Geostatistics for Environmental and Geotechnical Applications (Astm Special Technical Publication Stp)

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Geostatistics for Environmental and Geotechnical Applications Shahrokh Rouhani, R. Mohan Srivastava, Alexander J. Desbarats, Marc V. Cromer, and A. Ivan Johnson, editors

ASTM Publication Code Number (PCN): 04-012830-38

ASTM 100 Barr Harbor Drive West Conshohocken, PA 19428-2959 Printed in the U.S.A.

Library of Congress Cataloging-in-Publication Data Geostatistics for environmental and geotechnical applications/ Shahrokh Rouhani ... let al.l. p. cm. - (STP: 1283) Papers presented at the symposium held in Phoenix, Arizona on 26-27 Jan. 1995, sponsored by ASTM Committee on 018 on Soil and Rock. Includes bibliographical references and index. ISBN 0-8031-2414-7 1. Environmental geology-Statistical methods-Congresses. 2. Environmental geotechnology-Statistical methods-Congresses. I. Rouhani, Shahrokh. II. ASTM Committee 0-18 on Soil and Rock. III. Series: ASTM special technical publication: 1283. QE38.G47 1996 96-42381 628.5'01 '5195-dc20 CIP

Copyright © 1996 AMERICAN SOCIETY FOR TESTING AND MATERIALS, West Conshohocken, PA. All rights reserved. This material may not be reproduced or copied, in whole or in part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of the publisher.

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Peer Review Policy Each paper published in this volume was evaluated by three peer reviewers. The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications. To make technical information available as quickly as possible, the peer-reviewed papers in this publication were printed "camera-ready" as submitted by the authors. The quality of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of these peer reviewers. The ASTM Committee on Publications acknowledges with appreciation their dedication and contribution to time and effort on behalf of ASTM.

Printed in Ann Arbor, MI October t 996

Foreword This publication, Geostatistics for Environmental and Geotechnical Applications, contains papers presented at the symposium of the same name held in Phoenix, Arizona on 26-27 Jan. 1995. The symposium was sponsored by ASTM Committee on DIS on Soil and Rock. The symposium co-chairmen were: R. Mohan Srivastava, FSS International; Dr. Shahrokh Rouhani, Georgia Institute of Technology; Marc V. Cromer, Sandia National Laboratories; and A. Ivan Johnson, A. Ivan Johnson, Inc.

Contents OVERVIEW PAPERS Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred-MARc V. CROMER

3

Describing Spatial Variability Using Geostatistical Analysis-R. MOHAN SRI VASTA VA

13

Geostatistical Estimation: Kriging-sHAHROKH ROUHANI

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Modeling Spatial Variability Using Geostatistical Simulation-ALEXANDER J. DESBARATS

32

ENVIRONMENTAL ApPLICATIONS Geostatistical Site Characterization of Hydraulic Head and Uranium Concentration in Groundwater-BRUcE E. BUXTON, DARLENE E. WELLS, AND ALAN D. PATE

51

Integrating Geophysical Data for Mapping the Contamination of Industrial Sites by Polycyclic Aromatic Hydrocarbons: A Geostatistical Approach-PIERRE COLIN, ROLAND FROIDEVAUX, MICHEL GARCIA, AND SERGE NICOLETIS

69

Effective Use of Field Screening Techniques in Environmental Investigations: A Multivariate Geostatistical Approach-MIcHAEL R. WILD AND SHAHROKH ROUHANI

88

A BayesianiGeostatistical Approach to the Design of Adaptive Sampling ProgramsROBERT L. JOHNSON

102

Importance of Stationarity of Geostatistical Assessment of Environmental Contamination-KADRI DAGDELEN AND A. KEITH TURNER

117

Evaluation of a Soil Contaminated Site and Clean-Up Criteria: A Geostatistical Approach-DANIELA LEONE AND NEIL SCHOFIELD

133

Stochastic Simulation of Space-Time Series: Application to a River Water Quality Modelling-AMILcAR o. SOARES, PEDRO J. PATINHA, AND MARIA J. PEREIRA

146

Solid Waste Disposal Site Characterization Using Non-Intrusive Electromagnetic Survey Techniques and Geostatistics-GARY N. KUHN, WAYNE E. WOLDT, DAVID D. JONES, AND DENNIS D. SCHULTE

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GEOTECHNICAL AND EARTH SCIENCES ApPLICATIONS

Enhanced Subsurface Characterization for Prediction of Contaminant Transport Using Co-Kriging---CRAIG H. BENSON AND SALWA M. RASHAD

181

Geostatistical Characterization of Unsaturated Hydraulic Conductivity Using Field Infiltrometer Data-sTANLEY M. MILLER AND ANJA J. KANNENGIESER

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Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities Design at Yucca Mountain, Nevada-MARc V. CROMER, CHRISTOPHER A. 218

RAUTMAN, AND WILLIAM P. ZELINSKI

Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective after the 1994 Northridge, California Earthquake-JAMES R. CARR Spatial Patterns Analysis of Field Measured Soil

236

Nitrate-FARIDA S. GODERY A, M. F.

DAHAB, W. E. WOLDT, AND I. BOGARD!

248

Geostatistical Joint Modeling and Probabilistic Stability Analysis for ExcavationsDAE S. YOUNG

Indexes

262 277

Overview Papers

Marc V. Cromer l

Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred

REFERENCE: Cromer, M. V., "Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred," Geostatistics for Environmental and Geotechnical Applications. ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. J. Desbarats, A. I. Johnson, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Although successfully applied during the past few decades for predIcting the spatial occurrences of properties that are cloaked from direct observation, geostatistical methods remain somewhat of a mystery to practitioners in the environmental and geotechnical fields. The techniques are powerful analytical tools that integrate numerical and statistical methods with scientific intuition and professional judgment to resolve conflicts between conceptual interpretation and direct measurement. This paper examines the practicality of these techniques within the entitles field of study and concludes by introducing a practical case study in which the geostatistical approach is thoroughly executed. KEYWORDS: Geostatistics, environmental investigations, decision analysis tool

INTRODUCTION Although, geostatistics is emerging on environmental and geotechnical fronts as an invaluable tool for characterizing spatial or temporal phenomena, it is still not generally considered "standard practice" in these fields. The technology is borrowed from the mining and petroleum exploration industries, starting with the pioneering work of Danie Krige in the 1950's, and the mathematical formalization by Georges Matheron in the early 1960's. In these industries, it has found acceptance through successful application to cases where decisions concerning high capital costs and operating practices are based on interpretations derived from sparse spatial data. The application of geostatistical methods has since extended to many fields relating to the earth sciences. As many geotechnical and, certainly, environmental studies are faced with identical "high-stakes" decisions, geostatistics appears to be a natural transfer of technology. This paper outlines the unique characteristics of this sophisticated technology and discusses its applicability to geotechnical and environmental studies.

1 Principal Investigator, Sandia National Laboratories/Spectra Research Institute, MS 1324. P.O. Box 5800, Albuquerque, NM 87185-1342

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GEOSTATISTICAL APPLICATIONS

IT'S GEOSTATISTICS The field of statistics is generally devoted to the analysis and interpretation of uncertainty caused by limited sampling of a property under study. Geostatistical approaches deviate from more "classical" methods in statistical data analyses in that they are not wholly tied to a population distribution model that assumes samples to be normally distributed and uncorrelated. Most earth science data sets, in fact, do not satisfy these assumptions as they often tend to have highly skewed distributions and spatially correlated samples. Whereas classical statistical approaches are concerned with only examining the statistical distribution of sample data, geostatistics incorporates the interpretations of both the statistical distribution of data and the spatial relationships (correlation) between the sample data. Because of these differences, environmental and geotechnical problems are more effectively addressed using geostatistical methods when interpretation derived from the spatial distribution of data have impact on decision making risk. Geostatistical methods provide the tools to capture, through rigorous examination, the descriptive information on a phenomenon from sparse, often biased, and often expensive sample data. The continued examination and quantitative rigor of the procedure provide a vehicle for integrating qualitative and quantitative understanding by allowing the data to "speak for themselves." In effect, the process produces the most plausible interpretation by continued examination of the data in response to conflicting interpretations.

A GOAL-ORIENTED, PROJECT COORDINATION TOOL The application of geostatistics to large geotechnical or environmental problems has also proven to be a powerful integration tool, allowing coordination of activities from the acquisition offield data to design analysis (Ryti, 1993; Rautman and Cromer, 1994; Wild and Rouhani, 1995). Geostatistical methods encourage a clear statement of objectives to be set prior to any study. With these study objectives defined, the flow of information, the appropriate use of interpretations and assumptions, and the customer/supplier feedback channels are defined. This type of coordination provides a desirable level of tractability that is often not realized. With environmental restoration projects, the information collected during the remedial investigation is the sole basis for evaluating the applicability of various remedial strategies, yet this information is often incomplete. Incomplete information translates to uncertainty in bounding the problem and increases the risk of regulatory failure. While this type of uncertainty can often be reduced with additional sampling, these benefits must be balanced with increasing costs of characterization. The probabilistic roots deeply entrenched into geostatistical theory offer a means to quantify this uncertainty, while leveraging existing data in support of sampling optimization and riskbased decision analyses. For example, a geostatistically-based, costlrisklbenefit approach to sample optimization has been shown to provide a framework for examining the many tradeoffs encountered when juggling the risks associated with remedial investigation, remedial

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design, and regulatory compliance (Rautman et. aI., 1994). An approach such as this explicitly recognizes the value of information provided by the remedial investigation, in that additional measurements are only valuable to the extent that the information they provide reduces total cost. GEOSTATISTICAL PREDICTION

The ultimate goal of geostatistical examination and interpretation, in the context of risk assessment, is to provide a prediction of the probable or possible spatial distribution of the property under study. This prediction most commonly takes the form of a map or series of maps showing the magnitude and/or distribution of the property within the study. There are two basic forms of geostatistical prediction, estimation and simulation. In estimation, a single, statistically "best" estimate of the spatial occurrence of the property is produced based on the sample data and on the model determined to most accurately represent the spatial correlation of the sample data. This single estimate (map) is produced by the geostatistical technique commonly referred to as kriging. With simulation, many equally-likely, high-resolution images of the property distribution can be produced using the same model of spatial correlation as developed for kriging. The images have a realistic texture that mimics an exhaustive characterization, while maintaining the overall statistical character of the sample data. Differences between the many alternative images (models) provides a measure of joint spatial uncertainty that allows one to resolve risk-based questions ... an option not available with estimation. Like estimation, simulation can be accomplished using a variety of techniques and the development of alternative simulation methods is currently an area of active research. NOT A BLACK BOX

Despite successful application during the past few decades, geostatistical methods remain somewhat of a mystery to practitioners in the geotechnical and environmental fields. The theoretical complexity and effort required to produce the intermediate analysis tools needed to complete a geostatistical study has often deterred the novice from this approach. Unfortunately, to many earth scientists, geostatistics is considered to be a "black box." Although this is far from the truth, such perceptions are often the Achilles' heel of many mathematical/numeric analytical procedures that harness data to yield their true worth because they require a commitment in time and training from the practitioner to develop some baseline proficiency. Geostatistics is not a solution, only a tool. It cannot produce good results from bad data, but it will allow one to maximize that information. Geostatistics cannot replace common sense, good judgment, or professional insight, in fact it demands these skills to be brought to bare. The procedures often take one down a blind alley, only to cause a redirection to be made because of an earlier miss-interpretation. While these exercises are nothing more than cycling through the scientific method, they are often more than the novice is willing to commit to. The time and frustration associated with continually rubbing one's nose in the

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GEOSTATISTICAL APPLICATIONS

details of data must also take into account the risks to the decision maker. Given the tremendous level of financial resources being committed to field investigation, data collection, and information management to provide decision making power, it appears that such exercises are warranted. CASE STUDY INTRODUCTION This introductory paper only attempts to provide a gross overview of geostatistical concepts with some hints to practical application for these tools within the entitled fields of scientific study. Although geostatistics has been practiced for several decades, it has also evolved both practically and theoretically with the advent offaster, more powerful computers. During this time a number of practical methods and various algorithms have been developed and tested, many of which still have merit and are practiced, but many have been left behind in favor of promising research developments. Some of the concepts that I have touched upon will come to better light in the context of the practical examination addressed in the following suite of three overview papers provided by Srivastava (1996), Rouhani (1996), and Desbarats (1996). In this case study, a hypothetical database has been developed that represents sampling of two contaminants of concern: lead and arsenic. Both contaminants have been exhaustively characterized as a baseline for comparison as shown in Figures 1 and 2. The example scenario proposes a remedial action threshold (performance measure) of 500 ppm for lead and 30 ppm for arsenic for the particular remediation unit or "VSR" (as discussed by Desbarats, 1996). Examination of the exhaustive sample histogram and univariate statistics in Figures 1 and 2 indicate about one fifth of the area is contaminated with lead, and one quarter is contaminated with arsenic. The two exhaustive databases have been sampled in two phases, the first of which was on a pseudo-regular grid (square symbols in Figure 3) at roughly a separation distance of 50 m. In this first phase, only lead was analyzed. In the second sampling phase, each first-phase sample location determined to have a lead concentration exceeding the threshold was targeted with eight additional samples (circle symbols of Figure 3) to delineate the direction of propagation of the contaminant. To mimic a problem often encountered in an actual field investigation, during the second phase of sampling arsenic contamination was detected and subsequently included in the characterization process. Arsenic concentrations are posted in Figure 4 with accompanying sample statistics. The second phase samples, therefore, all have recorded values for both arsenic and lead. Correlation between lead and arsenic is explored by examining the co-located exhaustive data which are plotted in Figure 5. This comparison indicates moderately good correlation between the two constituents with a correlation coefficient of 0.66, as compared to the slightly higher correlation coefficient of 0.70 derived from the co-located sample data plotted in Figure 6. There are a total of 77 samples from the first phase of sampling and 13 5 from the second phase. The second sampling phase, though, has been biased because of its focus on "hot-

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GEOSTATISTICAL APPLICATIONS

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GEOSTATISTICAL APPLICATIONS

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GEOSTATISTICAL APPLICATIONS

spot" delineation. This poses some difficult questions/problems from the perspective of spatial data analysis: What data are truly representative of the entire site and should be used for variography or for developing distributional models? What data are redundant or create bias? Have we characterized arsenic contamination adequately? These questions are frequently encountered, especially in the initial phases of a project that has not exercised careful pre-planning. The co-located undersampling of arsenic presents an interesting twist to a hypothetical, yet realistic, problem from we can explore the paths traveled by the geostatistician.

REFERENCES Desbarats, A.J., "Modeling Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Rautman, C.A., M.A. McGraw, J.D. Istok, 1M. Sigda, and P.G. Kaplan, "Probabilistic Comparison of Alternative Characterization Technologies at the Fernald Uranium-InSoils Integrated Demonstration Project", Vol. 3, Technolo~y and Pro~rams for Radioactive Waste Mana~ement and Environmental Restoration, proceedings of the Symposium on Waste Management, Tucson, AZ, 1994. Rautman, C.A. and M.V. Cromer, 1994, "Three-Dimensional Rock Characteristics Models Study Plan: Yucca Mountain Site Characterization Plan SP 8.3.1.4.3.2", U.S. Department of Energy, Office of Civilian Radioactive Waste Management, Washington, DC. Rouhani, S., "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Ryti, R., "Superfund Soil Cleanup: Developing the Piazza Road Remedial Design," .Im!1:nill. Air and Waste Mana~ement, Vol. 43, February 1993. Srivastava, R.M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Wild, M. and S. Rouhani, "Taking a Statistical Approach: Geostatistics Brings Logic to Environmental Sampling and Analysis," Pollution En~ineerin~, February 1995.

R. Mohan Srivastava! DESCRIBING SPATIAL VARIABILITY USING GEOSTATISTICAL ANALYSIS

REFERENCE: Srivastava, R. M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. J. Desbarats, A. I. Johnson, Eds., American Society for Testing and Materials, 1996. ABSTRACT: The description, analysis and interpretation of spatial variability is one of the cornerstones of a geostatistical study. When analyzed and interpreted propp-r'j', the pattern of spatial variability can be used to plan further sampling programs, to improve estimates and to build geologically realistic models of rock, soil and fluid properties. This paper discusses the tools that geostatisticians use to study spatial variability. It focuses on two of the most common measures of spatial variability, the variogram and the correlogram, describes their appropriate uses, their strengths and their weaknesses. The interpretation and modelling of experimental measures of spatial variability are discussed and demonstrated with examples based on a hypothetical data set consisting of lead and arsenic measurements collected from a contaminated soil site.

KEYWORDS: Spatial variation, variogram, correlogram.

INTRODUCTION

Unlike most classical statistical studies, in which samples are commonly assumed to be statistically independent, environmental and geotechnical studies involve data that are not statistically independent. Whether we are studying contaminant concentrations in soil, rock and fluid properties in an aquifer, or the physical and mechanical properties of soil, data values from locations that are close together tend to be more similar than data values from locations that are far apart. To most geologists, the fact that closely lManager, FSS Canada Consultants, 800 Millbank, Vancouver, 13

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GEOSTATISTICAL APPLICATIONS

spaced samples tend to be similar is hardly surprising since samples from closely spaced locations have been influenced by similar physical and chemical processes. This overview paper addresses the description and analysis of spatial dependence in geostatistical studies, the interpretation of the results and the development of a mathematical model that can be used in spatial estimation and simulation. More specific guidance on the details of analysis, interpretation and modelling of spatial variation can be found in the ASTM draft standard guide entitled Standard Guide for Analysis of Spatial Variation in Geostatistical Site Investigations. DESCRIBING AND ANALYZING SPATIAL VARIATION

Using the sample data set presented earlier in this volume in the paper by Cromer, Figure 1 shows an example of a "variogam", the tool that is most commonly used in geostatistical studies to describe spatial variation. A variogram is a plot of the average squared differences between data values as a function of separation distance. If the phenomenon being studied was very continuous over short distances, then the differences between closely spaced data values would be small, and would increase gradually as we compared pairs of data further and further apart. On the other hand, if the phenomenon was completely erratic, then pairs of closely spaced data values might be as wildly different as pairs of widely spaced data values. By plotting the average squared differences between data values (the squaring just makes everything positive so that large negative differences do not cancel out large positive ones) against the separation distance, we can study the general pattern of spatial variability in a spatial phenomenon. Figure 2 shows an example of another tool that can be used to describe spatial variation, the "correlogram" or "correlation function". On this type of plot, we again group all of the available data into different classes according to their separation distance, but rather than plotting the average squared difference between the paired data values, we plot their correlation coefficient. If the phenomenon under study was very continuous over short distances, then closely spaced data values would correlate very well, and would gradually decrease as we compared pairs of data further and further apart. On the other hand, if the phenomenon was completely erratic, then pairs of closely spaced data values might be as uncorrelated as pairs of widely spaced data values. A plot of the correlation coefficient between pairs of data values as a function of the separation distance provides a description of the general pattern of spatial continuity.

SRIVASTAVA ON SPATIAL VARIABILITY

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Figure 2. An example of a correlogram using the sample lead data set described by Cromer (1996).

As can be seen by the examples in Figures 1 and 2, the variogram and the correlogram are, in an approximate sense, mirror images. As the variogram gradually rises and reaches a plateau, the correlogram gradually drops and also reaches a plateau. They are not exactly mirror images of one another, however, and a geostatistical study of spatial continuity often involves both types of plots. There are other tools that geostatistician use to describe spatial continuity, but they all fall into two broad categories: measures of dissimilarity and measures of similarity. The measures of similarity record how dif. ferent the data values are as a function of separation distance and tend to rise like the variogram. The measures of dissimilarity record how similar the data values are as a function of separation distance and tend to fall like the correlogram. INTERPRETING SPATIAL VARIATION

Variograms are often summarized by the three characteristics shown in Figure 3:

Sill: The plateau that the variogram reaches; for the traditional definition of the variogram - the average squared difference between paired data values - the sill is approximately equal to twice the variance of the data. 3 3The "semivariogram", which is simply the variogram divided by two, has a sill that is approximately equal to the variance of the data.

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GEOSTATISTICAL APPLICATIONS

Range : The distance at which the variogram reaches the sill; this is often thought of as the "range of influence" or the "range of correlation" of data values. Up to the range, a sample will have some correlation with the unsampled values nearby. Beyond the range, a sample is no longer correlated with other values. Nugget Effect: The vertical height of the discontinuity at the origin. For a separation distance of zero (i.e. samples that are at exactly the same location), the average squared differences are zero. In practice, however, the variogram does not converge to zero as the separation distance gets smaller. The nugget effect is a combination of:

• short-scale variations that occur at a scale smaller than the closest sample spacing • sampling error due to the way that samples are collected, prepared and analyzed

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Figure 3. Terminology commonly used to describe the main features of a variogram. Of the three characteristics commonly used to summarize the variogram, it is the range and the nugget effect that are most directly linked to our intuitive sense of whether the phenomenon under study is "continuous" or "erratic". Phenomena whose variograms have a long range of correlation and a low nugget effect are those that we think of as "well behaved" or "spatially continuous"; attributes such as hydrostatic head, thickness of a soil layer and topographic elevation typically have long ranges and low nugget effects. Phenomena whose variograms have a short range of correlation and a high nugget

17

SRIVASTAVA ON SPATIAL VARIABILITY

effect are those that we think of as "spatially erratic" or "discontinuous" j contaminant concentrations and permeability typically have short ranges and high nugget effects. Figure 4 compares the lead and arsenic variograms for the data set presented earlier in this volume by Cromer. For these two attributes, the higher nugget effect and shorter range on the arsenic variogram could be used as quantitative support for the view that the lead concentrations are somewhat more continuous than the arsenic concentrations. (b) Arsenic

(a) Lead

1600 1400 E 1200 ~ 1000

60000 50000 E 40000 ~

.2 30000

·2 CG

~ 20000

>

10000

o

o

20

40

60

80

soo

600 400 200 0

100 120

0

Separation distance (in m)

20

40

60

80

100 120

Separation distance (in m)

Figure 4. Lead and arsenic variograms for the sample data described by Cromer (1996).

(b) Northeast - Southwest

(a) Northwest - Southeast 60000

60000

50000

50000

~

~ 40000

40000

g, 30000 .c:

g, 30000 .c:

~ 20000

~ 20000

10000

10000

o

o

o 20

40

60

80

100 120

Separation distance (in m)

o

20

40

60

80

100 120

Separation distance (in m)

Figure 5. Directional variograms for the sample lead data described by Cromer (1996). In many earth science data sets, the pattern of spatial variation is directionally dependent . In terms of the variogram, the range of correlation often depends on direction.

18

GEOSTATISTICAL APPLICATIONS

Using the example presented earlier in this volume by Cromer, the lead values appear to be more continuous in the NW-SE direction than in the NE-SW direction. Geostatistical studies typically involve the calculation of separate variograms and correlograms for different directions. Figure 5 shows directional variograms for the sample lead data presented by Cromer. The range of correlation shown by the NW-SE variogram (Figure 5a) is roughly 80 meters , but only 35 meters on the NE-SW variogram (Figure 5b). This longer range on the NW-SE variogram provides quantitative support for the observation that the lead values are, indeed, more continuous in this direction and more erratic in the perpendicular direction. MODELLING SPATIAL VARIATION

Once the pattern of spatial variation has been described using directional variograrns or correlograms, this information can be used to geostatistical estimation or simulation procedures. Unfortunately, variograms and correlograms based on sample data cannot provide information on the degree of spatial continuity for every possible distance and in every possible direction. The directional variograms shown in Figure 5, for example, provided information on the spatial continuity every 10 m in two specific directions. The estimation and simulation algorithms used by geostatisticians require information on the degree of spatial continuity for every possible distance and direction. To create a model of spatial variation that can be used for estimation and simulation, it is necessary to fit a mathematical curve to the sample variograms.

(a) Northwest - Southeast 60000

60000

50000

50000

~ 40000

~ ·c

(b) Northeast - Southwest

~ 40000

0-

~

30000

30000

~ 20000

·c ~ 20000

10000

10000

o

o

o 20

40

60

80

100 120

Separation distance (in m)

o

20

40

60

80

100 120

Separation distance (in m)

Figure 6. Variogram models for the directional sample variograms shown in Figure 5.

SRIVASTAVA ON SPATIAL VARIABILITY

19

The traditional practice of variogram modelling makes use of a handful of mathematical functions whose shapes approximate the general character of most sample variograms. The basic functions - the "spherical", "exponential" and "gaussian" variogram models - can be combined to capture the important details of almost any sample variogram. Figure 6 shows variogram models for the directional variograms of lead (Figure 5). Both of these use a combination of two spherical variogram models, one to capture short range behavior and the other to capture longer range behavior, along with a small nugget effect to model the essential details of the sample variograms. In kriging algorithms such as those described later in this volume by Rouhani, it is these mathematical models of the spatial variation that are used to calculate the variogram value between any pair of samples, and between any sample and the location being estimated.

REFERENCES ASTM, Standard Guide for Analysis of Spatial Variation in Geostatistical Site Investigations, 1996, Draft standard from D18.01.07 Section on Geostatistics. Cromer, M.V., 1996, "Geostatistics for Environmental and Geotechnical Applications: A Technology Transfer," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Ed., American Society for Testing and Materials, West Conshohocken, PA. Deutsch, C.V. and Journel, A.G., 1992, GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York, 340 p. Isaaks, E.H. and Srivastava, R.M., 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 p. Journel, A.G. and Huijbregts, C., 1978, Mining Geostatistics, Academic Press, London, 600p. Rouhani, S., 1996, "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP ma, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Ed., American Society for Testing and Materials, West Conshohocken, PA. Srivastava, R.M. and Parker, H.M., 1988, "Robust measures of spatial continuity," Geostatistics, M. Armstrong (ed.), Reidel, Dordrecht, p. 295-308.

Shahrokh Rouhani 1 GEOSTATISTICAL ESTIMATION: KRIGING

REFERENCE: Rouhani, S., "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander 1. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Geostatistics offers a variety of spatial estimation procedures which are known as

kriging. These techniques are commonly used for interpolation of point values at unsampled locations and estimation of average block values. Kriging techniques provide a measure of accuracy in the form of an estimation variance. These estimates are dependent on the model of spatial variability and the relative geometry of measured and estimated locations. Ordinary kriging is a linear minimum-variance interpolator that assumes a constant, but unknown global mean. Other forms of linear kriging includes simple and universal kriging, as well as co-kriging. If measured data display non-Gaussian tendencies, more accurate interpolation may be obtained through non-linear kriging techniques, such as lognormal and indicator kriging. KEYWORDS: Geostatistics, kriging, spatial variability, mapping, environmental investigations. Many environmental and geotechnical investigations are driven by biased or preferential sampling plans. Such plans usually generate correlated, and often clustered, data. Geostatistical procedures recognize these difficulties and provide tools for various forms of spatial estimations. These techniques are COllectively known as kriging in honor of D. G. Krige, a South African mining engineer who pioneered the use of weighted moving averages in the assessment of ore bodies. Common applications of kriging in environmental and geotechnical engineering include: delineation of contaminated media, estimation of average concentrations over exposure domains, as well as mapping of soil parameters and piezometric surfaces (Joumel and Huijbregts, 1978; Delhomme, 1978; ASCE, 1990). The present STP offers a number of papers that cover various forms of geostatistical estimations, such Benson and Rashad (1996), Buxton (1996), Goderya et at. (1996), and Wild and Rouhani (1996). Comparison of kriging to other commonly used interpolation techniques, such as distanceweighting functions, reveals a number of advantages (Rouhani, 1986). Kriging directly incorporates the model of the spatial variability of data. This allows kriging to produce sitespecific and variable-specific interpolation schemes. Estimation criteria of kriging are based on IAssociate Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355.

20

ROUHANI ON KRIGING

21

well-defmed statistical conditions, and thus, are superior to subjective interpolation techniques. Furthennore, the automatic declustering of data by kriging makes it a suitable technique to process typical environmental and geotechnical measurements. Kriging also yields a measure for the accuracy of its interpolated values in the fonn of estimation variances. These variances have been used in the design of sampling plans because of two factors: (1) each estimate comes with an estimation variance, and (2) the estimation variance does not depend on the individual observations (Loaiciga et al., 1992). Therefore, the impact of a new sampling location can be evaluated before any new measurements are actually conducted (Rouhani, 1985). Rouhani and Hall (1988), however, noted that in most field cases the use of estimation variance, alone, is not sufficient to expand a sampling plan. Such plans usually require consideration of many factors in addition to the estimation variance. To use the estimation variance as a basis for sampling design, additional assumptions must be made about the probability density function of the estimation error. A common practice is to assume that, at any location in the sampling area, the errors are normally distributed with a mean of zero and a standard deviation equal to the square root of the estimation variance, referred to as kriging standard deviation. The nonnal distribution of the errors has been supported by practical evidence (Journal and Huijbregts, 1978, p. 50 and 60).

Ordinary Kriging Among geostatistical estimation methods, ordinary kriging is the most widely used in practice. This procedure produces minimum-variance estimates by taking into account: (1) the distance vector between the estimated point and the data points; (2) the distance vector between data points themselves; and (3) the statistical structure of the variable. This structure is represented by either the variogram, the covariance or the correlogram function. Ordinary kriging is also capable of processing data averaged over different volumes and sizes. Ordinary kriging is a "linear" estimator. This means that its estimate, Z', is computed as a weighted sum of the nearby measured values, denoted as z!, ~, ... , and Zy,. The fonn of the estimation is n

2:A;Z;

1

;=}

where Ai'S are the estimation weights. Z· can either represent a point or a block-averaged value, as shown in Fig. 1. Point kriging provides the interpolated value at an unsampled location. Block kriging yields an areal or a volumetric average over a given domain. • The kriging weights, Ai' are chosen so as to satisfy two suitable statistical conditions. These conditions are: ! (1) Non-bias condition: This condition requires that the estimator Z· to be free of any ! systematic error, which translates into

I !

i f

~

22

GEOSTATISTICAL APPLICATIONS

•

Zl

•

z· • Z4

.~

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• (a)

(b)

Fig. 1. Example of Spatial Estimation: (a) Point Kriging; (b) Block Kriging.

