filtering of Maps I
.
.
.
.
.
.
.
.
.
.
.
h __~__......
.
observation plane .
.
.
.
Layer # 1 .
.
.
.
.
.
.
.
.
.
.
.
.
Ah .
.
.
.
.
.
.
.
.
.
.
.
Layer#2
.
..............................
Figure 4.22. Two layer magnetic model.
near surface anomalies, the potential field is continued upward, away from the source. This statement appears to be rather surprising as one would expect that going closer to the source would enhance the anomalies. We show that in terms of the signal-to-noise ratio (SNR), upward continuation indeed improves the SNR. Consider the following simplified model of surface and deep seated anomalies. The surface source consists of a thin randomly (white) magnetized layer. The deep seated source is also a thin randomly (white) magnetized layer (see Figure 4.22). The radial spectrum of the magnetic field (pole reduced) due to a set of thin horizontal uncorrelated layers is given by Eq. (3.51). For simplicity we consider the direction of polarization to be vertical (that is, reduced to pole). The spectrum of the magnetic field follows from Eq. (3.51),
Rpole(S) -- or,l; 2 2 exp(-2sh)
E 2
or2 e x p ( - 2 s A h ) 1.0 + cr--~-
]
(4.28)
where cr2 and or22are the variances of magnetization in layer one and two, respectively. Consider the top layer of the magnetic sources as noise and the lower layer as a signal. We can now compute the signal-to-noise ratio (SNR) on the plane of observation.
2[}
02 h S N R - cr-~l h + Ah
A plot of SNR as a function of h is shown in Figure 4.23.
(4.29)
Digital filters for analytical operations
x1.0
183
......... "
i 0.5 r,,, z i,/'i
#
..../
0.0
....... i ......
6 -i
9
............. i.................. i................... i 9 :
! !
,
..J
0
5
h
10
IS
Figure 4.23. The signal-to-noise ratio (SNR) as a function of height of the observation plane. It is interesting to note that the SNR approaches the source SNR, as h ~ ~, that is, when the magnetic field is measured at a great height as in a satelliteborne magnetic survey. But at such a great height, the magnetic signal may be so feeble that the measurement errors and quantization noise may be more dominating.
4.5. Reduction to pole and equator 4.5.1. Reduction to pole The shape of the magnetic anomaly depends upon the shape of the causative body as well as the orientation of the polarization vector, which can be changed through a filtering operation. Often it is believed that the shape of the anomaly becomes simpler and more localized, thus reducing interference from the neighbouring anomalies, if the direction of polarization is vertical, that is, if the causative body is taken over to the north pole. Thus, the transformation of a magnetic signal into what it would be when the causative body is taken over to
Digital Filtering of Maps I
184
the north pole is known as reduction to pole. Remember that there is no physical relocation of the causative body nor is it necessary to know the shape of the body; however, the current direction of polarization must be known. We derive a more general result which shows how to change the direction of polarization from the current direction to another specified direction. The Fourier transform of the total magnetic field caused by a magnetic body which is uniformly magnetized is (see Eq. (2.44)) Fr(u
'
v, h) -
~
(1"c~u + j f l v - 3's)l(u v, w) . [l"Uex + j r % '
- Sez] exp(-hs) (s-jw)s dw
which may be rewritten in a different form
F r ( u , v, h) -
(l'au + j f l v - ~/s)(j'Uao + j v f l o - sT0)[ll
l+ ~K(u, v, w) exp(-hs) ,J
-- oo
dw
(s - j w ) s
(4.30) where [I[ is the magnitude of the inducing field, a0,/30, and ~0 are the direction cosines of the magnetization vector and K ( u , v , w ) is the Fourier transform of magnetic susceptibility. In Eq. (4.30) we have been able to separate out the factors which do not depend upon the causative body. Assume that a,/3 and ~, (which are the direction cosines of the earth's magnetic vector) as well as a0, rio, and 70 are known. To transform the total field given for one set of direction cosine parameters to another set, the required filter response is
(j'au + j f l v - 7S)(l'Uao + j v f l o - s70) A ( u , v) - ( j a u + j f l v - 7s)(j'UCto + j v f l o - sT0)
(4.31a)
For the special case of reduction to pole we have a ' = / 3 ' = 0, 3' ' = 1 and o~'0 =/3'0 = 0, ~'0 = 1 and Eq. (4.31a) reduces to [22,23]
2 Apole(u, v) --
s
(l'au + j f l v - "ys) (j'Uao + jvflo - s % )
The field reduced to pole can be expressed as
(4.3 lb)
Reduction to pole and equator
fpole(X,Y, h) -- ~
lff+
185
Apole(U, v)Fr(u, v,h)exp(j(ux + vy)) du dv oO
Let us express Eq. (4.3 lb) in polar coordinates. For this assume that the angle of declination is 6 measured with respect to geographical north. Then c~ = sin 6 and /3 = cos 6 and
Apole (s, 0)
1
(j" cos/sin(~ + 0) - 7)(/" cosI0 sin(~0 + 0) - 70)
(4.31c)
where 60 is the angle of declination and I0 is the angle of inclination of the magnetization vector. Similarly, 6 is angle of declination and I is angle of inclination of the earth's magnetic field vector.
4.5.2. Low latitude effect In high latitudes, the angle of inclination is close to 90 ~ or c ~ - / 3 ~ 0 and c~0-/3o ~ 0, the response of the reduction-to-pole filter is close to unity, Apole(U , v) ~ 1 but in low latitudes, 7 ~ 0 and 7o ~ 0, the filter response reduces to a real function
ms2 Apole(U , v) ~ (ctu +/3v)(uc~ o + Vflo)
(4.32a)
or in polar coordinates
A
[
~polet,S,
0)
1
~ sin(~ + 0)sin(~0 + 0)
which remains undefined (goes to infinity) on lines defined by ~0 + 0 ~5+ 0 - nTr (see Figure 4.24).
(4.32b)
nTr or
Example 4.11 An example of reduction to pole from low latitude (20~ to pole is shown in Figure 4.25. The total magnetic field anomaly of a vertical infinite prism (incli-
186
Digital Filtering of Maps I V
Il k
,|,
,,,
tl
Figure 4.24. The reduction-to-pole filter for low latitudes becomes infinite on the radial lines shown above. nation = 10 ~ and declination = 3 ~ is shown in Figure 4.25a. A filter with a response given by Eq. (4.22) was used to filter the magnetic signal and the result is shown in Figure 4.25c. For comparison the actual magnetic signal at the (north) pole is shown in Figure 4.25b. To this magnetic anomaly, pseudorandom white noise was added and a lowpass filter was used to remove the noise beyond a radial frequency. The noisy signal is shown in Figure 4.25d, the signal reduced to the pole without filtering is shown in Figure 4.25e and finally after filtering in Figure 4.25 f. 4.5.3. Reduction to equator !
!
At the magnetic equator 7 and 70 ~ 0 the response of the filter turns out to be +
Aequator(U , v)
+
(J'OzU+ j ~ v - 7s) (/'CtoU +j/3oV - 70s)
(4.33)
Note that in a low latitude the filter response is close to unity, A equator(U,V) ~ 1 but in a high latitude the filter response is just the reverse of Eq. (4.32), consequently Figure 4.25. Reduction to pole from low latitude. Use of a lowpass filter has improved the pole reduced data. See the text for explanation.
~r"
~
L
o
..Q
c)
0
o
Digital Filtering of Maps I
188
one may face similar difficulties as in reducing to pole from low latitude. It may be noted that reduction to pole and reduction to equator are not the only two possible transformations. Indeed it is possible to transform a given magnetic anomaly to any latitude.
Example 4.12 Here we give an example of reduction to equator. A magnetic signal was generated for a square prism located in low latitude (Bangalore, India as in Example 4.11). The computed field is shown in Figure 4.26a. The field is now reduced to equator (magnetic) using Eq. (4.33). The reduced field is shown in Figure 4.26b and for comparison, the actual magnetic signal at the equator is shown in Figure 4.26c.
4.5.4. Pseudogravity The magnetic field reduced to pole resembles the vertical derivative of the gravity field provided the density variation is proportional to the magnetic susceptibility variation. To verify this statement we multiply both sides of Eq. (4.30) by the reduction to pole filter given by Eq. (4.31). We obtain +oo
Apo,e(U, v)Fr(u v, h) - Ills
g(u, v, w) exp(-hs) dw oo
(4.34)
S ~jw
The right-hand side of Eq. (4.34) indeed corresponds to the vertical derivative of the gravity field. In Eq. (2.86) an expression for the Fourier transform of the gravity field is given, from which the Fourier transform of the vertical derivative is simply obtained by multiplying by the radial frequency, s. It then results in an expression which is the same as the right-hand side of Eq. (4.34), provided I~lK(u, v, w) =
GP(u, v, w)
(4.35)
If the susceptibility at every point is replaced by the density, numerically equal to
Figure 4.26. An example of reduction to the equator. (a) Magnetic field over a cube in low latitude; (b) reduced to the equator and (c) computed field for the same body at the equator. Contour interval is0.1 ~.
Reduction to pole and equator
189
(a) ~.
'"
(b)
(c)
190
Digital Filtering of Maps I
III GK(x,Y, z) the vertical derivative of the resulting gravity field will be exactly equal to the magnetic field reduced to the pole. By a similar argument, it is shown that the magnetic field reduced to the equator is equal to the horizontal derivative of the gravity field, provided Eq. (4.35) is satisfied.
4.5.5. Distortion analysis Reduction to pole requires a knowledge of polarization vector which often is taken to be the same as the earth's magnetic field, but when there is remnant magnetization, this is not true and it may lead to a distortion of the filtered anomaly. When there is an isolated anomaly caused by a simple geometrical object, for example, a prism (see Example 4.13) the distortion may be used to estimate the direction of polarization. One of the characteristic features of the distorted anomaly is the appearance of negative contours caused by the nonvertical polarization vector. This feature has been exploited for estimation of polarization direction [40]. The minimum value of the filtered anomaly is plotted as a function of the assumed declination and inclination of the polarization vector. The maximum in such a plot lies over the actual polarization declination and inclination. The method has been applied to both a synthetic model (see Example 4.13) and real data from Ischia Island (Tyrrhenian Sea, Italy) [40]. The results are reproduced in Figure 4.27. The above method of estimating the direction of polarization will work only when an isolated anomaly is available. The distortion of the filtered anomaly may be quantitatively described through a distortion filter. For this in Eq. (4.31b) let it be wrongly assumed that the direction of polarization is the same as that of earth's field. The result of applying such a filter is equivalent to applying a filter, called a distortion filter here, to the desired output (reduced to pole). The distortion filter is given by Adist(U, v)
juao +jv[3o -- sT0 jua +jvfl- s'y
In low latitudes 3, << l, then the above distortion filter can be expressed as Adist (U, V) "~Ju~
`770 (
1 +jua -+jv/3)
191
Reduction to pole and equator
(a)
45.0
km 36.0
.~27.0
18.0
9.0
Ol 0
I
I
1
I
I
18.0 27.0 36.0 45.0 km Easting HIGH PASS FILTERED MAGNETIC FIELD (nT) 9.0
-"~ .-.
......
(b)
-70 -60
z
-2o
-'2 30
35
40
45
50
55
60
INCLINATION ( ' )
Figure 4.27. (a) The total magnetic field over Ischia Island (Tyrrhenian Sea, Italy) [44] and (b) a plot of anomaly minimum as a function of assumed declination and inclination. Before applying a reduced-to-pole filter the magnetic field has been subjected to high pass filtering (the lower cut-off is 2~r/45 or all wavelengths above 45 km).
The factor in the denominator represents integration in the direction of assumed declination. This will result in smearing of the output in the direction of declination. This phenomenon, known as comet's tail in Ref. [41], is well developed in Figure 4.27a.
Example 4.13 Consider the total magnetic field due to a vertical prism of size 4 x 6 x 1 k m placed at a depth of 1 km below the observation plane and polarized in the direction, 6 o - 6 0 ~ and I 0 - 6 0 ~ The earth's field is in the direction 6 o - 0 ~ and
192
Digital Filtering o f M a p s I
16.0
16
~
12.0
~12
==
~ ".o--II\Y. ~
7///I ~' ,-
0
4.0
8.0
East ing
20.0' I
~
"~ 12.0 - I
"~
12.0
16.0
(km)
-
-
20,0
0
~
20.
--
v12
e.o
-~
I
0
4.0
B.O East ing
-v-
12.0 (km)
|6.0
ZO 0
0
4 0
8 0 Ea=t ing
12 0 (km)
16 0
20 0
(e)
20.0 -
16.0 t;
~e
-
v12.0
~
"8.0o ==
4.0
0
-
--0
4.0
8.0 East ing
12.0 (km)
16.0
20
Figure 4.28. Magnetic anomaly shown in (e) is subjected to a reduced-to-pole filter designed for different inclinations but with actual declination. (a) I - 20~ (b) I = 40~ (c) I = 60~ and (d) I = 80 ~ The value of the minimum contour increases, starting at -70 to -20 nT and then again decreases to -35 nT when I = 80 ~
193
Reduction to pole and equator
80 70 v
z c) F< Z
60 50
._J
o
LIJ
40 30
20
i
j
[
20
30
40
50
i
I
60
70
SO
INCLINATION (')
Figure 4.29. A plot of the anomaly minimum as a function of assumed declination and inclination. The maximum lies over true declination and inclination.
I0=60 ~ The computed magnetic field is shown in Figure 4.28e. Next the magnetic field is reduced to pole. While the declination is assumed to be the same as the true one, the inclination is varied from 20 ~ to 80 ~. The distorted anomalies are shown in Figure 4 . 2 8 a - d where one observes a changing anomaly minimum; it is maximum when the assumed inclination is equal to the true inclination. A plot of the anomaly minimum as a function of assumed declination and inclination is shown in Figure 4.29. The maximum occurs at true declination and inclination.
4.6. Reduction to a plane surface
All analytical operations described in the previous section assume that the potential field was measured on a horizontal plane, but in practice, neither in ground survey nor in aerial survey is it possible to meet this requirement. In particular, in hilly regions the potential field is often measured on a highly irregular surface, that is, a hilly topographic surface or undulating flight surface of a low flying light aircraft. Then, it is imperative that the measured field be reduced to the nearest horizontal plane above the topographic surface. This requires continuation of the potential field from the irregular surface to the horizontal plane. We describe two methods for reducing the observed field on an undulating surface. The basic idea is to determine a field on a horizontal plane
Digital Filtering of Maps I
194
lying entirely above the observation surface. This field when continued downward to the actual observation surface must agree with the observed field. We give two methods: a least squares approach [24] and an iterative digital filtering approach [25] to estimate the desired field on a horizontal plane. The filtering approach, although slightly restrictive, is found to be very fast.
4.6.1. Least squares approach We assume that all observation points are scattered in a three-dimensional space but they all lie on a smooth surface representing a topographic surface. The data points when projected onto a horizontal plane are scattered over the plane. Let f(x,y,h) be the potential field on a horizontal plane h units above the mean observation surface and F (u,v,h) be its 2D Fourier transform. We continue the unknown field on the horizontal plane downward to every point of observation,
f ( x i ' Y i ' z~Xz i ) = -4--~ j -oo a -oo
'
i = 0, 1 , . . . , N - 1
(4.36a)
In the above equation the unknown quantity is F (u,v,h) which may be discretized over a finite domain with the assumption that it vanishes outside this domain. For the sake of simplicity, we assume the domain is rectangular, 2a • 2b, and it is divided into (2p + 1) • (2q + 1) cells. We discretize the integral in Eq. (4.36a) and write
f(xi,Yi, Azi)---~ ~ k=-p
~
x exp
kxi+~lyi
where P -
2p + 1 and Q -
f
~-
col{f(xi,Yi, Azi)~
k,--Ql, h exp ( h - Azi)
F
k
+
1
l=-q
,
i--O, 1 , . . . , N - 1
2q + 1. Define the following matrices:
i~- O, 1,2,3, . . . , N - 1}
(Size: N x 1)
(4.36b)
Reduction to a plane surface
195
exp A
(-~ k) 2
exp
kxi + ~ lyi
1
-@x
i= 0,1,...,N-1;
l =-q,-q
k--p,-p+l,...,p-l,p;
+ l,...,q-
l,q
(Size: N x (2p + 1)(2q + 1) and F - col
-O
'
k--p,-p
+ 1 , . . . , p - 1,p,
1=-q,-q
+ 1, . . . , q - 1,q
(Size: (2p + 1)(2q + 1)) Eq. (4.36b) in the matrix version becomes f = AF. We should be able to solve the above system of linear equations for the unknowns contained in vector F. Naturally, we must have N ~> (2p + 1)(2q + 1); in the presence of noise a reliable solution is obtained only when N >> (2p + 1)(2q + 1). A least squares solution of Eq. (4.36b) is given by F-
(AHA)-IAHf
(4.36c)
where superscript H stands for the hermitian transpose.
4. 6.2. Iterative filtering We start with the data measured over an undulating surface such as an uneven topographic surface or flight surface. All three coordinates of a measurement station can be arbitrary; however, in our present discussion we assume that only the z coordinate is arbitrary and the x and y coordinates constitute a regular rectangular grid (where this is not possible we use a gridding procedure to reduce the data to a regular rectangular grid). The data points, when projected onto a horizontal plane, fall on a regular grid of points. Only the height of a data point is irregular. Let fo(x,y,2~z) be the potential field (gravity or magnetic) observed over a known but arbitrary surface 2~z(x,y) and f(x,y,h) be the desired field on a horizontal plane h units above the mean observation surface (h > 6z(x,y)).
196
Digital Filtering of Maps I
Now we continue the field downward to the observation surface and require that the downward continued field must satisfy the observed field,
fo(x, y, zXz) -- ~ 1 I f
+~ F(u, v, h)exp I sh( 1 - zXz(x'Y))]exp[l'(Ux + vy)] du dv h 7r
(4.37a) is the Fourier transform of the desired field where F(u,v,h) f (x,y,h) and s - ~/(u2 + v2). Note that the representation of the potential field given in Eq. (4.37a) satisfies the Laplace equation and the vanishing boundary condition at +oo. Next, we express -
where e < 1 and rt(x,y) is a normalized surface, that is, 1
x
4XY Z -X
Y
~
r/2(x'Y) -- 1,
X , Y --+ oo
-Y
Use the following expansions: oo
oo
e x p ( - s A z ( x y)) - ~ '
( - 1)ksk Azk k!
k=O
and
F(u, v, h) - ~
r Fi(u, , h)
i=0
in Eq. (4.37a). Express the right-hand side of Eq. (4.37a) as a polynomial in e. Then Eq. (4.37a) becomes
fo(x,y, Az) -- ~
ljj+
A(u, v, rl(x,y))exp(sh)exp)'(ux + vy)] du dv 7r
where
A(u, v, rl(x,y)) - Fo(u , v, h) - C{rl(x,y)sFo(u , v, h) - F 1(u, v, h)]
(4.37b)
197
Reduction to a plane surface
+C2[ rl2(x'y) s2F~
-- rl(X, y ) S F l ( U
--C 3 7]3(x'Y) s3Fo(bt v , h ) - ~(~'Y) 3!
'
,
v,h) + Fz(u, v,h)]
S2Fl(U
2!
v,
'
+ ~('Y)~. ~F~(,, ~,h) - F3(,, ~,h)
]
h)
+ ...
We set the coefficients of e to zero giving us the following set of interconnected equations from which we can obtain Fk(u,v,h), k - 0,1,2, .... For example,
Fo(u, v , h )
-
exp(-sh)Z Z f~ x
y
F1 (u, v, h) - exp(-sh) ~
x
~f
+ vy))
1
rl(x,y)exp(-j(ux + vy)) 47r2
Z x
2~)exp(-j(ux
y
+Tr
Fo(u, v,h)s exp(sh) exp(/'(ux + vy)) du dv
7"C
1 F2(u, v,h) - exp(-sh) Z Z rl(x,y) exp(-j(ux + vy)) 47r2 x
x
y
ff+ Fl(U,v,h)s
exp(sh) exp(/'(ux + vy)) du dv
7"(
-~-~ ~ x
1
r/2 (x, y) exp(-j(ux + vy)) 47r2
y
2 x f f+Tr Fo(u,v,h) s~exp(sh) exp(/(ux + vy)) du dr] and so on. The above system of iterating equations is easily evaluated using the FFT algorithm. The convergence is rapid, usually requiring about ten iterations, depending upon the amplitude of the surface undulations.
Digital Filtering of Maps I
198
.
"
" t,' zk'~. 9
-.
,7:~",~
.~
,
~.~6.~. 9 '
.,,:-, o-. " 0 ~ ~ . .9 ~ K.'.'~_"" )
~,,2,.t_v--, ' -
Figure 4.30. A map of an undulating surface along with observation stations (+) and an outline of magnetic targets.
Example 4.14 For the purpose of testing the above method of reduction to a horizontal surface, we have synthetically generated the field along a set of 16 almost parallel lines placed on the undulating surface, as well as at randomly scattered (2500) points (see Figure 4.30). On parallel lines, corresponding to undulating flight paths of an aircraft over hilly terrain, 160 samples were collected giving a total of 2560 samples. The data thus created were gridded using a commercially available software package (e.g., Data Plotting Services Inc., Toronto). For comparison with respect to reduction to a horizontal plane, we have also used the least squares method described in Ref. [24] which does not require gridding of scattered data prior to reduction. The contoured maps are shown in Figure 4.3 l a,b. The result of field reduction to a horizontal plane at 7.5 units above using the new method is shown in Figure 4.31c,d. The theoretically computed field on the plane of reduction is shown in Figure 4.31 e. The rms error between the reduced and the theoretical field is shown in Table 4.5. We note that in comparison with the least squares method, there is a marginal increase in the rms error; however there is a considerable saving in the
Reduction to a plane surface
(a)
(b)
(c)
(d)
199
\
i
(e) Figure 4.31. The contoured maps of a measured field on parallel lines (a) and at scattered points (b). Reduced field to a horizontal plane at 7.5 units using the iterative method is shown in (c) and (d). A theoretically computed field on the plane of reduction is shown in (e). Compare this figure with the reduced field shown in (c,d). computer time, by a factor of 1 0 - 1 2 (the least squares method required 125 min on the same machine [25]).
Example 4.15 As an example of application of t h e above method of reducing to a plane surface to real data, we consider the aeromagnetic data obtained in a low altitude (nominal flight height 50 m) aeromagnetic survey carried over a part of diamond bearing rocks, consisting of granites and granite gneisses belonging to the south Indian shield. The survey was carried out by the A M S E division of the Geological Survey of India using the Scintrex system mounted on a Twin Otter aircraft. In addition to the total magnetic field, the system provided continuous measurements of barometric height and the air column thickness which were used to reconstruct the actual flight surface. The navigation system consisted of Doppler
Digital Filtering of Maps I
200
TABLE 4.5 Computer time required and rms error between the reduced and the theoretical field for different types of data Nature of data (2500 points)
Cpu time in minutes on a Vax 750
rms error in %
Scattered data Parallel profiles Grid data
12.0 12.0 8.0
2.59 6.36 1.44
as well as a visual method based on photostrips. It is believed that the positional accuracy is better than 5 m. The raw aeromagnetic data after the application of the diurnal and instrumental drift corrections and subtracting a base value of 41 100 -y is shown in Figure 4.32a. Note the longitude and latitude of the area covered in this survey. The field reduced to a plane at a height of 678 m above sea level or 381 m above local base level is shown in Figure 4.32b.
4.7. Removal of the terrain effect
If the hills are magnetized, they would produce a magnetic anomaly characteristic of the shape of the hills and the distribution of magnetization. Naturally, a magnetic signal from a deeper source will be corrupted by the anomalies produced by the hills and valleys. The usual remedy to overcome this problem, particularly in gravity surveys, is to calculate the gravity effect of each hill or valley and subtract it from the observed field, a procedure often known as terrain correction. We describe a digital filtering approach for removing the terrain effect. In the first instance, we assume that the density or magnetization is uniform and that the topographic information is available. A digital filter is devised for removing the effect of the terrain. A basic deficiency of this approach is the assumption that the density or magnetization variation in the surface rocks is assumed to be constant. We describe a method where the knowledge of the variation of density or magnetization is not necessary. We estimate that component in the observed field which is strongly correlated with the terrain and then subtract it from the observed field. This is called correlation filtering [26]. The idea of removing the terrain effect through correlation was first suggested by Grauch [27]. In a different approach, the correlation between the Bouguer anomaly and topography was used to estimate the average surficial density [28].
Reduction to a plane surface
201
Figure 4.32. (a) The raw aeromagnetic data (after application of all corrections) over granites and granite gneisses belonging to the south Indian shield; (b) the field reduced to a plane at a height 678 m above sea level.
202
Digital Filtering of Maps I
4. 7.1. Filters to remove terrain effect When the topography effect can be modelled as a linear system, for example, when the undulations are gentle, we can express the topography effect in the frequency domain as follows:
drl(u, v) = A(u, v)dAZ(u, v)
(4.38)
where A(u,v) is the transfer function of the linear system connecting the topography and its potential field (see Eq. (2.70)). Let the observed field be given as a sum of the deeper signal and the topography effect, F(x, y) = fo (x, y) + r/(x, y). In the frequency domain, using Eq. (4.38), we get
dF(u, v) = dFo(u, v) 4- A(u, v)dAZ(u, v)
(4.39)
We can estimate the transfer function under the assumption that the topography is uncorrelated with the deeper signal [35]. Multiply both sides of Eq. (4.39) by dAZ(u,v) and carry out the expected operation, whereupon we obtain the transfer function,
A(u, v)
Sfzxz(U,v)
(4.40)
and from Eq. (4.40) we obtain the deeper signal
dFo(u, v) - dF(u ' v) - Sf M(u, v) v) dAZ(u, v) Szx~(u,
(4.41)
where Szxz(U, v) and Suez(U, v) are the spectrum of the topography and cross spectrum between the topography and the observed field, respectively (see Chapter 3). We have assumed that the density or magnetization contrast be constant but not necessarily known. The above method will not work when the topographic undulations are large and when the density or magnetization contrast is variable. We describe another method where the assumption of gentle undulations is removed. We have derived in Chapter 2 (Eq. (2.68)) the potential field (gravity) due to a random surface on a plane above it. We rewrite that equation in a slightly different form, oo
drl(u, v) = 27rGAph exp(-sh)
k! k=l
where
dAZ~ (u, v)
(4.42)
Removal of the terrain effect
lff§
203
d/kZk(u , v) exp(/(ux + vy))
4rr2
71-
Eq. (4.42) can be expressed in terms of a sequence of filters oo
drl(u, v) - Z Ak(u, v) dAZk(u, v)
(4.43a)
k=l
where
Ak(u , v) = Ak_ 1(u, v)
-sh
(4.43b)
with
Ao(U, v) - -2rrGAp
exp(-sh) s
as a starting value. It is essential that the density contrast is known so that Ao(u,v) can be computed and thereby the entire sequence of filters. When the topographic surface rises above the plane of observation the method described above will not work. A complication arises on account of the fact that the mass lying above the observation plane will produce attraction in the opposite direction. This problem is discussed in Refs. [46,47].
4. 7.2. Correlation filtering Finally, we consider a more general model where the magnetization, both its magnitude and direction, or density is spatially variable in a random fashion. The theory of correlation filtering for this model is developed in Refs. [26,29]. The basic idea in correlation filtering is to remove from the potential field that part which is strongly correlated with the topography. Let mx, my and mz be three components of magnetic polarization in the x, y and z directions, respectively. Further, let t~(x,y), ty(X,y) and g(x,y) be the total magnetic response (total magnetic field as measured in an aeromagnetic survey) of the terrain polarized in the x, y and z directions, respectively. Next we assume that there is a signal generated by a subsurface magnetic source which is uncorrelated with tx(X,y), ty(X,y) and tz(X,y) individually. The combined field as recorded in an aeromagnetic map may be modelled as follows:
fv(x, y) = mxtx(x, y) + myty(x, y) + mztz(X, y) + fo(x, y) whereJ~(x,y) is the signal generated by the subsurface magnetic sources. Since the
204
Digital Filtering of Maps I
topographic surface is known its magnetic responses tx(X,y), ty(X,y) and tz(X,y) can be calculated; however, the components of polarization are unknown. We now show how correlation processing may be used for estimation of the unknown components of magnetization. Let rh~, rhy and rhz be the estimates of the polarization components. From the observed field subtract the estimated contribution of the known topography. The residual is cross correlated with the computed topographic response, say t~(x,y).
E { tx (x, y) g~ (x, y) - mxtx (x, y) - m,t, (x, y) - mztz (x, y)] }
- (mx - rhy)E{t2 } + (my - thy)E{txty} + (mz - fnz)E{txtz} + E{tff~} Note
that
E{tff~} = 0 ,
by
assumption.
Further,
we
show
(4.44)
below
that
E{txty } - E{txtz } -- E{ tytz } -- 0 when the spectrum of the terrain has quadrant wise symmetry or radial symmetry. To show this, note that the Fourier transform of the total field may be written in terms of the Fourier transform of the surface (see Eq. (2.69b)). 27r
rx(~, ~) = - j - -s
27r
r,(,, ~) - -j.-
u(j'au +j/3v - ",/s) exp(-sh)AZ(u, v) v(1"c~u +j/3v - "75) exp(-sh)AZ(u, v)
s
Tz(u, v) - --s(j'au 27r +j~vs
"),s) exp(-sh)AZ(u, v)
(4.45)
where o~,/3, and 3, are the direction cosines of the earth's magnetic field, u, v are spatial frequencies in x and y directions, respectively and s = ((u 2 + v2), h is the height of data surface above reference plane, zXZ(u,v) is the Fourier transform of the topographic surface, ZXz(x,y). The cross correlations among tx(X,y), ty(X,y), and tz(X,y) may be written in terms of the corresponding cross spectrums:
E{tx(x,y)ty(x,y)} - 47r2 l f
f_ +~ E{Tx(U, v)Tf(u,
v)} du dv
7"f
f / + ~ 1 UV[laU +jflv - "ys]2exp( 2sh)Szxz(U v) du dv (4.46)
Removal of the terrain effect
205
where Szxz(U,v) = E{d&z(u, v)d&z* (u, v)} is the spectrum of the topographic surface. Similarly, for two other cross correlations we obtain
E{tx(X,y)tz(X,y)}-j f f+rr rr su ~l'C~u+jflv -- 7sl2exp(-2sh)S~z(U, v) du dv (4.47) and
E{ty(X,y)tz(X,y) } --j
f
f +r~ ~ sv ~'o~u+ jflv -- 7s[2exp(-2sh )SAz(U, v) du dv
(4.48)
Let us assume that SAz(U, v) = S/xz(-U, v) = Szxz(U, -v) = S/xz(-U, -v), that is, the spectrum of the topographic surface is qua&ant-wise symmetric. It may be seen from Eqs. (4.46)-(4.48) that all three cross correlation coefficients will be equal to zero whenever a or 13 or both = 0. In a given study area a or/3 can be set to zero by selecting the x or y axis in the direction of the horizontal component of the earth's magnetic field. Hence, from Eq. (4.44) we see that the correlation between the computed response, tx(X,y), and the difference between the measured aeromagnetic field and the estimated terrain effect becomes zero whenever mx - dnx. We can obtain a similar result for the two other components of polarization,
(my - m , ) U { t 2} --
(mz- rhz)E{tz2 } -- E{tzfo(x,y)}
(4.49)
In practice, in place of the expected operation in Eq. (4.44) we u s e arithmetic averaging, for example,
E{tx(X,y)fo(x,y)} -+ ~
~ tx(Xk,Yl)fo(xk,Y,) k
1
and so on. Further, E{tx(X, y)fo(x, y)} will not be equal to zero as the theoretical results above indicate but will attain some finite minimum value depending upon how closely the assumptions made here are satisfied in actual practice. Thus, rhx
206
Digital Filtering of Maps I
may be continuously varied keeping other components fixed until a minimum in the cross correlation is encountered. This procedure is then carried on for two other components, namely, rhy and rhz. The process of minimization is carried over a finite window, which is then slid step by step across the map. Finally, we list below different steps in correlation filtering. Step 1: Compute the total or vertical magnetic field caused by the topography. Let the magnetization of the top layer be unity and in the x direction throughout. Step 2: A small window is placed on the aeromagnetic map. The topographic effect within the window is estimated from the field calculated in step 1. It is then multiplied by the average magnetization within the window. Next we estimate the average magnetization. Step 3: Subtract the topographic effect estimated in step 2 from the aeromagnetic field. The residual field is correlated with the topographic effect. Compute the correlation coefficient for a series of assumed average magnetization values lying within a known range. Select that magnetization which produces the smallest correlation coefficient as the representative average magnetization for the selected window. Step 4: The average magnetization estimated in step 3 is assigned to a point at the centre of the window. Next, the window is slid by one unit and steps 2 and 3 are repeated until the whole map is covered. At the end of this step we have a matrix of magnetization in the x direction. Step 5: Now we compute the topographic effect taking into account the variable x component magnetization. The computed topographic effect is subtracted from the aeromagnetic field. The resulting field is free from topographic effect caused by the x component of magnetization. Step 6: To remove the effect of topography caused by the y and z components we repeat steps one to five.
Example 4.16 As an illustration of correlation filtering we consider a numerical example. A synthetic topography (Figure 4.30) was generated by placing 35 square hills in a random fashion on a horizontal surface (This surface is used as a reference plane). The height of the hills was varied between two and eight units as shown in Figure 4.33 A square shape was chosen so that the spectrum of the topography would have a quadrant-wise symmetry. The topographic map was digitized into a matrix of 128 x 128 digital topography. Each hill was magnetized randomly in the x, y, and z directions. The magnetization was varied uniformly in the range -180 to 320 -/ (using a random number generator on a computer). The vertical component of the
Removal of the terrain effect
207
cee i No 0 eee Figure 4.33. Synthetic topography consisting of 35 square hills, which are randomlypositioned. The hill height is also a random, uniformly distributed random variable. magnetic field at a height of ten units above the reference plane was computed. The maximum and minimum magnetic field values were 762 a n d - 4 5 1 -y. A magnetic signal due to a large vertical prism whose top is on the reference plane and its bottom is at 8 units below and whose magnetization vector has an inclination of 15 ~ a declination o f - 4 5 ~ and magnitude of 200 ~ was computed on a plane ten units above the reference plane. The magnetic signal and an outline of the prism are shown in Figure 4.34. The signal is then added to the computed topographic effect. The composite map is shown in Figure 4.35. This is the input to our correlation filtering procedure. The recovered signal, shown in Figure 4.36a, has a good resemblance to the actual signal shown in Figure 4.34. Note that in the method described above the direction of magnetization of the hills was varied randomly. Without this facility the results of filtering turn out to be poor. For example, if the topography is magnetized in the same direction as the magnetic target below, as assumed by Grauch [27], the performance of the method deteriorates .To demonstrate this we have assumed for the purpose of filtering that the direction of magnetization of the hills is same as that of the magnetic target below. The reconstructed signal is now shown in Figure 4.36b. Evidently, the output is much inferior to what was previously obtained in
Digital Filtering of Maps I
208
~
ism whose outline is also shown.
Figure 4.35. A composite map of a signal and terrain generated magnetic field.
Removal of the terrain effect
209
1
'i,~ ~ :~
bt
Figure 4.36. Recovered signal after correlation filtering. (a) The direction of the surface magnetization is random, (b) the direction of magnetization is the same as that of the magnetic target below.
Digital Filtering of Maps I
210
Figure 4.36a. In this numerical experiment we have used a window of size 15 x 15, slid by three units.
4.8. Wiener filters
4.8.1. Basic theory Wiener filters are optimal filters in the sense that the filtered output is close to the unknown signal in the least squares sense. The data are assumed to be the sum of signal plus noise, f (m, n) = qp(m, n) + r/(m, n) where ~p(m,n) is the signal and r/(m,n) is the noise, presumably stationary stoand S,(u,v), respectively. chastic processes characterized by their spectra, Sr It is shown, for example, in Ref. [30], that the optimum filter transfer function is given by
A(u, v)
S~(u, v)
- S,,~I v)
(4.50a)
When the signal and noise are uncorrelated the denominator in Eq. (4.50a) is simply given by Sr + S,(u,v). Note that when the signal and noise spectra are non-overlapping, the filter transfer function is equal to one where the signal spectrum is non-zero but it is equal to zero wherever the noise spectrum is finite; then a perfect separation of signal and noise is possible. When the spectra are partially overlapping, the Wiener filter will optimally suppress the noise with minimum loss of signal. This is illustrated in Figure 4.37. The loss of signal or mean square error (MSE) is given by
MSE --
S (u, v) +
v) du dv
(4.50b)
From Eq. (4.50b) it is clear that when the signal and noise spectra are not overlapping, then the numerator of the integrand becomes zero and the integral vanishes.
Wiener filters
E
211
l Frequency
E
I
x..
Optimum filter -.--__
Z
Frequency Figure 4.37. The optimum Wiener filter depends on the signal and noise spectrum. When these two are non-overlapping the optimum filter is a simple pass filter. When the spectra are overlapping the optimum filter has variable attenuation. For practical implementation of the Wiener filter, the signal and noise spectra must be known and for the filter to be effective the spectra must not be overlapping too much. In the context of potential field analysis, neither of the above requirements can be met. We can only attempt a suboptimal form of Wiener filter exploiting the theoretical models of the signal and noise.
4.8.2. Extraction of potential field signal One such suboptimal filter was first suggested by Strakhov [31], while an extension to two dimensions is given in Ref. [32]. In this approach the signal is modelled as a deterministic analytic function whose singularities are all confined to a finite volume in the lower half space (see Figure 2.5). Let h be the depth to the top most singularity, then the Fourier transform of the signal satisfies the following inequality [31 ]:
212
Digital Filtering of Maps I
IF(u, v)l < -
e x p ( - s ( h - e))
27r
(4.51)
where F(s) for a gravity field is equal to the total mass of the body which produced the signal and for a magnetic field it is a number in the range, 0 < F(e) < oo and it is a function of e, an arbitrarily small number. Define a 2D digital filter
am,n,
- M <_ m <_M
and
-N
<_ n _< N
where M and N are integers. The filter coefficients must satisfy the so-called quadrant symmetry am, n -- a_m,_n~
am,-n
(4.52)
-- a_m, n
so that the transfer function A(u,v) of the filter is real, and the signal upon filtering does not undergo phase distortion. The effect of the filter on the signal is given by
signal distortion - ~
1/'j
I:
, I: du dv
I[1 - A(u, v)] IF(u v) 71"
(4.53)
where signal distortion refers to the integral of the square of the difference between the original signal and its filtered version. We need to minimize the signal distortion or an equivalent expression
lff+
1[1- A(u, v)]21exp(-2sh)du dv
signal distortion _< 4~-g
(4.54)
7r
obtained by replacing the spectrum of the signal by its upper bound Eq. (4.51) valid for a potential field signal. It is assumed that the depth to the top most singularity h is measured in units of the sample interval, assumed to be unity on some undefined scale. Next, we would like to evaluate the effect of the filter on the noise. The variance of the filtered noise is given by [32]
var{rlfiltered} -- Z m
~
~-~ Zam,nam',n'C~(m -- mt, n -- n') n
m t
nt
(4.55)
Wiener filters
213
Naturally, it is desired that the variance of the filtered noise be as small as possible. In Strakhov's filter the noise reduction is a pre-defined constant less than one. The filter coefficients are now obtained by minimizing the signal distortion given by Eq. (4.54) under the constraint that the ratio of the variance of the filtered noise to the variance of the noise in the observed signal is equal to a pre-defined constant called the noise reduction factor (NRF). This constrained minimization may be carried out as described in Ref. [32] or by using any standard optimization package, such as the optimization tool box in Matlab.
Example 4.17 As an example, we give filter coefficients for h = 4 and NRF - 0.1. We have assumed, for the sake of computational simplicity, that the noise is spatially white. The filter size is 9 • 9, that is, M = N - 4. Only coefficients in the first quadrant are shown but the remaining may be obtained by simple mapping to satisfy the symmetry requirements (Table 4.6).
4.8.3. Signal distortion The effect of filtering on a pure potential field signal is approximately equivalent to vertically displacing the plane of observations or the signal causing body by the same amount in the opposite direction. The required displacement may be obtained from the filter coefficients themselves. Let J~lt(x,y) be the filtered noise free potential field signal and f (x,y; 6h) be the field caused by the body when displaced by 6h units downwards.