ROUHANI ON KRIGING

8

~

E

Q, Q,

.5 5os Cl 'g .., ....l

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23

24

GEOSTATISTICAL APPLICATIONS

tAi

=1

2

i==/

(2)

Minimum-variance condition: This requires that the estimator Z' have minimum variance of estimation. The estimation variance of Z', d-, is defmed as where Yio is the variogram between the i-th measured point and the estimated location and n

0.2

=

n

n

iLAiY io - 2:LAiAjYij + Y oo i:}

3

i:/ j : }

Yij is the variogram between the i-th and j-th measured points. The kriging weights are computed by minimizing the estimation variance (Eq. 3) subject to the non-bias condition (Eq. 2). The computed weights are then used to calculate the interpolated value (Eq. 1). As Delhomme (1978) notes: "the kriging weights are tailored to the variability of the phenomenon. With regular variables, kriging gives higher weights to the closest data points, precisely since continuity means that two points close to each other have similar values. When the phenomenon is irregular, this does not hold true and the weights given to the closest data points are dampened." Such flexibility does not exist in methods, such as distance weighting, where the weights are pre-defmed as functions of the distance between the estimated point and the data point.

Case Study: Kriging of Lead Data

As noted in Cromer (1996), a soil lead field is simulated as a case study as shown in Fig. 2. The measured values are collected from this simulated field. Similar to most environmental investigations, the sampling activities are conducted in two phases. During the first phase a pseudo-regular grid of 50x50 m is used for soil sampling. In the second phase, locations with elevated lead concentrations are targeted for additional irregular sampling, as indicated in Fig. 3. The analysis of the spatial variability of the simulated field is presented in the previous paper (Srivastava, 1996). Using this information, ordinary kriging is conducted. Fig. 4 displays the kriging results of point estimations. The comparison of the original simulated field (Fig. 2) and the kriged map (Fig. 4) shows that the kriged map captures the main spatial features of lead contamination. This comparison, however, indicates a degree of smoothing in the kriged map which is a consequence of the interpolation process. In cases where the preservation of the spatial variability of the measured field is critical to the study objectives, then the use of kriging for estimation alone is inappropriate and simulation methods are recommended (Desbarats, 1996). Each kriged map is accompanied by its accuracy map. Fig. 5 displays the kriging

25

ROUHANI ON KRIGING

• • • • • • • • • • • •• • • •• • • • • • • • • • • • • • ffII#fI' • fI' • • • • •••~ C • • • • •• •• • •

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:Il

oC I

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240

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Fig. 5. Kriging Standard Deviation of Soil Lead Concentration in ppm

Gi Z

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28

GEOSTATISTICAL APPLICATIONS

standard deviation map of soil lead data. zones of high versus poor data coverage.

This latter map can be used to distinguish between

Block Kriging In many instances, available measurements represent point or quasi-point values, but the study requires the computation of the areal or volumetric value over a lager domain. For instance, in environmental risk assessments, the desired concentration term should represent the Depending on the computed average average contamination over an exposure domain. concentration or its upper confidence limit, a block is declared impacted or not-impacted. This shows that the decision is based on the estimated block value, and not its true value. So there is a chance of making error in two forms: (1) Wrong Rejection: Certain blocks will be considered impacted, while their true average concentration is below the target level, and (2) Wrong Acceptance: Certain blocks will be considered not-impacted when their true average concentrations are above the target level. As shown in Iournel and Huijbregts (1978, p. 459), the kriging block estimator, Z', is the linear estimator that minimizes the sum of the above two errors. Therefore, the block kriging procedure is preferred to any other linear estimator for such selection problems.

Alternative Fonns of Kriging As noted before, ordinary kriging is a linear minimum-variance estimator. There are other folms of linear kriging. For example, if the global mean of the variable is known, the nonbias condition (Eq. 2) is not required. This leads to simple kriging. If, on the other hand, the global mean is not constant and can be expressed as a polynomial function of spatial coordinates, then universal kriging may be used. In many instances, added information is available whenever more than one variable is sampled, provided that some relationship exists between these variables. Co-kriging uses a linear estimation procedure to estimate Z' as n

"'

;~J

j~J

Z' = LA,;Z; + LWjY j

4

were Zj is the i-th measured value of the "primary" variable with a kriging weight of A;, and Yj is the j-th "auxiliary" measured value with a kriging weight of {OJ. Co-kriging is specially advantageous in cases where the primary measurements are limited and expensive, while

ROUHANI ON KRIGING

29

~auxiliary measurements are available at low cost. Ahmed and de Marsily (1987) enhanced their 1limited transmissivity data based on pumping tests with the more abundant specific capacity data. This resulted in an improved transmissivity map. The present STP provides examples of co~'kriging, such as Benson and Rashad (1996) and Wild and Rouhani (1996).

i

(Non-linea, Kriging

~'

i

The above linear kriging techniques do not require any implicit assumptions about the distribution of the interpolated variable. If the investigated variable is multivariate . :normal (Gaussian), then linear estimates have the minimum variance. In many cases where \:the histogram of the measured values displays a skewed tendency a simple transformation may ~,produce normally distributed values. After such a transformation, linear kriging may be used. If Ithe desired transformation is logarithmic, then the estimation process is referred to as lognormal ""kriging. Although lognormal kriging can be applied to many field cases, its estimation process ':requires back-transformation of the estimated values. These back transformation are complicated ,~and must be performed with caution (e.g. Buxton, 1996). ~t Sometimes, the observed data clearly exhibit non-Gaussian characteristics, whose log~transforms are also non-Gaussian. Examples of such data sets include cases of measurements i'with multi-modal histograms, highly skewed histograms, or data sets with large number of \~'below-detection measurements. These cases have motivated the development of a set of r~techniques to deal with non-Gaussian random functions. One of these methods is indicator ~'kriging. In this procedure, the original values are transformed into indicator values, such that ~l:they are zero if the datum value are less than a pre-defmed cutoff level or unity if greater. The ~.',I,'.,stimated value by indicator kriging represents the probability of not-exceedence at a location. "I::!,his technique provides a simple, yet powerful procedure, for generating probability maps "I~OUhani and Dillon, 1990). ~:"underlying

, I

:~~

:~i

~Recommended Sources

'l

i

~or more information on ~iging, readers are referred to Journel and Huijbregts (1978), Marslly (1986), Isaaks and Snvastava (1989), and ASCE (1990). ASTM Standard D 5549, titled: "Standard Guide for Content of Geostatistical Site Investigations," provides information on the various elements of a kriging report. ASTM DI8.0l.07 on Geostatistics has so drafted a guide titled: "Standard Guide for Selection of Kriging Methods in Geostatistical . ite Investigations." This guide provides recommendations for selecting appropriate kriging methods based on study objectives and common situations encountered in geostatistical site ;investigations .

~de

30

GEOSTATISTICAL APPLICATIONS

References

(1)

(2)

(3)

(4)

(5)

(6) (7)

(8)

(9) (10) (11)

(12)

ASCE Task Committee on Geostatistical Techniques in Geohydrology, "Review of Geostatistics in Geohydrology, 1. Basic Concepts, 2. Applications," ASCE Journal of Hydraulic Engineering, 116(5), 612-658, 1990. Ahmed, S., and G. de Marsily, "Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity," Water Resources Research, 23(9), 1717-1737, 1987. Benson, C.H., and S.M. Rashad, "Using Co-kriging to Enhance Subsurface Characterization for Prediction of Contaminant Transport," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Buxton, B.E., "Two Geostatistical Studies of Environmental Site Assessments," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Cromer, M., "Geostatistics for Environmental and Geotechnical Applications," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Delhomme, J.P., "Kriging in the hydro sciences , " Advances in Water Resources, 1(5), 251-266, 1978. Desbarats, A., "Modeling of Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Goderya, F.S., M.F. Dahab, and W.E. Woldt, "Geostatistical Mapping and Analysis of Spatial Patterns for Farm Fields Measured Residual Soils Nitrates," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Isaaks, E. H. and R M. Srivastava, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 p., 1989. Joumel, A. G. and C. Huijbregts, Mining Geostatistics, Academic Press, London, 600 p.,1978. Loaiciga, H.A., RJ. Charbeneau, L.G. Everett, G.E. Fogg, B.F. Hobbs, and S. Rouhani, "Review of Ground-Water Quality Monitoring Network Design," ASCE Journal of Hydraulic Engineering, 118(1), 11-37, 1992. Marsily, G. de, Quantitative Hydrogeology, Academic Press, Orlando, 1986.

ROUHANI ON KRIGING

(13) {14) (15)

. (16)

(17)

(18)

31

Rouhani, S., "Variance Reduction Analysis", Water Resources Research, Vol. 21, No.6, pp. 837-846, June, 1985. Rouhani, S., "Comparative study of ground water mapping techniques," Journal of ~ Ground Water, 24(2), 207-216, 1986. Rouhani, S., and M. E. Dillon, "Geostatistical Risk Mapping for Regional Water Resources Studies," Use of Computers in Water Management, Vol. 1, pp. 216-228, V/O "Syuzvodproekt", Moscow, USSR, 1989 . Rouhani, S., and Hall, T.J., "Geostatistical Schemes for Groundwater Sampling," Journal of Hydrology, Vol. 103, 85-102, 1988. Srivastava, R. M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Wild, M.R., and S. Rouhani, "Effective Use of Field Screening Techniques in Environmental Investigations: A Multivariate Geostatistical Approach," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996.

'7

Alexander J. Desbarats 1

MODELING SPATIAL VARIABILITY USING GEOSTATISTICAL SIMULATION

REFERENCE: Desbarats, J. A., "Modeling Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. I. Johnson, 1. A. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: This paper, the last in a four part introduction to geostatistics, describes the application of simulation to site investigation problems. Geostatistical simulation is a method for generating digital representations or "maps" of a variable that are consistent with its values at sampled locations and with its in situ spatial variability, as characterized by histogram and variogram models. Continuing the synthetic case study of the three previous papers, the reader is lead through the steps of a gebstatistical simulation. The si;nulated fields are then compared with the exhaustive data sets describing the synthetic site. Finally, it is shown how simulated fields can be used to answer questions concerning alternative site remediation strategies. KEYWORDS : Geostatistics, kriging, simulation, variogram

INTRODUCTION

In a geostatistical site investigation, after we have performed an exploratory analysis of our data and we have modeled its spatial variation structure, the next step is usually to produce a digital image or "map" of the variables of interest from a set of measurements at scattered sample locations. We are then faced with a choice between two possible approaches, estimation and simulation. This choice is largely dictated by study objectives. Detailed guidance for selecting between these two approaches and among the various types of simulation is provided in the draft ASTM Guide for the Selection of Simulation Approaches in Geostatistical Site Investigations. Producing a map from scattered measurements is a classical spatial estimation problem that can be addressed using a non-geostatistical interpolation method such IGeological Survey of Canada, 601 Booth st., Ottawa, ON KIA DES, Canada

32

DESBARATS ON SPATIAL VARIABILITY

33

as inverse-distance weighting or, preferably, using one of the least-squares weighting methods collectively known as kriging discussed in Ro~hani (this volume). Regardless of the interpolation method that is selected, the result is a representation of our variable in which its spatial variability has been smoothed compared to in situ reality. Along with this map of estimated values, we can also produce a map of estimation (or error) variances associated with the estimates at each unsampled location. This map provides a qualitative or, at best, semi-quantitative measure of the degree of uncertainty in our estimates and the corresponding level of smoothing we can expect. Unfortunately, maps of estimated values, even when accompanied by maps of estimation variances, are often an inadequate basis for decision-making in environmental or geotechnical site investigations. This is because they fail to convey a realistic picture of the uncertainty and the true spatial variability of the parameters that affect the planning of remediation strategies or the design of engineered structures. The alternative to estimation is simulation. Geostatistical simulation (Srivastava, 1994) is a Monte-Carlo procedure for generating outcomes of digital maps based on the statistical models chosen to represent the probability distribution function and the spatial variation structure of a regionalized variable. The simulated outcomes can be further constrained to honor observed data values at sampled locations on the map. Therefore, not only does geostatistical simulation allow us to produce a map of our variable that more faithfully reproduces its true spatial variability, but we can generate many equally probable alternative maps, each one consistent with our field observations. A set of such alternative maps allows a more realistic assessment of the uncertainty associated with sampling in heterogeneous geological media. This paper presents an introduction to the geostatistical tool of simulation. Its goals are to provide a basic understanding of the method and to illustrate how it can be used in site investigation problems. To do this, we will continue the synthetic soil contamination case study started in the three previous papers. We will proceed step by step through the simulation study, pausing here and there to compare our results with the underlying reality and the results of the kriging study (Rouhani, this volume). Finally, we will use our simulated fields to answer some questions that can arise in actual soil remediation studies. STUDY OBJECTIVES

The objective of our simulation study is to generate digital images or maps of lead (Pb ) and arsenic (As) concentrations in soil. We will then use these maps to determine the proportion of the site area in which Pb or As concentrations exceed the remediation thresholds of 150 ppm and 30 ppm, respectively. The maps are to reproduce the histograms and variograms of Pb and As in addition to observed measurements at sampled locations. Although the full potential of the simulation method is truly achieved only in sensitivity or risk analysis studies involving multiple outcomes of the simulated maps, we will focus on the generation of a single outcome. In many respects, even a sin~l~Il!~p of simulated concentrations is more useful than a map of kriged values. This is because a realistic portrayal of in situ spatial variability is often a sobering warning to planners whereas maps of kriged values are easily

'I

I

,I

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!

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I"~

iII'!.!; :

i

ii,

i!.

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1.

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34

GEOSTATISTICAL APPLICATIONS

misinterpreted as showing much smoother spatial variations. For our study, we have chosen the concentrations of Pb and As as the two regionalized variables to work with. This may seem like an obvious choice however we could have taken another approach based on an indicator or binary transformation of our original variables. The new indicator variables corresponding to each contaminant would take a value of 1 if the concentration exceeds the remediation threshold and a value of 0 otherwise. Proceeding in a somewhat different manner than shown here, we could then generate maps of simulated indicator variables for the two contaminants. From such maps, the proportion of the site requiring remediation is readily determined. The drawback with an indicator approach is that we have sacrificed detailed knowledge of contaminant concentrations in exchange for simplicity and conciseness. Should the remediation thresholds change, new indicator variables would have to be defined and the study repeated. Here, we will stick with the more involved but also more flexible approach of simulating contaminant concentrations. An application of indicator simulation is described in Cromer et al. (this volume). HISTOGRAM MODELS

The first step in our simulation study is to decide what probability distribution functions or, more prosaically, what histogram models are to be honored by our simulated concentrations. We would like these histograms to be representative of the entire site. Often, the raw histograms of sample data are the most appropriate choice. However, here this isn't the case: The sampling of our contaminated site was carried out in two stages. In the first stage, we obtained 77 measurements of Pb distributed on a fairly regular grid. In the second stage, we focused our sampling on areas identified in the first stage as having high Pb concentrations. Furthermore, by then we had become aware that arsenic contamination was present and we analyzed an additional 135 samples for both Pb and As. Thus, our Pb data consist of 77 values that are probably representative of the entire site area and another 135 values drawn from the most contaminated region. As for arsenic, our 135 samples were obtained exclusively from the most contaminated region and are probably not representative of the entire site. The raw histograms of Pb and As shown in Cromer (this volume) reflect the preferential or biased sampling procedure and do not provide adequate models for our simulation. The answer to this problem is to weight our sample data in such a way as to decrease the influence of clustered measurements while increasing that of more isolated values. In geostatistics, this exercise is known as "declustering" and can be accomplished several ways (Isaaks and Srivastava, 1989; Deutsch and Journel, 1992). Here we used a cell declustering scheme to find sample weights. This involved moving a 10 x 10 unit cell over N non-overlapping positions covering the study area. At a each cell position, the number n of samples within the cell was counted and each sample was then assigned a relative weight of liNn. This procedure may be expected to work well for Pb but for As there is no escaping the fact that our samples are restricted to a few small, highly contaminated patches and are hardly representative of the site as a whole. Obtaining a reasonably representative histogram is crucial for a simulation

35

DESBARATS ON SPATIAL VARIABILITY

Nunom.rofO.ta 212

~mbtrotD. 1~

m.." std. dri. coef. ITI4IlCimum upper (jUn.

300.6708 225.5126 0.7500 1003,0000 454.0000 machn 274.5673 w ... quam. 103.9253 rrinimum 0.0000

me.,

'4.3813 . .. day. 28.4787

0.600

of".

ooef. of " . 1.9803

maxitn.lm upper qu.-tll medi., lower quart. minimum

0.:500

g

0._

1

157.0000 11 .0000 1.9190 0 .0000 0.0000

0 .300

0 .200

0.100

o.

'0.

100.

1.0.

Figure 1: Declustered histograms of a) Pb and b) As . study therefore desperate measures are called for. Although no geostatistical method can truly compensate for lack of data, the following "fix" was attempted here: Using our knowledge of the correlation between Pb and As provided by the 135 samples of the second sampling campaign (Cromer, this volume), we filled in the missing As values at the 77 locations of the first campaign. For each of the 77 Pb values, we looked up the closest Pb value from the second campaign and read off the corresponding As value. Thus, all 212 sample locations have both Pb and As measurements and the same declustering weights can be used for both variables. The resulting histograms of weighted Pb and As samples are shown in Figures 1 a) and b), respectively. They should be compared with the un-declustered histograms shown in Cromer (this volume). We now have histograms that, we think, provide reasonable models of the exhaustive distributions of Pb and As that we are trying to replicate in our simulated fields. A peek at the true exhaustive distributions (Cromer, this volume) shows that our declustered Pb histogram does a fairly good job of reproducing the main statistical parameters whereas our As histogram does a rather mediocre job despite our best efforts. Further discussion of the declustering issue can be found in Rossi and Dresel (this volume). NORMAL-SCORE TRANSFORMATION OF VARIABLES

The next step of our study involves transforming our Pb and As sample values into standard Normal deviates (Deutsch and Journel, 1992). This "normal-score" transformation is required because the simulation algorithm we will be using is based on the multivariate Normal (or Gaussian) distribution model and assumes that all sam-

200.

36

GEOSTATISTICAL APPLICATIONS

pIe data are drawn from such a distribution. In simple terms, this transformation is performed by replacing the value correponding to a given quantile of the original distribution with the value from a standard Normal distibution associated with the same quantile. For example, a Pb value of 261 ppm correponding to a quantile of 0.50 (i.e. the median) in the sample histogram is transformed into a value of 0 corresponding to the median of a standard Normal distribution. In mathematical terms, we seek the transformations ZI and Z2 of Pb and As such that :

G(Zd

(1 )

where G( ) is the cumulative distribution function (cdf) of a standard Normal distribution and FI ( ) and F 2 ( ) are the sample cdfs for lead and arsenic, respectively. Implementation of this transformation is fairly straightforward except when identical sample values are encountered. In such cases, ties are broken by adding a small random perturbation to each sample value and ranking them accordingly (Deutsch and Journel, 1992). Here, this "despiking" procedure was required to deal with a large number of below-detection As values. In general however, it is good practice to avoid extensive recourse to this procedure. If, for example, large numbers of samples have values below detection limits, it is better to subdivide the data set into two populations, above and below detection, and analyze each group seperately, or adopt an indicator approach (Zuber and Kulkarni, this volume). PRINCIPAL COMPONENT TRANSFORMATION

Before we can proceed to the analysis of spatial variation, one last step is required. Our simulation algorithm can only be used to generate fields of one variable at a time. However, we wish to simulate two variables ZI and Z2, reproducing not only their respective spatial variation structures but also the relationship between them shown in Figure 2. We must therefore "decouple" the variables ZI and Z2 so that we can simulate them independently. To do this, we use the following principal component transformation which yields the independent variables Yi and 12 from the correlated variables ZI and Z2 :

(2)

12 = where p is the correlation coefficient between ZI and Z2 which is found to be 0.839.

DESBARATS ON SPATIAL VARIABILITY Pb

II'S As

37

: Declustered Normal Scores Nurrber 01 daIa 135 Nurrbertrlmmed n

3.00

X Varillb..: mean 0.887 otd. daY. 0.783

0 0

.;i-';.

2.00

0

1.00

~ 0.00

Y Varillb..: mean 0.80S Old. dey. 0.806

00

00

0

,

0 0

..

0

0

correlation 0.839 rank correlation 0.869

0

0

0

-1.00 0

0

-2.00

.3.00-h-T""I"''TTTT"'TTTT"......,..,...'''..,..,.....,..,..,..,....,..,..,..,...,.... -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z1

Figure 2: Scatter plot of Zl and Z2 for 135 sample values. The correlation coefficient is 0.839. VARIOGRAM MODELS

In this section, we examine and model the spatial variation structure of the two independent variables Y1 == Zl and l'2. The jargon and the steps involved in an analysis of spatial variation are described in more detail by Srivastava (this volume) so that only a summary of results is given here. Directional variograms, or more specfically correlograms, were calculated for Y1 using all 212 data values, and for 1'2 using the 135 values of the second sampling campaign. For each variable, eight directional correlograms were calculated at azimuth intervals of 22.5° using overlapping angular tolerances of 22.5°. Lag intervals and distance tolerances were 10 grid units and 5 grid units, respectively for Yi and 5 grid units and 2.5 grid units, respectively for 1'2. The purpose of these directional correlograms is to reveal general features of spatial variation such as directional anisotropies and nested structures. Results for Yi. and Y2 are shown in Figures 3 a) and b), respectively. These figures provide a planimetric representation of spatial correlation structure, displaying correlogram values as a surface, function of location in the plane of East~West (x) and North-South (y) lag components. For Yi., we observe, in addition to a significant nugget effect, what we interpret as two nested structures with different principal directions of spatial continuity. The first, shorter scale, structure has a direction of maximum continuity approximately North North-West, a maximum range of about 20 grid units and an anisotropy ratio of about 1.4 : 1. The second, larger scale structure has a direction of maximum continuity approximately West North-West, an indeterminate maximum range and a minimum range of at least 40 grid units.

38

GEOSTATISTICAL APPLICATIONS

--'§.s

40.0

20.0

"d .~

-

20.0

10.0

...:I

0.0

0.0

-20.0

-10.0

~

~

::s0 rn I

:E

Z

-40.0 -40.0 -20.0

0.0

20.0

-20.0 40.0 -20.0 -10.0

East-West Lag (grid units)

0.0

10.0

20.0

East-West Lag (grid units)

Figure 3: Directional correlograms (planimetric view) for a)

Yi and b) Y2.

Detailed experimental correlograms were calculated in the directions of maximum and minimum continuity of the larger scale structure. They are shown in Figures 4 a) and b), respectively. These correlograms were used in the fitting of a model to the spatial variation structure of Yi. The fitted model is the sum of three components: 1. A nugget effect accounting for 35% of the spatial variance.

2. A short-scale structure accounting for 50% of the spatial variance. It is represented by an exponential model with maximum continuity in the North NorthWest direction. It has a range parameter of 6 grid units and an anisotropy ratio of 1.43 : 1. 3. A large-scale structure accounting for 15% of the spatial variance. This structure is also represented by an exponential model with, however, maximum continuity in the West North- West direction. The model range parameter is 300 grid units with an anisotropy ratio of 10 : 1. Such a large maximum range ensures that the "sill" value of the structure is not reached in the direction of maximum continuity within the limits of the site. What we have done here is model a "zonal anisotropy" (Journel and Huijbregts, 1978) as a geometric anisotropy with an arbitrarily large range in the direction of greatest continuity. This model is also ·shown in Figures 4 a) and b) for comparison with the experimental results. For Y2 , there are less data and we are careful not to over-interpret the directional correlograms. Indeed, the apparent periodicity in the NNE direction is probably an artifact of the sampling pattern . With somewhat more confidence, we note a

39

DESBARATS ON SPATIAL VARIABILITY

0.800 Co".log,.m

,*,d:Z1

1I1/:Z1

dlrecHon 1

0.600 Comlog,.m

0.700

0.500

0.600

0.<400

f >

direction 2

0.300

0.500

~

/wd:Zf

II/I/:Z1

~

f

0.«10

>

0.300

0.200 0.100

0.200

0.000

0.100

.Q.loo .Q.2OO

0.000 0.0

10.0

20.0

30.0

40.0

SO.O

60.0

Dstanoe

0.0

10.0

20.0

30.0

40.0

SO.O

I

60.0

Dstanoe

Figure 4: Experimental correlograms for }'J in a) direction WNW; b) direction NNE. The fitted model is shown in dashed lines. strong nugget effect and a structure with maximum continuity in the West NorthWest direction, a maximum range of about 30 grid units and an anisotropy ratio of about 3 : 1. Detailed directional correlograms were calculated for the directions of maximum and minimum continuity and are shown in Figures 5 a) and b), respectively. The model fitted to these correlograms is the sum of two components: 1. A nugget effect accounting for 55% of the spatial variance. 2. A structure accounting for 45% of the spatial variance. This structure is represented by an exponential model with greatest continuity in the West North-West direction . The model has a range parameter of 8 grid units and an anisotropy ratio of 3 : 1. This mQdel is shown in Figures 5 a) and b) for comparison with the experimental results. SEQUENTIAL GAUSSIAN SIMULATION

We are now ready to simulate fields of the two independent standard Normal variables, Y1 and Y2 • The simulations of Y1 and Y2 are to be conditioned on 212 and 135 sample values, respectively. Both fields are simulated on the same 110 x 70 grid as the exhaustive data sets for Pb and As . To perform our simulations, we are going to use the Sequential Gaussian method. This method is based on two important theoretical properties of the multivariate Normal (or Gaussian) distribution: First, the conditional distribution of an unknown

40

GEOSTATISTICAL APPLICATIONS

0.600 Corre/ogram

IaIl:Y2

heIId:Y2

dlreclJon 1

0.600 Cotrelogram

0.500

direction 2

0.400

\

0.300

~

_:Y2

0.500

0.400

! 8'

IIIII:Y2

0.300

" "- "

0.200 0.100

! 8'

~

'"

0.200 0.100

---

0.000

"-

·0.100

-0.100

·0.200

-0.200 0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0

5.0

Distance

Distanct

Figure 5: Experimental correlograms for Y2 in a) direction WNW; b) direction NNE. The fitted model is shown in dashed lines. variable at a particular location, given a set of known values at nearby locations, is Normal. Second, the mean and variance of this conditional distribution are given by the simple kriging (SK) estimate of the unknown value and its associated error variance. Simple kriging is a variant of ordinary kriging (OK) described in Rouhani (this volume). Then, it follows that since the conditional distribution is Normal, it is completely determined by the mean and variance provided by simple kriging. The Sequential Gaussian Simulation algorithm is described in detail elsewhere (Deutsch and Journel, 1992; Srivastava, 1994) ; however, because of its simplicity, it is briefly outlined here: 1. Start with a set of conditioning data values at scattered locations over the field to be simulated. 2. Select at random a point on the grid discretizing the field where there is not yet any simulated or conditioning data value. 3. Using both conditioning data and values already simulated from the surrounding area, calculate the Simple Kriging estimate and corresponding error variance. These are the mean and variance of the conditional distribution of the unknown value at the point given the set of known values from the surrounding area. 4. Select at random a value from this conditional distribution. 5. Add this value to the set of already simulated values.

DESBARATS ON SPATIAL VARIABILITY

41

6. Return to step 2 and repeat these steps recursively until all points of the discretized field have been assigned simulated values. Thus, in many ways, the Sequential Gaussian simulation method is similar to the point kriging process described by Rouhani (this volume). The difference is that we are drawing our simulated value at random from a distribution having the kriged estimate as its mean, rather than using the kriged estimate itself as a "simulated" value. Intuitively, we see how this process leads to fields having greater spatial variability than fields of kriged values. BACK-TRANSFORMATIONS

We now have simulated fields of the two independent standard Normal variables Y1 and )12. In order to obtain the corresponding fields of Pb and As , we must reverse the earlier transformations. First, we reverse equations (2) to get the correlated standard Normal variables Zl and Z2 from Yi and Y2. Then we reverse equations (1) to get the variables Pb and As from the standard Normal deviates Zl and Z2. Finally, we are left with simulated fields of Pb and As on a dense 110 x 70 grid discretizing the site. Although here we are focusing on single realizations of each of these fields, multiple realizations can be generated by repeating the simulation step using different seed values for the random number generator. COMPARISON OF TRUE AND SIMULATED FIELDS

In addition to honoring values of Pb and As at sampled locations, the simulated fields should reproduce the histogram and correlogram models that we used to characterize contaminant spatial variability. These fields should also reproduce the correlation between Pb and As . Because this is a synthetic case study, we have exhaustive knowledge of Pb and As contamination levels over the entire site, something we would never have in practice. Therefore, we can conduct a postmortem of our study, comparing our simulated fields with the exhaustive fields described in Cromer (this volume). In order to compare the distributions of true and simulated values, we will use what is known as a Q-Q plot. This involves plotting the quantile of one data set against the same quantile of the other data set. Thus, we would plot the median (0.5 quantile) of our simulated values against the median of our true values. Therefore, if the histograms of the two data sets are similar, all points should plot close to the 45° line. Figures 6 a) and b) show Q-Q plots between exhaustive and simulated values of Pb and As , respectively. These results show that while we did a rather good job of reproducing the exhaustive histogram of Pb , we can claim no great success for As . Although we did our best to correct for the effects of a grossly unrepresentative sampling of As , in the end, this was not good enough. This failure serves as a reminder that geostatistics alone cannot compensate for a biased site sampling campaign. Next, we check how well our simulation reproduced the correlation between Pb and As concentrations. Figure 7 shows a scatter plot of simulated Pb versus As values. This figure is to be compared with the scatter plot of true values given by

42

GEOSTATISTICAL APPLICATIONS 200. Q-Q Plot for A. Slmul.tlon

1000. Q-Q Plot for Pb Slmul.tlon

160.