{
[[ft~lt(x,y) - f (x,y, ~Sh)[IL2_<
If
exp(-2sh)
exp(-~Shs) - ~
oo
ckt exp(j(uk + vl))
~ k
}2
du dv (4.56)
1
TABLE 4.6 -0.00800 -0.00984 0.02629 0.03852 0.04356
-0.00708 0.00904 0.02368 0.03436 0.03852
-0.00977 0.00366 0.01549 0.02368 0.02629
-0.01523 0.00494 0.00366 0.00904 0.00984
-0.02243 -0.01525 0.00977 -0.0070 -0.0080
214
Digital Filtering of Maps I
The right-hand side of Eq. (4.56) is now minimized with respect to 6h which will automatically ensure minimization of the difference between J~lt(x,y) and f (x,y; 6h) under L 2 n o r m . This results in a non-linear equation, 12
1 (1 -l-- _~)3
4 ~
~
k= - N l= - N
ak,, [4(1 + ~)2
k2
121,/2
(4.57)
_~__~ _qt_-~ l
which may be linearized for h >> N, where h is measured in units of Ax (=Ay),
~h h
1(
~
~
ak,t
k = - N I=-N N ) 1/3
ak,t
(4.58) -0.5
k = - N I=-N
Note that minimization of the right-hand side of Eq. (4.56) is equivalent to finding a least squares upward continuation digital filter. Thus, the Strakhov filter may be looked upon as an upward continuation filter such that the variance of the background noise is reduced to a pre-assigned noise reduction factor. The height of upward continuation, however, would depend upon the selected noise reduction factor.
Example 4.18 An example of application of Strakhov's filter to a 1D signal is given below [33]. In one dimension the filtering operation with a symmetric filter coefficients is given by N
ak[f (x + (k - 1)z2xx) + f (x - (k - 1)ZXx)]
f, ,t (.,,:) = k=l
A magnetic profile across the Great Slave Lake (Canada) aeromagnetic anomaly is shown in Figure 4.38a. The 1D filter coefficients are: 0.0986366, 0.096834, 0.0917995, 0.084477, 0.0760078. The filtered anomaly is shown in Figure 4.38b where the computed field due to two infinite dykes (vertically polarized and susceptibility contrast was assumed to be 0.0027 cgs units) is also shown. The
Wiener filters
215
0
4500 4400 ~4200 W4000 380O 3600' 0
S
15
tO
20
25
30
:35
40
4.5
50
x - - - ~ ( ' a uN,T : '/Z M,-E) b
i 4.ooo3600 ,K
3600340C 32000
i
5
!
i
I0
15
I i
|
20
|
25
J
30
i
35
-][1I
!
4,0
J
45
I
50
Figure 4.38. An example of application of Strakhov's filter to real data. depth correction on account of filtering turns out to be 1/4 h where h is the depth to the top of the dyke.
4.8. 4. Wiener filter for reduction-to-pole We have seen in Section 4.5 how the response of a filter for reduction-to-pole from low latitude becomes infinite along two radial lines. As a result, the background noise in the region where the filter response becomes very high gets amplified, drowning the signal. A simple remedy is to use a lowpass filter to
216
Digital Filtering of Maps I
remove as much noise power as possible before applying the reduction-to-pole filter. One can go a step further and devise an optimum Wiener filter, whose output is closest to the pole reduced magnetic anomaly, that is, fpole(X,y) given in Eq. (4.21). The desired Wiener filter may be derived in the same manner as derived in Ref. [30]. We obtain
S (u, v) Awiener(b/, v)
(4.59)
Sq)(u,v) -I- Sr/(U, v)Ap~ (/'/' v)
where Apole(U,V) is the reduction-to-pole filter defined in Eq. (4.3 l b). An equivalent result was derived by Hansen [34] who suggested estimation of the required signal and noise spectra from the radial spectrum (see Section 3.4) and their use in Eq. (4.59). Another approach is to devise a digital filter which will transform, in a least square sense, the magnetic response of some specific body at low latitude into the magnetic response of the same body at the pole [36].
4.8.5. Wiener filter for separation of fields from different levels The aim is to devise a digital filter to separate the potential fields arising from two different levels, for example, the field from the near surface sources and the field from the basement sources. The spectral characteristics strongly depend upon the level where the sources are located. Yet, it is not possible to achieve complete separation because of overlap of the spectra. The best that can be achieved is through an optimum Wiener filter. We consider a simple two layer case (see Figure 4.39) and study its performance. Let each layer be made of many
Layer One ah H Layer Two II
I
I
I
9
!
. . . . . . . . . . . . . . . . . . . . . . . . . . . I L,I l I , t l 1 ! 1 1 1 1 ! 1 | I ! I ! 1 I i i i ! i ! i I i I I I 1 ! i I i I I ! ! ! i 1 1 I ! i i I l l i ! l
1 IIIi!1111~
Figure 4.39. A model of two magnetic layers: magnetization is assumed to be vertical or the field is reduced-to-pole.
Wiener filters
217
uncorrelated thin layers (sandwich model on p. 116) and the polarization is in the vertical direction or the field is reduced to the pole. The spectrum of the observed magnetic field is given by (assume no measurement noise) Sz( ,
v) +
- s,
v)
where $1 (u, v) is the spectrum of the field due to the first layer,
S, (u, v) - 7rsS/x~, (u, v) exp(-2sh) [1 - exp(-2sAh)] and S2(u, v) is the spectrum of the field due to the second layer,
S2(u, v) - 7rs S/x,~2(u, v) exp(-2sH)[1 - exp(-2szXH)] S/x~, (u, v) and S/x~2(u, v) represent the spectrum of horizontal susceptibility variations in layer one and two, respectively (see Eq. (3.48d)). The Wiener filter for extraction of the magnetic field signal due to, say, the first layer is given by (see Eq. (4.50a) [1 - exp(-2sAh)] exp(-2sh) A(u, v) - [ 1 - e x p ( - 2 s A h ) ] e x p ( - 2 s h ) + [1 - e x p ( - 2 s s
exp(-2sH)
In deriving the above result we have assumed for convenience that S/x~, (u, v) -- S/x~2(u, v) = S/x,~(u, v). To study the performance of the Wiener filter we evaluate the mean square error (MSE). Note that the desired signal is the one from the top layer. Using Eq. (4.50b) for estimation of MSE, we obtain
l f+~176 MSE
--
47r2
a-oo
1 [~
Jo
a-oo
[1
--
e(-2sAh)]e (-2sh) +
[1
--
e(-2sZXH)]e(-2sH)
du dv
s 2 Six.(s)
ds exp(2sh) exp(2sH) [1-exp(-2sAh)] -~- [1-exp(-2sAH)]
To simplify further we invoke the fractal model for magnetization (see Section 3.6 for more details on the fractal model), that is, SA~(s)- Aerials 2 and
Digital Filtering of Maps I
218
Ah - zSH << h. The MSE normalized by the signal power reduces to MSE Signal power
4h2
f
oo
s exp(-2sh) ds 1 + exp(Zs(H - h))
The numerical values of the normalised MSE for different separations, ( H - h ) , are plotted in Figure 4.40. When the separation is zero, that is, the two layers are overlapping, the MSE is maximum and for separation greater than two units the MSE falls below 10%.
4.8. 6. Matched filter In the literature [37,38] a claim is made that a field from a level can be separated by means of the so-called 'matched filter' defined as follows. Consider a two layer case shown in Figure 4.39. The spectrum of the magnetic field may be expressed (see Eq. (3.48d) where it is assumed that Ah << h and M-/<< H) as 0.5
0.4
lla
0.3
O
r~
0.2
0.1
~ .
.
.
.
.
.
.
.
.
.
.
.
.
.
x,
".,., . . . . . . . . . . . . . . . . . . . . . . . . . .
0.0 1
2
3
4
Layer Separation Figure 4.40. MSE normalized by the signal power as a function layer separation. The measurement plane is placed at one unit above the top layer.
219
Wiener filters
Sz(u,
= s, (u, --
~-Ah
+ s2(u, o /2x ~ S 2
exp (
2 2 exp(-2sH) 2sh) + 7rAH crA~2s
where we have assumed thin two layers and the fractal model of susceptibility. The matched filter to extract the magnetic field signal from the first layer is defined as
A(u,v)
~/
~ 0"2A~2
1 + ~xh~ e x p ( - Z s ( H - h))
Note that all parameters required in the above filter can be obtained from the radial spectrum. The observed magnetic field is now passed through the matched filter given above. The output may be expressed making use of Eqs. (2.79) and (2.80b) which give the magnetic field (total component) due to a thin layer.
j~v(x,y) - -~-~1/f+~iosdKl(U, ~ 1 +v,h aI44~2Ah~lexp e(-sh) xp ( - 2 sexp (H(j(ux_ h))+ vy)) 91 /f+~IosdK2(u,v,H) -
e x p ( - s H ) exp(j(ux + .
.
.
.
.
.
vy))
1 + /~h~ 1 e x p ( - 2 s ( H - h)) Clearly the output contains signals from both layers, however the spectrum of the output is now equal to the spectrum of the signal from the first layer. But this is not enough, as there is important phase information missing. Thus, the socalled matched filter does not really separate the signals from different layers but only separates their spectra, which is already known.
References
[1] D.E. Dudgeon and R. M. Mersereau, MultidimensionalDigitalSignalProcessing,Prentice Hall, Englewood Cliffs, NJ, 1984. [2] W.R. Ford, Application of Green's theorem in two dimensional filtering, Geophysics,29, 693-713, 1967.
220
Digital Filtering of Maps I
[3]
W.A. Spirev and R. M. Thrall, Linear Optimization, Holt, Rinehart and Winston, New York, 1970. J.V. Hu and L. R. Rabiner, Design techniques for two dimensional digital filters, IEEE Trans., A U-20, 249-257, 1972. J. G. Fiasconaro, Two dimensional non-recursive filters, In: A. Rosenfeld (Ed.), Picture Processing and Digital Filters, Springer Verlag, Berlin, pp. 70-128, 1979. J.H. McClellan, The design of two dimensional digital filters by transformations, Proc. 7th Annual Princeton Conf. on Information Sciences and Systems, pp. 247-251, 1973. J.H. McClellan and T. W. Parks, Equiripple approximation of fan filter, Geophysics, 37, 573-583, 1972. R.M. Mersereau, W. F. G. Mecklenbrauker and T. F. Q. Quatrieri, Jr, McClellan transformations for two dimensional digital filtering: I -Design, IEEE Trans. Circuits and Systems, CAS-23, 405-414, 1976; W. F. G. Mecklenbrauker and R. M. Mersereau, McClellan transformations for two dimensional digital filtering: II -Implementation, IEEE Trans. Circuits and Systems, CAS-23, 414-422, 1976. M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Ser., 55, 1964. W.G. Mecklenbrauker and R. M. Mersereau, McClellan transformation for two dimensional digital filtering: II. - Implementation, IEEE Trans. Circuits and Systems, CAS 23, 414-422, 1976. R.E. Twogood, M.P. Ekstrom and S. K. Mitra, Optimal sectioning procedure for the implementation of 2D digital filters, IEEE Trans. Circuits and Systems, CAS-25, 260-268, 1978. J.D. Markel, FFT Pruning, IEEE Trans., AU-19, 305-311, 1971. G. Rivard, Direct Fast Fourier transform of bivariate function, IEEE. Trans., ASSP-25, 250252, 1977. B. Gold and C. H. Rader, Digital Processing of Signals, McGraw Hill, New York, 1969. B.R. Hunt, Minimizing the computation time by using the technique of sectioning for digital filtering of pictures, IEEE Trans. Comput., C-21, 1219-1222, 1972. R. E. Twogood, M. P. Ekstrom and S. K. Mitra, Optimal sectioning procedure for the implementation of 2-D digital filters, IEEE Trans. Circuits and Systems, CAS-25, 261268, 1978. J.A. Hamid, Differential reduction to the pole of regional magnetic anomalies, Geophysics, 53, 1592-1600, 1988. P. Embry, J. P. Burg and M. M. Backus, Wideband filtering- the pie slice process, Geophysics, 28, 946-974, 1963. R.H. Bamberger and M. J. T. Smith, A filter bank for the directional decomposition of images: theory and design, IEEE Trans. Signal Processing, 40, 882-892, 1992. V. W. Chandler, Interpretation of precambrian geology in Minnesota using low-altitude, high-resolution aeromagnetic data, in W. J. Hinze (Ed.), The Utility of Regional Gravity and Magnetic Anomaly Maps, SEG, Tulsa, OK, pp. 375-391, 1985. L. Cordell and V. J. S. Grauch, Mapping basement magnetization zones from aeromagnetic data in the San Juan basin, New Mexico, in W. J. Hinze (Ed.), The Utility of Regional Gravity and Magnetic Anomaly Maps, SEG, Tulsa, OK, pp. 181 - 197, 1985. P.J. Gunn, Linear transformation of gravity and magnetic fields, Geophys. Prosp., 23, 300312, 1975. K.L. Kis, Transfer properties of the reduction of magnetic anomalies to the pole and to the equator, Geophysics, 55, 1141- 1147, 1990.
[4] [5] [6] [7] [8]
[9] [10]
[11 ] [12] [13] [14] [15] [16]
[17] [18] [19] [20]
[21]
[22] [23]
References
221
[24] R.G. Henderson and L. Cordell, Reduction of unevenly spaced potential field data to a horizontal plane by means of finite harmonic series, Geophysics, 36, 856-866, 1971. [25] P.S. Naidu and M. P. Mathew, Fast reduction of potential fields measured over an uneven surface to a plane surface, IEEE Trans. Geosci. Remote Sensing, 32, 508-512, 1994. [26] P.S. Naidu and M. P. Mathew, Correlation filtering: a terrain correction method for aeromagnetic maps, J. Assoc. Explor. Geophys. (India), 13, 23-30, 1992. [27] V . J . S . Grauch, A new variable magnetization terrain correction method for aeromagnetic data, Geophysics, 52, 94-107, 1987. [28] Y. Murata, Estimation of optimum average surficial density from gravity data: an objective Bayesian approach, J. Geophys. Res. B, 98, 12097-12109, 1993. [29] P.S. Naidu and M. P. Mathew, Correlation filtering: a terrain correction method for aeromagnetic maps with application, J. Appl. Geophys., 32, 269-277, 1994. [30] J.G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithm and Applications, Prentice Hall, New Dehli, 1995. [31 ] V.N. Strakhov, The smoothing of observed strengths of potential fields I, Izv. Geophys ser., 10, 897-904, 1964; The smoothing of observed strengths of potential fields II, Izv. Geophys ser., 11, 1634-1653, 1964. [32] P. S. Naidu, Two dimensional Strakhov's filter for extraction of potential field signal, Geophys. Prosp., 15, 135-150, 1967. [33] P.S. Naidu, An example of linear filtering in aeromagnetic interpretation, Geophysics, 33, 602-612, 1968. [34] R.O. Hansen and R. S. Pawlowski, Reduction to the pole at low latitudes by Wiener filtering, Geophysics, 54, 1607-1613, 1989. [35] G . K . C . Clarke Linear filters to suppress terrian effects, Geophysics, 36, 963-966, 1971. [36] P.J. Gunn, An algorithm for reduction to the pole that works at all magnetic latitudes, Exploration Geophys., 26, 247-254,1995. [37] A. Spector and F. S. Grant, Statistical methods for interpreting aeromagnetic data, Geophysics, 35, 293-302, 1970. [38] D.R. Cowan and S. Cowan, Separation filtering applied to aeromagnetic data, Exploration Geophys., 24, 429-436, 1993. [39] J.B. Thurston and R. J. Brown, The filtering characteristics of least squares polynomial approximation for regional/residual separation, Can. J. Exploration Geophys., 28, 71-80, 1992. [40] M. Fedi, G. Florio and A. Rapolla, A method to estimate the total magnetization direction from a distortion analysis of magnetic anomalies, Geophys. Prosp., 42, 261-274, 1994. [41] I.N. MacLeod, K. Jones and T. F. Dai, 3D analytic signal in the interpretation of total magnetic field data at low magnetic latitudes, Exploration Geophys., 24, 679-688, 1993. [42] L.B. Pedersen, T. M. Rasmussen and D. Dyrelius, Construction of component maps from aeromagnetic total field anomaly maps, Geophys. Prosp., 38, 795-804, 1990. [43] S. Qin, An analytic signal approach to the interpretation of total field magnetic anomaly, Geophys. Prosp., 42, 665-675, 1994. [44] H.V. Ram Babu, An analytic signal approach to the interpretation of total field magnetic anomaly (discussion), Geophys. Prosp., 44, 495-497, 1996. [45] B . N . P . Agarwal and R. K. Shaw, An analytic signal approach to the interpretation of total field magnetic anomaly (comments), Geophys. Prosp., 44, 911-914, 1996. [46] R.L. Parker, Improved Fourier terrain correction, part I, Geophysics, 60, 1007-1017, 1995. [47] R.L. Parker, Improved Fourier terrain correction, part II, Geophysics, 61, 365-372, 1996.
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223 Chapter 5
Digital Filtering of Maps II
The ultimate cherished goal of potential field analysis is to produce a map or an image of the density or magnetization variation in two or three dimensions. While this may not be possible to achieve fully, owing to the inherent non-uniqueness, it is possible to approach the goal through a process of incorporating known information obtained through other sources into the problem. In this chapter we explore this line of approach. We start with straightforward mapping by means of inverse filtering where we assume that all sources are confined to a plane at a known depth. In the next two sections we look at imaging through linear inversion. It is here that much of the interesting work involving the use of 'other' information has been reported, but unfortunately, linear inversion does not fall under the discipline of modem signal processing. Yet, for the sake of completeness, a brief outline of linear inversion as applied to the potential field analysis is included in Sections 5.2 and 5.3. Finally, texture analysis of the potential field is covered in Section 5.4.
5.1. Inverse filtering In Chapter 2 we have shown that the potential field can be expressed as a convolution between the source distribution, for example, density or magnetization distribution in a layer or an undulating surface separating two homogeneous mediums and a filter function which is controlled by depth, polarization, etc. The purpose of inverse filtering is to remove the effect of the filter so that the output of the inverse filter is the desired source distribution.
5.1.1. Irregular interface We consider inverse filtering for determining the undulating surface separating two homogeneous mediums. A typical application of such a process is in the study of sedimentary basins where a thick layer of sediments rest on crystalline rocks beneath. The interface is an undulating surface with potential hydrocarbon
Digital Filtering of Maps II
224
accumulation. The gravity and total magnetic fields are related to the interface as given in Eqs. (2.70a,b) reproduced here for convenience:
Fz(u, v) = i-i (u, v) d F (u, v) =
(u, v)
I-I (u, v) aaZ(u, v)
(5.1)
where
He(u, v) = -2rcGAp exp(-sh) and
27r Hv(u, v) -- ----(jmxu +jmyv - mzS)(j'c~u + j ~ v - 7 s ) e x p ( - s h ) Note that d&Z(u,v) is the differential of the generalized Fourier transform of the random interface between two homogeneous media. The inverse filter is readily obtained from Eq. (4.57) provided we have some additional information on the depth to the interface, or density contrast or magnetization vector. The depth to the interface can be estimated from the radial spectrum (see Chapter 3, p. 114). Any error in the estimated depth will produce a distorted reconstruction of the interface. However, by comparing the field due to the reconstructed surface with the observed field it would be possible to come up with a correction to reduce the mean square error. Thus, we can adaptively arrive at the best depth estimate. The effect of imprecise knowledge of the depth on the reconstruction has been studied in Ref. [16]. While an error in the estimate of density contrast affects only the scale of the reconstructed interface, an error in the estimate of the magnetization vector can affect the shape of the interface [ 17]. The inverse filters, because of the term exp(sh), become unstable if the bandwidth of the interface is large, but this is rarely the case as sharp angularities in the basement surface would have been blunted by the natural forces of weathering. It is therefore essential that the inverse ill, ~ring is preceded by lowpass filtering. The method of inverse filtering described above is essentially limited to small undulations, that is, I&Z]max << h. There are situations when this assumption may not be true, for example, high resolution magnetic mapping of the ocean bottom, shallow sedimentary basins, etc. When the plane of observation is close to the undulating interface, the higher order terms in Eq. (2.68) must be taken into
Inverse filtering
225
consideration. Note that here we encounter higher powers of basement undulations. The inversion procedure becomes iterative. Let d2XZk(U,V) be the differential of the generalized Fourier transform of Az*(x,y). As in Section 2.5 it is assumed that undulating interface is a homogeneous random function of finite spatial extent. Hence, we rewrite Eq. (2.68) using the zXZk(u, v) 2D Fourier transform in place of dz~k(u, v),
O(5 (--S) k-1 ~-~
Fz(u, v) = -27rGAp e x p ( - s h )
k!
k=l
azk(u,
(5.2)
The convergence of Eq. (5.2) will depend upon the magnitude of zXz relative to h. The inverse filtering problem, namely, estimation of 2~z(x,y) given F~(u,v) proceeds as follows. Rewrite Eq. (5.2) as z~kZ1 (H V) --
,
Fz(bl' ~)
ec (_s)k-1
- 2~rGA--------fiexp(sh)- ~
k=2
-~.
AZk(u, v)
(5.3)
As a first approximation we ignore the second term on the right-hand side of Eq. (5.3) and compute the first approximation to the true interface z~l(x,y ) = -idft{F~(u, v)[exp(sh)/2~rGZXo]}. Use z~l (x,y) to compute ZkZk(u,v), k = 2,3,..., which are then used in Eq. (5.3) to obtain an improved estimate of Zkz(x,y). The process is continued until no significant improvement in the estimate of Z~z(x,y) is noticed. The convergence of the iterative scheme depends upon the ratio zXz/h. Further, there will be numerical difficulty in evaluating exp(sh) in Eq. (5.2), for large sh the exponential function can exceed the size of the computer word, for example, for sh - 32, exp(sh) - 7.8963 x 1013. Apart from the above numerical difficulty there is one other problem, namely, that the ubiquitous background noise will be amplified by the same factor as exp(sh). Indeed as shown in Example 5.1 even in the presence of low noise the result of inverse filtering can be completely mined for sh > 3. This places restriction on the resolvable fine detail in Az(x,y) or its bandwidth. It turns out that the smallest wavelength that can be resolved is of the order of 2 h.
Example 5.1 In this example we study, through computer simulation, the performance of inverse filtering. A smooth hill of height four units is placed on a flat basement at a depth of ten units as shown in Figure 5.1 a. The density contrast of the basement
Digital Filtering of Maps II
226
.J
J
(a)
Ce
(b)
....... ~ J
(d)
(c)
J
/
I
_J
(f)
Inverse filtering
227
rocks with respect to the sediments above was assumed to be 0.1 g/cm 3. The gravity field was computed over an area of 128 x 128 sq units. Next we considered two other models having two and three hills of the same type but they occupy the same area as the single hill in Figure 5. l a. The basement topography is shown in Figure 5.1c,e. The results of iterative reconstruction are shown in Figure 5.1b,d,f. The reconstructions shown in Figure 5.1b,d compare well with the assumed model in Figure 5.1a, c, respectively. There is about 10% reduction in the height of the hills. But the reconstruction of the three-hill model shown in Figure 5. I f has considerably deteriorated compared to the actual model shown in Figure 5.1e. Thus, the iterative inversion process has failed to capture the rapid variations of the interface. In fact the linear inversion was found to perform even more poorly as it could not resolve two hills shown in Figure 5.1c. The inversion may be extended to the magnetic case when the inducing magnetic vector is uniform over the entire interface (there is no remnant magnetization) and the susceptibility contrast is constant. This additional requirement of a uniform inducing magnetic vector adds more uncertainty to the estimate of the interface. Further, in low latitudes, on account of the presence of zeros of HM(u,v) (this function is the inverse of magnetic-to-gravity conversion filter, see p. 38) on two radial lines, we will encounter problems similar to those in the reduction to pole.
5.1.2. Density maps Lateral variation of density in a horizontal layer may be estimated from the gravity field. Recall the expression for a gravity field due to random variation of density in a thin sheet given by Eq. (2.81). From this result we can derive the gravity effect due to a sheet of finite thickness, Ah. For simplicity we assume that there is no vertical variation in density. Integrating Eq. (2.81) with respect to z0 we obtain
f (x, y)
lff§
G dP(u, v)exp(-sh)
(1 - exp(-szXh))
Ahs
exp(/'(ux + vy)) (5.4a)
It is now possible to express the (surface) density variation in terms of the gravity field Figure 5.1. Reconstruction of basement topography from the gravity data using the iterative technique described earlier.
228
Digital Filtering of Maps II
,/i
p(x, y) -- 27rG
o()
Ahs dFz(u, v)exp(sh) 1 - e x p ( - s A h ) exp(j(ux + vy))
(5.4b)
where O(x,y) is the surface density (mass per unit area of the layer). The generalized Fourier transform in Eq. (5.4b) is replaced by an ordinary Fourier transform when the density variations are deterministic. In practical applications, the generalized Fourier transforms are replaced by dft coefficients. The density map as a. lithological mapping tool was introduced by Gupta and Grant [ 18] who called it an apparent density map.
5.1.3. Susceptibility maps The magnetic map can be converted into a susceptibility map. Consider a thin randomly magnetized sheet. The total magnetic field is given by Eq. (2.83). For simplicity, we assume that the susceptibility variation is only in the horizontal direction. Carrying out integration with respect to z0 we obtain l
f T (x, y, h ) -- -f-s
oo (jul x +jvly - slz) (l'C~u + j/3v - 3's)
x exp(-sh)
1 - exp(-sAh)
exp(/(ux + vy))
dl((u, sAh (5.5a)
s
where dK(u, v ) = dK(u, v,z)Ah. From Eq. (5.5a) the surface susceptibility density is expressed in terms of the generalized Fourier transform of the total magnetic field
y)
1 j'j/_
-
•
sexp(sh) O'ulx +jvly - slz)(jau + j 3 v - -ys)
sAh dFr(u, v, h) exp(/'(ux + vy)) 1 - exp(-sAh)
(5.5b)
where K(x,y) is the surface susceptibility density. In practical applications, in place of a generalized Fourier transform one may use just an ordinary Fourier transform or specifically discrete Fourier transform as we will be dealing only with finite data. In obtaining the susceptibility maps, it is necessary to continue downward to the magnetic field close to the top of the magnetized layer. Further
Inverse filtering
229
it is assumed that the magnetization is only of induced type. Susceptibility maps are useful in lithological and structural studies as the effects of the direction of an induction vector as well as the earth's total magnetic vector have been removed, thus making the reduced map appear closer to the geology. A practical example is shown in Figure 5.2 in support of the above statement [ 19].
5.1.4. Undulating layer Next let us consider a more complex model, an undulating layer of constant thickness but with variable susceptibility. Such a model was analyzed in Chapter 2 (see Figure 2.22 and the derivations given there). Out of two functions of interest, namely, surface undulations and susceptibility variations, only one can be estimated from the observed magnetic field. We shall choose the susceptibility variations as the desired quantity and the surface undulations as a known function. The inversion algorithm is basically similar to that proposed for an undulating surface at the beginning of this section. First let us reproduce the basic equation, Eq. (2.88b),
1
fT(x'Y)--
2re
r ( u , v)~-~ ff~ +oo oo dQ~(u, v) (-s)~-' exp(-Shl ) exp(/'(ux 4- vy)) oo s n=O n-----T--.
v) (_,)n-, + ~
~
s
n=O
exp(-sh2) exp(/'(ux -+- vy)) n-----T--(2.88b)
For simplicity, we assume that K0 = 0 and simplify Eq. (2.88b)
f T(x' Y) --
1 27r
+o~P(u, v) ~ dQ],(u v) (--s)n-' ~ S n=0 ' n!
• [exp(-shl) - exp(-sh2)] exp(/'(ux + vy))
(5.6a)
Let FT(U,V) be the Fourier transform of the magnetic field. We use the Fourier transform in place of the generalized Fourier transform in Eq. (5.6a) which now reduces to
Digital Filtering of Maps II
230
:
!
I .....
i
mi
Figure 5.2. An example of a utility of susceptibility map. (a) Geology (1 dyke; 2 faults); (b) total field; (c) susceptibility (estimate) [19]. oo
_s)n-1
F r ( u , v) = - 2 r e r(u, v) (1 - e x p ( - s Z ~ h ) ) e x p ( - S h l ) ~
s
~(u, v ) ~ n! n=O (5.6b)
Inverse filtering
231
Rewriting the right-hand side of Eq. (5.6b) we bring out the zeroth order term to the left-hand side,
Q~(u, v)
sFT(u, v) 27r r~ S
~
(1 - exp(-szXh))exp(-Shl)
/~/z
n-
1
n! (5.7)
To start with, we assume that Qa(u, v ) = 0 for n = 1,2,3,... and estimate Q~(u, v) from Eq. (5.7). Following this step, the zeroth estimate of the susceptibility is computed from Eq. (2.88c), that is, ,Ms , v) = Q~(u, v). Since ZXZl(X,y) is assumed to be known, using Eq. (2.88c) we can compute the zeroth estimates of Qa(u, v) for all n. Going back to Eq. (5.7) we can compute the next estimate of the susceptibility, that is, ZXKl(x,y).The process is continued until there is no observable change between 2~Kp(X,y) and AKp_l(x,y) at the pth step. The question of uniqueness is of great interest. It was noted in Chapter 2 that if there is some susceptibility variation which satisfies Eq. (2.89), it will not produce any magnetic field. Such a susceptibility variation can indeed be computed from Eq. (2.89) using the above iterative algorithm. To show this, let us rewrite Eq. (2.89) as follows: oo
Q~(u, v)- ~(u)~(v) - ~ dQa(u, v) ( - s ) " n=l
n[
(5 8) "
As before, we start by assuming Qa(u, v)-0 for n - 1,2,3,... and estimate Q~(u, v) from Eq. (5.8) (which is simply a constant). From this zeroth order estimate and from the knowledge of ,Sz~(x,y) we obtain the zeroth estimates of Qa(u, v) for all n. These are used next to obtain the first order estimate of the susceptibility and this process is continued until there is no observable change. The susceptibility thus obtained will be consistent with Eq. (2.89) and it will produce no magnetic field. Indeed any amount of the above susceptibility may be added to the solution of Eq. (5.6) without affecting the measured magnetic field.
Example 5.2 This example shows the role of unobservable magnetization in the interpretation of marine magnetic anomaly. The marine magnetic observations are from the East Pacific rise [1]. The magnetic field reduced to a horizontal level 2475 m
Digital Filtering of Maps II
232
~
~
/~o
,,
~STANCE
I'~1
/
,o
,,
(kin)
\
Z.6 -"-
3.0 34
$
I0
iS
ZO
2"S"
DISTANCE (kin)
g . . . . . . . . . . . . . . . . . . . . . . . . . . .
N_ i-. lad Z
b
:E
01STANCE (kin)
r
Q
1.2,
Z
o_ N_ .e. I,-bJ Z (,9
Inverse filtering
233
below the sea surface and the sea bottom topography are shown in the top section of Figure 5.3. The magnetization derived from these data and an unobservable component are shown in the middle and bottom sections of Figure 5.3. The layer thickness is assumed to be 500 m following the topography. In the middle section of Figure 5.3 there are two curves. Curve (a) is produced by the inversion algorithm while curve (b) is produced by subtracting a certain amount of unobservable component but both are consistent with the observation. It is argued in Ref. [1] that curve (b) is a geophysically preferred solution.
5.2. Least squares inversion (2D distribution) In this section we explore an alternate approach for estimating density or magnetization distribution in a horizontal layer at known depth. The alternate approach consists of inverting the basic convolutional relationship between the density or magnetization and the corresponding potential fields. The inversion is readily achieved through linear least squares inversion as well as deconvolution (Backus-Gilbert approach).
5.2.1. Discrete model The model under consideration consists of a thin horizontal layer at a known depth. The layer is divided into many cells, each having uniform density or magnetization equal to the average of the actual density over the volume occupied by each cell whose size is roughly of the order of the observation grid interval. The discrete model is sketched in Figure 5.4. Consider the potential field at the (p,q)th point due to a cell located at (1Ah, mAh) where Ah (Ah << h) refers to the size of a cubical cell. We have assumed that the observation points lie on a square grid with unit separation between the nearest neighbours. Let alto be the density in the (/,m)th cell and fpq be the gravity field at (p,q). Following Eq. (2.35) we write N
N
fpq -- ~ ~
Gf/qpzm
(5.9a)
/=1 m = l
Figure 5.3. An example of unobservable magnetization and its role in the interpretation of marine magnetic data.
Digital Filtering of Maps II
234
Figure 5.4 The discrete model of a thin horizontal layer at known depth. The density or magnetization in any cell is constant, equal to the average density or magnetization over the volume of the cell. where
q _
Gh
&@
G~tm J-ah/Zl-ah/2 [(p -- (lAh + x) )2 + (q _ (mAh + y) )2 + h 2] 3/2
(5.9b)
is proportional to the gravity field at point (p,q) due to a cube of unit density placed at (lAh, mAh, h) and Plm is average density in the (/,m)th cell. Ideally, there ought to be an infinite number of cells but we shall assume that there are N 2 cells in the plane of the sheet. The gravity field is measured over a square grid with unit spacing. The size of the grid is P • P. Further we assume that P _> N. The gravity data are mapped into a vector, fz as shown in Figure 6.2. In analogous fashion we shall map Olin into a vector, j6. Finally, we map C~zq into a matrix, G and express Eq. (5.9a) in matrix form rz
(p2 • 1)
=
c
(p2 •
+
2)
(N 2 • 1)
(p2 • 1)
where 1/is a noise vector, and the size and type of the matrices are shown below each matrix. The noise includes observation errors, interference from the cells excluded from the model, geologic noise, etc. The rank of G will depend upon h and Ah. It is likely to be full rank only if h is not too large. In most practical situations the matrix G will be less than full rank.
5.2.2. Least squares solution It is desired to find a solution/5 which will minimize the error between the measured and the predicted, that is, it is required to minimize
Least squares inversion (2D distribution)
(fz--G
G x p) -- rain
x/~)T(fz-
235
(5.11)
with respect to iS. Differentiating Eq. (5.11) with respect to /5 and setting the derivative to zero we obtain GTG/5- GTfz
(5.12a)
By solving Eq. (5.12a) we obtain a least squares solution to an unknown density vector /~LS- [GTG]-IGTfz
(5.12b)
where [GTG] -1 stands for the inverse or the pseudoinverse of GTG depending upon whether the matrix GTG is full rank or singular. Note that the size of GTG is N 2 x N 2 (N x N for a one-dimensional source, that is, no variation in the y direction). The pseudoinverse is defined as follows. Let the eigenvector decomposition of GTG be UAU T where the columns of U are eigenvectors and A is a diagonal matrix with eigenvalues on the diagonal. For example, if GTG is of rank r, the eigenvalue matrix is A = diag[A1, A2, A3, ..., At, 0, ..., 0] The pseudoinverse is defined as U A - 1 U T where A_l_diag[A
1
1 --., -1- , 0 , ..., 0] Ar
1 ' A2' A3 '
When GTG is singular the solution given by Eq. (5.12b) is not unique, as any vector which is a linear combination of the eigenvectors corresponding to the null eigenvalues may be added to the solution and yet satisfy Eq. (5.12a). The least squares solution of Eq. (5.10), when G is a rank deficient matrix, may be expressed in terms of the singular value decomposition (SVD) of G [14, p. 414].The SVD of G is G - UY]V T
(5.13)
Digital Filtering of Maps II
236 where , Up2] is a p2 x p2 matrix
U -
[Ul, u2, u 3 , . . .
V -
[Vl, v2, v 3 , . . . , VN2 ] is a N 2 • N 2 matrix
and -- diag [O'1,0"2, O'3,... , OQ] where Q - min{P 2, N 2 } The or/are the singular values of the G matrix and the vectors ui and vi are respectively, the ith left singular vector and the ith right singular vector [14, p. 17]. Using the SVD decomposition of G (Eq. (5.13)) in Eq. (5.12b) we obtain
PLS -- ~
r u fz
.__
O'i Vi
(5.14)
The above lack of uniqueness can be reduced by incorporating some prior information on the density distribution. The average density of most rock types is fairly well known. When the target layer is a particular rock type, it is possible to use its average density as the prior information. Along with Eq. (5.11) we must minimise (/5 - p0) T (15 - P0) where P0 is the known average density. We must then minimize a linear combination (fz - GtS) T (fz -- GIS) + #(j6 - Po )T (/~ _/90) -- m i n
with respect to/~ where # is a constant 0 _< # _< 1. The solution is given by PdLS -- [G T G -+- ~I] -1 (GTfz -+- ~P0)
(5.15a)
This is often known as a damped least squares solution. The presence of/zI in
Least squares inversion (2D distribution)
237
[GTG + #I] -1 stabilizes the inversion even when GTG is singular or close to being singular. When the mean density is not known, we let P0 = 0 in Eq. (5.15) and obtain the minimum norm solution
Pmin-norm --.[ GTG
(5.15b)
q- #I]-lGTfz
Example 5.3 In this example we demonstrate the power of least squares inversion. Consider a horizontal layer of unit square cross-section rods at a depth of two units below the surface (see Figure 5.5a). Filled cells have variable density and the blank cells have zero density. The density variation actually used in 32 cells is shown in Figure 5.6a The gravity field (in arbitrary units) computed on the surface is shown in Figure 5.5b. Surface
~///~////~/~6~/~j?~/~.?/?~/~///~/~//~/////~/~//~/~///~/////~/~///~//~/~/~//~//~//~//`
!~I
I I ! ! I I IN!HN~NNI!N~NNNINI!N!N~!!~!!t!@!iN~!N (a) "0
! I ! I ! i ... I !
!
. m
~>
(.9 I,..
1.5
1
0.5
~
ib
2o
x-axis
a0
4o
(b)
Figure 5.5. The density variation model used here is shown in (a). shaded box, finite density cell; unshaded box, zero density cell. The gravity field (in arbitrary units) is shown in (b).
~,,,,J~
~
.
.
.
_.L
_A
0
0
o
.
0
.
o
.
0
0
0
0 0
0
0
0
0
.
.
.
0 ~
.
.
~--
0
.
0
..... i ..................................
.....
"', . . . .
0
Density ..~
--~
0
co
0~
......
....... o
o
i
i ......
i .....
:
i
:
i
i .....
0
i ......
o
! .....
i
!
:
i .....
.0
:
!
! .....
:I
I
I
I
, ....
...... ,'
. .....
!
0
:
:
'
i
i ....
~
-,..-"
0
CD
Least squares inversion (2D distribution)
239
The least squares inversion of the computed noise free gravity field was carried out; first the minimum norm solution given by Eq. (5.15b) with ~ = 0.1 followed by the damped least squares solution given by Eq. (5.15a) with the same value of but with a priori density information o0 given by the dashed line in Figure 5.6a. The results are shown in Figure 5.6b,c. While the broad features of the density variation have been captured, the fine variations in the density are lost in both inversions. The simulation study has shown that for the fine features to be reproduced, the depth to the layer should be of the order of one unit. 5.2.3. M e a s u r e m e n t error
The estimated density vector will differ from the true density vector on account of the noise present in the observed data. Let P - [GVG]-~G T be the projection operator which projects the field vector onto the space spanned by the density vector, that i s , / 5 - Pfz. Substituting for fz from Eq. (5.10) we obtain -
P ( C p + ,1) - p + P .
that is, the error in the estimate is given by t ) - P - P~7. It is easy to compute the covariance matrix of the error, cov{/)) = E ( ( / 5 - p)(/5- p)T} _ PE{r/qT}pT
(5.16a)
When the noise is uncorrelated, Eq. (5.16a) reduces to C o v { p } -- o-2~PP T -- o 2 [ G T G ] -1
(5.16b)
Notice that the error in the density estimates will be amplified whenever the matrix GTG is singular. 5.2.4. B a c k u s - G i l b e r t inversion
Instead of attempting to solve Eq. (5.10) by inverting G an inverse filter is sought. The inverse filter, in the form of a matrix, operates on the observed data
Figure 5.6. Density variation: (a) actual; (b) minimum norm estimate (Eq. (5.15b)); (c) damped least squares estimate (Eq. (5.15a)).