800.

]

600.

."

i

~ r/)

I

120.

1~

80.

j

~

400.

r/)

40.

200.

o.

200.

400.

600.

True valu••

800.

1000.

o.

40.

80.

120.

160.

200.

True vw..

Figure 6: Q-Q quantile plots of exhaustive and simulated data: a) Pb ; b) As Cromer (this volume). The comparison shows that we did quite a respectable job of reproducing the relationship between Pb and As in our simulated fields. Directional correlograms calculated on our two simulated fields are shown in Figures 8 a) and b). The main features of these correlograms compare favorably with those observed in the correlograms presented by Srivastava (this volume). Given the limited number of data and their spatial clustering, the models we fitted to the experimental correlograms were quite successful in representing the true spatial variation structures of Pb and As . No comparison of true and simulated fields would be complete without looking at images or maps of the simulated fields. Although qualitative, the visual comparison of simulated and true fields is in fact the most stringent measure of the success of our simulation. We must check how well we have captured the character of contaminant spatial variability at the site, its "noisyness", the grain of any spatial patterns, and any trends. We should also check to see what our simulated values are doing in areas far from conditioning data values. The spatial variability in such areas should be consistent in character with that observed in more densely sampled areas.'" Grey-scale digital images of simulated Pb and As fields are shown in Figures 9 and 10, respectively. Comparison with the corresponding true images in Cromer (this volume) shows that we have reason to be satisfied with our simulation. Discrepancies between simulated and true fields exist; however, these are manifestations of the uncertainty associated with our knowledge of site contamination as provided by the rather limited sampling data. It should be emphasized that we are looking at but one pair of images of contamination from amongst the many equally possible alternatives that would be consistent with sampling information. We can also compare Figure 9

DESBARATS ON SPATIAL VARIABILITY

43

vs As : Simulated values Number of data 7700 X Variable: mean 296.479 std. dey. 225.313

160.

Y Variable: mean 15.619 std. dey. 30.076 correlation 0.722 rank correlation 0.881

o.

200.

600.

400.

800.

1000.

Pb

Figure 7: Scatter plot of simulated Ph and As values

40.0 - -- - - "u-.::----,'-J,..,----.;:,-----, 20.0 t-=l

~

0.0

as~

-20.0

~ Z

20.0

-20.0 0

-40. 0 ~----r--"'r----Y'-----,""::"-fL-.----Y----, -40. 0 ~~--f.:""""';~,uLr--.-'-~..c:; -40.0 -20.0 0.0 20.0 40.0 -40.0 -20.0 0.0 20.0 40.0

East-West Lag (grid units)

East-West Lag (grid units)

Figure 8: Directional correlograms (planimetric view ) for simulated a) Ph and h) As .

44

GEOSTATISTICAL APPLICATIONS

0.0 .-.

.....s

§

~ 0 0 00

20.0

'1:l .~

.-.

S p.

'-'

til)

.S

40.0

p.

'-'

~ 0 0

~

Z

S!

~

60.0 0.0

20.0

40.0

60.0

80.0

100.0

Easting (grid units)

0

0

Figure 9: Grey-scale digital image of the simulated Pb field. with the kriged field shown in Rouhani (this volume) . We see that kriging smoothes spatial variations in a non-uniform manner: less in regions with abundant sample control, more in unsampled regions . This may lead the unsuspecting to conclude that large portions of the site are quite homogeneous! Simulation, on the other hand, preserves in-situ spatial variability regardless of the proximity of sampling points. APPLICATION

Now that we have simulated fields of Pb and As that we confidently assume are representative of the true yet unknown contamination at the site, we can use these fields to answer some simple questions . Perhaps the most basic question that we may ask is what fraction of the site requires remediation given the contamination thresholds of 150 ppm and 30 ppm for Pb and As , respectively? However, before we can attempt to answer that question, we must decide on a "volume of selective remediation" or VSR. Note that the concept of volume of selective remediation is identical to that of selective mining unit (smu) described in the mining geostatistics literature (chapter 6 of Journel and Huijbregts, 1978; chapter 19 of Isaaks and Srivastava, 1989). The volume, or in the present case, area of selective remediation is the smallest portion of soil that can be either left in place or removed for treatment, based upon its average contaminant concentration. The VSR may depend on several factors including the size of equipment being used in the remediation and the sampling information ultimately available for the selection process. It is an important· design parameter because the variance of spatially averaged concentrations decreases as the

DESBARATS ON SPATIAL VARIABILITY

45

0

0

0.0

to

,-..

~ .....

§

20.0 0

'0 .~

0

"
bO

.S

40.0

p.

'-"

1!

Z

,-..

S p.

'-"

~ 0

<

C"I

60.0 0.0

20.0

40.0

60.0

80.0

100.0

Easting (grid units)

0

0

Figure 10: Grey-scale digital image of the simulated As field. VSR becomes larger. This reduces the spread and alters the shape of the histogram of contaminant concentrations thereby affecting the proportion of values above a given threshold and the fraction of the site requiring remediation. Here, the original sample size or "support", as it is known in geostatistics, is a square of 1 x 1 grid units (5m x 5m). The corresponding standard deviations of Pb and As concentrations are 218 ppm and 35 ppm, respectively. If we were to consider a VSR with a support of 10 x 10 grid units (50m x 50m), the standard deviations of VSR-averaged Pb and As concentrations are reduced to 172 ppm and 18 ppm, respectively. In Tables 1 and 2, we compare fractions of the site requiring remediation for VSRs of 1 x 1 grid units and 10 X 10 grid units, respectively. Within each table, we also compare remediation fractions based on kriged, simulated and true values. For selection based on Pb concentration alone, results for both VSR sizes show good agreement between remediated fractions calculated on simulated and true fields. Remediated fractions based on kriged fields are overestimated for the smaller VSR. We note that the fraction of the site requiring remediation increases for the larger VSR. This is because the spatial averaging of Pb concentrations over a VSR smears high values over the entire block area thereby pushing its average over the remediation threshold. The same phenomenon may also happen in reverse, with low values diluting a few high values and thus lowering the average VSR concentration below threshold. In either case, it is obvious that the choice of VSR will have a significant impact on the fraction of the site requiring remediation. For selection based on As values alone, remediated fractions calculated on the simulated fields are almost half those calculated on the true fields. On the other hand, remediated fractions based on the kriged fields are much larger than those

46

GEOSTATISTICAL APPLICATIONS

Table 1: Fraction of site requiring remediation based on a VSR of 1 x 1 grid units. The Pb threshold is 150 ppm and the As threshold is 30 ppm.

Field

Kriged Simulated True

Pb cutoff

As cutoff

Combined cutoff

0.7956 0.6793 0.6998

0.4327 0.1692 0.2474

0.8360 0.6796 0.7026

Table 2: Fraction of site requiring remediation based on a VSR of 10 x 10 grid units. The Pb threshold is 150 ppm and the As threshold is 30 ppm.

Field

Kriged Simulated True

I'll

Pb cutoff

As cutoff

Combined cutoff

0.7975 0.7922 0.7792

0.4248 0.1688 0.3116

0.8011 0.7922 0.7792

based on the true fields. The cause of the poor simulation results can be traced back to our difficulties in obtaining a representative histogram for As concentrations. The poor kriging results are due to smearing of As values from the densely sampled highly contaminated zone to the surrounding area. Considering only the results for the true field, we see an increase in remediated fraction with the larger VSR size, as we saw previously with Pb. For selection based on either Pb or As threshold exceedance, results are similar to those for Pb selection alone : VSRs that would otherwise be misclassified based on their As value are correctly classified based on their Pb value. Although the simulated Pb field gave remediation fractions close to those obtained for the true field, this may be partly fortituous and, in any case, does not ensure that the blocks selected for remediation are the correct ones i.e. the same as in the true field. In practice, multiple simulations should be performed and, for each VSR within the site, a contamination threshold exceedance probability should be calculated from the resulting distribution of simulated concentrations for that location. The decision on whether or not to renlediate a given VSR would then be based on its threshold exceedance probability and not on a single concentration value.

DESBARATS ON SPATIAL VARIABILITY

47

CONCLUSIONS

In this paper, we have described the steps involved in a geostatistical simulation study, going from a small and possibly biased sample data set to a detailed numerical representation of contamination levels at a hypothetical site. We have shown that geostatistical simulation is a tool for producing maps of a variable that honor data values at sampled locations and models of the histogram and spatial variation structure that characterize the phenomenon. We have shown that maps of a variable produced by simulation are, in general, more useful than maps produced by kriging or other spatial interpolation methods because they provide a more faithful representation of in situ variability. We have shown that geostatistical theory and conditional simulation provide a powerful means of studying alternative remediation strategies based on the concept of volume of selected remediation. We have shown, perhaps unintentionally, that geostatistical methods are not a panacea. They cannot compensate for insufficient or grossly unrepresentative sampling. In the end, a geostatistical study is only as good as the sampling data that it is based on. Hopefully, through the case study, we have shown that the geostatistical approach is flexible, allowing for the incorporation of much collateral information and expert judgement concerning a site that might otherwise be neglected. Indeed, the tailoring of a geostatistical approach to specific site conditions is the hallmark of a successful study. With the overview of geostatistical simulation provide here, the reader should now be able to fully appreciate the subsequent papers on the topic contained in these proceedings. ACKNOWLEDGMENTS

The author wishes to thank Doug Hartzell, who coined the term "VSR", and one anonymous reviewer for their comments on the original manuscript. Geological Survey of Canada contribution no. 20995. REFERENCES

ASTM Standard Guide for the Selection of Simulation Approaches in Geostatistical Site Investigations, American Society for Testing and Materials, Philadelphia, draft submitted for Society approval by section DI8.01.07. Cromer, M., 1996, Geostatistics for Environmental and Geotechnical Applications: A Technology Transfer, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. , American Society for Testing and Materials, Philadelphia.

.1

!

I'

I

,, I I: ,,

I

:!

48

GEOSTATISTICAL APPLICATIONS

Cromer, M.V., C.A. Rautman and W.P. Zelinski, 1996, Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities Design at Yucca Mountain, Nevada, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. , American Society for Testing and Materials, Philadelphia. Deutsch, C.V. and A.G. Journel, 1992, GSLIB : Geostatistical Software Library and User's Guide, Oxford University Press, New York. Isaaks, E.H. and R.M. Srivastava, 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York. Journel, A.G. and C. Huijbregts, 1978, Mining Geostatistics, Academic Press, London. Rossi, R.E. and P.E. Evan Dresel, 1996, Declustering and Stochastic Simulation of Ground- Water Tritium Concentrations at Hanford, Washington, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. ,American Society for Testing and Materials, Philadelphia. Rouhani, S., 1996, Spatial Variability and Geostatistical Estimation : Kriging, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. , American Society for Testing and Materials, Philadelphia. Srivastava, R.M., 1994, An Overview of Stochastic Methods for Reservoir Characterization, in Stochastic Modeling and Geostatistics : Principles, Methods and Case Studies, J. Yarus and R. Chambers, Eds. , American Association of Petroleum Geologists, Computer Applications 3, p.3-16, American Association of Petroleum Geologists, Tulsa. Srivastava, R.M., 1996, Describing Spatial Variability Using Geostatistical Analysis, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. , American Society for Testing and Materials, Philadelphia. Zuber, R.D. and R. Kulkarni, 1996, A Geostatistical Analysis of Lake Sediment Contaminants at a Superfund Site, Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds. , American Society for Testing and Materials, Philadelphia.

, "

Environmental Applications

Bruce E. Buxton 1 , Darlene E. Wells 2 , Alan D. Pate 3 GEOSTATISTICAL SITE CHARACTERIZATION OF HYDRAULIC HEAD AND URANIUM CONCENTRATION IN GROUNDWATER

Bl!IFEBENCE: Buxton, B. E., Wells, D. E., Pate, A. D., "Geostatistical Site Characterization of Hydraulic Head and Vranium Concentration in Groundwater," Geostatistlos for Environmental and Geoteohnioal Applloatlons, ASTM STP 128:3, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, and Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: The first case study presented in this paper describes an assessment of the spatial distribution and temporal changes in hydraulic head pressure in the groundwater beneath a retired federal government uranium processing facility. Analysis of the hydraulic heads involved ordinary kriging which was found to be a better mapping method than such alternatives as inverse-distance weighting, mainly because kriging provides measures of estimation uncertainty. The objective of this kriging was to provide estimated steady-state head values for use in calibrating a groundwater flow model for the site. In the second case study, the spatial distribution of potential uranium contamination in the aquifer was assessed with lognormal kriging. Uranium measurements for this analysis were available at roughly three-month intervals across a four-year time period. The objective of the analysis was to assess where the uranium concentrations were highest. A second objective, not addressed in this paper, was to determine if the concentrations were changing significantly during the four-year time period. KEYWORDS: analysis

ordinary kriging, lognormal kriging, joint spatial temporal

Kriging is a statistical interpolation method for analyzing spatially and temporally varying data. It is used to estimate groundwater hydraulic heads (or any other important parameter) on a dense grid of spatial and temporal locations covering the region of interest. At each location, two values are calculated with the kriging procedure: the estimate of hydraulic head (in feet above sea level), and the precision of the estimate (also in feet above sea level). The precision can be interpreted as the half-width of a 95% confidence interval for the estimated head. The kriging approach includes two primary analysis steps: 1. Estimate and model temporal and spatial correlations in the available monitoring data using a semivariogram analysis.

lprogram Manager, Battelle, 505 King Avenue, Columbus, OR

43201.

2S en ior Data Analyst, Battelle, 505 King Avenue, Columbus, OR 43201.

3Research Scientist, Battelle, 505 King Avenue, Columbus, OH 43201.

51

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GEOSTATISTICAL APPLICATIONS

2. Use the resulting semivariogram model and the available monitoring data to interpolate (i.e., estimate) hydraulic head values at unsampled times and locations; calculate the statistical precision associated with each estimated value. Spatial Correlation Analysis The objective of the spatial correlation analysis is to statistically determine the extent to which measurements taken at different locations and/or times are similar or different. This section is written in terms of hydraulic head measurements; however, the analysis approach is similar for any measured parameter of interest. Generally, the degree to which head measurements taken at two locations are different is a function of the distance and direction between the two sampling locations. Also, for the same separation distance between two sampling locations, the spatial correlation may vary as a function of the direction between the sampling locations. For example, head values measured at each of two locations, a certain distance apart, are often more similar when the locations are at the same depth, than when they are at the same distance apart but at very different depths. Spatial/temporal correlation is statistically assessed with the semivariogram function, r(h), which is defined as follows (Journel and Huijbregts, 1981):

where Z(~) is the hydraulic head measured at location ~, h is the vector of separation between locations ~ and ~+h, and E represents the expected value or average over the region of interest. Note that the location ~ might be defined by an easting, northing, and depth coordinate, or for joint spatial/temporal data by an easting, northing, and time coordinate. Similarly, the vector of separation might be defined as a three-dimensional shift in space, or for joint spatial/temporal data as a shift in both space and time. The semivariogram is a measure of spatial differences, so that small semivariogram values correspond to high spatial correlation, and large semivariogram values correspond to low correlation. As an initial hypothesis, it is always wise to assume that the strength of spatial correlation is a function of both distance and direction between the sampling locations. When the spatial correlation is found to depend on both separation distance and direction it is said to be anisotropic. In contrast, when the spatial correlation is the same in all directions, and therefore depends only on separation distance, it is said to be isotropic. The spatial correlation analysis is conducted in the following steps using all available measured hydraulic head data: •

Experimental semivariogram curves are generated by organizing all pairs of data locations into various separation distance and direction classes (e.g., all pairs separated by 500-1500 ft (150-450 m) in the east-west direction ± 22.5°), and then calculating within each class the average squared-difference between the head measurements taken at each pair of locations. The results of these calculations are plotted against separation distance and by separation direction.

•

To help fully understand the spatial correlation structure, a variety of experimental semivariogram curves are generated by subsetting the data into discrete zones, such as different depth horizons or time periods. If significant differences are found in the semivariograms they are modeled separately; if not, the data are pooled together into a single semivariogram.

BUXTON ET AL. ON SITE CHARACTERIZATION

•

53

After the data have been pooled or subsetted accordingly, and the associated experimental semivariograms have been calculated and plotted, a positive-definite analytical model is fitted to each experimental curve. The fitted semivariogram model is then used to input the spatial correlation s~ructure into the subsequent kriging interpolation step.

In this study, the computer software used to perform the geostatistical calculations was the GSLIB software written by the Department of Applied Earth Sciences at Stanford University, and documented and released by Prof. Andre Journel and Dr. Clayton Deutsch (Deutsch and Journel, 1992). The primary subroutine used to calculate experimental semivariograms was GAMV3, which is used for threedimensional, irregularly spaced data. •

For three-dimensional spatial analyses, horizontal separation distance classes were defined in increments of 1000 ft (300 m) with a tolerance of 500 ft (150 m), while vertical distances were defined in increments of 20 ft (6 m) with a tolerance of 10 ft (3 m). Horizontal separation directions were defined in the four primary directions of north, northeast, east, and northwest with a tolerance of 22.5°.

•

For the joint spatial/temporal analysis, spatial separation distances and directions were defined in the same way as described immediately above, although there was no vertical direction associated with this analysis. For the temporal portion of this analysis, separation distance classes were defined in increments of 30 days with a tolerance of 15 days.

Interpolation Using Ordinary Kriging Ordinary kriging is a linear geostatistical estimation method which uses the semivariogram function to determine the optimal weighting of the measUrecfhydraulic head values to be used for the required estimates, and to calculate the estimation precision associated with the estimates (Journel and Huijbregts, 1981). In a sense, kriging is no different from other classical interpolation and contouring algorithms. However, kriging is different in that it produces statistically optimal estimates and associated precision measures. It should be noted that the ordinary kriging variance, while easy to calculate and readily available from most standard geostatistical software packages, may have limited usefulness in cases where the data probability distribution is highly skewed or non-gaussian. The ordinary kriging variance provides a precision measure associated with the data density and spatial data arrangement relative to the point or block being kriged. However, the ordinary kriging variance is independent of the data values themselves, and therefore may not provide an accurate measure of local estimation precision (e.g., appropriate width of estimation confidence interval). The kriging analysis was conducted in this study using the GSLIB computer software (subroutine KTB3D). The primary steps involved in this analysis were as follows: •

A three-dimensional grid was defined, specifying the locations at which estimated head values were required. The network included 112 blocks in the northern direction and 120 blocks in the eastern direction, and all blocks were 125 ft (37.5 m) square. For three-dimensional spatial kriging, the network included 30 vertical blocks 5 ft (1.5 m) thick. For joint spatial/temporal kriging, the network included 43 monthly blocks in increments of 30 days, starting at January 20, 1990.

•

At each block in the grid, the average hydraulic head across the block was estimated using all measured data found within a

54

GEOSTATISTICAL APPLICATIONS

pre-defined search radius. For three-dimensional spatial kriging of steady-state hydraulic head, the search radius was 6000 ft (1800 m) in all directions. For joint spatial/temporal kriging, the search radius was anisotropic and extended 6000 ft (1800 m) in space and 72 days in time. •

After the available data were identified for each grid block, the appropriate data weighting, estimated hydraulic head, and estimation precision were calculated using the appropriate semivariogram model.

•

Output from the kriging process was typically displayed in the form of contour maps, to represent spatial variations, and time-series graphs to represent temporal variations.

ANALYSIS OF HYDRAULIC HEAD Steady-state hydraulic heads were needed for calibration of a steady-state groundwater flow model. A two-step data analysis approach was used to estimate the steady-state heads. 1. A joint spatial-temporal kriging analysis was performed to estimate monthly hydraulic head changes at one depth horizon, and to select a single month representative of steady-state conditions. 2. A three-dimensional spatial kriging analysis was performed with data from the selected month at all available depth horizons to estimate steady-state hydraulic heads. Joint Spatial-Temporal Analysis The joint spatial-temporal kriging analysis was performed using monthly hydraulic head measurements in 177 wells (Figure 1) collected during the period from January, 1990 through July, 1993. Figure 1 shows the well locations where data were available for only the joint spatialtemporal analysis (denoted "JST" in the figure), for only the threedimensional steady-state analysis (denoted "55"), and for both of the analyses (denoted "Both"). There were a total of 3791 joint spatialtemporal measurements analyzed; and the minimum, maximum, mean, and standard deviation of these data were 493.7, 568.9, 519.9, and 5.3 feet (148.1, 170.7, 156.0, 1.6 m), respectively. The semivariogram curves, quantifying spatial and temporal correlation in these data, are shown in Figures 2 and 3. The spatial semivariograms in Figure 2 were calculated for four standard directions -- north, northeast, east, and northwest. These semivariograms show clear anisotropy with the highest variabilities directed north along the predominant flow direction, and the lowest variabilities directed east in the direction perpendicular to predominant flow. The corresponding temporal semivariogram for the monthly hydraulic heads is shown in Figure 3. Note that the units for separation distances between data locations are in days in Figure 3 and in feet in Figure 2. The semivariograms in Figures 2 and 3 were modeled with an anisotropic mathematical model containing three nested variance structures; the parameters of the model are listed in Table 1. Note in this table that three types of semivariogram models were used in various parts of these analyses: spherical, gaussian, and linear models. These models are fully described by Journel and Huijbregts (1981). In Figure 2 the bold line denotes the model in the north direction; the dashed line denotes the model in the northeast or northwest direction; and the dotted line denotes the model in the east direction.

55

BUXTON ET AL. ON SITE CHARACTERIZATION

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Fig. 1 -- Well locations where hydraulic head levels were monitored from January , 1990 through July, 1993. The monthly head measurements were used along with the semi v ariogram model to estimate via kriging monthly changes in the heads a c ross the entire groundwater modeling grid . The time period for this kriging analysis was taken as every 30 days starting January 20, 1990 and ending July 2, 1993 . The monthly heads (in feet above mean sea level) are depicted in Figure 4 for six locations uniformly spaced across the groundwater modeling grid. This figure shows that hydraulic heads during this period were relativ ely high in 1990, decreased in 1991 , were relatively low in 1 9 92 , and were increasing in 1993 .

56

GEOSTATISTICAL APPLICATIONS

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BUXTON ET AL. ON SITE CHARACTERIZATION

57

TABLE 1--Fitted semivariogram models of spatial and temporal correlation. (Note that 1 foot = 30 cm.) Data

Semivariogram Model

Joint Spatial-Temporal Hydraulic Head Pressure in ft.

K = 3 Nested Structures, Nugget Variance o ft.2 1. Geometric Anisotro~ic Spherical, variance = 1.5 ft. , Spatial Range 1200 ft., Temporal Range = 30 days 2. Geometric Anisotro~ic Spherical, variance = 5.5 ft. , Spatial Range 7000 ft., Temporal Range = 700 days 3. Zonal Gaussian in Spatial NS Direction, Variance = 22 ft.2, Spatial NS Range = 6060 ft.

Steady-State Hydraulic Head Pressure in ft. (June, 1993 Data)

K = 2 Nested Structures, Nugget Variance ft.2 1. Isotropic Linear, Slope = 0.00045 ft. 2/ft. 2. Zonal Gaussian in Horizontal NS Direction, Variance = 13 ft.2, NS Range = 8660 ft.

1990 Uranium Levels in flg/L

K = 1 Structure, Nugget Variance = 0.3 [In(flg/ L )]2 1. Geometric Anisotropic Spherical, Variance = 2.7 [In(flg/L)]2, Horizontal Range = 3000 ft., Vertical Range = 120 ft.

0

Steady-State Analysis One primary reason for performing the joint spatial-temporal analysis in the previous section was to select, for the steady-state analysis, a single month which was representative of average hydraulic head levels during the 1990-1993 time period. Examining the results in Figure 4, it appears that three months can be considered representative: January, 1990; November, 1991; and June, 1993. Hydraulic heads in each of these three months appear to be approximately equal to the average head levels across the entire 1990-1993 time period. However, several new wells were installed in the area in 1993, particularly in the southeastern part of the modeling grid. Therefore, a significantly greater number of head measurements were available for June, 1993 in comparison with January, 1990 and November, 1991. As a result, June, 1993 was selected as the month to represent steady-state conditions. Hydraulic head measurements for June, 1993 were available for the steady-state kriging analysis in 202 wells at various depths. The horizontal spatial semivariograms for these data are shown in Figure 5. As in the joint spatial-temporal analysis (Figure 2), horizontal semivariograms in Figure 5 were calculated in four primary directions. The fitted model is also shown in Figure 5, where the bold line denotes the model in the north direction; and the dashed line denotes the model in the east direction. A kriging analysis was performed with the June, 1993 data and the semivariogram model shown in Figure 5, as well as Table 1. This analysis estimated steady-state head levels across the groundwater modeling grid at regular 5 ft (1.5 m) vertical intervals from 390 to 540 ft (117 to 162 m) above sea level. The horizontal variability in steady-state head levels at the 490 ft (147 m) elevation is shown in

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GEOSTATISTICAL APPLICATIONS

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Fig. 4--Temporal Profile of Six Selected Estimation Locations. Note that 1 foot = 30 cm . Figure 6 . This figure shows a general trend of decreasing head pressure to the south associated with the predominant flow direction, along with a hydraulic head depression caused by two pumping wells in the eastern portion of the grid. Note in the extreme south-central part of Figure 6 that an unrealistically abrupt transition is shown between the uniform head values to the west and the pumping depression to the east . Unrealistic kriging features like that are possible when estimates are calculated for areas beyond the coverage of the data locations (see Figure 1). Figure 7, which presents the statistical uncertainty (in feet) associated with the estimates in Figure 6, shows that the steadystate heads are generally estimated to within 1 or 2 ft ( . 3 or . 6 m) . For areas beyond the spatial coverage of the data, the uncertainties increase to 3 ft. (.9 m), or more . These uncertainties can be interpreted as half-widths of a 95\ confidence interval for the estimates. That is, the confidence interval for the hydraulic head at any location in the grid is

BUXTON ET AL. ON SITE CHARACTERIZATION

59

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HEAD ± PREC where HEAD is the estimated hydraulic head from Figure 6 and PREC is the estimation uncertainty from Figure 7 . ANALYSIS OF URANIUM LEVELS Estimated uranium levels were needed for calibration of a groundwater solute transport model. Separate three - dimensional spatial kriging analyses, similar to that for steady-state hydraulic head, were performed using average uranium levels measured during 1990, 1991, and 1992 . One important difference between these uranium analyses and the head data analysis was that a logarithmic transformation of the uranium data was performed prior to the semivariogram and kriging analyses. This transformation was required to reduce the extreme variability seen in the uranium concentrations, making the semivariogram analysis more reliable . That is, the semivariograms calculated with untransformed data showed extreme variability which would have been difficult to model, while the semivariograms calculated with transformed data exhibited less variability to which a model could more reliably be fit. However, this transformation of the data leads to possible complications when the subsequent kriging results are back-transformed. The most direct back-transformation is a simple inverse-logarithmic transformation of the kriging estimate and estimation uncertainty. However, this back-transform corresponds to estimation of the median uranium concentration across a grid block, instead of the mean uranium concentration. In addition, the 95% confidence intervals for uranium

60

GEOSTATISTICAL APPLICATIONS

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Fig. 6--Estimated steady-state hydraulic head (feet) . Note that 1 foot = 30 cm. concentrations are multiplicative, rather than additive, in format. That is, the confidence interval at any location in the grid is [CONC / PREC, CONC*PREC] where CONC is the back-transformed estimated median uranium concentration and PREC is the back-transformed estimation uncertainty. In the alternative approach, the kriging estimate and estimation uncertainty can be back-transformed using analytical relationships between the mean and variance of the normal and lognormal probability distributions (Journel and Huijbregts, 1981, p. 572), resulting in an

61

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Fig. 7--Statistical uncertainty (width of 95\ confidence interval) in feet for estimated steady-state hydraulic head. Note that 1 foot ; 30 cm. estimate of the mean uranium concentration rather than the median. In this case, as with the analysis of water levels discussed earlier, the confidence intervals are additive, although there is no guarantee that the lower confidence bounds will be greater than zero. 1990 Uranium Levels The spatial kriging analysis was performed using average uranium concentrations (~g / L) measured during 1990 in 169 wells at various depths (Figure 8). The mean of these measurements was 29.3 ~g/L, although the maximum concentration (691 ~g / L) was considerably higher.

62

GEOSTATISTICAL APPLI CATIONS

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o T<"-'rrITT"-.rrTT"~rr"""-'"""-'rr"",,-r,,,,,,~ o 1500 3000 4500 6000 7500 9000 10500 12000 13500 15000 East Fig. 8--Wel l locations where uranium concentrations were meausred in 1990. The overall variability in the uranium data, as measured by the coefficient of variation , was also relatively high (2.90), particularly in comparison with that of the hydraulic head data (0.01). Horizontal semivariograms were calculated with the log-transformed data in the four primary directions (Figure 9), using horizontal separation distance classes defined in increments of 500 ft (150 m) with a tolerance of 250 ft (75 m), and vertical distance classes in increments of 20 ft (6 m) with a tolerance of 10 ft (3 m). The horizontal semivariograms indicated no significant anisotropy; that is, all four directional curves exhibited the same shape and variability . The vertical semivariogram (Figure 10) was found to plateau at the same overall variance as the horizontal semivariograms; however, the vertical semivariogram reaches it plateau at a separation distance of about 1 2 0 ft (36 m) wh ile the horizontal semivariograms reach their plateau at a separation distance of about 3000 ft (900 m) . As a result, a geometric an i sotropic semivariogram mode l was fi t ted to these curves, as shown in Figures 9 and 10, as well as Table 1.

BUXTON ET AL. ON SITE CHARACTERIZATION

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separation distance (feet) Fig. 9--Horizontal semivariograms for log-transformed 1990 average uranium concentrations. Note that 1 foot; 30 cm. A three-dimensional kriging analysis was next performed using the log-transformed 1990 average uranium concentrations and the semivariogram model discussed above. In this analysis, a data search radius of 12,000 ft (3600 m) was used. The resulting estimated spatial distribution of the median uranium concentrations is depicted in Figure 11 which is a horizontal cross-section at a depth of 512 ft (153.6 m). above sea level. The most significant uranium concentrations, those above 70 ~g / L, occur in a northeast oriented area extending about 2500 ft (750 m) by 900 ft (270 m) horizontally, and about 40 ft (12 m) vertically . A surrounding area, about 5-10 times larger, contains lower uranium concentrations between 10 and 70 ~g / L. Figure 12, which presents the statistical uncertainty associated with the estimates in Figure 11, indicates that these uranium concentrations are typically estimated to within a multiplicative factor of 10; that is, the true concentrations could be 10 times higher or lower . In contrast, Figure 13 presents the mean 1990 uranium concentrations calculated from the same lognormal kriging, but using the second back-transform described above. Note that because the estimated mean concentration is more strongly affected by high uranium data values than is the estimated median concentration, the mean estimates in Figure 13 exhibit greater spatial variability than the median estimates in Figure 11. This is particularly true in the northwest, northeast, and southeast corners of the figure where the kriging is extrapolating beyond the spatial coverage of the available data. Kriging estimates in those areas should not be trusted, and are probably best excluded from the final map. However, they have been retained in Figure 13 to point out this common problem . The corresponding statistical uncertainty associated with the mean estimates is presented in Figure 14; the uncertainties range from about 150 ~g / L to 240 ~g / L. Qualitatively, the uncertainty results in Figures 12 and 14 are similar and result in

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GEOSTATISTICAL APPLICATIONS

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similar upper confidence bounds. However , as noted earli e r , the uncertainties for the mean estimates (Figure 14) imply lower confidence bounds which are often below zero ~g / L while the uncertainties for the median estimates (Figure 12) are not plagued b y this problem .

CONCLUSION This paper presents four variations of the ordinary kriging methodology which was found useful for environmental characterization of groundwater at a potentially contaminated site . When estimating hydraulic head pressure in the groundwater aquifer, three - dimensional ordinary kriging was used in two different ways: (1) to assess temporal changes in the two-dimensional spatial distribution of head pressures, and (2) to estimate the three-dimensional spatial distribution of head pressures at a fixed point in time. In both cases ordinary kriging was applied directly to the head data, and the resulting kriging variances were used to construct statistical confidence intervals for the estimated head values. Ordinary kriging could be used directly in these cases because the head data exhibited relatively low overall variability and a symmetric probability distribution. In contrast, kriging of uranium concentrations in the groundwater, which exhibited much greater variability and a skewed probability distribution, required modification of the standard ordinary kriging procedure. In this case, ordinary kriging was performed after making a natural logarithmic transformation of the uranium data to help reduce the variability and make the subsequent semivariogram analysis more reliable. The major complication with this approach is related to the back-transformation which must be performed after kriging to convert the estimates back into the original scale of measurement. Two approaches

65

BUXTON ET AL. ON SITE CHARACTERIZATION

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were presented in this paper . The first approach leads to estimated median (rather than mean) uranium concentrations, and mUltiplicative (rather than additive) confidence bounds. The second approach results in estimated mean uranium concentrations and additive confidence bounds. However, there is no guarantee that the lower confidence bounds will be non - negative, and the widths of the confidence intervals are independent of the local data, which may not be appropriate for highly skewed data distributions .

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REFERENCES Deutsch, Clayton V., and Andre G. Journel, GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York, 1992, 340 pp. Journel, A.G., and Ch . J . Huijbregts, Mining Geostatistics, Academic Press, reprinted with corrections, 1981, 600 pp.

BUXTON ET AL. ON SITE CHARACTERIZATION

67

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Fig. 13 - -Mean estimate of 1990 uranium concentrations

(~g / L).

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GEOSTATISTICAL APPLICATIONS

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Fig . 14--Additive uncertainty factor (width of 95% confidence interv al) in ~g / L for mean estimates of 1 9 90 uranium concentrations .

Pierre Colin/ Roland Froidevaux/ Michel Garcia3 and Serge Nicoletis4

INTEGRATING GEOPHYSICAL DATA FOR MAPPING THE CONTAMINATION OF INDUSTRIAL SITES BY POLYCYCLIC AROMATIC HYDROCARBONS: A GEOSTA TISTICAL APPROACH

REFERENCE: Carr, J. R., "Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective After the 1994 Northridge, California Earthquake," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, and Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: A case study is presented of building a map showing the probability that the concentration in polycyclic aromatic hydrocarbon (P AH) exceeds a critical threshold. This assessment is based on existing PAH sample data (direct information) and on an electrical resistivity survey (indirect information). Simulated annealing is used to build a model of the range of possible values for P AH concentrations and of the bivariate relationship between P AH concentrations and electrical resistivity. The geostatistical technique of simple indicator kriging is then used, together with the probabilistic model, to infer, at each node of a grid, the range of possible values which the P AH concentration can take. The risk map is then extracted for this characterization of the local uncertainty. The difference between this risk map and a traditional iso-concentration map is then discussed in terms of decision-making. KEYWORDS: polycyclic aromatic hydrocarbons contamination, geostatistics, integration of geophysical data, uncertainty characterization, probability maps, simulated annealing, bivariate distributions, indicator kriging.

IHead, Environmental and Industrial Risks, Geostock, Rueil-Malmaison, France 2Manager, FSS Consultants SA, Geneva, Switzerland 3Manager, FSS International r&d, Chaville, France 4Head, Geophysical Services, Geostock, Rueil-Malmaison, France

69

70

GEOSTATISTICAL APPLICATIONS

Steelwork and coal processing sites are prone to contamination by polycyclic aromatic hydroccu"bons (P}Jl), SOnte of which are known to be carcinogenic. Consequently, local and state regulatory agencies require that all contaminated sites be characterized and that remediation solutions be proposed. The traditional approach for delineating the horizontal and vertical extent of the contamination is to use wells and boreholes to construct vertical profiles of the contamination at several locations. This approach, however, is both time consuming and expensive. Recent work has shown that, in some situations, electrical conductivity and resistivity surveys could be used as a pathfinder for delineating the contaminated areas. These geophysical surveys, which are both expedient and cost effective, could be used to reduce the number of wells and boreholes to be drilled. Geophysical data, however, do not provide direct information on soil chemistry. They are indicative of the ground nature, which in turn may reflect human activities (backfill material, tar tanks) and potential sources of ground pollution. These geophysical data have to be treated as indirect and imprecise infonnation. The mapping of the contamination therefore requires that imprecise geophysical data be correctly integrated with precise chemical analyses from wells and boreholes. Geostatistics offers an ideal framework for addressing such problems. Different types of information can be integrated in a manner which takes into consideration not only the statistical correlation between the different types of infonnation, but also the spatial continuity characteristics of both. Using this approach, it is possible to provide maps showing the probability that the P AH exceeds some critical level. A case study from an industrial site in Lorraine (northern France) is used to compare the geostatistical approach to the traditional approach of directly contouring the data from wells and boreholes. AVAILABLE DATA The available information consisted of chemical measurements of P AH concentrations (in ppm) from 51 boreholes. Figure 1 shows the location of these boreholes. As can be seen, the coverage is not even, and the lower right quadrant of the map is undersampled. In terms of distribution of the concentration values, it is seen that the highest concentrations are located toward the middle, where the cokeworks were located. The geophysical information included both conductivity measurements (electromagnetic survey) and resistivity measurements (dipole-dipole electrical measurements). The electromagnetic survey made it possible to investigate the overall site area and to identify anomalous zones where more accurate resistivity measurements were carried out. These resistivity measures are average values over large volumes of soil and are directly related to the soil nature (recent alluvium, slag deposits and other backfill material). They depend also on soil heterogeneities and the spatial arrangement of these heterogeneities. Although the physical phenomena that govern P AH transport, and the reactions between P AH and soils of different nature are not yet well understood (they are the subject of ongoing research projects), the presence of P AH in significant amounts has been found to be be

COLIN ET AL. ON GEOPHYSICAL DATA

71

associated with low resistivity values (i.e it increases, locally, the soil conductivity). This relationship, however, remains site specific and cannot be considered as a general law. The available resistivity measurements (in ohm-meter) come from 14 electrical lines, tightly criss-crossing the contaminated area. Gaps between electrical lines were filled, first, by sequential simulation to produce the full resistivity map shown on Figure 2.

500 rn

o

0

0 00 0 0 0

••

PAH concentration :

0 0

0

• " 0 0" " 0 0 " 0 • " 0 o a- "· 0 0 0 " • • " • 0

·.

•

...

0 0

0

> 200 ppm 40 - 200 ppm < 40 ppm

0 0

" 0

0 0

0 0

600rn

Figure 1 : Sample Location Map

500

15 ohm· m 25

~a~

• •• •• II

0 0

600

Figure 2 : Electrical Resistivity Map

40 95 250 375 750 1500 3000

If

72

GEOSTATISTICAL APPLICATIONS

OBJECTIVE OF THE STUDY The problem at hand is to delineate (on a 10 by 10 meter grid) areas where the risk that the P AH contamination is in excess of a critical threshold is deemed large enough to warrant either remediation or further testing. The critical threshold used for this study is 200 ppm PAH, and three classes of risk were considered: Low Risk :

The probability that the concentration in P AH exceeds 200 ppm is less than 20 percent.

Medium Risk: The probability of exceedance is in the 20 to 50 percent range. High Risk:

The probability of exceedance is over 50 percent.

From a methodological point of view, assessing the risk of exceedance implies that, at each node of the grid, the range of possible values for the P AH concentration, along with their associated probability, be available. The challenge therefore is to take advantage of both the direct measurements of P AH concentration and the indirect information provided by electrical resistivity to infer, at each grid node, the range of possible values that P AH concentration could take. These ranges of possible values, called also local conditional distribution functions, can be viewed as measures of the local uncertainty in the P AH concentration. Once these local uncertainties are established, the risk maps can be produced.

EXPLORATORY DATA ANALYSIS Given the objectives of the study, we need to understand the following critical features from the available data: 1- The range of possible values for the PAH concentrations which may be encountered away from sampled locations; 2- The relationship which exists between P AH concentrations and electrical resistivity. In other words, knowing the resistivity value at a particular location, what can we say about the possible range of P AH concentration values at the same location? 3- The spatial correlation structure of P AH concentrations?

Univariate distribution ofPAH concentrations Any geostatistical estimation or simulation algorithm requires a model describing the probability distribution function of the variable under consideration, i.e. an enumerated list of possible values with their associated probabilities. Traditionally, this probability distribution function is based on the experimental histogram built on the available data. Figure 3 shows the experimental histogram, and the corresponding summary statistics, of the available PAH

COLIN ET AL. ON GEOPHYSICAL DATA

73

concentration data. We can see that: -

The number of data available to construct the histogram is fairly limited, resulting in a lack of sufficient resolution : the class frequencies tend to jump up and down erratically and there are gaps between data values.

-

The histogram is extremely skewed to the left, the bulk of the values being below 300 ppm, with some erratic high values extending all the way up to 6500 ppm (Figure 3a). Not surprisingly, the coefficient of variation is very high (3 .03).

-

The mean and variance of the data are severely affected by this high variability and they cannot be established at any acceptable level of reliability: removing the two largest values reduces the average by a factor 3 and the variance by a factor 65!

-

Ifwe use a logarithmic scale to visualize the same histogram (Figure 3b), we see clearly the existence of three populations: a first one below 50 ppm and accounting for 61 percent of the total population, a second one in the range 50 to 500 ppm and including 34 percent of the population, and a third small (5 %) population characterized by extreme P AH values ranging from 600 to 6500 ppm.

f

Summary statistics Number of data : Mean : Standard deviation : Coer variation : Minimum : 1st quartile : Median: 3rd quartile : Maximum :

0.5

0.25

0.0

-Jlllll:ilib:tlhl...frlliL..Illl.....IDlla,.__ 250

51 382 ppm 1159 ppm 3.03 2 ppm 9 ppm 33 ppm 206 ppm 6500 ppm

PAH (ppm)

500

Figure 3a : Histogram and summary statistics for P AH concentrations (in ppm)

74

GEOSTATISTICAL APPLICATIONS

f

0. 1

0.0

..,....""'-'''''-+ PAH wpm] 50

500

Figure 3b : Histogram ofPAH concentrations (logarithmic scale)

The first population can be interpreted as representing the background concentration level on the site. The second population seems clearly related to contamination itself, with the bulk of it above the critical threshold of 200 ppm. The third population is a little more dubious to interpret, primarily because it is represented only by three samples. Although it is obviously associated with the contamination, it is not entirely clear whether it represents a different factor of contamination or if it is merely the tail end of the second population. From this analysis it is clear that the experimental histogram cannot be used as is as a model of the distribution function of P AH concentrations over the site area. This probability distribution function should, instead, be modelled and have the following features :

It should not be based on parameters like the mean and variance, which are highly affected by extreme values and are, as a result, not known with any degree of reliability; It should provide probabilities for the entire range of possible values, from the absolute minimum to the absolute maximum and fill the gaps between existing data values;

It should reproduce the existence of the three populations and their respective frequencies.

Bivariate Distribution The cross-plot shown on Figure 4 describes the relationship which exists between PAH concentrations and electrical resistivity.

COLIN ET AL. ON GEOPHYSICAL DATA

75

PAH [ppm[ 1000

100

•

•

•• • ••• • • •• •••

• • • • •• • • • • •

10

10

100

Resist . ohm. m

1000

Covariance : -0.799 Correlation (pearson) : -0.360 Correlation (Spearman) : -0.266

Figure 4 : Cross plot and bivariate statistics ofPAH vs Electrical Resistivity

The most important feature ofthis plot is the existence oftwo distinct clouds of points, which is a direct consequence of the multi-modality of the PAH distribution, and of the bimodality of the electrical resistivity: An upper cloud where P AH values are in excess of 3 5 ppm and the electrical resistivity ranges from 15 to 150 ohmomo Within this cloud, the correlation between the two attributes is positive.

A lower cloud with PAH values below 35 ppm and with electrical resistivities in the 30 to 1600 ohmom range. The correlation, again, appears to be positive, but less significantly so . From this cross-plot it seems that high concentrations ofPAH (over 35 ppm) are associated to rather low resitivity values. One possible explanation of this feature, which stilJ remains to be confirmed, is that P AH, which are viscous fluids, tend to flow down through backfilJ materials until they reach the top of the natural soil. At this level they filJ up the soil pore volume, thus creating a flow barrier to water.

76

GEOSTATISTICAL APPLICATIONS

Traditionally, bivariate distributions are parametrized by the mean and variance of their n;arginal distributions together with the correlation coefficient. Such an approach is inapplicable in our case, since it will fail completely to reflect the most important feature of the cross-plot which is the existence of the two populations. The solution adopted for this study consists of using directly a bivariate histogram to describe the bivariate distribution model. Because of the sparsity of data, the experimental cross plot is not sufficient to inform all the possible bivariate probabilities: it is spiky and full of gaps. The required bivariate histogram, therefore, will be obtained by an appropriate smoothing of this experimental cross plot, making sure that the two clouds of points are correctly reproduced. Spatial Continuity Analysis The variogram analysis performed on the natural logarithm ofPAH concentrations (Figure 5) shows that the phenomenon is reasonably well structured, with a maximum correlation distance (range) of approximately 70 meters. There was no evidence of anisotropy and the shape of the variogram was exponential.

y(h) 6

... .. ,.. .. .. i · .. · .. ... .... ..................... .......... .. ~

..

~

~

'

.. ~' 3

0r-----------r----------.-----------r~ hIm) so

100

ISO

Figure 5 : Experimental variogram for Ln(P AH) concentrations

COLIN ET AL. ON GEOPHYSICAL DATA

77

BUILDING THE PROBABILISTIC MODEL Based on the results of the exploratory analysis, the probabilistic model to be used in estimation and uncertainty assessment will consist of the following: -

A smooth univariate histogram approximating the marginal distribution of the P AH concentration, and

-

A smooth bivariate histogram describing the bivariate distribution of P AH concentration and electrical resistivity.

Several approaches have been proposed to produce smooth histograms and cross-plots: quadratic programming (Xu 1994), fitting of kernel functions (Silverman 1986; Tran 1994) and simulated annealing (Deutsch 1994). The technique selected for this study is simulated annealing, because it was perceived to be the most flexible to accommodate all the requirements of the probabilistic model. Simulated annealing is a constrained optimization technique which is increasingly used in earth sciences to produce models which reflect complex multivariate statistics. A detailed discussion of the technique can be found in (Press et al. 1992; Deutsch and Journel 1992; Deutsch and Cockerham 1994). In this study the modelling of the bivariate probabilistic model was done in two steps: first the marginal distribution of the P AH concentration was modelled, and then the cross-plot between PAH and electrical resisitivity (there was no need to model the marginal distribution of resistivity, since it was directly available from the resistivity map).

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The modelling of the marginal distribution ofPAH via simulated annealing was implemented as follows: 1- An initial histogram is created by subdividing the range of possible values into 100 classes, and assigning initial frequency values to each of these classes by performing a moving average of the original experimental histogram and then rescaling these frequencies so that they sum up to one. 2- An energy function (Deutsch 1994) is defined to measure how close to desired features of the final histogram the current histogram is. In the present case the energy function takes into consideration the reproduction of the mean, variance, selected quantiles and a smoothing index devised to eliminate spurious spikes in the histogram. 3- The original probability values are then perturbated by choosing at random a pair of classes, adding an incremental value b.p to the first class and substracting it from the second, hence ensuring that the sum of the frequencies is still one. 4- The perturbation is accepted if it improves the histogram, i.e. if the energy function decreases. Ifnot, the perturbation may still be accepted with a small probability. This will ensure that the process will not converge in some local minimum. 5- This perturbation procedure is repeated until the resulting histogram

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78

GEOSTATISTICAL APPLICATIONS

satisfactory (the energy function has reached a minimum value) or until no further progress is possible. The modelling of the cross plot followed a similar general approach: 1- An initial bivariate histogram is created by subdividing the range of possible values along both axes into 100 classes, and assigning initial bivariate frequency values to each of these cells by performing a moving average of the original cross-plot followed by a rescaling of the frequencies to ensure that they sum up to one. 2- An energy function is defined to measure the goodness of fit of the current bivariate histogram to desired features of the final one. In the present case the energy function takes into consideration the reproduction ofthe marginal distributions defined previously, the correct reproductions of some critical bivariate quantiles and, again, a smoothing index devised to eliminate spurious spikes in the cross plot. 3- The original bivariate frequencies are perturbated by randomly selecting a pair of cells, and adding an incremental probability t,p to the first class and subtracting it from the second, leaving therefore the sum of frequencies unchanged. 4- As before the perturbation is accepted if it decreases the energy function and accepted with a certain probability if not. 5- The perturbation mechanism is iterated until the energy function has converged to some minimum value. A detailed discussion on how to use simulating annealing for modelling histograms and cross plots can be found in Deutsch (1994). The result of this modelling is shown on Figure 6 (experimental histogram of P AH concentrations and smooth model) and Figure 7 (experimental cross plot of P AH versus resistivity and corresponding smooth bivariate histogram). As can be seen, all the important features appear to be well reproduced: the multi-modality of P AH, the bi-modality of the resistivity and the existence of the two clouds on the cross plot.

ESTIMA TING THE LOCAL DISTRIBUTION FUNCTIONS Having developed the bivariate probabilistic model for P AH and electrical resistivity, we will now use it to infer the local conditional cumulative distribution function (ccdt) of the P AH concentration. This inference (see Appendix I) involves two steps: 1- The local a priori distribution function (cdt) of the P AH, given the local resistivity value, is extracted from the bivariate histogram. This local cdf characterizes the uncertainty of the P AH value based on the overall relationship existing between P AH and resistivity, but

COLIN ET AL. ON GEOPHYSICAL DATA

before using the local P AH data values themselves.

f

0.1

...,...~~--+

0.0

50

PAH [ppm]

500

f

...,...--.... PAH[ppm]

0.0

50

500

Experimental Figure 6 : P AH Concentration histogram and smooth histogram model (logarithmic scale)

79

80

GEOSTATISTICAL APPLICATIONS

2- The local a priori cdf is then conditioned to the nearby P AH data values via simple indicator kriging (Journel \989) . This ccdf now describes the uncertainty on the P AH concentration once the local conditioning information has been accounted for. Note that simple indicator kriging calls for a model of the spatial continuity of the residual indicators (see Appendix I). This model is shown on Figure 8.

y

... .

·f

01) = 0.01 + 0.25Gaus 70 (h)

.. ." ........................... " ......... " ....... " .. •

r-----.---~r---~----~-----r-----r~h[ml 70

140

210

Figure 8 : Variogram model for the indicator residuals

In this approach, the two types of information (direct measurements ofP AH concentrations and electrical resistivities) are mixed in a smooth, transitional fashion: when there is abundant nearby sample data, the simple indicator kriging system will put a lot of emphasis on this conditioning information and down play the influence of the indirect information, whereas when there is little or no conditioning data, the range of possible outcome for P AH will be primarily controlled by the local resistivity value.

PROBABILITY MAPS Having inferred the local distribution functions of P AH concentrations, we can now build the probability map (Figure 9c) showing the risk that this concentration exceeds the critical threshold of200 ppm .

COLIN ET AL. ON GEOPHYSICAL DATA PAH (ppm )

• 1000

•

100

..... •

• •• •••

• • • • • •• • • ••

10

10

100

ResilL (ohm-m 1

1000

PAH (pp m(

Freguency 0 0.01 0.02

1000

;:m:-:::*

• •• f.~~~*

II

100

••

10

0.03 0.04 0.05 0.06 0.07 0.08 0.09

=="-~ Resistivity [o1un . m] 10

100

1000

Figure 7 : P AH vs Electrical Resistivity Experimental cross-plot and bivariate histogram model

81

82

GEOSTATISTICAL APPLICATIONS

500 m

a) Iso-concentration map

II

PAH > 200 ppm

[j

PAH <200 ppm

o 500m

~1I1I1I~------------~

b) Probability map (PAH only)

High risk Mediwn risk Low risk

0

600 m

500 m c) Probability map (PAH & ResistivM

II

High risk

~

Mediwn ri sk

II

Low risk

0 0

600m

Figure 9 :Iso-concentration map and probability maps

COLIN ET AL. ON GEOPHYSICAL DATA

83

For the sake of comparison, this probability map is compared with two other maps: - A probability map built by indicator kriging also, but taking into account the P AH concentrations only (Figure 9a), and - A map showing the area where the estimated PAH concentration (expected value) exceeds the critical threshold. This map was obtained by ordinary kriging (Figure 9a). If we look first at the estimated map (Figure 9a), we see that the area where the P AH estimated value exceeds 200 ppm forms a rather homogenous, smoothly contoured, zone concentrated around the cokeworks, where all the high sample values are located. If this map was used for decision-making, one could come to the conclusion that this zone, representing a surface of 135,900 square meters, is the only one requiring attention. Looking now at the probability map produced by indicator kriging based on the P AH concentrations only (Figure 9b), we see that the picture gets more complex: the center zone is still high risk but less homogeneously so, and the medium risk zone extends further to the south. The medium and high risk zones, now, represent 190,700 square meters. However, because oflack of direct information, the periphery appears mostly as a low risk zone. Finally, by including the information provided by the electrical resisitivity, we see that there is a significant probability (medium risk) that peripheral areas to the south, but also to the north and east, are contaminated. These areas would probably warrant further testing to confirm the level of contamination. In this case, the medium and high risk areas total 271,900 square meters. From these results it is clear that estimated maps are inadequate for delineating risks of contamination: they provide information on the expected value of the concentration, but not on the local uncertainty. And because of their intrinsic smoothing properties, they may either vastly overestimate or underestimate the extent of the risk zone. In this case the contour map indicates a risk area (estimated concentration greater than 200 ppm) which is half the size of the medium and high risk area as shown on the probability map inferred from both the P AH concentrations and the electrical resistivity. It is worth remembering also that the objective of such study is to provide a classification of whether or not the soil, at a particular location, is likely to be affected by the contamination, and not to provide a good estimate of the concentration at that location. The challenge is to come up with probabilities and not with estimated concentrations. One may even argue that the latter is irrelevant to the task at hand: high estimated values may correspond to a low probability of exceedance and, conversely, moderate estimated values may be associated with high probabilities of exceedance.

CONCLUSION Prediction of contaminant concentration can be improved by taking into consideration indirect, related information such as geophysical data. To integrate effectively this indirect

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84

GEOSTATISTICAL APPLICATIONS

information into the estimation process it is crucial that the bivariate relationship existing bp,tween the direct and indirect information be correctly rendered. Very often this relationship cannot be captured by the classical parametric description based on the means and variances of the marginal distributions and the correlation coefficient. A more general alternative consists in using a full bivariate histogram to describe the relationship between the two attributes, and it is proposed that this bivariate histogram be modeled by simulated annealing. This bivariate probabilistic model can then be used to infer the local a priori distribution function ofthe contaminant concentration given the known value of the secondary attribute. This local a priori distribution function is finally conditioned to the existing local contaminant data by simple indicator kriging. This approach is very general and can be used to address many different situations. It should be stressed, however, that the relevance of its results depends heavily on how physically meaningful is the relationship between the main and co-attribute as described by the bivariate probabilistic model.

APPENDIX I Simple Indicator Kriging with Bivariate Histogram This paper is concerned with P AH concentration and electrical resistivity. The technique however is completely general and can be used each time that secondary information is provided under the form of a bivariate histogram. We will use the following notation: Z(u)

random variable describing the attribute of main interest, informed by N data values, z(uu), a= 1, ... , N

Y(u)

random variable describing the secondary attribute, informed by M data values, p= 1, ... ,M

y(u~),

u

location coordinates vector

Prob{

Zj_1 ::;

Z(u) < z; and Yj-J::; Y(u)

<

yj}

= bivariate histogram frequency with: Zj,

i =1, .. , M = number of thresholds discretizing the range of values [zmm' zmar]

Yp j= I, ... , ~ = number of thresholds discretizing the range of values [ymm. YmaJ At a given location u, the a priori distribution function of the main attribute will be given by:

COLIN ET AL. ON GEOPHYSICAL DATA

85

It

EtcU; F O(u; zit

I y(u»

= Prob{Z(u) ~ zit

I y(u)}

ZI;

y(U»

1=1

(1)

N,

EtcU;

ZI;

y(U»

1=1

and the estimated local ccdf (posterior distribution) by:

n

F*(u; zit Iz(u •.) a=l, ... , n) =

L

A. .. ·r(u .. ; zit) + FO(u; zit

I y(u»

(2)

.. =1

I~i

~;: I I,

I:

Ii" ,~

,

I",

I,

I, 1

~II I

with Z(U~),

a= 1, ... , n being the n local conditional data

and rU!~;

zJ being the residual indicator value:

(3)

; otherwise

Simple kriging with a bivariate histogram, therefore, involves the following steps: 1- Select the number thresholds~, k=1, ... ,K required to provide an adequate discretization of the local ccdf This number K of thresholds depends on the goal of the study and needs not to be as large as the number of thresholds ~ used to build the bivariate histogram; 2- For each datum z(u a ), define the residual indicator values r(ua,zJ, k=1, ... , K, using the local a-priori distribution function FO(Ua,Zk) ;

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86

GEOSTATISTICAL APPLICATIONS

3- Establish the variogram model y /h; z.) of the residual indicator values; 4- Then, at each grid node u and for each threshold

~:

- determine the local a priori distribution of the main attribute Z(u), given the local secondary attribute value y( u) using equation I; - estimate, by simple kriging, the local ccdf of Z( u) using equation 2; - check and correct for potential order relation problems 5- Process the estimated ccdf to extract the required probabilities, quantiles or estimated values. In principle, this involves solving, at each grid node u, K simple kriging systems since variogram models may be different for each threshold z.. Ifit can be shown that the variogram model does not change significantly from one threshold to another, then it is sufficient to solve a single system, and to use the same weighting scheme for every threshold. This approach, called Median Indicator Kriging - or "mosaic-model" (Lemmer 1984; Journel 1984), can simplifY and speed up the whole estimation process.

REFERENCES Wenlong, Xu, 1994, "Histogram and Scattergram Smoothing Using Convex Quadratic Programming," SCRF proceedings, Stanford University. Silverman B., 1986, "Density Estimation for Statistics and Data Analysis," Chapman and Hall, New-York. Tran, T., 1994, "Density Estimation using Kernel Methods," SCRF proceedings, Stanford University. Deutsch, c.v., 1994, "Constrained Modelling of Histogram and Cross-Plots with Simulated Annealing," SCRF proceedings, Stanford University. Press W., et aI., 1992, "Numerical Recipes in C, The Art of Scientific Computing," CambridgeUniversity Press. Deutsch, C.V. and Journel, AG., 1992, "GSLIB Geostatistical Software Library and User's Guide," Oxford University Press. Deutsch, C.V., and Cockerham, P.W., 1994, "Practical Considerations in the Application of Simulated Annealing to Stochastic Simulation," Mathematical Geology, Vol. 26, No.1, pp. 67-82.

COLIN ET AL. ON GEOPHYSICAL DATA

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Journel, A. G.,1989, "Fundamentals of Geostatistics in Five Lessons," Volume 8, Short Course in Geology, American Geophysical Union, Washington D.C. Lemmer, I. c., 1984, "Estimating Local Recoverable Reserves via Indicator Kriging, "in G.Verly et aI., Geostatistics for Natural Resources Charaterization, pp. 349-364, Reidel,Dodrecht, Holland. Journel, A. G., 1984, "The Place of Non Parametric Geostatistics," in G.Verly et aI., Geostatistics for Natural Resources Charaterization, pp. 307-335, Reidel, Dodrecht, Holland.