240
Digital Filtering of Maps II
and estimates the density or magnetization in each cell. A method of computing such an inverse matrix was provided by Backus and Gilbert [20]. Let cPmq, (p, q = 1 , . . . , P and m, n = 1 , . . . ,N) be a set of inverse filter coefficients with which we will multiply both sides of Eq. (5.9) and sum over p and q. We obtain P
P
N
N
P
P
(5.17a) p = l q=l
l=1 m=l p = l q=l
or N
Pl'm' -- ~
N
~
(5.17b)
Sl'm'lmPlm
l=1 m=l
where P
P
P
-
Sl,
p = l q=l
,lm
P
-
p = l q=l
tSt'm' is an estimate of density of the l'm' cell. The closeness of this estimate to the actual value depends upon Sl, m,lm. In particular, if Sz'm'tm - 6Z't,m'm where (Sl,l,m,m -- 1
= 0
when l ' -
l and m' - m
otherwise
^
Pl'm' -- Plm
This is not likely to be achieved in practice; however, one may attempt to achieve this result through optimum estimation of inverse filter coefficients. For this a measure of closeness to a delta function (Kronecker delta in the discrete case) is required. In Ref. [ 15] the following measure was used: N Ql'm' - ~
N ~--~Jl'm' lm[Sl'mt lm -- (Sl'l,m'm] 2
(5.18a)
/=1 m=l
where Jz'm'Z,,,- 1 2 [ ( l ' - / ) 2 + matrix form
( m ' - m ) 2 ] . It is possible to write Eq. (5.18a) in
Least squares inversion (2D distribution)
Ql, m,
-
-
el,Tm, ]~Ii, m, Cl,m, + constant
241
(5.18b)
where N
N
{ ml, m, } p, q,,p q - ~--~ ~ ' ~ Jl, m, lm GtTl'mq'Cfflmq 1=1 m = l
and Cl,m, is a column vector of length p2 Note that the inverse filter for observations on a plane is a two-dimensional filter. This must be mapped into a column vector by vertically stacking all the columns of the two-dimensional filter into a single column. More details on such a mapping scheme along with an illustration are given in Chapter 6. Further, it is required that N
N
Z s,,m,,o-1 /=1 m = l
which may be written in matrix form T
Cl,m , U -
(5.19)
1
where N
N
1
m
We will now minimize Eq. (5.18b) subject to Eq. (5.19) with respect to Cl, m,. The Lagrange expression for this is i it
-- Cl,Tm, Ml, m, Cl,m, nt- constant + A(1 - c~,~,u)
(5.20)
where X is a constant. Differentiating Eq. (5.20) with respect to Cl'm' and setting the resultant derivative to zero we obtain Cl'm' - AM~I'u-Further, on account of Eq. (5.20)we have
A
T -1 U Ml, m,U
242
Digital Filtering of Maps II
Thus, we obtain -1 Cl'ml --
Ml'm'U T - 1 [U M l , m,U
(5'21)
The minimum value of Ql'm' is given by 1
Ql'm'Jmin - uTM/_;_I, u + constant
(5.22)
which gives a measure of closeness of Sl, m,lm to ~l'l,m'm, also a measure of resolution of density variation. When there is background noise as shown in Eq. (5.10) the operation of inverse filtering will modify the noise in the output. The variance of the noise in the estimate of film is given by 2
O'il -- c~mCrlClm
(5.23a)
where C, is the covariance matrix of the noise in the observed data. It may be observed that for observations on a plane, the covariance function is a twodimensional function. It is necessary that it be mapped into a covariance matrix with block Toeplitz symmetry. A procedure for such a mapping is explained in Chapter 6. When the background noise is white, Eq. (5.23a) simplifies to 2 O'rl -- C~mClm
4
(5.23b)
5.2.5. Resolution The size of the cell used to discretize the layer is a measure of the resolution. This is intimately related to the rank of GTG. If it is full rank the unknown densities can be retrieved without any loss, except for the loss due to background noise. The rank of GTG depends purely on the geometry of the model. No analytical estimate of the rank of GTG is known. However, it is recommended that the depth to layer should be about twice the cell size [10]. To verify this claim, a simple experiment was performed to study the dependence of the rank of
243
Least squares inversion (2D distribution) TABLE 5.1 Rank of the GTG matrix as a function of depth to the layer Depth
Effective rank
Condition no.
1.0 1.5 2.0 3.0
Full Full 13 9
100 2018 > 10000 > 10000
Effective rank was computed by counting all eigenvalues greater than 0.001% of the largest eigenvalue. GTG on the depth to the layer. Referring to Figure 5.4, a layer containing 16 cells (unit cross-section squares) at a variable depth was selected. Thirty-two observations were taken at unit distance apart. For the sake of simplicity, we have replaced each cube by a point mass located at its centre. All depths are with reference to the position of point mass. In Table 5.1 the rank and condition number (ratio of maximum eigenvalue to minimum eigenvalue) are shown.
5.3. Least squares inversion (3D distribution)
In this section we extend the concepts of linear inversion of the previous section to three-dimensional distribution of density or magnetization. Conceptually, the modelling ideas are similar to those used in tomographic imaging. The discrete model will result in a large system of linear equations where the number of unknowns far exceeds the number of observations (that is, the number of equations). Evidently, it is not possible to uniquely solve for the unknown cell densities or magnetizations without imposing some additional constraints on the solution and use of a priori information about the distribution.
5.3.1. Discrete model (3D) The entire source space below the survey area and up to a specified depth is divided into many tiny cubes or cells. In each cell the average density or magnetization is assumed to be unknown. An illustration of division of source space into cells is given in Figure 5.7. Within the domain of the survey, the field at any point is equal to the sum total of contributions from all cells. When the point lies close to the survey boundary, it is likely that the field at this point may have contributions from the cells which were not considered in the model. Consider the potential field at the (p,q)th point
244
Digital Filtering of Maps II
Source ;, l llS
~ A Z
Ay
Figure 5.7. The space filled with variable density or magnetization (source space) is divided into many cubical cells. The density or magnetization in each cell is assumed to be constant, equal to the average density or magnetization over the volume of the cell. For simplicity the cell size,
zXx-Ay--az--Ah. due to a cell located at (lAh, mAh, nAh) where Ah (= 1) refers to the size of a cubical cell. We have assumed that the observation points lie on a square grid with unit separation between the nearest neighbours. Let 01m, be the average density in the (l,m,n)th cell and fpq be the gravity field at (p,q) on the surface. Following Eq. (2.35) we write N
N
fPq -- Z ~
N
~ C~lqnPlmn
(5.24a)
1=1 m=l n=!
where qH
~--
Ah/2J-Ah/2J-Ah/2 [GO _ (lAh + x))2 + (q _ (mAh + y))2
nt-
(nAh + h + 2) 2] 3/2
is proportional to the gravity field at point (p,q) due to a cube of unit density placed at (lAh, mAh, nAh + h). The gravity data are mapped into a vector, fz,
245
Least squares inversion (3D distribution)
shown in Figure 6.2. In analogous fashion we shall map Otto, into a vector, p. Finally, we map c~lq~ into a matrix, G and express Eq. (5.24a) in a matrix form, fz
=
(p2 x 1)
G
p
(p2 x N 3)
(X 3 x 1)
+
r/ (N 3 x 1)
(5.24b)
where r/is the noise vector. The size and type of the matrices are shown below each matrix. The noise includes observation errors, interference from the cells excluded from the model, geologic noise, etc. There are N 3 cells or unknowns and p2 observations or knowns. In most cases of practical interest there are many more unknowns than knowns, that is, N 3 >> p2. This is particularly true of 3D distribution. Therefore, the system of equations (Eq. (5.24b)) is highly underdetermined without any unique solution. One can, however, attempt some kind of constrained least squares solution incorporating reasonable constraints on the model and use all available geological information. Some useful constraints are listed below: (a) density or susceptibility at every point must be positive, (b) the range of density or magnetization variation is limited to a finite interval, (c) density or susceptibility variations are probably confined to some domain and zero outside that domain [9]. An equivalent condition is to minimize the volume of the source [8], (d) a smoothness condition may be imposed [6]. The available geological information is in terms of the rock types and the structure of the near surface strata. To each rock type we can associate certain density or susceptibility distribution. This naturally calls for extensive field measurements coupled with the information drawn from the existing data bases. Structural maps and vertical sections of the near surface strata are often prepared by structural geologists. All such information may be pooled together to create an initial estimate of the source vector,/9o. For the constrained least squares method to succeed it is imperative that the initial source vector P0 must be close to the true source vector, as the available information in the form of the observed potential field is generally inadequate. 5.3.2. Constraint least squares
It is required to minimize of two functionals, namely
Ilfz -
Gpll
2
-
min
(5.25a)
246
Digital Filtering of Maps II
and (5.25b)
(P - P0) TW (P -- P0) -- min
with respect to o. In Eq. (5.25b), W is a symmetric weight matrix defined so that the resulting solution has certain properties such as smoothness (see, for example Ref. [13]). The simplest type of weight matrix is a diagonal matrix where the diagonal elements weigh various elements of the ( p - P0) vector according to their importance. To solve for p we must minimize a linear combination /'I/ -- ][fz -- GP[I 2 -Jr-~ ( P -
p0)TW(p-
P0) -- m i n
(5.26)
with respect to p. The result is
- [ c + c + u w ] - ' {#wp0 + C+fz} : [GTG + #W] -l {#Wp0 + GT(f~ - f0) + GTGpo}
: p0 + [G+G + # w ] - ' c + ( f ~ - fo)
(5.27)
where fo is the field due to the known mean density information.
5.3.3. Linear programming The problem of solving Eq. (5.24) for density distribution under constraints may be reduced to linear programming. It is assumed that the background noise is limited to ~min to ~/max. Eq. (5.24) then reduces to r/rain --< fz -- Gp < ~max
(5.28a)
The prior information is now given as bounds on the density in each cell, Pmin ~ P _~ Pmax
(5.28b)
A method to generate all solutions of Eq. (5.28b) using a simplex algorithm has been given in Refs. [11,12]. In N-dimensional space the set of inequalities
Least squares inversion (3D distribution)
247
given by Eq. (5.28) defines a convex polyhedron with N + P sides. Every point lying either inside or on the polyhedron satisfies Eq. (5.28), and is therefore a possible solution. To get a particular solution it is necessary to minimize (maximize) a cost function which is defined as a linear function of p, -- gYp
(5.29)
As the cost is varied, the hyperplane representing the cost function spans the entire N-dimensional space. The zero cost function goes through the origin. The cost function will first come in contact with the polyhedron solution space at one of its comers. This is the minimum cost solution satisfying Eq. (5.28). Thus, to find the optimal solution all one has to do is to compute the cost function at all comers of the polyhedron and then select the comer where the cost function is minimum. The same procedure is followed to find a comer where the cost function is maximum. This seemingly simple problem is computationally highly intractable as there are a very large number of comers to be located and the cost function to be evaluated. The number of comers is less than or equal to
(N + P)! N!P! (e.g. for P = 10 and M = 5 this number is 252) The hyper plane representing the cost function may intersect the polyhedron at a comer or along an edge or over one of the faces of the polyhedron. In the case of the last two possibilities, there is no unique solution to the minimum problem. Note that the cost function will represent average density if the coefficients column, c, is chosen to represent the relative volume of different cells. It is possible to compute new bounds on the density.
Example 5.4 This example, taken from Ref. [ 13], shows how to improve upon the bounds on the density of different ore bodies using surface gravity data. The gravity data are from Neves Corvo in the Baixo Alentejo, Portugal, over a massive sulphide ore deposit. The density information was obtained from the density logs carried out on drill holes. From this informationthe lower and upper bounds on the density were computed. The model consisted of two sets of prisms having square crosssection of 100 x 100 m 2 variable depth and thickness. The first layer had 222 prisms and the second layer had 261 prisms. There were 254 gravity stations. In
Digital Filtering of Maps II
248
TABLE 5.2 Ore body
A priori bounds
Neves Corvo Zambujal Graca Total
4.35 4.35 4.35 4.35 4.35
A posteriori bounds 4.95 4.95 4.95 4.95 4.95
4.37 4.35 4.35 4.35 4.36
4.48 4.43 4.54 4.55 4.43
As a result of linear inversion of the gravity data, the bounds on the densities of different ore bodies were narrowed down. A priori bounds were obtained from the density logs. Similarly, the bound on the mass of each ore body could be narrowed down (but are not shown above) [ 13]. Table 5.2 we reproduce the results showing how the bounds on the densities of different ore bodies could be narrowed down after linear inversion of the gravity data.
5.4. Texture analysis In this section we describe some concepts of texture analysis and the related topics pertinent to aeromagnetic maps. The motivation is to look for changing patterns in the aeromagnetic field variations which are an expression of underlying rock type. If we can map regions of similar pattern, what is known as texture, we should be able to map the rock type beneath, identify the boundaries between rock types, trace linear features, etc. There are two approaches to texture analysis, namely, the structural method and the statistical method. In the structural method, the texture is considered to be a repetition of some basic primitive pattern, for example, a tile on a tiled floor. In the statistical method, the texture is considered as a spatial variation of some stochastic property. Texture analysis is a filtering operation and like digital filtering works on a small sliding data window. The output of a texture filter is a number that quantifies the spatial variation of a magnetic field inside the window. Texture analysis is close to a derivative filter, for example, a vertical derivative filter which we considered in Chapter 4 but it differs in one important manner, that is, it may be a non-linear filter where a nonlinear operation is employed to enhance the small differences but suppress the large differences.
5.4.1. Non-linear transformations We describe two non-linear methods under this category. The first method is
Texture analysis
249
63
28
45
88
40
35
67
40
21
Windowed data
Four level map
Mapping rules: <30 =0
31-51 =1 51-70 =2 >70 =3
Figure 5.8. Mapping of windowed data into a four level map. based on measuring the relative frequency of occurrence of two field values (pixels or picture elements) of given amplitude separated by a given distance in a given direction. The method is known as grey level co-occurrence matrices (GLCM) [2]. A 3 x 3 window of a magnetic field is taken, that is, a point surrounded by its three immediate neighbours. The first step is to map the magnetic field values inside the window to four levels (or a small number of levels). The mapping rule may vary from window to window but the number of levels remain unaltered. Let us illustrate this step through an example taken from Ref. [3]. A 3 x 3 windowed set of data is mapped into a four level map using a set of mapping rules listed in Figure 5.8. The next step is to compute the relative frequencies along four possible directions, namely, horizontal (H), vertical (V), left diagonal (LD) and right diagonal (RD). There are 16 possible combinations of two levels taken with replacement from a set of four. These are conveniently represented on a 4 x 4 GLCM divided into 16 cells. In each cell we represent the number of times the levels represented by the coordinates of the cell occur side by side. Consider the shaded cell in Figure 5.9. The coordinates of the cell are (20). This pair of levels is found to occur side by side only once when you horizontally scan the four level map in Figure 5.8. The same number is repeated in cell (02) and hence the entries on the diagonal are doubled. In this manner three other GLCM matrices are generated. A contrast parameter for each GLCM is now defined in terms of the frequency of occurrences
Digital Filtering of Maps II
250
Level s 1
0
2 9. : , : , : . : . : , : , : . : . : , : , : . : . : . : . : : : .
-
.:.:+:.:.:.:.:+:.:.:.:,:,:.:,
2
0 L e
i',Ni',i!!!i!iii ,iil .,.,.,,..-.,.......
v.....v....
2
2
1
1
1
0
0
1
0
V e
1 S
3
G L C M (PH) Figure 5.9.
3
contrast - ~-~n 2 n=0
3
Z Z i=0
3
j=O
ps(i'j) R
(5.30)
,1- li-jl
where R stands for the total number of entries in GLCM. There are other types of contrast parameters, all of which are based on intuitive considerations.
Example 5.5 This example shows an application of textural filtering using the concepts of grey level co-occurrence matrices (GLCM) and a textural spectrum on real data from granitoid-greenstone terrain in Western Australia [3]. The aeromagnetic field map and its vertical derivative are shown in Figure 5.10, top left and right frames, respectively. The aeromagnetic survey was carried out at a height of 60 m with line spacing of 200 m. Both maps are as grey level images with artificial illumination from the east. Textural units were computed using different sizes of window and grey level mapping to generate the GLCM. The results are shown in Figure 5.10, left and right bottom frames. The left frame is the result of using a 7 x 7 window and eight grey levels. The right
Texture analysis
251
Figure 5.10. Example of textural filtering of real aeromagnetic data [3]. flame is the result of applying a textural spectrum using a 3 x 3 window and four grey levels. Notice that all major linear features present in the vertical derivative map are also present in the texturally filtered maps. However, very weak features found in the vertical derivative map are now emphasized in the texturally filtered maps, a direct result of non-linear mapping rules used in the generation of GLCM and in the definition of the textural unit.
5.4.2. Textural spectrum A different method of textural filtering has been suggested where a new quan-
Digital Filtering of Maps II
252
tity called a texture unit (TU) is defined in terms of relative difference between the central pixel and its eight immediate neighbours [4]. For example, in a window of 3 x 3 shown in the Figure 5.11, the central pixel when compared with its immediate neighbours as per the mapping rules results in a three level map, a textural unit, which is written in vector form TU -- [2,0,2,0,0, 1,2,2] A textural unit number is next defined as a single number 8
NTU -- ~ 3 i- 1Xi i--0
(5.31)
The range Of NTu is from 0 to ~-~81 3i-12 -- 6560. Note that NTU is dependent on the order in which the elements of the TU vector are arranged, for example, there are eight NTuS obtained by clockwise rotation of the numbers listed in the TU vector. For every window position we get one NTU and thus there are as many NTuS as
63
28
45
2
88
40
35
2
67
40
21
2
Windowed data
0 0 1
Three level map Mapping rules: f, < f o x,-O fi - fo
x, - I
f, > f o i=1,2
Figure 5.11. Mapping rules.
x, ....
8
=2
Texture analysis
a
253
b
c
h
d
g
f
e
Figure 5.12. Eight clockwise successive ordering ways of the eight elements of texture units. The first element may take eight possible positions from a to h.
the number of pixels. A frequency distribution or histogram of all Nsus is called the textural spectrum. Very often a spatial distribution of NTu in the form of a map is quite adequate. This is a kind of textural filtering.
5.4.3. Textural features For texture characterization, it is useful to define the textural features. A few examples of textural features derived from the textural spectrum S(i), i = 0, 1, . . . , 6560 are given below [4]. (1) Black-white symmetry (BWS) 3279
IS(i) - s(3281 - i)l BWS -
1 -- i=0
6560
X 100
S(i) i--0
BWS measures the symmetry between the left half (0-3279) and right half (3281-6560) of the texture spectrum (2) Geometric symmetry (GS)
I ~4 GS--
1 -4
6560 ~
/~= 0 ~
j=l
]Sj(i)
-
Sj+ 4
6560
2 y ] Sj(i) i=0
(i) 1 x 100
Digital Filtering of Maps II
254
where Sj(i), i = 0,...,6560 andj - 1,...,8. The subscriptj represents the particular ordering of TU elements. There are eight possible options. GS measures the symmetry between the spectra under the ordering options a and e, b and f, c and g, and d and h. It reveals information about the shape regularity of images. (3) Degree of direction (DD) 6560
DD-
1 3 1-g~~
4
Z
m=l n=m+l
~ i=0
ISm(i) - Sn(i)[
xl00
6560
2 ~
Sm(i )
i=0
DD measures the linear structure within an image. A high value of DD indicates that the texture spectrum is sensitive to the orientation of the image. These textural features were successfully used to classify surface rocks from airborne synthetic aperture radar (SAR) images [4].