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Michael R. Wild i and Shahrokh Rouhani 2

EFFECTIVE USE OF FIELD SCREENING TECHNIQUES IN ENVIRONMENTAL INVESTIGATIONS: A MULTIVARIATE GEOSTATISTICAL APPROACH

REFERENCE: Wild, M. R., Rouhani, S., "Effective Use of Field Screening Techniques in Environmental Investigations: A Multivariate Geostatistical Approach," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Environmental investigations typically entail broad data gathering efforts which include field screening surveys and laboratory analyses. Although usually collected extensively, data from field screening surveys are rarely used in the actual delineation of media contamination. On the other hand, laboratory analyses, which are used in the delineation, are minimized to avoid potentially high cost. Multivariate geostatistical techniques, such as indicator cokriging, were employed to incorporate volatile organic screening and laboratory data in order to better estimate soil contamination concentrations at a underground storage tank site. In this work, the direct and cross variographies are based on a multi-scale approach. The results indicate that soil gas measurements show good correlations with laboratory data at large scales. These correlations, however, can be masked by poor correlations at micro-scale distances. Consequently, a classical direct correlation analysis between the two measured values is very likely to fail. In contrast, the presented multi-scale co-estimation procedure provides tools for a cost-effective and reliable assessment of soil contamination based on a combined use of laboratory and field screening data. KEYWORDS: geostatistics, cokriging, multivariate, field screening, volatile organics

Assessing the extent of soil contamination can be very costly. Laboratory analysis of common environmental contaminants can range from $200 to $1000 per sample for standard method testing. Consequently, many investigations first use field screening techniques to help identify relative levels of contamination and then select a few samples for laboratory analysis. In many cases, the validity of field data are questioned and such data are rarely used in the actual delineation of source contamination. This paper presents a geostatistical technique for an optimal and defensible incorporation of field screening and laboratory data. This approach is intended to

iProject Environmental Engineer, Dames & Moore, Inc., Six Piedmont Center, 3525 Piedmont Road, Suite 500, Atlanta, Georgia 30305. 2Associate Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0355.

88

WILD AND ROUHANI ON FIELD SCREENING

89

accomplish the following objectives: Perform site characterization in a cost-effective and information-efficient manner, Minimize the need for additional environmental investigations, and Employ defensible approaches based on rigorous mathematical techniques for the analysis of spatial data. Several screening devices or kits are available to measure various contaminants, such as volatile organics, metals and pesticides. The most commonly used devices are portable soil-gas probes, xray fluorescence spectrometers, and immunoassay kits. The measured results of these tools are mostly qualitative in nature and may not correlate well to actual laboratory measurements. Almost all environmental investigations require procedures to determine the extent of contamination. Investigators would prefer to employ all types of available information, including field and laboratory data, to perform this task. However, federal and state regulatory agencies discourage the direct use of field data. Consequently, a large portion of useful information is either neglected or under-utilized. This paper provides a timely solution to extract the maximum amount of information from available data. For this purpose, multivariate, non-linear techniques, such as indicator cokriging, are used. This paper applies these concepts to a site whose soil has been contaminated by several underground storage tanks over a period of 20 to 30 years. The tanks were used primarily for storage of kerosene, gasoline and diesel fuels and various industrial solvents. Although the site was extensively investigated with over 300 samples, limited laboratory confirmation of volatile organic compounds (VOCs) was performed. This attempt to save money in laboratory cost actually prohibited the delineation of contamination extent.

BACKGROUND INFORMATION Screening tools for VOCs, such as photoionization detectors (PID) and organic vapor analyzers (OVA), can provide an inexpensive alternative to laboratory testing, especially for large, multilayer investigations. Manufacturers of these instruments advocate the use of these devices as an effective method of measuring VOCs for preliminary site characterization, including the delineation of subsurface contamination. Unfortunately, the reliability of these devices has proven to be dependent on weather conditions, soil type and actual contaminant concentrations. Several published case studies exist that incorporate data gathered using these instruments. One study performed by Marrin and Kerfoot (1988) used a portable gas chromatograph and PID to predict the extent of groundwater contamination by measuring volatile organics in the soil gas. Thompson and Marrin (1987) also measured soil gas concentrations at 49 locations to estimate groundwater contamination. The results of this estimation, however, were verified by an inadequate number of groundwater samples (five). Crouch (1990) used gas detection tubes to estimate contaminant concentrations in soil vapor. According to Crouch, gas detection tubes were used because the OVA and PID are not compound specific and only provide a total VOC measurement. None of the above investigations performed any correlation analyses between soil gas probe

,,''" ,,'

90

GEOSTATISTICAL APPLICATIONS

readings and laboratory measurements and therefore assumed that the measured results from these s<::reening devices were reasonably accurate. Seigrist (1991) compared gas chromatography to two PIDs of varying ionization potentials to measure volatile organics in a controlled environment. The results of the comparison showed poor correlation between both PIDs and the gas chromatograph and demonstrated that the PIDs were very sensitive to water vapor and responded to natural organics including methane, ethylenes and alcohols. Smith and Jensen (1987) tried correlating OVA and PID readings to laboratory measurements for total petroleum hydrocarbons (TPH) in soil. Again, poor correlation prohibited using screening tools to estimate actual TPH, and the authors cautioned against using screening tools as a sole criterion for determining soil contamination. The above works clearly indicate that the screening tools which are widely used and accepted in the environmental field provide only qualitative information on total VOCs. The use of these tools in delineating contamination is cautioned against because their effectiveness and reliability remain unverified. On the other hand, comprehensive laboratory analyses of environmental investigations can be prohibitively expensive. This study presents geostatistical procedures, such as cokriging, to link the two measurement techniques and produce accurate maps of contamination. Cokriging is defined as the estimation of one variable using not only observations of the variable but also data on one or more additional, related variables defined over the same field (Olea 1991). Cokriging is suitable for cases where the targeted variable is not sampled sufficiently to provide acceptably precise estimates of that variable over the entire investigated field (Joumel and Huijbregts 1978). Such estimates may then be improved by correlating this variable with bettersampled auxiliary variables as a function of their separation distance. This approach is fully compatible with actual field conditions where VOC screening data are extensively collected but only relatively few samples are verified by laboratory analysis. However, it must be emphasized that the utility of cokriging depends on the level of spatial cross-correlation between the primary and auxiliary variables. To obtain an adequate cross-correlation between investigated variables, it may become necessary to apply data transformations prior to the actual co-estimation process. This may require use of non-linear cokriging techniques. The use of non-linear geostatistical techniques is preferable if one or both of the variables exhibit non-gaussian tendencies. Such distributions are commonly observed in contamination assessments of VOCs in soil. VOC data sets are usually characterized by a few significant outliers and a majority of very low or non-detectable samples. Indicator kriging has been found to be useful for highly variant phenomena where data present long-tailed distributions (JoumelI983). This characterization also applies with respect to mineral deposits data sets (lsaaks and Srivastava 1989). This type of kriging uses a non-parametric approach that does not suffer from the impact of outliers since the original values are transformed to either a 0 or 1 based on cutoff or threshold limits (Isaaks and Srivastava 1989). The transformed values can then be used to estimate the spatial distribution of the data.

GEOSTATISTICAL METHODOLOGY Geostatistics provides tools for the analysis of spatially correlated data and is well-suited to the

WILD AND ROUHANI ON FIELD SCREENING

91

study of natural phenomena (Journel and Huijbregts 1978). The theory of geostatistics has been well-documented over the years; therefore, only a general description of techniques applicable to this investigation are provided. These techniques are cokriging and indicator kriging. For more information on geostatistics, see Journel and Huijbregts (1978). Geostatistics allows for the estimation of values at unsampled locations. This estimation approach is commonly known as kriging and is a linear combination of known nearby values, as shown by (1)

where Zo* = the estimated value ofZ (an arbitrary parameter) at location xo; Zj = the measured value at location Xj; A. j = the kriging weight of the parameter value at Zj; and n = the number of nearby sample points to be used in the estimation. The weights are calculated to produce the lowest estimation error or variance and to satisfy the unbiasedness condition (LA.j = 1, I = 1 to n). The minimized variance for ordinary kriging can be written as n

Lj=! Ajy~

v'o

+ Jl

(2)

where Vo' = the minimum variance of estimation error; yjjZ = the variogram between Zj and Zj; and f.J- = the Lagrange multiplier. Cokri~in~

Cokriging is the estimation of one variable based on measured values of two or more variables. This procedure can be regarded as a generalization of kriging in the sense that, at every location, there is a vector [Z(Xj), Y(Xj)' ... ] of many variables instead of a single variable Z(x) (Olea 1991). The variable to be estimated is denoted as the target or primary variable while all other variables are categorized as auxiliary or secondary variables. The secondary variable is cross-correlated with the primary variable. The cokriging procedure is especially advantageous in cases where abundant secondary values are more abundant than primary variables. The co-estimation of the primary variable is calculated as n

Z~

=

m

Lj=! AjZj +L vjYj j=!

(3)

where Vj = the weight factor for the secondary variable, Y, measured at Xj; and m = the number of secondary-variable measurements (which is typically much greater than n).

92

GEOSTATISTICAL APPLICATIONS

Minimizing the variance of estimation error, Vo', subject to cokriging unbiasedness conditions (~;\'i =1, ~Vj = 0) results in

V;

where

=

n

m

i=l

j=l

L Ai y~ + L Vj y~Y

+

(4)

III

yl

=the variogram of the primary variable; and YijZY = the cross-variogram of the primary and secondary variables.

This technique of cokriging improves the estimation and reduces the variance of estimation error (Ahmed and De Marsily 1987). Indicator Kriging Indicator kriging is a non-parametric technique used in probability theory when parametric assumptions are not appropriate to describe the distribution of the data set. Instead, indicator kriging provides an estimate of the cumulative distribution of the data set by calculating conditional probabilities. These probabilities can be estimated by transforming the variables to a one or zero, depending upon whether they fall above or below a cutoff level (Sullivan 1984):

.

l(X;Z)

=

{I

if z(x) S Zk 0 if z(x) >Zk

(5)

where Zk is the cutoff level. By using kriging, the interpolated indicator variable at any point Xo can be estimated by n

I: (xo)

=

L

(6)

AjOik(x)

j=l

where

1*k = the estimate of the conditional probability at Xj

'\0 = the kriging weight for the indicator value at point Xj

(Rouhani and Dillon 1989).

The conditional probability in this case is defined as:

(7) By varying zk' the cumulative probability can then be constructed.

CASE STUDY The case site has been in operation for over 50 fifty years and is currently a commercial site. The area under investigation has a total of nine underground storage tanks (USTs) situated around the site. Some of the tanks have been in operation for up to 30 years and were suspected to have

WILD AND ROUHANI ON FIELD SCREENING

93

leaked for an unknown number of years. The site is approximately 90 to 95 percent covered with concrete and is relatively level. The investigated data and some site characteristics have been altered to maintain confidentiality. The horizontal and vertical contamination at the site were assessed by three boring programs. A total of 82 borings were advanced. All USTs were eventually excavated and removed after the environmental investigation. A confirmatory campaign was performed after the tanks were removed and produced 12 additional sample locations. Figure 1 shows the locations of the 94 borings and the previous locations of the tanks. A Foxboro® OV A3 was used to screen for VOCs in each boring for all four investigations. The OV A was used to screen over 300 samples from these borings.

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iii

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B.lidirg UST Sci BOfirg

Figure 1-- Site Map. In addition to the OV A screening, laboratory analysis was performed on 35 of the 300 samples. Typically, laboratory testing is performed on high OVA readings if one wants to identify the highest concentrations for risk assessment. Alternatively, when OVA readings are neither high nor low, laboratory results can be used to determine potential contamination of a sample. In this case study, the boring campaigns collected VOC information over the entire site. Laboratory samples were severely under collected for the size of the site. From an agency standpoint, this site characterization would be incomplete and unacceptable. Figure 2 demonstrates the collected laboratory samples in the surficial soils.

3A Foxboro® OVA uses the principle of hydrogen flame ionization for detection and measurement of total organic vapors. The OVA meter has a scale from 0 to 10 which can be set to read at 1 X, 10 X or 100 X or 0-10 ppm, 10-100 ppm and 100-1000 ppm, respectively. The OVA is factory-calibrated to methane.

94

GEOSTATISTICAL APPLICATIONS

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. 1520

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.

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20'-'

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Figure 2-- Laboratory analyzed sample locations; ethylbenzene concentrations are shown. This paper salvages this extensive investigation by providing means to extract the maximum level of information from the existing data.

ANALYSIS OF DATA SET

Cross-correlations were computed between the laboratory analyzed compounds in order to determine applicable indicator compounds, if any. The available data were also analyzed for correlation between the laboratory-analyzed data and the OV A samples. Next, the characteristic distribution of the data was determined to select the most appropriate geostatistical technique for the spatial or structural analysis. Only benzene, ethylbenzene, toluene and xylene (BTEX) were investigated. All other compounds detected in the samples, together, made up less than eight percent of the total volatile compounds detected (EPA Method 8240 or Priority Pollutant compounds were tested) . These compounds, which were mostly methylene chloride measurements, were consistently measured at low levels. Both the OVA and VOC measurements were grouped into 3-foot intervals. Because the three boring campaigns did not always collect data at consistent depths, the intervals had varying amounts of data and spatial distribution. However, these measurements were distributed over various depths. Figure 2 shows only 13 samples in the surficial layer which was the most impacted layer in terms of horizontal extent. Cross-Correlation Analysis A cross-correlation analysis was performed between each variable to determine applicability of

95

WILD AND ROUHANI ON FIELD SCREENING

indicator compounds. For this purpose, the entire data set of 35 samples, which spanned various depths, was used. Ethylbenzene had the highest average correlation value with the other BTEX compounds, R2 = 0.92 (R2 is the correlation coefficient). In addition, ethylbenzene correlated well with total BTEX (R2 = 0.94). As mentioned previously, the soil-gas probes measure total VOCs. Therefore, ethylbenzene was used as an indicator of the other three parameters and for total VOCs . The correlation analysis perfonned for each BTEX compound versus the corresponding OVA reading produced poor correlations (Figure 3). The highest correlation coefficient was for the

-

1~.--------------------------------.

1400

--

ill-

1200

8' WOO E; 800 ' ~

o

•

600 400 200

o~

°

10,000

20,000

30,000

40,000

Ethylbenzene (Ppb)

Figure 3-- OVA to ethylbenzene correlation analysis. complete OVA data set versus ethylbenzene yielding an R2 = 0.37 . This low correlation coefficient indicates that direct correlation between laboratory and OVA measurements could not be justified. As discussed previously, similar results were also found by many investigators. Structural Analysis of OVA Measurements A structural analysis was perfonned on the surficial soil OVA measurements to determine their spatial correlation. Due to the qualitative nature of the OV A, the structural analysis exhibited a high degree of variability. It was therefore concluded that, given the non-gaussian shape of the histogram of OVA measurements, an indicator transfonnation was preferable. Two approaches were considered for this analysis. The first approach, suggested by Isaaks and Srivastava (1989), uses the median value of the data as the cutoff. Given the qualitative nature of the OVA data, the median cutoff value may have no real significance. Therefore, a second approach was developed. This approach identifies an OVA cutoff that would provide a high degree of confidence that the soil is less contaminated than an established regulatory threshold for petroleum hydrocarbons. This threshold was based on a review of a number of government guidelines on petroleum-contaminated soils. The threshold or target value could then be used to develop the conditional probability Prob[Ethylbenzene ~ Target

/ OVA]

(8)

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GEOSTATISTICAL APPLICATIONS

where Target =the cleanup or suggested maximum-allowable hydrocarbon contamination level. Calculation of the conditional probability for varying target levels (Figure 4) showed that there was a greater than 95 percent chance that the ethylbenzene level in the soil was equal to or less than a 100% 95%

,

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80% 75% 70%0

200

400

600

800

1000

OVA Readings (ppm) Figure 4-- Conditional probability based on OV A readings and regulatory cleanup standards. 20 parts per billion (ppb) target level, given an OVA reading of 20 parts per million (ppm) or less. Therefore, a target level of 20 ppb for ethylbenzene was selected as a conservative cleanup standard. The 20 ppm cutoff value for the OVA readings was similar to the median values for the surficial soils. Using the surficial soils data, the indicator variogram of OVA at 20 ppm (Y) is shown as Figure 5. 0.4 r - - - - - - - - - - - - - - - - - - - - - ,

•

0.3

•

0.3

!

0.2

o

~ 0.2 0.1 0.1

Distance (meters)

Figure 5-- Indicator variogram of OVA measurements at 20 ppm threshold

WILD AND ROUHANI ON FIELD SCREENING

97

As shown by this figure, the variogram demonstrated a well-defined spatial structure.

Structural Analysis of Ethylbenzene Recalling Figure 2, only 13 surficial ethylbenzene measurements were available for mapping soil contamination. As detennined in exploratory data analysis, the ethylbenzene measurements exhibited a tendency toward a log normal distribution. To account for this tendency, the natural log of the ethylbenzene measurements were taken. Furthermore, in order to avoid the possibility of numerical errors in the cokriging process, the log-transformed values were then normalized (mean =0, standard deviation =1). This made the latter data set numerically consistent with the indicator OVA values, thus minimizing the chance of numerical errors in cokriging. Unlike the OVA measurements, the variogram for the standardized, log-transformed ethylbenzene measurements (Z) demonstrated a relatively poor spatial structure (Figure 6). This short range prohibited accurate mapping with the current ethylbenzene data set. 1.4,-------------------,

•

1.2

~ 1.0 ~

0.8 0.6

10

20 30 Distance (meters)

40

50

Figure 6-- Direct variogram of normalized ethylbenzene measurements . All the above direct variographies were performed using the U.S. Environmental Protection Agency (EPA) public domain program, GEO-EAS (Englund and Sparks 1988). Cross-Variography of OVA and Ethylbenzene Measurements Cross-variography between the above two variables was conducted based on the linear model of co-regionalization (Rouhani and WackernageI1990). These computations were conducted using EPA's program, GEOPACK (Yates and Yates 1989). In this approach, the relationship between the direct and cross variograms is defined as

,i'

98

GEOSTATISTICAL APPLICATIONS + ~g2 +

yZ = algI yY = bIg] yZY = c]g]


(9)

such that (10) where yZ = the direct variogram of standardized, log-transfonned ethylbenzene, Z; y Y = the direct variogram of indicator OVA, Y; yZY =the cross-variogram of Z and Y; gi = the ith basic variogram model (sill = 1); '\ = the sill of gi in the nested variogram model of Z; bi = the sill of gi in the nested variogram model of Y; c i = the sill of gi in the nested cross-variogram model of Z,Y; and k = the number of basic models used in the nested variograms. The ratio of the fitted ci2/,\b i represents the correlation coefficient (Ri 2) at a scale consistent with the range of the ith basic variogram. Figure 7 depicts the cross-variogram between the standardized, log-transfonned ethylbenzene and 0.1.....---------------------,

~ -0.1

5b 0-0.2

'la ~

-0.3

•

-0.4

•

-O.5L-~-L-~-L-~-L-~-L_~_L_~~

o

10

20 30 40 Distance (meters)

50

60

Figure 7-- Cross-variogram of standardized ethylbenzene and indicator OVA. the indicator OVA. Table I summarizes the variogram models used for all structural analyses.

WILD AND ROUHANI ON FIELD SCREENING

99

TABLE 1 -- Summary ofvario&raphy.

Y

Spherical

0.03

0.07 0.2

21 40

Z

Spherical

0.6

0.4 0.25

21 40

Z,Y

Spherical

0.05

0.167 0.223

21 40

The scale-specific correlation coefficients displayed in Table 1 indicate that at micro-scales associated with the nugget effect, the correlation between the log-transformed ethylbenzene and the indicator OVA values is ratherlow (R,2 = 0.14). This can be attributed to measurement fluctuations. However, at larger scales, associated with the spherical models with ranges of 21 and 40 meters, the correlation between the above variables improves significantly (R22 = 0.996, R/ = 0.995). These correlations, however, can be masked by the poor correlations at the micro-scale distances. Consequently, a classical direct correlation analysis between the two measured values is very likely to fail.

CONCLUSIONS The mapping of the under-sampled ethylbenzene measurements is made feasible by incorporating the well-sampled OVA measurements. Figure 8 demonstrates the target 20 ppb contour for ethylbenzene. While the ethylbenzene measurements alone do not provide a basis for delineation of the contaminated soil, the cokriged map allows us to define the extent of contamination. However, specific areas of the site still require additional laboratory confmnation. These areas are south of the 26 ppb ethylbenzene measurement and between the two major tank pits. The above results show that field screening of contaminated sites can provide valuable information for characterization and mapping. This objective is accomplished by: Indicator transformation of OVA measurements based on a site-specific and regulatory-dependent conditional probability analysis; Multi-scale direct variography of transformed data; and Multi-scale cross-variography of the transformed OVA and laboratory measurements.

100

GEOSTATISTICAL APPLICATIONS

•

•

•

•

2.1'

•

• •• • •

•

_•

BUldng UST

• •

Soil Booing E.b<mooe • aJ ppb

Figure 8-- Results of co-estimation; ethylbenzene contamination extent In conclusion, attempting to directly correlate field screening and laboratory data is usually prone to failure. Instead, the auxiliary data is subjected to indicator transformation which is consistent with the qualitative nature of field screening data. This transformation, however, requires that a cutoff or threshold value be determined. In this work, the cutoff value is computed based on analysis of conditional probabilities of soil samples passing various regulatory criteria. Such an approach provides a flexible, site-specific algorithm for the transformation of field screening data and their eventual use for co-estimation. This combined information can then be cokriged with laboratory measurements to produce an information-efficient assessment of the extent of contamination.

REFERENCES Ahmed, S. and G. Marsily, 1987, "Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity," Water Resources Research. 23(9), pp 1717-1734 Crouch, M. S., 1990, "Check soil contamination easily," Chemical Enl:'ineerinl:' Prol:'ress, pp 41-45 Englund, E., and A. Sparks, 1988, "GEO-EAS (Geostatistical Environmental Assessment Software) Use's Guide," EPA600/4-88/033, ENMSL, Environmental Protection Agency, Las Vegas, NV Isaaks, E. H. and R. M. Srivastava, 1989, Applied Geostatistics, Oxfords University Press, New York Journel, A. G. and C. J. Huijbregts, 1978, Mininl:' Geostatistics, Academic Press, London

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Journel, A. G., 1983, "Non-parametric estimation of spatial distributions," Mathematical Geology, 15(3), pp 445-468. Marrin, D. L. and H. Kerfoot, 1988, "Soil gas surveying techniques," Environmental Science and Technology, 22(7), pp 740-745 Olea, R. A., 1991, Geostatistical Glossary and Multilingual Dictionary, International Association of Mathematical Geology Studies in Mathematical Geology No.3, Oxford University Press, 1991 Rouhani, S. and M. Dillon, 1989, "Geostatistical risk mapping for regional water resources studies," The Use of Computers in Water Management, in International Water Resources Association - Technical Session, Moscow, pp 216-228 Rouhani, S. and H. Wackernagel, 1990, "Multivariate geostatistical approach to space-time data analysis," Water Resources Research, 26(4), pp 585-591 Siegrist, R. L., 1991, "Volatile organic compounds in contaminated soil. The nature and validity of the measurement process," Conference - Characterization and Cleanup of Chemical Waste Sites, Washington D.C., Journal of Hazardous Materials, 29(1), pp 3-15 Smith, P. G., and S. Jensen, 1987, "Assessing the validity of field screening of soil samples for preliminary determination of hydrocarbon contamination," Superfund '87, Hazardous Materials Control Resources Institute, pp 101-103 Sullivan, J., 1984, "Conditional recovery estimation through probability kriging theory and practice," in G. Verly ~ aI..., eds., Geostatistics for Natural Resource Characteristics, Part I, D. Reidel Publishing Co., Dordrecht, pp 365-384 Thompson, G. M. and D. L. Marrin, 1987, "Soil gas contaminant investigations: a dynamic approach," Groundwater Monitoring Reyiew, 7(3), pp 88-93 Yates, S.R., and M.V. Yates, 1989, "Geostatistics for Waste Management: A User's Manual For the GEOPACK (Version 1.0) Geostatistical Software System," EPA, R.S. Kerr ERL, Ada, OK

Robert L. Johnson l

A BayesianJGeostatisticaJ Approach to the Design of Adaptive Sampling Programs

REFERENCE: R. L. Johnson, "A BayesianJGeostatisticaJ Approach to the Design of Adaptive Sampling Programs," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, and Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Traditional approaches to the delineation of subsurface contamination extent are costly and time consuming. Recent advances in field screening technologies present the possibility for adaptive sampling programs---programs that adapt or change to reflect sample results generated in the field. A coupled Bayesianlgeostatistical methodology can be used to guide adaptive sampling programs. A Bayesian approach quantitatively combines "soft" information regarding contaminant location with "hard" sampling results. Soft information can include historical information, non-intrusive geophysical survey data, preliminary transport modeling results, past experience with similar sites, etc. Soft information is used to build an initial conceptual image of where contamination is likely to be. As samples are collected and analyzed, indicator kriging is used to update the initial conceptual image. New sampling locations are selected to minimize the uncertainty associated with contaminant extent. An example is provided that illustrates the methodology.

KEYWORDS: adaptive sampling program, indicator kriging, Bayesian analysis, site characterization, sampling strategy

INTRODUCTION Characterizing the nature and extent of contamination at hazardous waste sites is an expensive and time-consuming process that typically involves successive sampling programs. The total cost per sample can be prohibitive when sampling program

lStaff engineer, Environmental Assessment Division, Argonne National Laboratory, Bldg. 900, 9700 S. Cass Ave., Argonne, IL 60439.

102

JOHNSON ON SAMPLING PROGRAMS

103

mobilization costs, drilling or bore hole expenses, and sample analysis costs are all included. For example, the Department of Energy (DOE) estimates that it will spend between $15 and $45 billion dollars for analytical services alone over the next 30 years to support environmental restoration activities at its facilities (DOE 1992). One of the primary products of a site characterization study is an estimate of the extent of contamination. Traditional characterization methodologies rely on pre-planned sampling grids, off-site sample analyses, and multiple sampling programs to determine contamination extent. Adaptive sampling programs present the potential for substantial savings in the time and cost associated with characterizing the extent of contamination. Adaptive sampling programs rely on recent advances in field analytical methods (FAMs) to generate real-time information on the extent and level of contamination (McDonald et al. 1994). Adaptive sampling programs result in more .cost effective characterizations by reducing the analytical costs per sample collected, by limiting the number of samples collected by strategically locating samples in response to field data, and finally by bringing characterization to closure in the course of one sampling program. Adaptive sampling programs can result in characterization cost savings on the order of 50% to 80% (Johnson, 1993). Supporting adaptive sampling programs requires the ability to estimate the extent of contamination based on available information, to measure the uncertainty associated with those estimates, to determine the reduction of uncertainty one might expect from collecting additional samples, and to direct sample collection so that sample locations maximize information gained. Two key characteristics of contaminated sites must be taken into account. The first is that spatial autocorrelation is often present when samples are collected. The second is that there may be abundant "soft" information regarding the location and extent of contamination, even if little "hard" sample data are initially available. Soft data refers to information such as historical records, non-intrusive geophysical survey results, preliminary fate and transport modeling results, aerial photographs, past experience with similar sites, etc. A number of geostatistical approaches to the design of sampling programs for characterizing hazardous waste sites have been proposed in the past. Early methods focused on minimizing some form of kriging variance (e.g., Olea 1984 and Rouhani 1985). More recent work has centered on stochastic conditional simulation techniques, Bayesian implementations of geostatistics and more complex decision rules (for example, Englund and Heravi 1992; McLaughlin et al. 1993; James and Gorelick 1994). In practice, site characterization sampling program designs tend to blend rigid sampling grids with selective sampling based on best engineering judgement. Typically there is little quantitative analysis to support the final sampling program design. A combined Bayesianlgeostatistical methodology is well suited to quantitative adaptive sampling program support. Bayesian analysis allows the quantitative integration of soft information with hard data. Geostatistical analysis provides a means for interpolating results from locations where hard data exists, to areas where it does not. A general Bayesianlgeostastical approach to merging soft and hard data is the Markov Bayes model described by Deutsch and Journel (1992). The Markov Bayes model estimates conditional cumulative density functions by developing covariance relationships between soft and hard data sets, and pooling the two different data sources through a form of indicator cokriging.

1".,

I·r' 'Ii

'II

:r,

104

GEOSTATISTICAL APPLICATIONS

The methodology described in this paper exploits the fact that environmental indicator sampling resembles binomial sampling. Binomial sampling events allow for the derivation of conjugate prior and posterior probability density functions, which in turn greatly simplifies computational effort. By incorporating soft information into an initial conceptual model that is subsequently updated as hard sampling data becomes available, the development of covariance models between soft and hard data is avoided. The classification of areas as clean or contaminated, and the selection of additional sampling locations, is based on a form of Type I and Type II error analysis, an approach consistent with the Environmental Protection Agency's (EPA) Data Quality Objectives approach to environmental restoration decision making.