References R. L Parker and S. P. Huestis, The inversion of magnetic anomalies in the presence ot topography, J. Geophys. Res., 79, 1587-1593, 1974. [21 R. M. Haralick, K. Shanmugam and I. Dinstein, Textural features for image classification, IEEE Trans., SMC-3, 610-621, 1973. [3] M. Dentith, Textural filtering of aeromagnetic data, Exploration Geophys., 26, 209-214, 1995. [4] L. Wang and D. C. He, A new statistical approach to texture analysis, Photogramm. Eng. Remote Sens., 56, 61-66, 1990. [51 W. M. Newman and R. F. Sproull, Principles of lnteractive Computer Graphics, 2nd edn., McGraw-Hill, Tokyo, 1979. [6] Y. Li and W. Oldenburg, 3-D inversion of magnetic data, Geophysics, 61,394-408, 1996. [7] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1983. [8] B. J. Last and K. Kubik, Compact gravity inversion, Geophysics, 48, 7130-721, 1983. [93 W. R. Green, Inversion of gravity profiles by use of a Backus-Gilbert approach, Geophysics, 40, 763-772, 1975. [10] M. H. P Bott and M. A. Hutton, Limitation on the resolution possible in the direct interpretation of marine magnetic anomalies, Earth Planetary Sci. Lett., 8, 317-319, 1970. [11] P. C. Sabatier, Positive constraints in linear inverse problem. I - General theory, Geophys. J. R. Soc., 48, 415-422, 1977. [12] P. C. Sabatier, Positive constraints in linear inverse problem. II -Applications, Geophys. J. R. Soc., 48, 443-469, 1977. [13] V. Richard, R. Bayer and M. Cuer, An attempt to formulate well-posed questions in gravity: [1]
References
[14] [ 15] [ 16] [ 17] [18]
[19]
[20]
255
application of linear inverse techniques to mining exploration, Geophysics, 49, 1781-1793, 1984. G.H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1983. G. Backus and F. Gilbert, The resolving power of gross earth data, Geophys. J. R. Astr. Soc., 16, 169-205, 1968. H.J. Lehman, Examples for the separation of fields of magnetic sources in different depths by harmonic analysis method, Boll. Geofis. Teor. Appl., 71, 97-117, 1970. D.C. Mishra, Magnetic anomalies - India and Antarctica, Earth Planetary Sci. Lett., 71, 173-180, 1984. V. K. Gupta and F. S. Grant, Mineral exploration aspects of gravity and aeromagnetic surveys in the Sudbury-Cobalt area, Ontario, in W. J. Hinze (Ed.), The Utility of Regional Gravity and Magnetic Anomaly Maps, SEG, Tulsa, OK, pp. 392-411, 1985. S. Yungsheng, D. W. Strangeway and W. E. S. Urquhart, Geological interpretation of high resolution aeromagnetic survey in the Amos-Barrayte area of Quebec, in W. J. Hinze (Ed.), The Utility of Regional Gravity and Magnetic Anomaly Maps, SEG, Tulsa, OK, pp. 413425, 1985. G. Backus and F. Gilbert, Numerical application of a formalism of geophysical inverse problem, Geophys. J. R. Astr. Soc., 13, 247-276, 1967.
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257 Chapter 6
Parameter Estimation
A potential field signal is caused by a region of anomalous mass or magnetization. The object of analysis is to find out whether a signal is present at all, and if the answer is positive, how to extract the size and shape of the anomalous region under a simplifying assumption of uniform density or magnetization contrast confined to a region bounded by plane surfaces. In signal processing the first part is known as detection and the second part as estimation. The parameters of interest are the points of intersection of three surfaces (or two lines in 2D space). Each intersection or a comer is described by three parameters, namely, x, y and z coordinates. An elementary model is a point or a dipole with three parameters, in addition to a parameter representing the strength of point or dipole source. A model of some practical interest is a prism with eight comers, making a total of nine parameters. Quite often a geological structure will have to be modelled as a super position of two or more prisms, making the count of parameters to be estimated very large. The signal is often accompanied by noise which introduces an element of error in the estimation. This chapter aims at characterizing the nature of this error. We use the framework of maximum likelihood (ML) estimation which enables us to asymptotically reach the lower bounds on the errors, often known as Cramer-Rao bounds. A note of caution for the traditional geophysicist engaged in the analysis of potential fields is that the concepts presented in this chapter may appear somewhat obscure. Yet, he/she should be aware that the quantities estimated from the observed field are not without errors and he/she should understand how to characterize them. The first two sections deal with the ML method and its application to some simple source models. In the last section we consider the least squares (non-linear) method which is shown to be a special case of the ML method valid when the background noise is white Gaussian.
6.1. Maximum likelihood (ML) estimation The basic requirements of any estimation procedure are: on average the estimate must lead to the actual value of the parameter, that is, the estimation procedure is unbiased and the mean square error, that is, the variance, must be
Parameter Estimation
258
minimum. Such an estimate is known as the minimum variance unbiased (MVU) estimate [1, p. 15].To find such a MVU estimate one must first be able to find a sufficient statistic, that is, one which uses all the data efficiently, either by trial and error or by following a procedure of factorization of the probability density function and use the sufficient statistic for the estimation of the unknown parameter [1]. But in practice an MVU estimate may not exist at all or cannot be found even if it does exist. Hence an alternate popular approach, the maximum likelihood estimator, which asymptotically achieves the Cramer-Rao lower bound, has become popular.
6.1.1. Basic detection theory The decision theory deals with the question of making a decision based on the observed data as to which one among all possible hypotheses is true. Let B be our observed event (data) and let A l,A2,...Ak be mutually exclusive events; one of the events must have occurred to give rise to the observed event. Now let A1,Az,...Ak form a set of all possible hypotheses and our aim is to find out which one is the most likely. For this we must form the conditional probability, Pr(Ak[B), k = 1,2,.... The joint probability of occurrence of B and Ak is expressed as follows: Pr(B,A~) = Pr(B)Pr(Ak[B) = Pr(A~)Pr(B[Ak) and hence Pr(Ak[B) =
Pr(Ak)Pr(BIAk) Pr(B)
(6. la)
We note that the probability of B to occur may be written as
Pr(A,)Pr(B[Ak)
Pr(B) - ~
(6.1b)
k
Substituting Eq. (6.1b) in Eq. (6.1 a) we obtain
Pr(AklB )
Pr(Ak)Pr(B[Ak) }-~ Pr(A k ) Pr(BIA k ) k
(6.1c)
Maximum likelihood (ML) estimation
259
Pr(Ak[ B) is called the a posteriori probability of hypothesis Ak, Pr(Ak) is called the a priori probability of hypothesis A~, and Pr(B [Ak) is called likelihood function. With reference to signal detection, the question is whether the observed data f are a mixture of signal and noise or f is a pure noise. Let us represent the first event by s and the second event by o. From Eq. (6.1c) it follows that the a posteriori probability of the event being a mixture is given by Pr(s)Pr(fls) P r ( s l f ) - P r ( s ) P r ( f l s ) + Pr(o)Pr(f[o ) or
Pr(slf)
PrOc s) PrOc o) Pr(f[s) Pr(o) Pr(f[o) -~- Pr(s)
(6.2a)
where the ratio Pr(fls) Pr0Clo ) is called the likelihood ratio which we represent by A and rewrite Eq. (6.2a) as A Pr(s]f) - A + er(o) er(,)
(6.2b)
We note that the a posteriori probability of hypothesis 's' is a monotonic function of the likelihood ratio A. As A --~ oo, Pr(s If) --~ 1 which means that occurrence o f f implies s with probability 1. From a practical point of view the likelihood ratio can be more easily computed than the a posteriori probability, because the latter requires a knowledge of a priori probabilities which is hard to obtain. Detection of a signal is based on the magnitude of the likelihood ratio, for example, A > Ahold
signal 4- noise
A < Ahold noise
260
Parameter Estimation
where Ahold is the threshold which is determined from the noise statistics. For each choice of Aholdthere is associated error, that is, the probability of missing the signal; or a false alarm, that is, the probability of saying the signal is present when it is actually not there. In the literature on detection there is extensive discussion on the probability of error and false alarm but this may not interest us here as much of the effort seems to be in the estimation of parameters.
6.1.2. Parameter estimation If the signal depends upon a parameter 0 the likelihood ratio takes the form A(0)
Pr(fls(0))
Pr0Clo )
(6.3)
where the parameter 0 is unknown but not random. Let us now assume that in the observed datafthere is a signal with unknown parameter, s(O). Evidently the joint probability density function p(f,s(O)) will be maximum only when 0 equals its true value. Thus, estimation of 0 involves maximization of the joint probability density function. Let 0 be an estimate of 0 for which p(f,s(O)) is maximum, that is, p(f,s(O)) = max. When this happens we see from Eq. (6.3) that the likelihood ratio also becomes maximum leading to the maximization of the a posteriori probability density function, p(s(O) IJ). Intuitively, this may be understood as, for the choice of parameters based on the maximum likelihood ratio, the probability of the presence of a signal in the observed data is maximum. 0 - 0(1) is called the maximum likelihood estimator. Instead of maximizing the likelihood ratio A(0) we can maximize any convenient monotonic function of A(0). A commonly used monotonic function is the natural logarithm, lnA(0). The maximum likelihood estimate is now given by
OlA(0) 00
=0 0=0
The parameter estimation can be done either in the space domain or in the frequency domain depending upon how the parameters occur in the expression which needs to be differentiated in order to maximize the likelihood function. It turns out that in potential field analysis, the frequency domain parameter estimation is more convenient as the unknown parameters often occur inside exponential functions, which are easily differentiated and evaluated.
Maximum likelihood (ML) estimation
261
6.1.3. Cramer-Rao bound A maximum likelihood estimator is generally efficient and always asymptotically efficient, which means that as the sample length is increased, the variance of the estimate asymptotically approaches the minimum variance set by the following inequality known as the Cramer-Rao lower bound
1
var 0 > -
(6.4a)
s
which holds good when the estimate is unbiased. The above inequality is known as the Cramer-Rao lower bound [2,3]. For proof and discussion the reader is referred to van Trees classic book [4, p. 67]. We can express Eq. (6.4a) in a slightly different form
var 0 > --
1
f 0 2 lnp(f 0)
(6.4b)
oo-------r---p(flO) df
An estimate is said to be efficient when it satisfies the bound with equality. This is possible only when the likelihood ratio satisfies for all f and 0,
OlnpfflO) 00
= (O(f) - O)k
(6.4c)
where k is a constant. Above condition is satisfied whenever the unknown parameter occurs in a linear form. When we have a collection of p parameters, 00, 01,02, ..., Op_l, the CR lower bound on each parameter is given by
var
0i ~
Ji-il,
i -- O, 1,2, ...,p -- 1
(6.5)
where j/~-l is the ith diagonal element of the inverse of Fisher' s information matrix [5] defined as follows:
262
Parameter Estimation
E{O2lnp(f10)o902}
. "*
"'~l ]y?.j"O02 ]np~ ~ 0) }
9..
E{
a = (-1)"
0 2 lnp0r 0)
E{ oo,_,OOo}
lnp(f 0)
}
where 0 - {00, 01,..., Op_l } is a vector representing all parameters. In Eq. (6.5) sign of equality becomes effective when the derivatives of the likelihood function are a linear combination of estimation errors, that is, O l
p(flO)
00~
-
.j
(4Or)
-
o
for all f and 00, 01,..., Op_l. When the unknown parameters occur linearly, for example, three components of the magnetization vector, the estimates satisfy the CR bound with equality.
6.1.4. Properties of ML estimates There are several nice properties of the maximum likelihood estimate which makes it a most sought after estimate. These properties are listed below: (1) A maximum likelihood estimator is always asymptotically unbiased, that is, as data length increases the mean of an estimate tends to its true value. (2) A maximum likelihood estimator is efficient by which we mean that the variance of an ML estimate will be the lowest possible. (3) A maximum likelihood estimator is asymptotically efficient, that is, as the data length increases the variance of the estimate approaches the CR bound. (4) A maximum likelihood estimator is asymptotically normally distributed. (5) It satisfies the property of invariance, that is, if 0 is an ML estimate of 0, then g(0) is an ML estimate of g(O) where g( ) is a function having a single valued inverse [6]).
6.1.5. ML estimation and Gaussian noise Maximum likelihood estimates can be derived for any distribution of back-
263
Maximum likelihood (ML) estimation
ground noise but the most commonly considered noise model is the Gaussian distribution for which more definite results can be derived. Furthermore these results may be reduced to popular least squares estimates whenever the background consists of white noise. As in Eq. (6.3) we assume that the observed data consists of a deterministic signal s(O) and Gaussian noise. The conditional Gaussian probability density functions used in Eq. (6.3) may be written as
1
Pr{f I s(0)}
Pr{f I 0}
v/(ZTr)~ IC [
V/(2~_)n[C i
exp{ 1 {C} 2-~
exp{
f- s(0)
(f - s(0)) T
{C}
f
fT
0
0
where f is the observed data vector of length n, s(0) is the signal vector and C is the covariance matrix of the background noise [16]. Note that the notation IA[ stands for the determinant of a matrix A. The likelihood ratio may now be expressed as
Pr(fls(0)) { 1 [ {C} A(0)- Pr(flo) exp 2[CI (f - s(0)) T
f- s(0) fT
0
fll}
0
or the log likelihood ratio as
'{
In A(0) - 21CI
{C}
f-
(f - s(0)) T
s(0) 0
{C}
f
fT
0
Let us now maximize the log likelihood ratio with respect to 0. By setting the derivative of ln(A(0)) with respect to 0 to zero, we obtain
0[S(0)TC-ls(0)
0[S(0)Tc-lf]
0[fTc-' IS(0)]
00
00
00
-0
When the background noise is white noise whose covariance matrix is equal to C - azI, the above relation reduces to
0[S(0)Ts(0)] O[S(0)Tf] 0[fTs(0)] 00
00
00
0
(6.6)
264
Parameter Estimation
Now consider the least square estimate obtained by minimizing the error power, (f-S(0))T(f - S(0))= min. Upon differentiating with respect to 0 and setting the derivative to zero, we obtain O[S(0)Ts(0)] -- o[fTs(0)] -- O[S(0)Tf] = 0
00
00
00
which is exactly the same as Eq. (6.6). Thus, the ML estimate in the presence of Gaussian white noise background reduces to the least squares estimate. Since, in practice, we have very little knowledge of the background noise, it is common to assume the Gaussian white noise model and follow the least squares approach. It should be borne in mind that the ML estimate is superior to the least squares estimate.
6.2. M L e s t i m a t i o n
source parameters
We consider ML estimation of parameters of simple source models, for example, a point mass, dipole and a vertical prism. These three models are sketched in Figure 6.1. The background noise is Gaussian but not necessarily white. 6.2.1. P o i n t m a s s
Let us consider the simplest model of potential field, namely, a point mass. The ,,
,,
h
i
Point
.
.
.
.
.
.
.
h
,-
l
Dipole
H
Prism Figure 6.1. Three simple source models are considered for parameter estimation. The simplest model has four unknowns, mass and three location parameters. Both ML estimates and CR bounds are discussed. The dipole model has six parameters but we attempt only three parameters of the polarization vector. Finally, the prism is the most complex model having eight parameters, of which we attempt four parameters, namely (a, h, H, m:).
265
ML estimation source parameters
unknown parameters are mass, m and three location coordinates, x0, Y0, h (see Figure 6.1). The mass parameter occurs in linear form, hence its ML estimation and CR bounds are easily obtained. We first show how to obtain the ML estimate of m given other parameters. In fact it turns out that to estimate m one does not require x0, Y0, h if the entire gravity field is available for estimation. The signal model is (6.7)
f: (x, y) = mJ~ (x, y) + r/(x, y) where
Gh fo(
,y) =
{(X- X0)2 -~- ( y - y0) 2 q- h2] 3/2
We assume that the location coordinates, x0, y0, h0 are known and the unknown parameter is mass m of the point source. The background noise rl(x,y) is assumed to be homogeneous (stationary) and Gaussian distributed. The likelihood function given in Eq. (6.3) can be expressed as follows:
A(m) - exp
1 2-~
{C} (fz - mf0) T
fz - mf0
0
{c}
fz
fz
0
(6.8)
where fz is the data vector, f0 is the signal vector and C is the covariance matrix of ~(x,y). These quantities are defined below. For convenience consider 2D gravity data consisting of N x N points mapped into N column vectors each of size N as shown in Figure 6.2. All these vectors are now stacked one below the other to form a large vector fz of size N 2
el e2 fz -
.
_ eN
(6.9)
_
Let the f0 vector be defined in the same fashion with the only difference being
266
Parameter Estimation
el
ei
eN
Gravity data matrix
Figure 6.2.2D gravity data is mapped into columns, el,e2 ..... eN each of length N.
that the data has only a signal component. The covariance matrix C is a block matrix of covariance and cross covariance matrices of noise. This is illustrated in Figure 6.3. Each block in the block covariance matrix is a covariance or cross covariance matrix between two noise vectors defined in the same manner as in Figure 6.2. For example, C k , ; - E{ekef} To maximize the likelihood ratio we differentiate Eq. (6.8) with respect to m and set the derivative to zero. We obtain the following equation:
ML estimation source parameters
267
0(1,1) C(2,1) , C(k,I)
,
i m
i
i i
C(N,1)
i
Figure 6.3. Block covariance matrix o f the data matrix. Each block is a covariance or cross covariance between two column vectors shown in Figure 6.2. The size o f each block is N • N and there are N 2 blocks, thus making N 4 entries in the block covariance matrix.
{c}
fz
(f0) T
0
{C}
fo
(fo) T
0
(6.10a)
m
In practice it is impossible to get an estimate of the noise covariance matrix from the observed data, for there is rarely a pure noise sample. Then, one resorts to noise modelling and estimating the parameters present in the noise model from the data. One such elementary model is a zero mean white noise model where the unknown parameter is the variance. The covariance matrix C is now simply a diagonal matrix whose diagonal elements are the same, equal to cry. The estimate of mass reduces to N
N
Efz(k,/)fo(k, l) rh _ k-] t=] U
E
(6.10b)
N
k--1 l=1
(k,/)
268
Parameter Estimation
which is indeed the least squares estimate of m obtained from Eq. (6.7). Conversely, a least squares estimate of a parameter is an ML estimate whenever the background noise is a white Gaussian homogeneous process. Note that the variance of the noise does not appear in the final expression, Eq. (6.10b). Let us now estimate the mean and variance of rh. From Eq. (6.10) it is straightforward to show that E{rh} = m, that is, the estimate is unbiased. To estimate the variance, we use the CR bound given by Eq. (6.4b) with the sign of equality. Note that the required conditional probability density function in Eq. (6.4b) is given by
1 P~lm)-
exp {
V/(27r)N[cI
1 2-~
c
fz - mf o
(fz - mfo) T
0
(6.11)
Using Eq. (6.1 l) in Eq. (6.4b) we obtain
1
var rh >_ rT
1
(6.12a)
10 C - f0
We next show that Eq. (6.4c) is satisfied. For this, differentiate the logarithm of Eq. (6.11) with respect to m and then use Eq. (6.10) to obtain
-1 C
O lnp(fi.lm ) Om
f=
m
ICl (fo) m 0
-- - ( r h - m)
C
fo
(fo) T
0
C
fo
foT
0
which is indeed in the same form as Eq. (6.4c). Hence, in the CR bound we can now take the sign of equality, var rh -
fTc-lfo
For the white noise case, Eq. (6.12a) simplifies to
2 var rh - f~'fo en
(6.12b)
269
ML estimation source parameters
Note that Eq. (6.12b) represents a quantity inverse of the signal-to-noise ratio where the signal power is that due to a unit mass. Further, note that the denominator of Eq. (6.12b) will become independent of the location parameters, which were assumed to be known, if a sufficiently large map of the field is used. Ideally, the map size should be large enough so thatj~(x,y) vanishes outside the map area, then fTf 0 will attain its maximum value.
6.2.2. Point mass - location parameters The location parameters, (xo,Yo,h), appear in a non-linear form. The frequency domain approach seems to be most attractive for the estimation of these parameters. The Fourier transform of the signal due to a point mass of unit magnitude is given by
Fo(u, v) = 27rG exp(-sh) exp(-j(uxo + Vyo))
(6.13a)
The signal model is same as in Eq. (6.7) but is now expressed in the frequency domain as
dFz(u, v) = 2rcGm exp(-sh) exp(-j(uxo + vyo)) + dN(u, v)
(6.13b)
where dFz(u,v) and dN(u,v) are the generalized Fourier transforms of the observed gravity field and background noise, respectively. The probability density function of dFz(u,v)assumes a simple form as the background noise in the frequency domain becomes uncorrelated (assuming that it is homogeneous, see Chapter 2, p. 50 for the properties of Fourier transform of stochastic field)
I
([dFz(u'v)-mF~ 2S~(u, v)
(6.14)
p(dFz(u, v)[0) -- v/ZrcS~(u, v)exp -
where 0 represents the unknown parameters (xo,yo,h). In practice the Fourier transform is evaluated over a finite data matrix, therefore, the continuous frequencies become discrete,
u-kAu-k2rr
and N
v--lAv--12rr N'
(k,l)-0
+1 '
'
N 2
270
Parameter Estimation
The discrete Fourier coefficients become approximately independent for large data size (see Chapter 3, p. 105). The joint probability density function of all Fourier coefficients may be written as a product of terms which are similar to that in Eq. (6.14).
p(dFzlO) ~
N/2 1 ( [dFz(k,l)- mFo(k,l)]2.~ k,2S,7(k, l) ] rI v/27rS~(k, l)exp k,l=-U/2
(6 15a)
and when there is no signal, the joint probability density function is given by
p(dFzlO) ~
N/2 1 (IdFz(k'l)12) H v/ZTrS,7(k l)exp - 2S~(k l) k,l---N/2 ~
(6.15b)
We are now ready to write down the likelihood function but we need a prior distribution of the unknown parameters. In the absence of this information, it is recommended to assume uniform distribution. Using Eq. (6.15a) and Eq. (6.15b) in Eq. (6.3) we obtain
A(0) ~
]-I
exp - I d F z ( k '
k,l=-N/2
l) - mFo(k, l)12-1dFz(k, 1)12 2S'7(k' l)
(6.16a)
or in logarithmic form
-Z
N/2
k,l=-N/2
[[(dFz(k, l) - mFo(k, l)12-ldFz(k, 2S,7(k, l)
l)12]
(6.16b)
The likelihood function is maximized with respect to the unknown parameters, x0, y0, h. Differentiating Eq. (6.16b) with respect to x0, Y0 and h,
ML estimation source parameters
271
Oln{A(xo,Yo,h)} Oxo N/2
Z ['-J~ kdFz(k, l)mF~(k, l) +j ~ kdF;(k, l)mFo(k,/)] 2Sv(k, l) k,t=-N/2 Oln{A(xo,Yo,h)} Oyo N/2
Z
2Sv(k, 1)
k,t=-N/2 and
Oln{A(xo,Yo,h)} Oh _ ~N/2 ~ i~ v/k2 + 12[dFz(k'l)mF~
+ dFf(k, 1) 1)mFo(k,1) - 2[mFo(k,1)[2].1
k,l=-N/2
After equating the derivatives to zero we obtain the following system of equations, where we have now explicitly shown the unknown parameters: N/2 k,l=--N/2
Im e--~V~+/2h {kdFz(k , l)ei(-~(~~+ly~
2Sv(k, l)
" kdF, (k, l)e-J(z~(kx~176) }1
- 0
(6.17a) N/2
k,t=-u/2
Ime-z~~+12h[ldFz(k, /)e/(-~(kxo+tyo)) l dF; (k, l)e-J(Z~(~~176 2Sv(k, 1)
0
(6.17b) and
Parameter Estimation
272
N/2
Z ImV/k2_Jr_12e-~X/~+12h[dFz(k,2s,7s-~(kx~176 (k,l)l)e + dF~(k, l)e-JZ~(kx~1761
k,l=-N/2
-- ZN j2 ' ~
,
~,,~,~
]
v/k2 q- 12e-2-~~h]
(6.17c)
k,l=-N/2
The solutions of Eqs. (6.17a) and (6.17b) are weakly dependent on h and hence it may be set to zero, for convenience, and then simplify these equations N/2
N/2
k,l=-U/2
k,l=-U/2
N/2
N/2
k,l=-N/2
l -
[kdP(k,l)ed(-~("~176
*
y~ [ldP(k,l)eJ(~(kx~176 *
k,l=-N/2
(6.18) Solution of Eq. (6.18) involves 2D Fourier transformation of
ldP(k,l) and of their conjugates where
kdP(k,1) and
dFz(k,l)l) exp ( - -~2rr~/(k2+12)h) dP(k,l) - 2-~,1(-~, Wherever the equality of Fourier transform is satisfied we get the desired solution. The solution is not affected by the assumed value of h because the equality occurs when the phase terms are equal but opposite in sign. The selection of h does not contribute to the phase matching. After estimating x0 and Y0, the centre of the coordinates is relocated at that is, the anomaly is centred. This step simplifies Eq. (6.17c) to
(xo,Yo),
N/2
k,t=-u/2
lZh)[dFz(k,l)+ dFz(k,1)]l 2S,7(k,l)
I v/k + / 2 e x p ( - - ~ x/k 2 +
ML estimation source parameters
273
N/2
Z
(6.19)
k,l=-N/2 where
dFz(k, l)
is now the discrete Fourier transform of the centred anomaly.
6.2.3. Point mass CR bounds Next we evaluate the CR bounds for three unknown parameters. The first step is to evaluate the Fisher matrix. N
Jll - ~
2
2rr
~-(~-~/2rl
(k, ~/
k, l---} N l(k,/) k, l= -N N
J,+ - Z
~ -J( I~kv/k~ + t2e-~ ~e~-~v,(k, 0
k,l- N
J2,k,l=-S/2 N/2
(l)2Vl (k, t)
J22k,l=-N/2
~3-
1 v/k 2 +/2exp
- - ~27r - v/k2 +
lZh) Pl (k, l)
k,l=-N/2 N/2
J31-
F+ k,t=-N/2
F(k'l) -~1 -J( N )2kv/k2 + 12exp (27rv/kZ+lZh)
Parameter Estimation
274
N/2
Y~ - j
J32 ~-
(2-~)2l v/k 2 +/2exp ( -~27rv/k2q-12h)Fl(k~')
k,l=-u/2
N/2 y~ _
J33 --
(k 2 +/2)exp
27r v/k2 +
12h {El (k, l) - 2Fz(k,/)}
k,t=-u/2
where
r, (k, l) - E{dFz(k, l)mF~ (k, l) - dF; (k, l)mFo(k, l)} 2S,(k,l) F2(k,
l) - I.ImF~ l)
Since the background noise is zero mean, E{N(k,I)} = 0 for all k and 1. Hence, Pl(k,/)= 0 for all k and l. Thus, all elements except J33 of the Fisher matrix become zero. The CR lower bounds on the variance of ~0 and )90 are therefore equal to zero, var{s
> 0,
var{~0} > 0
and that on h is given by
var{J~} _>
N/2 2 y~ ~ k,l=-N/2
(6.20a)
1 (--~)2(k2--[-12)
e x p ( - - ~ v/k2-[ -
12h)~2(k,l)
The denominator in Eq. (6.20a) does not converge to a finite value as N becomes large. The double sum in the denominator may be expressed as a double integral var{h} ~ N2
271-2
1
f f.q--Trs2exp(_sh)C2(u,
v) du dv
(6.20b)
ML estimation source parameters
275
where
[mFo (u, v) [2 1~2 (U, V) --
S (u,
Assuming the noise spectrum is constant, that is, equal to its variance and using Eq. (6.13a) we can evaluate Eq. (6.20b) 4
var{h} _>
/37rN2 (Gin)2
(6.2 la)
or by rearranging the terms
var{:} >
-- 371" (Gm) 2
(6.21b)
where/3 ~ 8/27. From Eq. (6.21) we note that the CR lower bound is proportional to the fourth power of the depth to the source. As the source goes deeper, unless the map size and/or signal-to-noise ratio is increased quadratically, it becomes increasingly difficult to accurately estimate the depth to the source. However, the estimates of x0 and Y0 are unaffected as long as the background is zero mean, which implies that regional field has been removed fully.
6. 2.4. Dipole The next model we consider is a magnetic dipole source. There are six unknown parameters, three for polarization vectors and three for location. In order to keep the mathematical complexity to a minimum, we consider the estimation of the polarization vector only. The signal model remains the same as in Eq. (6.7) where the signal part is now given by Eq. (2.39) in the frequency domain,
dFr(u , v) - (l'mxU +jmyv - Smz) x [27r(/'c~u +j/3 v
7s) exp(-Sh)s exp(-j(xou + yov))] + dN(u, v)
276
Parameter Estimation
(6.22)
- O'mxU +jmyv - Smz)Fo(u , v) § dN(u, v)
where the unknown parameters are (mx, my, mz), the three components of the polarization vector and Fo(u,v) is the known part of the signal. There is a strong similarity between Eq. (6.22) and Eq. (6.7). The unknown parameters occur linearly, therefore, the mathematical analysis closely follows that derived in the previous section for point mass. The likelihood function is given by ln{A(0)}
N/2
.[dFT(k,I - -y 2rrel.rexk _.{_jmyl - v/k2 --}-12mz)Fo(k ,
l)[2 idF(k,Oi2] -
2S,7(k,l )
k,l=-N/2
(6.23) Differentiating Eq. (6.23) with respect to the three unknowns (mx, my, mz) and setting the derivatives to zero, we get three linear equations,
N/2
2rrk I1-'2 (k' l) -- 2 2U(mxk
Z k,t=-u/2 Y'~ ---N--
myl)lF~
l)[21 - 0
2 &, (k , l )
N/2
2roll jF2(k' l) - 2 2~ (mxk myl)lF~ ~ ~ N 2&,(k, l) k,l=-N/2 N/2 k,l=-N/2
Z
2rrv/(k2 +/2)
N
I,
- 0
2rrv/(+/2) k2
F, (k, l) + 2mz u 2S~(k,l)
where Pl (k, l) - dF~(k, 1)Fo(k, l) + dFv(k, l)F~ (k, l) Fz(k , l) - dF~(k, 1)Fo(k, l) - dFT.(k, l)F~ (k, l)
IFo( k, l)l
21
-0
ML estimation source parameters
277
whose solution yields following estimates of (mx, my, mz), DC - EB
thx -- A C - B 2 '
thy
DB - EA -- A C - B 2
(6.24a)
where N/2
A
B
F~ F~
[.IF~
B-
L S~l(k, 1) J'
k,t=-u/2
~ k, ,=-N/2
~
kt I s.(k,
1
t) I
N/2 C
m
Z
L s,,,(k, ,,,)
k,l=-N/2 N/2 O
~
k,t=-N/2
27rk ).F2 (k, 1)] 2S~I~ ' -
N/2
E--
~
k,l:-X/2
27rl FFz(k,/)1 ~ ~ L Z S ~ ( k ,~)
and N/2 y: l~nz -- --
~ [27rV/~ +/2)2Sn(k,l)J ~1~,,~.1
k,l=-N/2 N/2~ ~
[.
k,l=-N/2
(kZ+lZ)
(6.24b)
So(k,l)
It can be shown with some effort that the estimates are unbiased, that is, E{rhx} - mx, E{rhy} -- my, and E{rhz} - mz. 6.2.5. D i p o l e C R b o u n d s
Next we evaluate the CR bounds. The information matrix is obtained by taking the second derivative of the likelihood function and computing the expected value of each element in the information matrix. We obtain
II
bJ
+
t,J
t
I
bJ
y
i ~,,.
v
I
11~
,,....
t,J
t,J
I
i
t,J
II
I
t~
t~
,.,.,
I
t,J
t~
i
t~
II
II
I
!
+
~L"J
v
L~
II
o
~==~~
t,J .,..i
ML estimation source parameters
279
In order to evaluate the above expression, we replace the double sum by a double integral under the assumption that N is large. In effect the entire signal is assumed to be available free from contamination from other neighbouring sources. In practice this is a very ideal situation, hence the closed form expressions given below are indeed the lower bounds on the variance of the estimates. Further, we assume that the background noise is white, with variance, (r2~. /32 2-cos(2~o) 4 _+_ ,.)/21 h 4 cr~ 2
var{rhx}
8 ,.)/4 + ,.)/2/32 + T(} 3t~2 3~N2{ e 2 2+cos(2~p) 4 __[_,.)/2] h 4 cr~ 2
var{rhr}
N 2 {,.)/4 @ ,.)/2t32 +
16 J
4 2 h or,7 var{rhz} - 3~N2(1 + ,.)/2) 8
(6.26)
where e 2 -- (y2 ~_/~2 and (p -- tan -1/3/c~ (Note that o~, [3 and 3" refer to the direction cosines of the Earth's magnetic field). From Eq. (6.26) it is possible to show a simple relationship among (rhx, rhy, rhz), var{rhx} + var{rhy}
(2 - 82) 2
var{rhz}
1 - T 13 g t32
(6.27)
The ratio of variances as a function o f ~/(o~2 -~-/~2) is shown in Figure 6.4. It is interesting to note that when s = 0, that is, when only vertical component is used in the estimation, Eq. (6.26) reduces to a simple form
var{thx} - var{rhy} - 2var{thz}
2 h 4 cr~
(6.28)
6.2.6, Vertical prism In a time series context, a sum of sinusoids or damped sinusoids is a useful
Parameter Estimation
280 8,
h! iill..... ....... i
1
.........
4
A
_
,L
.L . . . . . . . .
9 v
3 0.0
0.2
0.4
9.........
0.6
9. . . . . . . . .
;___l
b.
9
. . . . . . . . .
]
~_J
I
0.8
1
E ------]P
Figure 6.4. Ratio of variances, [var rhx + var rhy]/var rhz, as a function of ~/(c~2 +/32). model. Many powerful techniques have been developed, including ML techniques, for estimation of parameters of a sinusoid; namely, frequency, phase, and amplitude. Some of these techniques can be used in the analysis of potential fields. However the principal difference is that the sinusoidal model is applicable only in the (spatial) frequency domain and the parameters of a sinusoid now refer to the physical shape of the source. To illustrate the point let us consider the model of vertical prism. We rewrite Eq. (2.51) in a slightly different manner as shown below: exp(j(au + by) - sh) - e x p ( / ( - a u + by) - sh) - e x p ( j ( a u - by) - sh) + e x p ( - j ( a u
+ by) - sh)
( s u v ) F z ( u , v, h) - 27rGpo
- e x p ( j ( a u + bv) - Shl ) + e x p ( / ( - a u + by) - s h l ) +exp(j'(au - by) - sh~) - e x p ( - j ( a u
+ by) - s h l )
(6.29) where we find on the right-hand side a sum of eight damped sinusoids. Each damped sinusoid corresponds to one comer. The exponents of the damping factor are h or h l (=h + 2c) and the parameters o f the (two-dimensional) sinusoidal factor are proportional to two sides of the top of the prism. Interestingly, the
ML estimation source parameters
281
expression for a horizontal cylinder whose cross-section is a rectangle (2a • 2c) may be obtained from Eq. (2.33a),
(u2)Fz(u) -j2rcGpo
} e x p ( - j u a - uh) - exp(/'ua - uh) +exp(/'ua - uh,) - e x p ( - j u a uh,) '
u > 0
(6.30a)
which is also a sum of four complex (one-dimensional) sinusoids. A slice of Eq. (6.29) along any radial line (other than u - 0 or v - 0) is also a sum of eight complex (one-dimensional) sinusoids. exp(j(a
(x/2uZ)Fz(u, h) - 27rGpo
+
-
-
exp0"(-a
+
-
-exp(/'(a - b)u - sh) + e x p ( - j ( a + b)u - sh) - e x p ( j ( a + b)u - shl) + exp(/'(-a + b)u - Shl) +exp(/(a - b)u - Shl)
-
-
e x p ( - j ( a + b)u - Shl) (6.30b)
Thus, the problem of parameter estimation for both a vertical prism and a horizontal cylinder seems to belong to the class of parameter estimation for damped sinusoids. We now describe Prony's algorithm for estimation of parameters of damped sinusoids in a time series analysis [7]. Consider a problem o f p damped sinusoids, p
F o ( u ) - ~-~Akexp(--qSku)
(6.31)
k=l
where 4~k,k = 1,2,...,p are complex parameters of damped sinusoids, for example, 491 = (h - j a ) when referred to Eq. (6.30) and Ak, k = 1,2,...,p are complex amplitudes of sinusoids. Let Fo(u) be known at 2p discrete equispaced frequencies. From Eq. (6.31) we can define a system of 2p linear equations, P
Fo(Au) -- Z A k
exp(--qSkAu)
k=l p
F0(2Au) -- ~--~Ak exp(--2qSkAu) k=l
Parameter Estimation
282
P
F0(3Au)- ZAkexp(-3q~kAu) k--1
p
Fo(2pAu) -- ~
Akexp(-2pOkAu)
(6.32)
k=l
Linearly combine the first p equations from Eq. (6.32) with weighting coefficients O~p,O~p_1,...,o~ and subtract it term by term from the (p + 1)th equation. We obtain p
p
F0((p + 1)Au) - y ~ O~p_t+lFo(lAu) - Al (Zl (p+l) - Z l=l
C~p_t+lZ-(')
/=1
p
p
C~p_t+lZ2t) + . . . + Ap(zp (p+I) - y ~ C~p_,+lZpt)
+Az(z2 (p+l) - ~ l--I
(6.33)
/=1
where Zl = exp(4~lAU), 2 2 - - exp(4)zAu) and so on. The weighting coefficients are so chosen that the terms in parentheses in Eq. (6.33) are equal to zero. For this to be true Zl,Zz,Z3,...,Zp should be the roots of the following polynomial" p
(1 - ~
c~tz') - 0
(6.34)
l--1
Given this, Eq. (6.33) reduces to p
F0((p -+- 1)Au) - ~
c~,F0((p -+- 1 - l)z2xu) - 0
(6.35)
/=1
Next, from Eq. (6.32) we form a weighted sum ofp equations, from 2 to p + 1 and subtract it from the (p + 2)th equation. We get yet another equation of the type Eq. (6.35). In this manner we obtain a set of p linear equations with p unknowns. In practice it may be necessary to have a larger set of equations and seek a least squares solution.
ML estimation source parameters
283
P
Fo((p + 1)Au) - ~,~,Fo((p
+ 1 - t)~Xu) - o
l=1 p
F0((p + 2)Au) - ~
oLIFo((p + 2 - - / ) A u ) -- 0
/=1
p Fo(2pAu ) -- ~ oLIFo(2p -- l ) A u ) -- 0 /=1
(6.36)
Prony's algorithm now consists of solving Eq. (6.36) for O/1,a2,...,O/p and then finding the roots of Eq. (6.34). From the computed roots we get the unknown parameters, 4~1,4~2,...,4~p,of the sinusoids. The amplitudes of the sinusoids may be then estimated by going back to Eq. (6.32), where now 4'1,4~2,...,4~p are assumed to be known. The modem version of Prony's algorithm such as that based on the subspace approach may be found in Ref. [7]. E x a m p l e 6.1
We illustrate an application of Prony's algorithm in potential field analysis. Consider the gravity field due to a square horizontal prism as shown in Figure 6.5 ( a - 1/2, h - 1, and hi = 2).We assume no noise. We have four complex sinusoids.
I Figure 6.5. Estimation of the comers of a square horizontal prism using Prony's algorithm.
Parameter Estimation
284
TABLE 6.1 An illustration of an application of Prony's algorithm Actual parameters of sinusoids
Estimated parameters, ~Au
Estimated parameters, 4)
-1.0 -1.0 -2.0 -2.0
-0.5236 -0.5236 -1.0472 -1.0472
-1.0 -1.0 -2.0 -2.0
+ + -
0.5j 0.5j 0.5j 0.5j
+ + -
0.2620j 0.2620j 0.2620j 0.2620j
+ + -
0.5004j 0.5004j 0.5004j 0.5004j
F r o m (6.30) we obtain e x p ( - u - j I u) - e x p ( - u + j I u)
]
Fo(u ) - j G p + e x p ( - Z u + j 8 9u) - e x p ( - Z u - j 8 9u) Sixteen samples of Fo(u) at interval ~r/6 were obtained and a least squares solution of the resulting system of equations following the method described in Ref. [7] was obtained. The results were as shown in Table 6.1, column 2. After dividing the above result by 7r/6 we get the estimates of the desired coefficients of the complex sinusoids as shown in column 3.
Example 6.2 In this example we illustrate how Prony's algorithm can be used to estimate the comers of a vertical prism shown in Figure 6.6. The gravity field (in the frequency domain) on a radial line, e.g. on u = v may be obtained from Eq. (6.29), exp(j'(a + b)u - x/~uh) - e x p O ' ( - a + b)u - x/~uh)
u3Fz(u, h) - 27rGp
- e x p ( j ( a - b)u - x/2uh) + e x p ( - j ( a + b)u - x/~uh) - e x p ( j ( a + b)u - x,/-2uh, ) + e x p ( j ( - a + b)u - x/-2uhl) + e x p ( j ( a - b)u - x/~uhl) - e x p ( - j ( a + b)u - x/2uhl)
W h e n a = b = 0.5 the above equation reduces to a sum of six sinusoids; out of these, two sinusoids are of zero frequency. A radial slice (u = v) of the Fourier transform is shown in Figure 6.7. The true and estimated parameters of the sinusoids are given in Table 6.2.