METHODOLOGY Classical statistics estimates the most likely value for 1t, the probability of encountering contamination, by using hard sample data results. For example, if 20 random locations were sampled at a site and 5 of these samples returned contamination levels above an action threshold, then an unbiased estimator of the true probability of observing contamination above that threshold for any random location at the site would be the number of hits divided by the number of samples, or 0.25. In classical statistics one could carry the analysis one step further and develop confidence intervals around this estimator with some basic assumptions about the underlying probability distribution. Kriging provides similar results for individual points in space, accommodating spatial autocorrelation as well. Neither classical statistics nor geostatistics provide a means of quantitatively accommodating soft information in the analysis. For the design of sampling programs to characterize contamination extent and subsequent analysis of sampling program results, soft information often plays a crucial role. A Bayesian approach differs from classical statistics by assuming that parameters (such as the presence of contamination at a node) are unknown initially, but have some known probability distribution called the prior probability density function (pdf). As additional information becomes available (such as results from new sampling locations), these prior pdfs can be updated quantitatively using Bayes' rule to produce posterior probability density functions:

P(XIY)

oc

P(X)P(YIX)

P(XIY) is the posterior pdf for X, P(X) is the prior pdf for X, and P(YIX) reflects the probability distribution associated with observing Y given the prior pdf of X. From a Bayesian perspective, a two parameter beta distribution Be(o:,~) is a conjugate prior in the context of Bernoulli trials and the binomial distribution (Lee 1989). Be( o:,~) ranges between zero and one, and can assume a variety of shapes depending on the values of 0: and~. For a random variable 1t that follows a beta distribution, the expected value of 1t is given by:

(1)

JOHNSON ON SAMPLING PROGRAMS

£(11:)

(ex

+

P)

105 (2)

where: ex,P = parameters associated with the beta pdf for 11:, ex,P >= O. The variance of 11: is given by:

Var{11:)

(3)

Binomial distributions provide the probability of observing a specified number of successes within a specified number of trials. Conjugate priors are priors that retain their same underlying pdf after the application of Bayes' rule. In the case of a binomial trial with an unknown underlying probability 11: of seeing a success in any given trial, if X successes are obtained in N trials, a prior for 11: of the form Be( ex,P) becomes the posterior Be(ex+X,p+N-X). N functions as the total amount of additional information supplied to the prior. As N grows large, E(11:) approaches the classical maximum likelihood estimator for 11:, XIN, and the Var(11:) decreases monotonically. When one considers only the presence or absence of contamination above some threshold, environmental sampling resembles a binomial trial---N samples collected, X of which encounter contamination above the threshold. The primary difference is that environmental samples are not independent, as required in a traditional binomial sampling sequence. Sample values, even at an indicator level, are spatially autocorrelated. The issue is how to update a prior beta distribution at a given point in space with results from samples nearby that is consistent with the derivation of beta distributions as conjugate priors for binomial distributions and that recognizes their spatial autocorrelation. Two pieces of information are required from the set of samples: N' .0' the total amount of information represented by the set of samples appropriate for that point in space, Xo, and p' .0' the probability of encountering contamination at Xo based on the samples' results. Indicator kriging provides a means for deriving these two pieces of information. An unbiased estimator of p' at Xo is given by: N

Lwl(x) i=l

where Xi

Z(x)

=

locations where samples have been collected; 0 or I, depending on whether the sample at Xi encountered

(4)

1/ q

;, "

106

GEOSTATISTICAL APPLICATIONS

contamination below or above the threshold; kriging weights. The set of kriging weights, W, can be derived by solving the following set of simultaneous linear equations: Wi

=

N

E i= 1

for j= 1, .... ,N

CijW i + WN*l

(5)

(6)

where Cij CjO

= =

covariance between sample locations Xi and Xj; covariance between sample location Xj and the point where the interpolation is taking place, "0. N* at Xo can be tied to N, the number of samples taken, through the following relationship:

Varestim

- 1

(7)

N

Varestim

where Varestim Coo ~

=

=

Coo -

(E WiCiO

+

f.L)

(8)

i= 1

the estimation variance associated with the interpolation of p* at location xo; the variance of the indicator values; the average of the indicator values for the sample locations involved in the updating.

Equation (7) is heuristically based. When the sampled locations are all "distant" from the point of interest (i.e., greater than the spatial autocorrelation range), N* goes to zero, implying that the sampled locations contribute no information at the point of interest. As a sampled location comes close to the point of interest, N* goes to infinity, indicating that the sample information has specified the probability at the point of interest exactly. The methodology begins by defining a uniform grid over the region of interest. Grid nodes are designated as Decision Points (DPs). At each DP, a pdf based on the two

JOHNSON ON SAMPLING PROGRAMS

107

parameter beta distribution Be(ct,p) is defined. The beta pdf associated with each DP describes the probability of encountering contamination above a pre-selected threshold level at that DP. Initial values for ct and p are selected to represent a synthesis of any soft information available for a site, using equations (2) and (3). For a particular DP, the values of ct and p relative to each other determine the expected probability of contamination at that DP. The absolute sizes of ct and p determine the certainty associated with the beta distribution at that DP. For example, both ct =P=O.4 and ct =P=40 result in an expected probability of contamination equal to 0.5. However, in the latter case, the variance as calculated in equation (3) is much less. In the unlikely case where no information is available at a particular DP, a "non-informative" prior can be selected that sets ct and p equal to one at that DP. Updating the set of decision points with hard sampling data requires knowledge of the variogram or covariance function for the site. Because the values of p and N at x are independent of Coo' the primary covariance function parameters of concern are its shape, or functional form, and its range. If sufficient hard data exist, one can estimate the covariance function from an experimental variogram analysis. A simple measure of the uncertainty associated with contamination extent is to categorize decision points as either "clean", "contaminated", or state uncertain at a given certainty level, where the probability of contamination being present at any given decision point is based on equation (2) using the posterior beta pdf parameters that are associated with that decision point. For example, if one wishes to be 90% certain that the classification is correct when a decision point is classified as either clean or contaminated, then decision points with E(n) ranging between 0.1 and 0.9 would be classified as state uncertain. This definition of uncertainty parallels the use of uncertainty by the EPA in its Data Quality Objectives approach to decision-making. This method for handling uncertainty also leads naturally to measures of benefit one might expect from additional data collection. For example, one might wish to sample those locations that would be expected to maximize the number of decision points classified as "contaminated" or "clean" at a given certainty level, or to minimize the number classified as state uncertain.

EXAMPLE APPLICATION A simple example illustrates this methodology in action. Figure 1 provides a plan view of a hypothetical site with surface soil contamination. The site contains a waste lagoon that was breached during a storm. The owner's property is bounded by two secondary roads. The demarcated area indicates where surface soil contamination actually exists (7 940 m2) ___ an area unknown to the site owner. The owner acknowledges that contamination exists, and that portions of the site will require remedial action. The purpose of the characterization effort is to determine the extent of contamination so that the soils can be removed and treated off-site. The responsible regulator wants all contaminated soils identified and removed. The regulator wants to ensure that the sampling program is designed so that soils that are contaminated are not erroneously classified as clean. The owner will have to pay for the characterization, excavation and remediation of all soils believed to be contaminated.

108

GEOSTATISTICAL APPLICATIONS

)(

x

x

x

IPrope rty Llne l

x x

FIG. l--Example site

The owner wants to avoid remediating soils that are actually clean, and also to minimize his characterization costs. After negotiations, the regulator agrees to tolerate a 20% chance that a soil volume identified as clean is actually contaminated. The owner will be responsible for removing and remediating all areas that have greater than 20% chance of contamination being present. There is no initial hard sampling data for this site. The available soft information includes the location of the lagoon, scattered survey points from which a terrain model can be built to indicate the probable direction of overland flow and hence contaminant migration, the location of a utility building on site that would have been a barrier to flow, and the location of roads with embankments that would have also blocked flow . This soft information is used to construct the initial conceptual image of where contamination likely is, and where it likely is not. A grid is superimposed over the site that consists of 625 decision points (Figure

JOHNSON ON SAMPLING PROGRAMS

109

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2). At each decision point, a beta distribution is defined, with parameters selected to reflect the soft information available. For decision points that are in the building, IX is set equal to zero and Pto a very large number to reflect the fact that the interior of the building is known clean. For decision points within the lagoon, IX is set equal to a very large number and p equal to zero, to reflect that fact that the lagoon is known to be contaminated. For the balance of the decision points, IX and Pare set to values less than 0.5, with their relative sizes selected so that equation (2) reflects the initial probability of the presence of contamination. Figure 3 shows the gray-scale representation of the initial conceptual model once the beta distribution parameter values have been selected, along with a set of terrain contours based on-the available survey points. As is shown in Figure 3, the initial conceptual image is faithful to the location of the lagoon, building, and land surface contours. The area demarcated with the heavy black line indicates soil with contamination probability greater than 0.2 based on this initial conceptual modeL At this

110

GEOSTATISTICAL APPLICATIONS

Contam. Probability

x

I

o

1.00

FIG. 3--Initial conceptual model

point, without any sampling, the owner would have to clean up 34 440 m2 of soil, more than four times what is actually contaminated. Before the adaptive sampling program can begin, the methodology requires a covariance function. At the outset there is no hard data upon which to base a covariance function choice. If the covariance function were selected to honor the initial conceptualization, a range of approximately 200 meters would be used. The larger the assumed range, however, the fewer the samples that would be required to characterize the site. As a conservative start, for this example an isotropic exponential covariance function is assumed with range of 50 meters. A traditional sampling program for a site such as this would probably rely on a pre-planned, regular sampling grid. As a point of comparison for the subsequent adaptive sampling examples, Figure 4 shows an example pre-planned sampling program based on a triangular grid pattern. The gray-shaded surface contained in Figure 4 shows the results when a non-informative initial conceptual model is updated with the

JOHNSON ON SAMPLING PROGRAMS

I.

Sampling Poinl

111

I

x

x

Contam. Probability

0.011

0.99

FIG. 4--Standard sampling program results

information that would have been derived from this sampling program. The underlying beta distribution parameters for each decision point were set to a = P=0.1. In this scenario, the 14 samples result in classifying 23230 m2 of soil as requiring remedial action (i.e., the probability of contamination for these soils is greater than 0.2). This captures 87% of the soils actually contaminated, and includes 16230 m2 of uncontaminated soil. The classification of much of the clean area in Figure 4 as being contaminated is a product of the uncertainty associated with the use of an "ignorant" or non-informative prior during the updating process. If one uses the initial conceptual model shown in Figure 3, and updates it with the results from the sampling program shown in Figure 4, one obtains a different interpretation of the site. Figure 5 shows the results graphically. Using an initial conceptual model that reflects what is known at the outset about the site results in classifying 22 000 m2 of soil as requiring remedial action. This captures more than 98% of the soil actually contaminated, and includes 14 190 m2 of uncontaminated

112

GEOSTATISTICAL APPLICATIONS

x

'Ie

Sampling Point

I

x

FIG. 5--Standard sampling grid with initial conceptual model

soil. If one incorporates the underlying soft information available for the site, as displayed in Figure 3, and then sequentially selects 14 sampling locations that maximize the area that would be classified as clean at the 80% certainty level, then one obtains the pre-planned sampling program shown in Figure 6. The sequential selection of sampling locations proceeded as follows. First, a set of potential sampling locations based on a tight grid was established. Second, each potential sampling point was evaluated based on the impact sampling that point would have on the categorization of soils as requiring remedial action or not. If a potential sampling location had already been selected for sampling, then it was discarded. In this evaluation, it was assumed that the sampling result observed would be the most likely result based on the initial conceptual model conditioned with any locations either already sampled, or already selected for sampling. The potential sampling location that provided the greatest increase in the area of soil classified as clean would be added to the list of locations to sample. This process was

JOHNSON ON SAMPLING PROGRAMS

113

x

I_

Sampling Point

I

x

Contam. Probability

x

!

o

1.00

FIG. 6--Preplanned sampling program with initial conceptual model

used interatively until 14 locations had been selected. Figure 6 also shows the results from updating the underlying conceptual model with the results that would actually have been obtained from this pre-planned sampling program. These 14 samples reduce the amount of soil classified as requiring remedial action from 34 440 m2 in the original conceptual model to 15 120 m2 , a reduction, on average, of 1 380 m2 of soil reclassified as clean per sample collected. This captures more than 97% of the soil that is actually contaminated, and includes 7 395 m2 of uncontaminated soil. The selection of sampling locations for the pre-planned program was based on what was assumed would have been the results from sampling each of those locations. As the number of sampling locations included in a pre-planned program increases, the probability that at least one sample will encounter results that are unexpected also grows. In an adaptive sampling program, the results from previously selected sampling locations are available when the decision is made where to sample next. While the same process is

114

GEOSTATISTICAL APPLICATIONS

x

)(

I •. Sampling Point I x

x

Contam. Probability

x

I

o

1.00

FIG. 7--Extended adaptive sampling program

used for identifying the next sampling location, the difference is that the decision is conditioned on actual sample results, not assumed sample results as in the selection process for the pre-planned program. An adaptive sampling program at this site, driven by the objective of maximizing the area classified as clean at the 80% certainty level, would initially follow the same course as the pre-planned program shown in Figure 6. The reason is that for the fourteen samples collected as part of the pre-planned sampling program, all encountered what was expected---no contamination. In the case of an adaptive sampling program, one has the additional option of continuing sampling until the goals of the program have been met. Figure 7 shows the locations of an additional 14 samples for this site, along with the results from updating the underlying conceptual model with their results. The additional 14 samples reduced the area classified as requiring remedial action to 10070 m2 • This included 96% of the soil actually contaminated, and 2 460 m2 of uncontaminated soils. Each sample reclassified, on average, 350 square meters of soil, a significantly smaller amount than

JOHNSON ON SAMPLING PROGRAMS

115

obtained from the first 14 sampled. There are two reasons for this: first, there is simply less area available for reclassification to clean. The second is that the sampling has begun to encounter the unexpected---contaminated soil.

CONCLUSIONS Adaptive sampling programs provide the opportunity for significant cost savings during the characterization of a hazardous waste site. The challenge for adaptive sampling programs is providing real-time sampling program support that both incorporates the typically significant amounts of soft information available, and that accounts for the spatial autocorrelation that is omnipresent. A.joint Bayesian analysis/indicator geostastistical method can be used to guide the selection of sampling locations, to estimate the extent of contamination based on available data, and to determine the expected benefits to be gained from additional sampling. The example illustrates how the addition of soft information to the design of a sampling program can result in a more directed sampling strategy. When the ability to guide the program while in the field is added, the potential for cost savings is great. l.\tlll

ACKNOWLEDGEMENTS The work presented in this paper was funded through the Mixed Waste Landfill Integrated Demonstration, funded by the Office of Technology Development, Office of Environmental Restoration and Waste Management, U.S. Department of Energy through contract W-31-109-ENG-38.

REFERENCES Department of Energy, Analytical Services Program Five-Year Plan, Laboratory Management Division, Office of Environmental Restoration and Waste Management, Washington, D.C., January 29, 1992. Deutsch, C. V. And A.G. Joumel, GSLIB: Geostatistical Software Library and User's Oxford University Press, New York, NY, 1992.

~,

Englund, EJ. and N. Heravi, "Conditional Simulation: Practical Application for Sampling Design Optimization", Geostatistics Troia '92, A. Soares, ed., Kluwer Academic Publishers, Dordrecht, 1992, pp. 631-624. James, B. R. and S. M. Gorelick, "When Enough is Enough: The Worth of Monitoring Data in Aquifer Remediation Design", Water Resources Research, Vol. 30, No. 12, December, 1994, pp.3499-3514. Johnson, R. L., Adaptive Sampling Strategy Support for the Un!ined Chromic Acid Pit.

i

III JII

j" I'

116

GEOSTATISTICAL APPLICATIONS

Chemical Waste Landfill. Sandia National Laboratories. Albuquerque. New Mexico, Argonne National Laboratory ANLIEADffM-2, Argonne National Laboratory, Argonne, IL, November, 1993. Lee, P. M., Bayesian Statistics; An Introduction, Oxford University Press, New York, NY, 1989. McDonald, W. C, M. D. Erickson, B. M. Abratam, and A. R. Robbat, "Developments and Applications ofField Mass Spectrometers", Environmental Science & Technology, Vo!. 28, No.7, 1994, pp. 336-343. McLaughlin, L. B., L. B. Reid, S.-G. Li, and 1. Hyman, "A Stochastic Method for Characterizing Ground-Water Contamination", Ground Water, Vo!. 31, No.2, 1993, pp. 237-249. Olea, R. A., "Sampling Design Optimization for Spatial Functions", Mathematical Geology, Vo!. 16, No.4., 1984, pp. 369-392. Rouhani, S., "Variance Reduction Analysis", Water Resources Research, Vo!. 21, No.6, June, 1985,pp. 837-846.

Kadri Dagdelen 1 and A. Keith Turner 2 IMPORTANCE OF STATIONARITY FOR GEOSTATISTICAL ASSESSMENT OF ENVIROMENTAL CONTAMINATION ----~------

----------

REFERENCE: Dagdelen, K., Turner, A. K., "Importance of Stationarity for Geostatistical Assessment of Environmental Contamination," Geostatlstlcs for Environmental and Geotechnical Appliaations, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. I. Johnson, A. J. Desbarats, Eds., Ameriaan Society For Testing and Materials, 1996.

ABSTRACT: This paper describes a geostatistical case study to assess TCE contamination from multiple point sources that is migrating through the geologically complex conditions with several aquifers. The paper highlights the importance of the stationarity assumption by demonstrating how biased assessments of TCE contamination result when ordinary kriging of the data that violates stationarity assumptions. Division of the data set into more homogeneous geologic and hydrologic zones improves the accuracy of the estimates. Indicator kriging offers an alternate method for providing a stochastic model that is more appropriate for the data. Further improvement in the estimates results when indicator kriging is applied to individual subregional data sets that are based on geological considerations. This further enhances the data homogeneity and makes use of stationary model more appropriate. By combining geological and geostatistical evaluations, more realistic maps may be produced that reflect the hydrogeological environment and provide a sound basis for future investigations and remediation. KEYWORDS: Geostatistics, environmental contamination, kriging, second order stationarity.

~ ::

INTRODUCTION

"

,~

Determination of the extent of contamination at a site is usually based on the collection and analysis of a limited number of samples. Accurate assessment of these sample values requires knowledge of the geologic conditions and correct application of geostatistical methods to extend the sample values over the entire site area. These two requirements are mutually supportive. Geological conditions control the movement of contaminants; therefore evaluations of existing information concerning contamination are dependent on a clear and unambiguous understanding of the geologic framework. Site contamination patterns may be affected in significant ways by the geologic framework in regions surrounding the site. Thus, geologic studies should extend appropriate distances beyond the immediate site boundaries. Geostatistical methods are frequently employed to convert sampled values into a complete description of the contamination pattern at a site.

Assistant Professor, Mining Mines, Golden, CO. 80401. Professor, Geological Golden, CO. 80401.

Engineering

Engineering

117

Dept.,

Dept.,

Colorado

Colorado

School

School

of

of

Mines,

:

", t,: "

118

GEOSTATISTICAL APPLICATIONS

Ordinary kriging is recognized as the best linear unbiased estimator (II.L.U.E.) that minimize the variance of error in determining the average contaminant concentration at unsampled locations. The mechanics of the kriging process are relatively straightforward. However, kriging is based on several assumptions concerning the character of the model, and kriging produces a B.L.U.E. only as long as these assumptions are not violated. Violation of these assumptions may result in strongly biased kriged estimates and a flawed site assessment. Kriging procedures are based on a random function model that is second order stationary. The stationarity of the model is the chief assumption of the kriging procedure that is often violated. A random function is said to be stationary if the probability distribution of each of its random variables is the same. A random function is first order stationary if the expected mean value of each of its random variables is the same. A random function is said to be second order stationary if, in addition, the covariance between pair of random variables exist and is the same for all points separated by a distance ~h (Journel and Huijbregts 1975). The unbiasness condition of kriged estimates are based on a random function model that is first order stationary. The ability to view a particular sample data set as an outcome of a first order stationary random function is directly related to the ability of a set of samples to represent a local population whose expected value is the same at all locations of the search neighborhood. This paper describes why a data set coming from an environmental site may not be suitable to be analyzed under the assumptions of stationary random function model. It shows how ignoring this condition leads to biased kriged estimates and documents approaches to address the stationarity issue, hereby producing more accurate estimates of contaminant distribution.

SITE DESCRIPTION The site is located on 464 acres of land in the foothills of the Colorado Front Range 20 miles south-southwest of the city of Denver. Since 1957, activities at the site consisted of missile assembly, engine testing, and research and development for the Titan I, II, and III missile programs, and included fuels development, purification, and testing in support of the Titan III program.

Geologic Framework The site straddles the eastern margin of a portion of the Colorado Front Range. The western portions are dominated by Precambrian high-grade metamorphic and igneous intrusive rocks. Younger sandstone formations are found to the east of the Precambrian rocks. These now dip away from the mountain front at relatively steep angles, up to 500. Consequently, the eastern portions of the site are entirely restricted to the lower and middle portions of the Fountain Formation. Large and small fractures, faults, and shear zones, some over a mile wide and extending for many tens of miles are common in the Precambrian rocks. Renewed movements along several of these zones of weakness introduced fractures and faults within the younger sedimentary rocks. These sandstones are partly covered by unconsolidated Quaternary and Holocene deposits, composed of silty sandy gravels with substantial proportions of clay. However, the older of these units represent pediment surface deposits and are distinct from the younger units, which are valley-fill-alluvium deposited at lower elevations in more geographically restricted areas

DAGDELEN AND TURNER ON STATIONARITY

119

following a period of valley down-cutting. The Data Set The bedrock at the site is penetrated by about 80 drill holes extending to various depths (Figure 1). Figure 1 shows these borehole locations and highlights the five samples with the highest TeE concentrations. TeE concentrations were reported for III samples, but 31 of these samples were duplicates. The sampled TeE concentrations, range between 0 and 10,000 ppm and are skewed, with an arithmetic average of 328.8 ppm and a coefficient of variation of 3.96. As shown if Figure 2, considerable numbers of samples show "non detect" conditions, and only 50% of the samples exceed 3.0 ppm TeE. Evaluation of Hydrogeologic Conditions Three distinct hydrologic regimes are obvious at the site: the older Precambrian rocks, younger sedimentary rocks, and overlying unconsolidated deposits. Each has distinctive characteristics and interactions between these regimes are relatively complex. Ground-water flow in the Precambrian rocks may be characterized as a system governed by fracture flow. The intrinsic permeability of these rocks is so low as to be negligible, but fractures and foliation planes are common and pervasive. Numerous studies have demonstrated the importance of fractures in controlling ground-water flow within these otherwise relatively impermeable Precambrian rocks. A large zone of sheared rock is mapped along one major trend that crosses the western boundary of the site. Within such regions, there may be substantial hydraulic interconnection between surface water, ground water in the relatively thin, spatially-confined and discontinuous alluvial deposits, and the regional ground-water flow systems. Water movement through the Fountain sandstone is primarily controlled by its relatively low matrix permeability. These rocks contain considerable silt and clay which reduces and clogs the pores between the sand and gravel particles. Within the Fountain, the highest permeabilities are generally oriented parallel to the inclined bedding. Fractures are often sealed by calcite, which further reduces their ability to transmit water. The Fountain thus appears to have a relatively consistent and generally low value of effective hydraulic conductivity, especially in the direction normal to the PrecambrianFountain contact. This value is lower than the effective rock-mass permeability of the fractured portions of the Precambrian terrane. At least in some areas, it seems probable that the Fountain sediments may act as a "permeability blanket" to the regional groundwater flow system in the Precambrian terrane. The presence of a leaky hydraulic barrier in the lower Fountain would be manifested by artesian or confined ground-water conditions within the lower Fountain sediments, and springs and seeps or recharging streams along or near the Precambrian-Fountain contact. The presence of such seeps and recharging streams has been reported. The unconsolidated deposits have a generally similar texture and hydraulic conductivity values, but their recharge and discharge characteristics and inter-connections are highly variable throughout the site. The ability of these deposits to act as a single shallow aquifer system is uncertain. Movement of contaminants through upper portions of weathered Fountain bedrock may provide hydrologic connections, but the existence of such flow paths does not yet appear convincing.

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Blldrock Tee Data (log s<:ale)

Number 01 Oala 111 mean 328 .7676 SId. dev. 1304.3257

roe!. 01 va. 3.9673

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Histogram and descriptive statistics defining the sample distribution of the TeE values in the bedrock.

DAGDELEN AND TURNER ON STATIONARITY

121

Available well measurements were adequate to allow the construction of two potentiometric surface maps, one showing the distribution of heads in the bedrock aquifers and the second showing conditions within the shallow ~alluvialH aquifer represented by the entire suite of unconsolidated deposits. Potentioemetric contours for the ~bedrockH groundwater system suggest water flows from the west and discharges toward the east and southeast. Shallow aquifer contours also show a general west to east flow direction over much of the site. A difference map was created by subtracting values of the heads in the alluvial aquifer from those in the bedrock. On this map positive difference values correspond to areas where the potential flow is upward, from bedrock to the alluvial aquifer. Similarly, negative difference values correspond to areas where the heads in the alluvial aquifer are higher than in the bedrock aquifer and the potential for downward flow exists. Areas of upward flow are found mostly in the lower portions of the Fountain Formation, supporting the concept that significant ground water flow from the Precambrian rocks along fracture systems may be partially blocked, causing increased pressures within the lower Fountain. From this difference map, three distinct zones were defined: • a zone of downward flow where ground water may flow from the shallow alluvial units into the bedrock; • a zone of upward flow where ground water may flow from the bedrock units into alluvium; and • a zone where neither upward nor downward flow gradients are strong and there are no apparent preferred directions of vertical groundwater movements. Definition of Subregions at Site

In those zones where upward flow from the bedrock into the alluvium appears to dominate, most wells monitoring the alluvial units report TCE contamination. Yet, in this same zone, the majority of wells monitoring the bedrock ground-water flow system show no TCE contamination. In contrast, in those zones where downward flow dominates, alluvial wells, with only a few exceptions, report no TCE contamination at locations where most bedrock wells report TCE contamination. In the zone where neither upward nor downward flow appears to dominate, many bedrock and alluvial wells report TCE contamination. TCE contamination of the Fountain bedrock thus appears to be mostly restricted to those portions of the site where downward ground-water movement from the alluvial units may be occurring. In these locations, small groups of bedrock wells reporting TCE contamination are surrounded by non-contaminated bedrock wells. The reported TCE contaminants in the bedrock thus appear to be directly related to downward movement of contaminants from the overlying unconsolidated materials. Based on such hydrogeologic evidence, the site was divided into subregions. Four initial subregions were defined by examining the shallow unconsolidated deposits in terms of: (a) the hydrogeological setting, (b) the directions of ground-water flow, and (c) the location of known contaminant sources. Their boundaries were largely defined by interpreted ground-water flow directions and ground-water divides identified by analysis of the potentiometric contours. Comparison of these initial subregions with the zones of potential vertical ground water movements, defined by the methods described earlier, and with known major contaminant source areas, yielded six subregions (Figure 3).

122

GEOSTATISTICAL APPLICATIONS

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Figure 3 . Site map showing the six subregions. Each subregion represents an area that which is believed to contain a distinct combination of surface and bedrock geologic conditions, and a cornmon contaminant source or sources . Thus each should have a distinct population of contaminant values , and sample values from within each subregion should be considered as an outcome of a stationary random field. Each subregion should be evaluated independently by geostatistical methods , when there are sufficient samples within the area , or by visual inspection when there are too few samples to allow for geostatistical analysis. THE ASSUMPTION OF STATIONARITY IN ESTIMATION The theoretical derivation of the kriging procedure is based on the assumptions that the data observations can be conceptualized as an outcome of a second- order stationary random function. That is , the variable being measured has the same mean value at all locations and the same spatial covariance or variogram function between all points separated by a distance ~h.

DAGDELEN AND TURNER ON STATIONARITY

123

In practice, the assumption concerning stationarity of the mean values requires the sample set being evaluated to be derived from the domain under study in such way that, at any point in the domain, the expected values of samples surrounding this point should be the same. In other words, the probability of sampling high values within any local region will be the same throughout the domain, as will the probability of sampling low values. In a similar fashion, second order stationarity requires that the expected value of the squared difference between pairs of points ~h apart in a given direction to remain the same throughout the domain. Site-wide sampling campaigns may provide data sets that may be inappropriate for analysis by stationary random function models, especially if sampled locations preferentially represent locations of contaminated zones within a larger domain (Isaaks and Srivastava 1989). When data sets representing zones of different concentration levels are mixed to form a single data set, the stationary random function model may no longer be justified. When this combined data set is analyzed by kriging, samples coming from one stationary domain will influence the estimation of unknown concentrations in other domains, violating the assumptions of the stationary random function model and resulting in biased estimates. Indicator kriging determines, by using the samples in the neighborhood, the probability of data values in a given area being greater than a defined threshold value(Journel 1983; Isaaks 1984). To conduct indicator kriging, data values are transformed into indicator values: original values which exceed the chosen threshold value are coded 1, and those below the threshold value are coded O. These indicators are then analyzed to determine their spatial directional variability with a series of experimental variograms. By inspection of these variograms, orientations of greatest and least spatial continuity are selected. Variogram models are fitted to the experimental variograms corresponding to these two directions. Then the indicator data are kriged using these variogram models to determine the probability of exceeding the threshold value at a series of desired grid locations. Though actual values from multiple local zones of contamination cannot be combined to provide unbiased estimates of the average contamination at a given unsampled location, experience has shown that it may be appropriate to combine the median indicator values and treat them as an outcome of stationary random function model. ANALYSIS PROCEDURES Limitations to the use of ordinary kriging are illustrated with the data from this site. For these analyses, a threshold limit of 3.0 ppm TeE contamination was selected, because it seemed to be about the lowest reported value in any of the wells and was slightly less than the drinking water standard of 5 mg/l. When data from the entire site were combined and evaluated by ordinary kriging procedures, the resulting bias produced over-estimation of the observed concentration values over much of the site(Table 1). The data were then divided into subregions defined by careful interpretation of geologic conditions at the site, as described previously. These subregional data sets were individually analyzed with ordinary kriging procedures, and although a lower degree

124

GEOSTATISTICAL APPLICATIONS

of over--estimation of the observed values was observed (Table 1). The j,ias in these estimates was still considered unacceptable. Thus 'r.dicat:or kriging methods were used to determine if assumptions of stationarity could be achieved by this method, hereby providing more accurate ("unbiased") estimates (Table 1). Table 1 Summary of Cross Validation Results, Showing Over- and UnderEstimation Rates Achieved by Different Analysis Methods. Procedure Ordinary Kriging on entire data set (Fig.7) Ordinary Kriging on Subregions (Fig.9) Indicator Kriging on entire data set (Fig.ll) Indicator Kriging on Subregions (Fig.13)

# False Positives 25 (35%)

# False Negatives 5 (7%)

21 (29% )

4 (5%)

19 (26% )

17 (24%)

l3

12 (17%)

(18%)

Ordinary Kriging Using Data from Entire Site Pairwise relative variograms were created using routines in GSLIB (Deutsch and Journel 1992) to determine directional anisotropies within the entire data set. Figure 4 shows the contour map of the resulting variogram surface. The main axis of anisotropy is aligned along the Azimuth of 112.5°(see Figure 4), and the anisotropy ratio is 0.5. Figure 5 shows eight directional variograms oriented at 22.5° intervals. The modeled variogram uses a spherical model with a range of 700 ft, a sill of 1.2 ppm, and a nugget of 0.2 ppm. Figure 6 shows results of ordinary block kriging of the entire data set using the above parameters, and a minimum of 3 and a maximum of 16 samples. The map suggests that almost all the areas covered by drill holes are contaminated at levels exceeding 3.0 ppm, although examination of the observational data revealed that only 43.5% of the drill holes exceed this value. Considerable bias toward over-estimation has apparently occurred (Table 1). Figure 7 shows the results observed by cross-validation of kriged estimates and sampled values. Cross validation allows testing of the estimation method at locations of existing samples. The sample value at a particular location is temporarily discarded from the sample data set; the value at the same location is then estimated using the remaining samples. The procedure is repeated for all available samples (Isaaks and Srivastava 1989). On Figure 7 (and also in Figures 9,11,and 13, a circle enclosing a plussign represents locations where the sample value is below 3.0 ppm, yet the kriged estimate is

125

DAGDELEN AND TURNER ON STATIONARITY

greater than 3.0 ppm. Thirty six percent of the locations (25 of the 72 locations that had at least 3 samples within the search window) were estimated as contaminated (over 3.0 ppm) when , in reality , the sampled value was below 3.0 ppm. These results are summarized in Table 1 .

500

400

LEGEND

300

1.80

200

1.60 1.40

100

1.20

0

1.00

Y

0 .80

-100

0 .60 0 .40

-200 0.20

0.00

-300

-400

-500+"---P---""f----t=="'---t--=="'--+--'--+-----'t-----¥-----''-fL----''----+_ -50

Figure 4.

X Contour map of the pairwise relative variogram surface for the TCE data set for the entire site .

Such over-estimation has important consequences; the kriging suggested that 86% of the area may be considered contaminated while only 43.5% of the samples showed such contamination . I f the kriged va l ues are accepted as correct , substantial remediation costs can be expected . Figure 7 also shows that 5 sample locations were estimated as not contaminated (under 3.0 ppm) when in reality , the sampled value was above 3.0 ppm.

126

GEOSTATISTICAL APPLICATIONS

'~ !,---:::-----:::--.. :::--

= - = - -=,

DIoc.r-,h

..'"

01, - . " ,

...,

Figure 5. Eight directional experimental relative variograms and the fitted model for the data set for the entire site.

Ordinary Kriging Using Data by Sub-regions

In order to produce a data set that can be viewed as an outcome of a second-order stationary model for the kriging purposes, the entire data set was partitioned according to the six subregions described previously. Ordinary kriging procedures were applied to these individual subregions using the global variogram model given earlier . Figure 8 shows the results of this process , while Figure 9 shows the cross-validation plot of these results . By comparison with Figure 7 , it can be seen that the overestimation bias has been somewhat reduced, yet 21 (29%) sample locations remain over-estimated (Table 1).

DAGDELEN AND TURNER ON STATIONARITY

127

N612500

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N608000

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128

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.,.

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Eas1ing Map showing cross-validations for ordinary kriging applied to the data sets for the subregions.

DAGDELEN AND TURNER ON STATIONARITY

129

Indicator Kriging Using Da·ta from Entire Site

To further explore applicability of estimators based on second-order stationary models for the data, indicator kriging was used to analyze the entire data set. The TCE data values were transformed into 0 and 1 values, depending on their values relative to the 3.0 ppm (the median of sample values)TCE threshold. Directional variograms were produced. Indicator kriging was then applied to produce a map of probabilities of any location exceeding the 3.0 ppm threshold (Figure 10). Figure 11 shows the cross-validation plot of these probability estimates against the actual occurrence of sample values greater than 3.0 ppm ( using 50% or greater probabilities). There are 19 locations (26%) with false positives and 17 locations (24%) with false negatives (Table 1). The bias toward over-estimation has been further reduced, but additional reduction in the numbers of false positive and false negative locations appeared desirable. Indicator Kriging Using Data by Sub-regions

The indicator kriging process was then independently applied to the individual sub-regional data sets. Figure 12 shows the estimated probability of exceeding 3 ppm in each block. Figure 13 shows the cross-validation plot for these resulting probability estimates against the actual occurrence of sample values greater than 3.0 ppm. A further reduction in the degree of overestimation of the area of contamination exceeding the 3.0 ppm threshold is evident. There are 13 false positives, (18%)and 12 false negatives (17%) (see also Table 1).

ttlll 11,11

CONCLUSIONS

Mis-interpretation of the extent and degree of contamination at a site is likely to occur when traditional kriging is applied to a sample data set that does not consider the geological complexity. This result is likely because: •

Kriging should not be applied to data sets having a coefficient of variation greater than 1.0, since a few high concentration samples in such skewed data sets make the model assumptions inappropriate for the data at hand, resulting in biased estimation of local averages.

•

One of the important assumptions of geostatistics is second-order stationarity. In order to be able to apply kriging, a given data set must combine samples so that they can be conceptualized as an outcome of a second-order stationary of random function. This means that the data being processed by geostatistical kriging should come from a single consistent population. Only data from similar contaminant sources and geologic environments are likely to satisfy the stationarity assumption of the model.

For the example site discussed in this paper, the coefficient of variation for the bedrock TCE data is approximately 3.96, much greater than the limit of 1.0 defined above. At this site, because the TCE contamination appears to come from multiple point sources and to be

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N60BOOO

, 2,U$a7'-.--'---'-'-->'2f'Oh50r"----"--->'2fO*'2•.---"---'---.,2fl

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DAGDELEN AND TURNER ON STATIONARITY

N608000

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,---J----'-'-21~ ...1~25,-c-...J''-.,2roo1soo,---J-'-J'-~'''''5--'---'-'~2~1"'!o6~---'-' --'-'~21-.Jo8s:!-~---'----'-~L

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132

GEOSTATISTICAL APPLICATIONS

,~CClt rollEd cy dLO!ererlt ground-water flow regimes, the assumption of stationarity was not satisfied. Hence, application of the ordinary kriging technique to the entire site without subdivisions gave biased and erroneously high estimates of local TCE values, "smearing" high TCE values into locations where they do not actually occur. Such a "smearing" effect presents a false impression of widespread TCE contamination throughout the site, and suggests the presence of a large contaminant plume. Cross validation analysis provides a means for assessing the degree of bias and therefore the appropriateness of the kriged estimates at existing sample locations.

Indicator kriging was used to analyze bedrock TCE contamination data with a threshold limit of 3.0 ppm. (median of the data values). This method indicates that many areas within th~ site have low probabilities of being contaminated with TCE above this very low threshold level. Application of indicator kriging at higher threshold levels will define even more restricted areas as having significant probabilities for higher levels of TCE contamination. Analysis of the entire data set by indicator kriging procedures still resulted in slightly biased estimation; better results were obtained when indicator kriging was applied to subregional data sets. These results are summarized in Table 1. Indicator kriging is thus proposed as an appropriate method for developing realistic estimates of contamination levels at many geologically complex sites. It provides a mechanism for substantially meeting the underlying assumptions of stationarity in the model. Coupled with a complete conceptualization of the geological and hydrological framework for the site, optimal estimates may be achieved by applying indicator kriging methods to subregional data sets that reflect geologic controls. This approach will identify location of the misclassification bias both with respect to overestimation and underestimation and provide more accurate assessment of contamination limits. REFERENCES

Journel, A.G., and Huijbregts, C.J., 1975, Mining Geostatistics, Academic Press, New York, NY. Isaaks, E., and Srivastava, R., 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, NY. Journel, A.G., 1983,"Non Parametric Estimation of Spatial Distribution". Mathematical Geology: Vol. 15, No.3; 1983, pp. 445-468. Isaaks, E., 1984,"Risk Qualified Mappings for Hazardous Waste Sites: A case study in Distribution-free Geostatistics". Master's thesis, Stanford University, CA. Deutsch, C.V., and Journel, A.G., 1992, GSLIB: Geostatistical Software Library and User's Guide. Oxford University Press, New York.

Daniela Leonte 1 , and Neil Schofield 2

EVALUATrON OF A sorL CONTAMrNATED A GEOSTATrSTrCAL APPROACH

srTE

AND

CLEAN-UP

CRrTERrA:

REFERENCE: Leonte, D. and Schofield, N., "Evaluation of a Soil Contaminated Site and Cleanup Criteria: A Geostatistical Approach," Geostatistics foI' EnviI'onmental and Geotechnical Applications, ASTM STP 128:5, R. Mohan SI'ivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

,II>

Ii" I<

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ABSTRACT: A case study of soil contamination assessment and clean-up in a site proposed for residential development is presented in this paper. The contamination consists mainly of heavy metals of which lead is the most important contaminant. The site has been sampled on an approximately 25 x 25 square meter grid to between 1 and 3 meters depth to evaluate the extent of the contamination. Three hotspots were identified based on eyeballing the lead sample values and a crude contauring. A geostatistical approach is proposed to map the lead cont~ination and provide an alternate evaluation. The results suggest a significantly different evaluation of the area for clean-up based on the probability of the lead concentration exceeding allowable levels.

KEYWORDS: soil contamination, hotspots, thresholds, indicator kriging, conditional probability

geoscatistics,

rNTRODUCTrON The issue of contaminated land has only recently become of importance in Australia, although chemical contamination of land and groundwater has a long history, going back to the first years of European settlement. The actual extent of the problem is yet to be accurately determined, with some predictions placing the number of

lSenior Environmental Scientist, McLaren Hart Environmental Engineering (Australia), 54 Waterloo Road, North Ryde, NSW 2113. 2Manager, FSSI Consultants (Australia) Pty. Ltd., Suite 6, 3 Trelawney Street, Eastwood NSW 2122.

133

" '"'

134

GEOSTATISTICAL APPLICATIONS

c:mtaminated sites around 10 000 [.ll. Much of the regulatory framework deal;.ng with the management of contaminated sites has been developed during the last decade. The Australian and New Zealand Guidelines for the Assessment and Management of Contaminated Sites, prepared jointly by the Australian and New Zealand Environment and Conservation Council (ANZECC) and the National Health and Medical Research Council (NHMRC), were released in January 1992 (2]. This document provides the unified framework within which individual States are developing their own legislation and guidance. Pollution control requirements are administered directly by the Environment Protection Authorities in New South Wales, Western Australia and Victoria, by the Department of Environment and Planning in South Australia, and by the Departments of Environment in Queensland and Tasmania. Specifically for the soil medium, the present lack of a unified legislative approach results from a combination of factors, the most important being: 1. More than twenty different soil profiles exist in Australia, including many where there is a sharp distinction between various horizons; as a result the natural levels and range of chemical components vary significantly throughout the country. 2. A myriad of different plant and animal species are unique to this continent [~l. 3. The value of land is still driven by commercial rather than environmental factors. Consequently, criteria-based standards which involve predetermined clean-up levels are not entirely favoured by both the public and various regulatory bodies. The ANZECC/NHMRC document, recognising the need for flexibility, concluded that "the most appropriate approach for Australia is to adopt a combination of the site-specific and criteria-based standards". This methodology incorporates, at a national level, a general set of management principles and soil quality guidelines which guide site assessment and clean-up action, obviating, where appropriate, the need to develop costly site-specific criteria. However, this approach also recognises that "every site is different" and that "in many cases, sitespecific acceptance criteria and clean-up technologies will need to be developed to reflect local conditions". As a result, the national guidelines provide a set of criteria to assist in judging whether investigation of a site is necessary. Soil quality guidelines, based on a review of overseas information and Australian conditions, give investigation threshold criteria for human health (Table 1) and the environment (Table 2). Levels refer to the total concentrations of contaminants in dry soil, and have been defined from a small number of investigations in both urban and rural areas. Background criteria pertaining to the level of natural occurrence for various chemical components in soils are also specified in the guidelines. Site data with levels less than the criteria indicate that the quality of soil may be considered as acceptable irrespective of land use, and that no further contamination assessment of the site is In cases where the contaminant concentration exceeds the required.

LEONE AND SCHOFIELD ON CONTAMINATED SITE

135

criteria, a contamination problem may exist at the site and requires further assessment. As the guidelines do acknowledge that Table 2 is "conservative and has been set for the protection of groundwater", most state environmental regulatory agencies use these levels as a starting point for further investigation, and determination of clean-up levels specifically for each site. TABLE 1--Proposed Health Investigation Guidelines [21.

Substance

Health LeveL, mg/kg

Lead Arsenic (total) Cadmium Benzo(a)pyrene

300 100 20 1

'III

TABLE 2--Proposed Environmental Soil Quality Guidelines [21.

1111

'I

III' ~ III

ti,'

Substance

Background

Antimony Arsenic Barium Cadmium Chromium Cobalt Copper Lead Mercury Molybdenum Nickel Zinc

4 - 44 0.2 - 30 20 - 200 0.04 - 2 0.5 - 110 2 - 170 1 - 190 <2 - 200 0.001 - 0.1 < 1 - 20 2 - 400 2 - 180

Env. investigation

20 20 3 50 60 300 1 60 200

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I

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These criteria have been applied to the geostatistical case study discussed below.

THB

DATA

SBT

Site Description The site considered in this study is an almost rectangular parcel of land of some 70 000 m2 , having a general flattish topography with a slight fall to the centre. Its entire history is not well recorded and the site is only known to had been occupied by a brewery from 1885 to 1910. The site is now vacant with all buildings having been removed between 1984 and 1986. The majority of the land in the region was used by timber merchants for milling of timber from 1928 to 1980. It is

Ii' ,:::1

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136

GEOSTATISTICAL APPLICATIONS

also known that the whole area, being a long strip of land along a ~') 7"ner wharf, is heavily contaminated. The contamination is associated with old practices of dumping, both domestic and industrial residues, in times when legislation controlling waste disposal in Australia was nonexistent and mudflat "reclamation" practices of this manner were actually encouraged. These residues are known from other nearby areas, to be of both Australian ane overseas sources. A development proposal to use the site for a medium density residential development comprising some 200 residential units and a retirement village complex initiated a site assessment as an initial evaluation of the potential for soil contamination. The site was sampled by taking 54 mrn diameter continuous cores from boreholes located on a grid of approximately 25 x 25 m2 . The boreholes were sampled every 500 mrn to a depth of 3 m and submitted for chemical analysis. The 1 000 - 1 500 mrn and 1 500 - 2 000 mrn layers were alternated between boreholes. The depth to which samples were taken was based on the depth to groundwater and natural soil, which was recorded on the borelogs. Samples were taken by splitting the core down the middle. The remaining core was retained for reference and in the cases of "hotspots", was used for further testing. Sampling of Hotspots All samples collected initially were analysed for a suite of parameters which included pH, Cadmium (Cd), Chromium (Cr), Copper (Cu), Nickel (Ni), Zinc (Zn), Lead (Pb), Mercury (Hg), Sulphur (S), Arsenic (As), Cobalt (Co), phenols, cyanides and total hydrocarbons. The chemical analysis revealed certain boreholes where concentrations of heavy metals, of which lead was the most important, were significantly higher than the global mean for the site. These groups of holes defined hotspots which were investigated in detail to map the extent of the contamination. Four additional boreholes were drilled around each hotspot at approximately 12.5 m spacing. Discussion of Chemical Analyses Chemical analysis was carried out on a total of 378 samples. Elevated levels of lead were found in approximately 150 samples. Calculation of the arithmetic mean across the site for various contaminants, showed that in the case of lead, the mean was 572 ppm, with a high standard deviation which reflects the highly variable nature of the contamination. In hotspot areas, lead levels as high as 3.7% (37 000 ppm) were measured. Analysis of the lead levels across the site and down the individual boreholes showed the following arithmetic mean and standard deviation values (in ppm) for the number of data points (n) in each layer:

layer layer layer layer layer

1 2 3 4 5

(0 - 500 ( 500 - 1 (1 000 (1 500 (2 000 -

mrn) 000 mrn) 1 500 mrn) 2 000 mrn) 2 500 mrn)

Mean 963.96 589.55 387.44 83.16 11. 00

St. deviation 4 269.92 1876.59 746.76 148.34 8.32

n 126 126 88 52 18

LEONE AND SCHOFIELD ON CONTAMINATED SITE

137

These statistics suggest a decrease in contamination concentration with depth, especially bellow 1.5 m. The number of samples in each layer is also decreasing with depth. Findings of the Initial Site Assessment Following chemical analysis of lead concentrations, results were used to delineate the hotspots by considering the midpoint between two nearby samples which generally followed the "rule" of one sample showing a lead concentration above the acceptable limit and the other one below this limit. However, this rule has not been obeyed for all sampling points. Specifically in bore 109 (see Figure 2), lead levels of 2 450 ppm in the first 500 mm and 1 640 ppm in the 500 - 1 000 rom soil depth intervals were considered as being isolated, and mixing during the excavation of soil was recommended as a clean-up method. However, bores 76 and 75, located to the south and north of bore 109, indicated a much lower concentration in layer 1 and a higher concentration in layer 2, as shown below:

111'

'"

1i! 1:1

I' rd,

B 76

o-

500 mm 470 ppm 500 - 1 000 mm 2 650 ppm 1 000 - 1 500 mm 180 ppm

B 75 14 ppm 3 150 ppm not sampled

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1:11

1'1

Furthermore, Figure 2 shows that 2 of the hotspots on site are located immediately to the east of bores 76 and 75. These details indicate that the extent of contamination in the north, could be larger than that estimated in the first site assessment study. A similar situation is encountered in the south-east corner of the site, where boreholes on easting 295 m and 320 m showed very high lead concentrations at one or several depth levels. Because high concentrations did not appear clustered on all levels, it was concluded in the earlier study that no contamination existed there.

GBOSTATZSTZCAL

ANALYSZS

OP

LEAD

DATA

A file of the lead data was created to be used for a geostatistical study. The analysis was restricted to samples from the first three layers, where sampling has been relatively uniform. Statistical Analysis of Lead Data Figure 1 shows the declustered histogram of lead concentration in some 335 samples from the boreholes. The data have been declustered using the method of Schofield 1992(1] to account for the clustering of sampling in the hotspots. Lead shows a strongly positively skewed histogram with a very high coefficient of variation related to the presence of a few extreme values in excess of 10 000 ppm. The mean of lead is 793 ppm, well in excess of the third quartile and over twice the limit of 300 ppm above

"I

138

GEOSTATISTICAL APPLICATIONS

which investigation is recommended. For lead concentrations between 100 and 1 200 ppm, the histogram may be well fitted with a lognormal distribution model. Table 3 compares the cumulative probabilities for the declustered data and for a lognormal distribution with the same mean and log variance.

Statistics

...

# data: 323

Mean: 693.8

.J!!

(.)

Variance: 8190539

~

Coet.Var: 4.12

~ Cl

Minimum: 4.0

'0

1st Quart: 20.000

c::

0 .€

Median: 105.0

8.

3rd Quart: 430.0

0.

Maximum: 40800.0

e

mAD: 665.5

0.0

200

400

600

800

1000

Lead ppm

Figure I--Histogram of declustered lead concentrations in 500 mm samples.

Table 3--Comparison of the cumulative histogram of the declustered data and a lognormal distribution with the same mean and log variance. Lead, ppm

"I

'I

100 300 500 700 900 I 200

Cumulative Prob. Lognormal Distance

Cumulative Prob. Declustered Data

0.48 0.70 0.78 0.83 0.86 0.89

0.49 0.68 0.78 0.84 0.86 0.90

Spatial Distribution of Lead and Local HotsootS Figure 2 presents a contour map of lead concentration generated from a moving average of the lead sample values. The moving average method is described by Isaaks and Srivastava 1989 [~l as a useful tool for identifying local anomalies in the variability of data values. The map also shows the locations of the boreholes and the three hotspots

LEONE AND SCHOFIELD ON CONTAMINATED SITE

139

shaded in grey that were previously identified for clean-up. The contours show chree or possibly four areas of significant lead contamination. The contours do not show any preferred directional structure to the lead contamination but the lead concencration in the most easterly area is significantly higher than that in the other areas. The five samples with lead concentrations above 10 000 ppm all occur in the most eastern hotspot. Global Lognormal Model for Lead Swane et al. [~] has suggested that some regulatory agencies in Australia may favour defining acceptable clean-up criteria in terms of the 75th percentile of a lognormal model. This means that if the 75th percentile of a lognormal distribution model applied to the global histogram of lead concentrations is higher than the recommended limit of 300 ppm lead, further investigation and possible clean-up would be recommended. For the present lead data set, the 75th percentile of a lognormal model is 470 ppm. By removing the five samples with the highest lead values above 10 000 ppm, the 75th percentile is reduced to 296 ppm which may indicate that an acceptable site clean-up has been achieved. Therefore the global lognormal model would only require remediation of small areas of extreme contamination in order to reduce the global level of contamination and satisfy the acceptance threshold of 300 ppm lead for the site. Variogram Analysis of Lead

I

,

: I

The spatial continuity of lead concentration in the soil has been characterised by a set of ornni-directional indicator variograms for a range of relevant indicator thresholds. The actual spatial continuity measure used is the correlogram of Srivastava and Parker, 1988[ft]. Analysis of directional variograms did not indicate any preferred orientation to the lead contamination. There is no reason to believe that some scructured pattern of dumping lead contaminated waste would have been used at the site. Figure 3 shows a plot of the ornni-directional horizontal variograms. The limited vertical extent of sampling does not allow reasonable inference of the vertical continuity of lead contamination. The horizoncal variograms indicate an ornni-directional structure at all thresholds and an increasing nugget with increasing threshold. In all cases, the sample variograms have been reasonably fitted with a nuggeC and a single exponential model. Indicator Kriging Model The Indicator Kriging approach was used to map locally the probabilicy that the lead contamination in soil exceeds certain threshold concentrations some of which are used as clean-up criteria. The approach follows that of Isaaks 1984[~] in mapping lead concentration in soil around a lead smelter in Dallas, Texas and that discussed by Journel 1988[Q].

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Figure 2- Con kour map of lead con cencrations in the first soil lay er (0 SOO rom) showing bo r ehole and hocspo t l ocak i ons in grey shading

LEONE AND SCHOFIELD ON CONTAMINATED SITE

141

Lead Variog,a",

:2 ):'

~

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Model Parameters

co: 0.1 1 nugg C1: 0.91 exp range: 52 O.

0.0

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La Distance h IndicalDr 300 ppm

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0 O.

0.0

.4

120.5

1SO.7

La Distance h

Figure 3--Qrnni-directional horizontal sample yariograrns for l ead and for several indicator thresholds. Estimates of the local conditional probability for the lead concentration to exceed given thresholds were made using indicato r point kriging on a 10 meter square grid. Contours of the probability of exceedance for the 300 ppm lead thre shold for layer 1 and layer 2 are shown in Figure 4. The previously identified hotspots of high lead contamination are also shown with grey shading. The maps indicate

142

GEOSTATISTICAL APPLICATIONS

: . E. .~£f-= areas wi th a probability of at least 70 percent that the lead concentration in samples exceeds 300 ppm. The hotspots identified for clean-up lie close to the contaminated centres of two of these areas. However, a large area of high lead contamination in the south-eastern part of the site (20 m x 120 m northing and 270 m x 320 m easting) has been ignored completely, most likely because remediation of the previously identified hotspots would ensure compliance under the 75th percentile of a global lognormal criteria for lead at the site.

Layer 1, Pr (Pb > 300 ppm)

220

I CI t:

:E

120

~

o

Z

20 20

120

220 Easting (m)

320

Layer 2, Pr (Pb > 300 ppm)

220

E CI t:

:E

120

~

o

Z

20 20

120

220 Easting (m)

320

Figure 4 Cont ours of the conditional probability for lead concentration to exceed the recommended level of 300 ppm. Hotspots are shown by grey shading.

LEONE AND SCHOFIELD ON CONTAMINATED SITE

143

Figure 5 presents contour maps 0;: t he: '=s ·,:il r,at. e~1 pr::>(.clinl i ty for lead concentration to exceed 500 ppm in soils for layers 1 and 2. On these maps, the areas with very low probability of contamination are clearly shown. The areas with potentially high contamination are also clear with the southern area (20 m x 120 m northing and 270 m x 320 m easting) again standing out as unrecognised by the previous investigation.

Layer 1, Pr (Pb > 500 ppm)

220

~, 0.1-,-1

1 7 -- - , 1

I 01 ~

:E

120

t

o

Z

20 20

120

220 Easting (m)

Layer 2,

Pr (Pb > 500 ppm)

120

220 Easting (m)

320

220

I

01 ~

:E

120

t

o

Z

20 20

320

Figure 5 Contours of the conditional probability for lead concentration to exceed the recommended level of 500 ppm. Hotspots are shown by grey shading .

144

GEOSTATISTICAL APPLICATIONS

CONCLtl'SrONS

'I '" I:~

:( I'~

'", , '" ,,'

The recommendation of the ANZECC/NHMRC document for the use of both criteria-based and site-specific standards to assess soil contamination and clean-up is supported by the authors of this paper. The use of a universal or blanket standard for assessment of all sites appears inappropriate. This conclusion is supported by the outcome of applying the 75th percentile of a lognormal model criteria to the site in question in this paper. The cleaning of small areas of extreme contamination may often reduce the global level of contamination below some acceptance threshold. However, large areas carrying a significant risk of contamination above the acceptance threshold may go unrecognised and uncleaned. The application of geostatistical methods to analyse and model the lead contamination at this site appears appropriate. The dumping of lead contaminated material at the site does not seem to have been highly organised introducing considerable uncertainty as to the exact location of the contamination. Subsequent migration of the lead in soil due to natural processes has likely modified the spatial distribution of lead introducing greater complexity and uncertainty into its spatial geometry. Indicator kriging has enabled a mapping of the lead contamination at a local scale which permits an assessment of the risk associated with certain levels of contamination. When compared to previous attempts to identify areas of significant contamination (hotspots), the IK mapping indicates much larger areas associated with those hotspots where the risk of contamination is high. In addition, a large area of significant contamination which has previously gone unrecognised due to a naive decision rule, has been identified through geostatistical analysis. Although other techniques would have enabled estimation of the global lead contamination at the site by accounting for its specific directions of spatial continuity, the IK tool uniquely introduces the risk factor through the quantification of the uncertainty associated with the estimation process. Therefore making decisions on the extent and nature of remedial action to be implemented becomes a more informed process in which clean-up cost and potential liability associated with it can be evaluated.

REFERENCES

Lll

[~l

M.G. Knight. "Scale of the hazardous waste problem in Australia and disposal practice," Symposium on Environmental Geotechnics and Pyoblematic Soils and Rocks, Bangkok: Asian Institute of Technology, South-east Asian Geomechanics Society, 1985. Australian and New Zealand Environment and Conservation Council, and National Health and Medical Research Council (ANZECC/NHMRC). Australian and New Zealand Guidelines for the Assessment and Management of Contaminated Sites, January 1992.

LEONE AND SCHOFIELD ON CONTAMINATED SITE

U.l

[~]

[~]

[.§.]

[1]

[.a]

[.2.]

145

J. Daffern, C.M. Gerard and R.McFarland. "Regulatory and nonregulatory control of contaminated sites," Geotechnical Management of Waste and Contamination, Fell, Phillips and Gerrard (editors), Balkema, Rotterdam, 1993. E.H. Isaaks. Risk Oualified Mappings for Hazardous Wasce Sites; A Case Study in Distribution Free Geostatiscics. Master's thesis, Stanford University, 1984. E.H. Isaaks and R.M. Srivastava. An Introduction to Applied Geostatistics. Oxford University Press, 1989. A.G. Journel. "Non-parametric geostatistics for risk and additional sampling assessment," Principles of Environmental Sampling, L. Keith (ed.), American Chemical Society, 1988. N.A. Schofield. "Using the entropy statistic to infer population parameters from spatially clustered sampling," Proceedings of the 4th International Geostatistical Congress, Troia 92, pages 109-120, Kluwer, Holland, 1992. R.M. Srivastava and H. Parker. "Robust measures of spatial continuity," M. Armstrong (ed.), Third International Geostatistics Congress, D. Reidel, Dordrecht, Holland, 1988. M. Swane, I.C. Dunbavan and Riddell P. "Remediation of contaminated sites in Australia," Fell, Phillips and Gerrard, (editors), Geotechnical Management of Waste and Contamination, Balkerna, Rotterdam, 1993, pp 127-141.

,"

I'I!