ML estimation source parameters
285
surface
l Figure 6.6. The gravity field of a vertical prism is expressed as a sum of damped complex sinusoids. A slice along a radial line u = v is used for parameter estimation using Prony's algorithm.
exp(/'u- v~u) - 2exp(-x/~u)
u3Fz(u,h) -- -27rGp
+ e x p ( - j u - x/~u) - exp(j'u - S u ) + 2 e x p ( - 5 u) - e x p ( - j u - 5 u)
It may be pointed out that as shown in Refs. [ 14,15] even for complex models such as polyhedrons with triangular facets the Fourier transform can be expressed as a sum of complex sinusoids with the coordinates of the comers as parameters of the complex sinusoids. An approach for estimating the coordinates similar to that described above but in the continuous domain has been suggested in Ref. [8].
6.2. 7. D a m p e d sinusoids CR bounds The C r a m e r - R a o bounds for damped sinusoids have been worked out in Ref. [9]. We use those results for deriving the CR bounds on estimates of the comers of a prism. There is an important difference between the signal model arising in our problem and that studied in Ref. [9]. In our signal model the damped sinu-
Parameter Estimation
286
0.3
0.2 r r
o., 1"4
XIII
i!.:.i
.....t l ...........
0.1
.........
~
:
;,
0 LL'
0.0 0
'"' 2
-
.
. . . . . . . . . . .
-'4
6
8
Frequency Figure 6.7. A diagonal slice (u = v) of the Fourier transform is shown above. Thirty-two samples of this function (sampled at intervals Au = 7r/16) are used in this example.
soids appear in the frequency domain with unknown parameters as coordinates of the comers of the prism (see Eq. (6.30)) while in Ref. [9] the sinusoids are in the time domain with parameters as frequencies and damping coefficients. Define a matrix Q of size (4P • 4P) where P stands for the number of damped sinusoids
Q - [2Re { Z Z
I-I }]-'
TABLE 6.2 An illustration of application of Prony's algorithm to the estimation of the comers of a vertical prism True parameters of sinusoids
Estimated parameters, 4~z~u
Estimated parameters, 4)
- v / 2 - 1.0j -v/2+l.0j -v/2+0.0j - 6 / v / 2 - 1.0j 6/v/2 + 0.0j 6/v/2 + 1.0j
-0.2777-0.2777 + -0.2777 + -0.8331 -0.8327 -0.8331 +
-1.4142-1.4142 + -1.4142 + -4.2432 -4.2411 + -4.2432 +
0.1964j 0.1964j 0.0j 0.1963j 0.0j 0.1963j
1.0j 1.0j 0.0j 0.9997j 0.0j 0.9997j
ML estimation source parameters
287
where
Z--
-oN
,
(4P•
ON jON
1 e~ e2Ol... e(N1
e 02 e2~
1)01
-
e (N-l)02
,
ON --
1 e ~ eZ~
(P•
e (N-1)0P
0 e 01 2e2~
( N - 1)e (N-l)~
0 e ~ 2e2~
( N - 1)e (N-1)02
(P • N)
~N
_0 e ~ 2e2~
(N - 1)e (N-1)0P
01,02, ..., Op are parameters (complex) of the damped sinusoids and N is the number of data points. The parameters of the damped sinusoids may be expressed in terms of body parameters, for example, for a semi-infinite rectangular prism the parameters of damped sinusoid are given by (see Example 6.3)
O1 --j(a + b) - x/2h 02 - - j ( - a + b) - x/2h 03 - j ( a -
b) - v ~ h
04 -- - j ( a + b ) - x/2h 05 --j(a + b)
-
v/2hl
06 - - j ( - a + b) - x/2hl 07
j ( a - b) - x/2hl
288
08 -
Parameter Estimation
+ b) - x / 2 h ,
The CR bounds for the real and imaginary parts of 01,02, ..., Op are given by var{Im[0i]} > ~ Qii,
i - - 1,...,P
-- S N R i
var{Re[0i]} > Qi+P,i+P -
SNRi
i-
1, . . . , P
(6.37a)
,
where (27rGp0) 2 S N R i --
Further, it is shown in Ref. [9] that Qii = Qi+P,i+P, i = 1, ...,P, hence the CR lower bounds on the real and imaginary parts of the parameters of the damped sinusoids are equal. If we have just a single damped sinusoid, the CR bound expression is greatly simplified for large data length,
>
xp( 2h!/
[var{Im[0i]}, var{Re[0i]}] _ 2 e x p ( - 2 h ) S N R i
(6.37b)
Eq. (6.37b) may be used as a measure of error when the comers of the body are well separated. A plot of Eq. (6.37b) is shown in Figure 6.8; we notice that the CR bound increases rapidly for depth greater than four sample units.
6.3. Least squares inverse (non-linear) We have noted earlier in Section 6.1 that the maximum likelihood estimation reduces to the least squares estimation whenever the background is white Gaussian noise. In practice, the white Gaussian noise model is widely accepted as a default model, justifying the use of least squares estimation. In this section we describe the space domain least squares methods for estimation of parameters of source models. Use of the word 'inverse' in the section title follows the practice in the geophysical literature while 'parameter estimation' is more commonly used in the signal processing literature. We use both terms with the same meaning. As
289
Least squares inverse (non-linear)
1 2 0 0
o
,
looo
9
-
.......................
:
....
i.......................
.
; ..........
~oo ........... io........... i........ i ........... i ....... t ~oo I ........... i- ........... ~........... -i........... ~..... / ,ool ........... i. ........... i. .......... i ........... ~....~....
/ ~oo . . . . . . . . . . .
0
~. . . . . . . . . . . .
~o. . . . . . . . . .
/ ~...--..--..~.,, ........
I
2
3
Depth
4
5
Figure 6.8. CR lower bounds on the estimate of parameters of a single damped sinusoid as a function of depth (in units of sample interval). SNR = 10. in the previous section, the source model consists of a well defined geometrical body such as a prism whose coordinates of the comers (geometrical parameters) are the unknown parameters, which are to be estimated by minimizing the mean square difference between the observed and the computed field. The density or magnetization distribution within the body is uniform but unknown. The problem may be reduced to a standard function minimization problem which has been extensively studied in mathematics (for example, see Ref. [12]). We briefly describe the commonly used Gauss-Newton method followed by Levenberg [10] and Marquardt [11] modifications. It may be noted that although the unknown density or magnetization appears in a linear form (see Section 6.2), on account of the assumed uniform distribution of density or magnetization but fixed geometrical shape of the causative bodies, the parameters of the model appear in a non-linear form, which requires a non-linear least squares inverse. 6.3.1. G a u s s - N e w t o n m e t h o d
Consider a functionf(x,y,O) representing the gravity or magnetic field due to an object whose geometrical parameters are 0z, i = 1,2,...,p and 00 is the density or magnetization (vector).The field is measured at N points, fn = f ( x n , Y n , O ) -t- tin ,
n = 1, . . . , N
290
Parameter Estimation
where rln is noise in the observation. The mean square difference between the observed and the estimated is given by N
Q(O) _ y ~ ~ , , - f ( x , , y , , , O, ) ]
2 _ fTf
(6.38)
n=l
as a function of 0 and assumed values of the unknown parameters, where f -- fn -- f(Xn,Yn,O),
n -- 1, . . . , N
The gradient vector g of Q(/~) is given by g-
2jTf
(6.39)
where J is a Jacobian matrix whose (n,p)th element is given by
Of(Xn,y.,O)
The size of J is size N x P. We define one other matrix called the Hessian matrix, G, whose (p,q)th element is defined as OzQ(O) {G}p,q-
OOpOOq
and is of size (P x P). At the point of minimum of Q(/~), that is, at stationary point we must have g(/~min) -- 0
(null vector)
(6.40a)
and in the neighbourhood o f 0min an additional property must be satisfied,
x0vG xt > o
(6.40b)
Least squares inverse (non-linear)
291
that is, G must be positive definite. The minimization scheme works as follows: (I) Start at some point close to the true solution and determine a search vector, v. Let the starting point be 01 and the search vector be vl.The search vector may be any vector which satisfies a condition, g~vl < 0 where gl is the gradient of
Q(O) at 01(IX) Along the search vector, a minimum of O(0) is reached at t~2 where gzgvl- 0. At 02 another search vector, v2, is selected satisfying the condition given in I. (III) The process is continued until the gradient vector becomes a null vector. In the close neighbourhood of the stationary point, say at the kth iteration, the search vector will be approximately given by gk + Gkvk ~ 0
(6.41 a)
or
vk ~ - G k -1 gk
(6.41 b)
Eq. (6.41) may be used to estimate the search vector at each iteration provided the Hessian matrix is positive definite. The Hessian matrix may be expressed as G-
2 j T j + 2S(/~)
(6.42)
where N
S(/~) -
~-~f(x,,y,,/~)T(x,,y,,/~) n=l
where T(xn,Yn,O)is the Hessian matrix ofj'(Xn,Yn,O).In the Gauss-Newton method, for the sake of reducing the computational load, S(0) is completely ignored.
6.3.2. Levenberg-Marquardt mod~cation The Gauss-Newton method of object function minimization produces satisfactory results so long as the starting values of the unknown parameters are close to the actual values. Otherwise the Hessian matrix as approximated in the GaussNewton method may not remain positive definite. To overcome this problem, the Hessian matrix at the kth iteration is approximated as
292
Parameter Estimation
epicenter . . . . . . .
l T x
I
I
Figure 6.9. Vertically polarized rectangular vertical prism. The parameters of the model are listed below. There are nine parameters including the coefficients of the planer regional field. Gk -- 2J~Jk + 2"/kl
(6.43)
where 3'k is a constant selected with a view to keeping Gk positive definite. Using Eqs. (6.43) and (6.39) in Eq. (6.41a) we obtain [J[Jk + ")'kl]vk -- -J~'fk
(6.44)
Example 6.3 We consider a synthetic example of a vertical rectangular prism as shown in Figure 6.9. The assumed parameters are as follows: size of the rectangular top, 4 x 8 units; depth to the top, 5 units; depth to the bottom, infinity; epicentre (16, 15) units; magnetization, 90 units; polarization, vertical.
Least squares inverse (non-linear)
293
TABLE 6.3
Initial values After three iterations Initial values After three iterations
Magnetization
Width
Length
Epicentre x
Epicentre y
Depth to top
100 87.1 130 37
5.6 4.0 12 9.0
10.6 8.0 18 7
14 16.0 10 15.9
16 15.0 12 15.0
4 4.99 9 4.5
The top row represents the initial values used in the Levenberg-Marquardt minimization scheme. The estimated parameters after three iterations are shown in the second row. The value of is reduced from 17.737 to 0.114. Similarly, the third row is another set of initial values (further removed from the actual). The estimated parameters after three iterations are shown in the bottom row. The value of is reduced from 54.678 to 2.9.
~/[Q(O)/N]
~/[Q(O)/N]
The magnetic field (vertical component) produced by the model was mixed with random errors lying in the range of +5 nT and also with the regional field given by 0 . 2 5 x - 0.4y - 5. The total field was in the range of 21 to -45.5 nT. A data matrix of size 32 x 32 was thus prepared. In Table 6.3 we list the parameters estimated by the Levenberg-Marquardt method for two sets of initial values. The first set of initial values is close to the true values, the estimated values are therefore practically error free but the second set is further removed from the true values and so are the estimated values. Thus, this example stresses the importance of the initial values in getting correct estimates. The choice of constant "Ykprovides an additional degree of freedom. The lower limit is set by the fact that the Hessian matrix must remain positive definite. When 3'k is very large, then the Hessian matrix is dominated by its diagonal terms. The search vector will be approximately given by
Vk ,~,
J~fk 89
which is now in the negative direction of the graciient, that is, steepest descent. The length of the search vector, however, becomes very small and hence has slow convergence. This sets an upper limit to 3'k. By changing the value of 3'k at each iteration it is possible to control the rate of convergence; convergence is slow when ")'k >> 1 and rapid convergence when 3'k << 1. It is claimed that the Levenberg-Marquardt method not only assures convergence of iterations, it also results
294
Parameter Estimation
in the best fit in the least square sense [ 13]. There is, however, one problem, that is, the minimum reached may not be the global minimum and hence the best fit least squares estimate may not be the correct answer. One possible solution is to start the iteration process from different points. If the minimum reached is the global minimum the result must be independent of the starting value.
References S.M. Kay, Fundamentals of Statistical Signal Processing, Prentice Hall, Englewood Cliffs, NJ, p. 15, 1993. [2] H. Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ, 1946. [3] C.R. Rao, Information and the accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc., 37, 81-91, 1945. [4] H.L. van Trees, Detection, Estimation, and Modulation Theory, Part I, Wiley, New York, 1968. [5] R.A. Fisher, On the mathematical foundations of theoretical statistics, Phil. Trans. R.I Soc., A222, 309-368, 1922. [6] I.C. Hancock and P. A. Wintz, Signal Detection Theory, McGraw-Hill, New York, p. 131, 1966. [7] P.S. Naidu, Modern Spectrum Analysis of Time Series, CRC Press, Boca Raton, FL, 1996. [8] X. Wang and R. O. Hansen, Inversion of magnetic anomalies of arbitrary three dimensional bodies, Geophysics, 55, 1321-1326, 1990. [9] Y.-X. Yao and S. M. Pandit, Cramer-Rao lower bounds for damped sinusoidal process, IEEE Trans. Signal Proc., 43, 878-885, 1995. [ 10] K. Levenberg, A method for the solution of certain non linear problems in least squares, Q. Appl. Math., 2, 164-168, 1944. [ 11 ] D.W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math., 11, 431-441, 1963. [12] L.E. Scales, Introduction to Non Linear Optimization, Macmillan, London, 1985. [ 13] C.C. Ku and J. A. Sharp, Werner deconvolution for automated magnetic interpretation and its refinement using Margquardt's inverse modeling, Geophysics, 48, 754-774, 1983. [ 14] L.B. Pederesen, Wavenumber domain expressions for potential fields from arbitrary 2-, 2 1/ 2-, and 3-dimensional bodies, Geophysics, 43, 626-630, 1978. [ 15] R.O. Hansen and X. Wang, Simplified frequency-domain expressions for potential fields of arbitrary three dimensional bodies, Geophysics, 53, 365-374, 1988. [16] V.S. Pughachev, Theory of Random Functions, Pergamon Press, Oxford, 1965. [ 1]
295
Subject index
a priori information 243 aliasing error 83, 166 definition 83 analytic continuation 20 analytical signal 25, 178 angular spectrum 128 definition 79 direction of a fault 130 estimation 130 peaks 130 polarization vector 129 annihiliation 41 asymptotically efficient 262 autocorrelation function 78 Backus Gilbert approach 233 Backus-Gilbert inversion 239 band limited spectrum 83 bias and variance 109 bilinear interpolation 130 block covariance matrix 266 co-occurrence matrices 249 coherence analysis 136 coherence estimate 111 coherence function definition 81 estimation 111 complex signal 24 condition number 243 conditional probability 258 constrained least squares solution 245 conversion filter gravity to magnetic 47 gravity to magnetic potential 38 convex polyhedron 247 correlation filtering 200, 203, 207 cost function 247 covariance function definition 76 properties 76
CR lower bound 261 depth of source 275 Cramer representation 49 Cramer-Rao bounds damped sinusoids 285 Cramer-Rao lower bound 258 definition 261 cross correlation function 78 definition 77 cross spectrum properties 78 cyclic autocorrelation 109 cylinder polygonal cross 31 polygonal cross section 45 rectangular singularities 35 damped sinusoids 279, 280 decision theory 258 deconvolution 233 deconvolution filtering 6 degree of freedom 5, 293 density maps 227 design of 2D Filter 153 dft coefficients statistical properties 105 digital filter analytic continuation 174 reduction to equator 186 reduction to plane surface 193 reduction to pole 183 terrain correction 200 vertical derivative 176 digital filtering 8, 145 digital filters 145 digital gravity 3 directional filtering 172 discrete Fourier transform 2D 92 discrete potential fields 83 distortion analysis 190
Subject index
296
doubling algorithm 98 in 2D 101 downward continuation 175 dsp some reservations 11 dynamic range 91 earth noise 7 eigenvector decomposition 235 elliptical passband 162 ergodic random field 77 excitation function 62 exploration gravity and magnetic (G & M) 1 mineral 5 seismic 1 fan shaped filter 149 fast convolution 166 fast Fourier transform 97 fault 34 finite impulse response 146 FIR filter 146 Fisher's information matrix 261 folding of spectrum in one dimension 83 in two dimensions 85 Fourier integral representation 21 Fourier transform 3D magnetic moment 40 density distribution 45 generalized 21, 51 in two dimensions 20 symmetry properties 21 fractal character 117 fractal model 217 fractal models of susceptibility 116 frequency numbers 94 frequency of occurrence 249 full rank 235 Gauss-Newton method 289 Gaussian distribution 263 Generalized sampling 86 Gibb's oscillations 149 example 151 gravitational potential 27
half plane filter 146 half space model 66 hessian matrix 290, 291,293 Hilbert transform pair 24 Hilbert transforms 24 hilly regions 193 horizontal derivative 178 IIR filter 146 ill-conditioned 9 image processing 11 implicit periodicity 166 impulse response function 82 inequalities 246 infinite impulse response 146 information content 5 mformation processing 7 initial values 293 mterface model 124 interpolation 3 mterpolation filters 90 reverse filter 10 inverse filtering 6, 12, 223 example 225 instability 224 undulating surface 223 isostatic compensation 142 Lagrange interpolation 4 Laplace equation 19 Laplace transform 41 density log 61 Levenberg-Marquardt method example 292 Levenberg-Marquardt modification 291 likelihood function 259 likelihood ratio 260 line source 30 linear equations 9 linear inversion 223 discrete model (3D) 243 example 248 linear least squares inversion 233 linear programming 246 linear system 82 low latitude effect 185
Subject index
lowpass filtering 6, 146 magnetic potential 25 mapping rules 249 matched filter 218 maximum likelihood algorithm 9 maximum likelihood estimate properties 262 maximum likelihood estimator 258 definition 260 efficient 261 mean square error 257 measurement noise 8 minimum variance 257 ML estimation simple models 264 model error 8 noise 8 " non-uniqueness 41 Nyquist criterion 4 Nyquist sampling interval 3, 5 optimal filters 210 orientation of a fault 130 parallel profiles 3 parameter estimation 9, 114, 260 physical significance of 'spectral' depths 125 pixels 252 Poisson relation 23 polygonal filter 156 polygonal support 147 positive definite 292 potential field integral representation 21 random 50 singularity 35 spectral representation 51 potential field in free space 19 potential field signal 19 power spectrum 50 prism model 57 Prony's algorithm 282, 283 example 283 pseudo inverse 235
297
pseudo-magnetic 71 pseudo-random numbers 78 pseudogravity 71, 188 quadrant filter 146 quantization error 91 finite word length 91 quantization noise 92, 128 radial spectrum 114, 117 definition 79 depth estimate 120 estimation 123 quantization error 128 random fields two dimensional 76 random interface 53 random layer 67 randomly scattered profiles 3 rank deficient matrix 235 reduction to plane surface 193 remnant magnetization 190 removal of regional fields 170 resolution 210 roots 242 sandwich model 115, 124 side lobes contribution from 110 single layer spectrum 114 singular points 20 singular value decomposition 9, 235 singularities gravity field due to a prism 44 south Indian shield 132 space filled with sources 27 spatially varying filters 171 spectral depth 125, 126 spectral representation 112 spectrum 50 definition 78 properties 79 spectrum analysis 10 spectrum estimation 92 Welch method 92 statistical variability 109 stochastic field spectral representation 49
298
Subject index
stochastic model random interface 49 random medium 62 stochastic models 19, 49 stochastic process 6 spectral theory 75 Strakhov's filter 213 suboptimal filter 211 sufficient statistic 258 sum of sinusoids 279 susceptibility maps 228 example 229
time series 279 total derivative 178 transformation of 1D filters 158 transition band 153 two-dimensional (2D) filter 145
terrain correction 200 textural features 253 textural filtering example 250 spectrum 251 texture definition 248 texture analysis 223, 248 structural method 248 texture filter 248 thick layer model 64 thin horizontal layer 62
vertical dyke 33 vertical prism 279 damped sinusoids 279 visualization 11
unbiased 257 underdetermined 245 uniqueness 231 unobservable magnetization 233 example 231 upward continuation 175
white noise process 78 Wiener filter 210 reduction-to-pole 251 separation of fields 216 Wiener filters 210 window function 95, 112