III(

1111

'I

I

I I

Amilcar O. Soares', Pedro 1. Patinha2, Maria J. Pereira

2

STOCHASTIC SIMULATION OF SPACE-TIME SERIES: APPLICATION TO A RIVER WATER QUALITY MODELLING

REFERENCE: Soares, A. 0., Patinha, P. J., Pereira, M. 1., "Stochastic Simulation of Space-Time Series: Application to a River Water Quality Modelling," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

ABSTRACT: This study aims to develop a methodology to simulate the joint behaviour, in space and time, of some water quality indicators of a river, resulting from a mine effluent discharge, in order to enable the prediction of extreme scenarios for the entire system. Considering one pollutant characteristic measured in N monitoring stations along the time T, a random function X(e, t), e=l, ... ,N, t=1, .. . ,T, can be defined. The proposed methodology, a data driven approach, intends to simulate the realisation of a variable located in station e in time t, based on values located before e and t, and using a sequential algorithm. To simulate one value from the cumulative distribution F{X(e,t) I X(e-J,t), ... ,xU, t),x(e,t-I), ... ,x(e, I)}, the basic idea of the proposed methodology is to replace the e.t conditioning values by a linear combination of those:

* [x(e, t)1

=

e-I

(-I

u=1

~=I

L: au X( u, t} + L: brt X( e, ~) which allows the values to be drawn sequentially I-'

from bidistributions. The final simulated time series of pH and dissolved oxygen reproduce the basic statistics and the experimental time and spatial covariances calculated from historical data recorded over 15 months at a selected number of monitoring stations on a river with an effluent discharge of a copper mine located in the south of Portugal.

KEYWORDS: stochastic simulation, space-time series, water quality

'Professor, CVRM - Centro de Valoriza<;:ao de Recursos Minerais, Instituto Superior Tecnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal. 'Research Fellow, CVRM - Centro de Valoriza<;:ao de Recursos Minerais, Instituto Superior Tecnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal.

146

SOARES ET AL. ON SPACE-TIME SERIES

147

RIVER WATER QUALITY MODELLING The main objective of this study is to develop a methodology to simulate the behaviour, in space and time, of some water quality indicators of a river resulting from a mine effluent discharge, in order to enable the prediction of extreme scenarios for the entire system. The first results of the model application, in a case study of a mine located in the south of Portugal, will be presented. The time series of pH and dissolved oxygen were simulated, using historical data recorded over time at a selected number of monitoring stations on a river at Neves Corvo mine. The occurrence of extreme situations from the joint simulated time series: simultaneous high spikes of the parameters and durability in time and space of one extreme situation - can be visualised.

MODELLING METHODOLOGY: STOCHASTIC SIMULATION OF SPACETIME SERIES The proposed methodology is a data driven approach of sequential simulation of a random vector. Defining N dependent random variables Xj, X 2,... ,xN , the simulation of the cdf F( Xl, X2, ... ,xN) can be generated by a conditional distribution approach (Law and Kelton 1982, Johnson 1987, Ripley 1987), involving the following sequential procedure: · draw the first value Xl from the marginal distribution F(Xl ) of the RV Xl' · draw the second value X2 from the distribution ofRV X 2 conditioned onXl=xl: F(X2 I Xl=Xl) · draw the}/h value of the conditional distribution: F(XNI X l =Xl,x2=X2,"., XN-l=XN-l) The random variables Xj, X 2,... ,xN could be the same attribute of water quality measured in different time and spatial locations. The main limitation of the practical implementation of this algorithm is the calculation of all these conditional distributions, considering that, in most applications of earth and environmental sciences, just one realisation of Xj, X2, ... ,xN is available. Journel and Gomez Hernandez (1989) presented one solution for spatial realisations of F(Xj, X 2,... ,xN), in the geostatistical framework, using the indicator formalism (Sequential Indicator Simulation-SIS) or the multigaussian approach (Sequential Gaussian Simulation-SGS) (Deutsch and Journel 1992). Both methods apply for spatial stationarity: the stationarity of the indicator random function in SIS method or the stationarity of the gaussian transform function in SGS. Some cases, like a pollutant along the river coming from one point source, can not be considered spatially stationary. As a matter of fact, in the case study presented here, some sharp drops in the pollutant concentration are found between two monitoring stations. Thus, the application of the mentioned methodologies is not straightforward. Considering the pollutant characteristic measured in N monitoring stations over time T, the random function X( e,t), e= 1,.. ,N, t= 1,.. ,T, can be defined. The previous sequential procedure is written for X(e,t) : Monitoring Station I t=1 draw a value Xl of F(X(I,I))

148 t=2 t=T

GEOSTATISTICAL APPLICATIONS

draw a value X2 of F(X(I,2)1X(1,1)=xl) draw a value xrof F(X(I,1)IX(I,l)=xb .... ,x(I,T-I)=xr.l)

Monitoring Station N t=1 draw a value Xl of F(X(N,I)) t=2 draw a value X2 of F(X(N,2)IX(N, 1)=Xl) t=T draw a value Xr of F(X(N, 1)1X(N, 1)=x J, ••• ,x(N,T-l )=Xr-l) The basic idea of the proposed methodology is to substitute the NxT conditioning values by a linear combination of these: N-l

T-l

[X(e,t)]*= La"X(a,t)+ "

•

" j

Ij

:f.. :~"

"

.,

Lb~X(e,p) ~-l

With this approximation to simulate one value of X in monitoring station e in time t, instead of using the cdf - F[X(e,t)IX(l,l), ... ,x(e-l,t-l)] - one uses the bidistribution:

1

X(e,t) I[ X(e,t)] * =

~aaX(a,t -1) + t,bpX(e -1, (3)]

Thus, for all monitoring stations we have: Monitoring Station 1 t=1 draw a value Xl of F(X(I,I)) t=2 draw a value X2 of F(X(I,2) I [X(I,2)]* =X(I,I))=Xl) t=T draw a value XT of F(X(I,1) I [X(I,1)]* = b I X(1,I)+ ...+b r _l X(1,T-l)) = = F(X(1, 1)lbIXl+H+bT-I-xr-l)

I

I ":1

,Ii

il

."I

Monitoring Station N t=1 draw a value Xl of F(X(N,I)) t=2 draw a value X2 of F(X(N,2) I [X(N,2)]* = X(N,I)) = Xl) t=T draw a value xrof F(X(N,1) I [X(N, 1)]* = a\X(1,T-l)+ ...+b r_lX(N-l,T-l)) = F(X(N, 1)laIXl+H+br-l.xr-d

Practical Implementation of the Algorithm /

,,''" 'I

If the same pattern of neighbourhood values (in space and time) are used to calculate all [X(e,t)]*, e=1 ,N, t=1 ,T, the bidistributions functions F(X(e,t) I [X(e,t)]*) can be inferred from the historical data. Using the same neighbourhood pattern one can calculate the pairs (X(e,t), [X(e,t)] *) for all t=1,T and e=I,N. The bi-plots (X(e,t),[X(e,t)]*) can thus be calculated for each monitoring station and for homogeneous periods of time (see Fig.1). To simulate one value for X(e,t) conditioned on the known value [X(e,t)]* =x* (calculated with the previously simulated values X( e-l ,t-l)), first we need to select all pairs which belong to the class of [X(e,t)]* and, afterwards, one value ofX(e,t) is drawn randomly from them. Once the value Xs is simulated, X(e,t)=xs will be part of the next estimators [ (X(e+1,t)]* or [X(e,t+1)]*.

SOARES ET AL. ON SPACE·TIME SERIES

F(x(e,t)

I x· =

[X(e,t»)'

X

149

• • • • •• I.' . • I.

.:~.'~ ••

X S(o.')

•

'C"•• '" •

~---t_--Jr----::-.~,t-'

• •~ •

... .. ... ' :•

'1#' :

~. ~ ; : ~'. ~

•• :

:

• j

Fig 1. Illustrative representation of the simulation procedure from the bi-distribution.

STATISTICAL CHARACTERISATION OF SIMULATED TIME-SERIES With this ty'pe of data driven approach usually one wishes to mime the real data in some statistics of fundamental variables, which measure their spatial and time structures. The proposed algorithm generates a time series in each monitoring station with some statistics identical to the historical data: means, marginal histograms and spatial and time correlations. Considering for each monitoring station e, the statistics of historical data: m(e)- mean of the time data Fx(X;e)=prob{X(e,t)<x}- the marginal cdf of e Ce(h) = E{X(e, t)X(e, t+h}- m(e)2 - the time covariance in e, and the spatial covariance C(h)=E{X(e,t)X(e+h,t}- m(e).m(e+h)' Ce(h) and C(h) are measures ofthe spatial and time structure of the variableX(e,t). If the simulation sequence starts with a limited set of conditioning values with the marginal distribution FxCX;e), the bidistribution F(X(e,t)1 [X(e,t)]*) will generate a simulated time series with the same mean: m'(e) = m(e), and same marginal cdf F/ (X;e) = FxCX;e) . The simulated values Xs(e,t) reproduce the covariances between X(e,t) and [X(e,t) ]*:

150

GEOSTATISTICAL APPLICATIONS /-1

e-J

La"cov [X(e,t)),X(u,t)] + Lb~cov[X(e,t),X(e,l3)l ,,~l

~~l

[1] The simulated values Xs(e,t) reproduce an average of the time and space covariances. Consequently, the weights au and bl3 must be chosen in such way that the individual covariances are represented in final time series.

DEFINITION OF THE LINEAR COMBINATION [X( e, t)] N-I

*

)'-1

To define [X(e, t )]* = La" X(u, t) + Lb~X(e, (3) one could choose a centered H

.,

l "

and minimum variance estimator: E{X(e,t)}= E{[X(e,t)]*} and min (var{X(e,t)-[X(e,t)] *}), which entails to a kriging system written in terms of time and space covariances. For simplicity sake let us use the same notation for the weights and for the spatial and time location. The estimator of any location Xo is written:

[X(xo)r =

L A"X(X,,) "

J~ A"C(X",x~) + Jl = C(x",xo)

l~ A~ = 1

'I

The solution vector A combines two distinct effects: i) The proximity of the samples to the estimated point Xo determine their influence nd in terms of weights (2 member of the kriging system). ii) The declusterizing effect of the first member of kriging system leads usually to underweighting of clustered samples. Now the problem is that the samples in time series are usually clustered. Thus, due to the declusterizing effect, this minimum variance estimator tends to overweight the influence of the nearest sample and underweight the others influence. Consequently, the average covariance of [1] represents mainly the short distances structures. To avoid this drawback, in this study an estimator has been chosen which accounts only for the proximity effect, i.e., the weights are directly proportional to the correlation coefficient (p) between any sample Xu and the estimated point Xo : A" = Pu,o +

1I

l

Nl 1- ~P~o J

Note: this is the solution of the kriging system when there is a null correlation between the conditioning samples Xu .

SOARES ET AL. ON SPACE-TIME SERIES

151

Obviously, this is no longer a minimum variance estimator, but it assures the representativity of all co variances (in space and time) in the simulated time series. N- I

In summary, the estimator

T-I

[X(e,l)r = LaaX(a ,I) + Lb p X(e ,.8)

is defined

P- I

by the weights:

r

1 ["

Ja" = p".• + N.

lI

Tl I -

1 [ bp = Pl oP + N . T

[,-1

I-I]l J

£:P;oe + ~P/.j

e-I I-I]] l-l[" £:P;o. + ~P/.j

Note - This estimator was chosen for this particular case to avoid the overweighting of the small distances structures resulting from the ordinary kriging of clustered string data. The solution consisted of putting all covariances between samples equal to zero. However, other corrections of the covariances between samples can be adopted ( for example Deutsch 1994, suggests a kriging estimator with a 10umel ' s redundancy measure correction of the 1s l member of the equations system, for similar purpose).

MODEL VALIDATION WITH A CASE STUDY The data set used to implement the stochastic simulation consisted of monitored values of pH (daily analysis in lab) and dissolved oxygen at 4 monitoring stations located along a river, with a effluent discharge of a mine (Fig. 2), during a period of 15 months. The monitored data are shown in Figs.3a, 3b and FigsAa, 4b representing the time series, histograms and time variograms of 15 months period for pH and dissolved oxygen.

Mine Site ;

..

Ir,. '

Fig 2 - Layout of the mine site and the 4 monitoring stations along the river.

152

GEOSTATISTICAL APPLICATIONS

FF===~====~====~====~==o=n=it=o~riRng~tart~i~on~I__

- r____

pH 13

~r-________~-"

200

12 160

"

10

120 80

100

200

300

400 10 pll

days ~onitoring

12

14

Station 2

pH R===~====~============R F ~----~~~~--~~--~ 13 200

12

"

16 0

10

120 80 40

14

~====~===============~ ~o=n=it=o~riRng~tartrio~n~3______ . . __-,____________-.,

pH 13

200 _ ..

12

,-160

"

10

120 80

~ ..

40

soo

o

r--

10 pit

pH

12

14

~==========~==~=====~ ==o=n=it=o~riRng~tartrio_n_4____r -__________- '______" -

13 200 12 160

" 10

120 80 40

100

200

300

400 10 pll

Fig. 3a - Time series, histograms of pH historical data.

12

14

SOARES ET AL. ON SPACE-TIME SERIES

Monitoring Station 2

Monitoring Station I

y(d) , - - - - - - - - - - - - - - - - - ,

y(d) 1.60

153

,

....'.

.

..... .

3.20

1. 20

2.40

..

0.80

'.

,'

.... .... ,

0.40

"

1. 60

",

........

, ,., . '.'

0.00 100

200

300

days

Monitoring Station 4

Monitoring Station 3 y(d) , - - - - - - - - - - - - - - - - - ,

120

090

..

0.16

..

'

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

0.12

"'"

11.08

OJO

O.()4

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

. ." ..

y(d) , - - - - - - - - - - - - - - - - - - ,

II II

_ ,"

.

•••••••

+-.a.----',-,WO.. -.~-~-------l

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

Fig. 3b - Time variograms of pH historical data.

Two different simulations were implemented: a conditional simulation in which the input is the real time series of the first station corresponding to a mine effluent, and a non-conditional simulation where all time series (including the mine effluent) were simulated. Conditional Simulation The estimators [X(e, t)]* were calculated with four samples before time t and two samples spatially located before e: [X(e, t)]*=a, X(e-l ,t)+ a2 X(e-2,t)+ b, X(e,t-l)+ b2 X(e,t-2)+ b3 X(e,t-3)+ b4 X(e,t-4) Based on experimental bi-plots {X(e, t), [X(e,t)]*} for each monitoring station, the simulation procedure has been initialised with real time series of the first station. The resulting simulated time series of the remaining 3 stations are shown in Figs. 5, 6 and 7a, 7b, for the two elements studied. The time correlations of the real data are quite satisfactory reproduced in the variograms of the simulated values.

154

GEOSTATISTICAL APPLICATIONS

Monitoring Start-ri_ o n~l~~~_ _~~-.-,--.-,--.-,-~~..--, F

00

.0

90 80

60

70 40

60 50

20

10

30

20

40

50 100

40

days

Monitoring Sta_t~i;c o;c n .::2;.,.-~~_ _~~-.--,-~-.-,-~-.-,-..-.--,

00

RF==~==~============R F

90

r

80

80 60

70 40

60 50

20

90

40

days

100

Monitoring Station 3 00 RF==~====~==~======~ 90

F ~--~--~~~--~~--~ 80

80 60

40

20

40

days

50

60

90

100

Monitoring Station 4

00

R==========t=I Fr====:-:r:::::::==:-:r:::::::==::::q

90

80

80 60

70 40

60 50

................ _.. ".. __ .....

20

40

100

200

300 days

500 40

90

Fig. 4a - Time series, histograms of Dissolved Oxygen historical data.

100

155

SOARES ET AL. ON SPACE-TIME SERIES

Monitoring Station 2

Monitoring Station 1 y(d)

64

..

.

48

y(d)

.. ... .'. ." .. . . .' '. •

64

'0

"

.'0

'.'

48

.. ,....

0

'

32

32

16

)6

......" .,'

.'." . ..

..

..... '

'.

0

0

100

0

200

300

400

100

0

Monitoring Station 3

96

..'

......

....

....... ...... . '

Monitoring Station 4

..

y(d)

128

400

300

200 d.ys

dlY'

64

.......

"

. . . ..... '.'

y(d)

64

48

.......... . . ..'. '. . ...... .'. . .. . . . . I.

I,

'

32

32

'0

16

100

200 days

300

400

100

200

300

400

days

Fig. 4b - Time variograms of Dissolved Oxygen historical data.

Non-Conditional Simulation The non-conditional simulation is presented only for pH. The time series of all monitoring stations were simulated including the first one corresponding to the mine effluent, based on experimental bi-histograms {X(e, t), [X(e,t)]*} . The simulated time series of the four stations are reproduced in Figs. 8a, 8b as well as the time covariances of the simulated values. Discussion of the Results The proposed simulation methodology presented very satisfactory results regarding the main objectives: generation of a time series with the same basic statistics, time and spatial correlation as the observed historical data. With the first simulation, as it is conditioned to experimental data of the first monitoring station, it reproduces, at the remaining stations, not only the variograms but also the main features of the experimental time series. The non-conditional simulation generated a time series of pH with excellent reproduction of the spatial and time correlation of historical data.

156

GEOSTATISTICAL APPLICATIONS

Monitoring Station 2 pH P===~========~======~

13

__..-.. . _ -..

~--~~~

_ ~.-

~~~---r;

200

12 16 0

II

10

120

~::

9

.::: •• •• • • •

80

40

6

o

100

200

300

400

500

14

days

Monitoring Station 3 pH ~==~=============q

13

~~~~--~~~--~ 200

12 160

II

10

•••••

~• '::

6

o

100

200

300

400

500

10

days

12

14

pH

Monitoring Station 4 pH ~==========~====~

13

~--~--~--------~ 20 0

12 160

II

10

•• •• •• • ••••••••...•••••• •••••••• • •• • ••• . •••• •

120 80 40 ~ ..

o

100

200

300 days

400

500

n

10

12

pH

Fig. 5 - Conditional Simulation of pH time series and histograms

14

SOARES ET AL. ON SPACE·TIME SERIES

Monitoring Sta;::.t:;:i;o:.:n;.:..:::2~~~~~~~~_......~~..,

DO ~====~~====~~====~

~ .•••••~••

80

-

40

20

40 b===~==~====~======~ 100 200 300 400 500

40

days

90

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

1--1-

60

'0

70

60

80

90

100

90

100

DO

80

60

40

rh--

20

40 b=====================~ 100

200

300

400

0

500

"40

days

'0

70

'0

80

DO

~==========~=========M~on=i=to=rii:ngSta~t~io~n~4~~-r......--'~~_~-r_~

DO I-

90

80

1-·

60

I--

40

20

100

200

300 days

400

500

40

'0

60

70

80

90

100

DO

Fig. 6 . Conditional Simulation of Dissolved Oxygen time series and histograms.

157

158

GEOSTATISTICAL APPLICATIONS

Monitoring Station 2

Monitoring Station 1

y(dl r---------..::~-----_,

y(dl

,.......,

.......

3.2 2.4

....

1.2

..

0.9

",

..··0 ," ,

.......

0.6

1.6

............ "

0.3

0.8

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

0.0 ~~~~~~~~~~~~~~~~~~ o 100 300 400

100

200

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

300

400

days

Monitoring Station 3 y(dl 0 16 0. 12

"

0.08

, , ...',..... ....... .......

.....

.........

,-

0.04

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SOARES ET AL. ON SPACE-TIME SERIES

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CONCLUSIONS The presented stochastic data driven approach aims to generate a set of realisations reproducing some basic statistics regarding the contiguity in space and time of relevant variables of a river water quality. With the stochastic realisations of time-series one can visualise the joint behaviour of the water quality characteristics and to predict extreme scenarios in the environmental system. The crucial point of the proposed methodology consists on the definition of the bidistributions between real and estimated values. They should generate posterior timeseries reproducing the same space and time covariances and marginal histograms for each monitoring station as the equivalent statistics of historical data. It means that the pattern of neighbourhood values of a space and time location to be simulated must be chosen in order to represent the relevant covariances in space and time. This simple and easy to implement data driven approach has one limitation: one can generate time-series only in the spatial locations (and with the time periodicity) of the historical data, corresponding to the monitoring stations ofthe presented case study. However, in these situations it typically is useless to simulate the time behaviour of a pollutant between two monitoring stations. Unless one has another pollutant source

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between them, any expected value in the middle of two stations belongs to the interval of its values.

REFERENCES Deutsch, C., and Journel, A , 1992, GSLIB : Geostatistical Software Library and User 's Guide,Oxford University Press, New York Deutsch, C. , 1994, "Kriging with Strings of Data", Mathematical Geology, Vol 26, n- 5, pp 623-638. Johnson, M., 1987, Multivariate Statistical Simulation, John Wiley & Sons, New York Journel, A, and Gomez-Hernandez, 1., 1989, Stochastic Imaging of the Wilmington Clastic Sequence, SPE paper # 19857 Law, A, and Kelton, D., 1982, Simulation Modeling and Analysis, McGraw Hill Int.Ed., New York Ripley, B., 1987, Stochastic Simulation, John Wiley & Sons, New York

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Gary N. Kuhn , Wayne E. Woldt , David D. Jones , Dennis D. Schulte3

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SOLID WASTE DISPOSAL SITE CHARACTERIZA nON USING NON-INTRUSIVE ELECTROMAGNETIC SURVEY TECHNIQUES AND GEOSTATISnCS

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REFERENCE: Kuhn G. N., Woldt W. E., Jones D. D., Schulte D. D., "Solid Waste Disposal Site Characterization Using Non-Intrusive Electromagnetic Survey Techniques and Geostatistics," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. 1. Johnson, A. 1. Desbarats, Eds., American Society for Testing and Materials, Philadelphia, 1995. ABSTRACT: Prior to the research reported in this paper, a site-specific hydrogeologic investigation was developed for a closed solid waste facility in Eastern Nebraska using phased subsurface characterizations. Based on the findings of this prior investigation, a surface based geoelectric survey using electromagnetic induction to measure subsurface conductivity was implemented to delineate the vertical and horizontal extent of buried waste and subsurface contamination. This technique proved to be a key non-intrusive, cost-effective element in the refinement of the second phase of the hydrogeologic investigation. Three-dimensional ordinary kriging was used to estimate conductivity values at unsampled locations. These estimates were utilized to prepare a contaminant plume map and a cross section depicting interpreted subsurface features. Pertinent subsurface features were identified by associating a unique range of conductivity values to that of solid waste, saturated and unsaturated soils and possible leachate migrating from the identified disposal areas. KEYWORDS: Geoelectrics, Electromagnetics, Geostatistics, Conductivity, Leachate, Hydrogeology, Vadose

IGraduate Student, University of Nebraska - Lincoln, Department of Biological Systems Engineering 2Assistant Professor, University of Nebraska - Lincoln, Department of Biological Systems Engineering 3Professor, University of Nebraska - Lincoln, Department of Biological Systems Engineering

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INTRODUCTION

Past landfill management and operational practices in the United States have created environmental problems and are commonly associated with soil and groundwater contamination. These landfills have either been upgraded to meet current State and Federal legislation or have closed. As a result, the number of operational landfills has decreased from over 20,000 in 1978 to approximately 3,300 in 1994. Because of strict State and Federal legislation passed dealing with closure of these landfills, the severe impact that past operations had on the environment is becoming more apparent. For example, in 1987 the State of Nebraska required the Nebraska Department of Environmental Quality (NDEQ) to conduct a comprehensive assessment of all community solid waste disposal sites (SWDS). The purpose of this assessment was to ascertain compliance of SWDS to standards established by the Nebraska Environmental Protection Act (NEPA) and the Federal Resource Conservation and Recovery Act Subtitle D (RCRA Subtitle D)(SCS 1991). In 1991, nearly 210 landfills in Nebraska had ceased operations or were recommended for closure by NDEQ. This recommendation was based on insufficient capacities or significant constraints imposed on owners to maintain compliance with RCRA Subtitle D requirements (SCS 1991). Due to the risk posed to Nebraska's surface and groundwater by the existence of unlicensed landfills, the NDEQ focused on closure activities. Information compiled by NDEQ identified sites located near public or private drinking water sources, that were underlain by a shallow water table surface, or were located in a 100 year flood plane~ The survey revealed that a significant number of SWDS should have further investigation. Based on this study, the NDEQ directed most of its efforts at the currently unregulated sites utilized primarily by rural communities (NDEQ 1990). These sites have not been subject to regulation since 1972 when the Whitney Amendment to the Nebraska Environmental Protection Act specifically exempted all cities within the second class (5,000 population or less) and villages from state solid waste rules and regulations (NDEQ 1990). Although these sites were exempt from state solid waste regulations, NDEQ revoked this amendment in 1991 when RCRA Subtitle D was reauthorized. As solid waste disposal sites are forced to close over the next few years, hundreds of millions of dollars will be spent in the United States to identify, characterize and remediate sites contaminated with hazardous materials. Traditional site investigation techniques typically include compiling hydrogeologic and contaminant fate and transport information from testhole and groundwater monitoring well data. Commonly, background information is limited until the results of the first round of groundwater samples is available. Only then does it become apparent that the plume may not be completely delineated and additional monitoring wells are required.

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To address this problem, non-intrusive field methods and geostatistical analysis tools were utilized to gather preliminary subsurface information pertaining to hydrogeologic features, horizontal and vertical extents of wastes and suspected leachate plumes. This information can be utilized to optimize and minimize testhole or permanent monitoring well locations.

LITERATURE REVIEW Electromagnetic (EM) surveying techniques, combined with appropriate geostatistical analyses, are rapid and non-intrusive methods of characterizing subsurface environments. The non-intrusive nature of this technique reduces drilling or other intrusive investigative tools. The technique of EM surveying is based on the principle of utilizing varying subsurface conductivity measurements as an indication of differing geologic and or other subsurface constructs. Surface electrical methods have been used successfully in many types of subsurface investigations. Kelly (1976) showed that the d-c resistivity method can be effective in delineating a plume moving off-site from a landfill. The use of EM data sources for delineation of contaminated groundwater has been described by Greenhouse and Siaine (1986). French et al. (1988) utilized geoelectric surveying to identify anomalous regions to focus subsequent boring and sampling activities. Hagemeister (1993) identified potential waste volumes and suspected contaminant migration present at an unregulated landfill. In each case, differing electrical conductivity was interpreted as an indication of changes in the systems being investigated. Geostatistics has been utilized in numerous investigations to estimate expected values at unsampled locations. Cooper and Istok (1988) utilized geostatistics to estimate and map contaminant concentrations and estimate errors in a groundwater plume from a set of measured contaminant concentrations. Cressie et al. (1989) prepared kriged estimate and error maps to predict a migration pathway of radionuclide contaminants from a potential high-level nuclear waste repository site. Woldt (1990) mapped the location of a suspected contaminant plume based on observed geoelectric measurements and geostatistics. Hagemeister (1993) utilized geostatistics to map subsurface electrical conductivity in two dimensional cross sections across a site. In each case, kriged estimate and error maps were prepared to assist in the interpretation of the measured data.

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A three dimensional data set, developed by obtaining readings at several sounding depths across a gridded area, was subjected to geostatistical analyses. This data set was utilized \ in conjunction with available background data to identify pertinent subsurface features \ and approximate their general locations. The background data included boring logs, \ groundwater analytical reports and industrial waste disposal permits. These permits

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allowed the disposal of industrial wastes until the late 1970' s. The methods utilized during this study consisted of establishing a sampling grid, completing an electromagnetic survey, and performing a geostatistical analysis. Each procedure was directed towards non-intrusive characterization of the subsurface environment. Existing testhole data was correlated with the predicted locations of pertinent site features for validation purposes. Site Description Based upon information obtained from the NDEQ, the study site was operated as a "trench and fill" SWDS from 1975 to 1987. During this time, the owner accepted domestic and industrial waste from nearby rural communities. Information pertaining to the actual quantities received are not available. The site and the surrounding areas are located near the eastern most edge of the Nebraska Sandhills region. The topography of this region is mostly undulating to rolling. The elevation of the site is approximately 460 to 466 meters above mean sea level (MSL) near the northeast and southwest corners, respectively (NDEQ 1990). The surface geology consists of approximately 38 to 42 meters of fine to medium grain sands interbedded with coarse sand and fine gravel deposits, which is characteristic of this region. The uppermost monitorable aquifer is located in sand and gravel deposits of the High Plains aquifer system. The water table is approximately 11 and 15 meters below grade level (BGL) in the northeast and southwest corners of the site, respectively, and the saturated thickness of the unconfined aquifer is approximately 27 meters (NDEQ 1990). Based on regional bedrock maps for this area, it appears that the top of the uppermost confining unit is the Niobrara Shale formation that underlies the water table aquifer at an approximate elevation ranging from 422 to 424 meters above MSL near the northeast and southeast corners of the site, respectively. Under a Multi-Site Cooperative Agreement with Region VII of the Environmental Protection Agency (EPA), the NDEQ performed a Preliminary Assessment (PA) at the site to assess the threat posed by the site to human health and the environment. The NDEQ concluded that leachate from the site resulted in a leachate contaminant plume migrating in an east-northeast direction towards a river 2.5 kilometers away. Because of the low human and livestock population in the area, no evidence was found indicating that the site posed an immediate threat to human health and the environment (NDEQ 1990). Interviews with the site owner revealed that the standard operating procedures involved excavating a 5 meter deep cell with a backhoe, depositing refuse at the toe of the working face, compacting the refuse and providing 15 centimeters of daily cover material. After each cell was completely filled, 1 to 1.5 meters of silty clay was placed on top of the waste as a final cover. Based on this information, MSL elevations were assigned to the pertinent subsurface features and are presented in Table 1.

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Table 1. Approximate Elevation of Pertinent Site Features (meters) Site Feature Southwest Corner Northeast Corner Surface 466 460 Bottom of Trench 461 455 Water Table 451 449 Top of Bedrock 424 422 Sampling Grid Sampling point locations were established based on minimizing data collection efforts, maintaining minimum measurement support volumes of each instrument, and spatially defining the study area. The sampling point spacing utilized at the site was approximately 30 meters in the north and east directions and extended nearly 30 meters beyond all four property boundaries. The horizontal extent was selected based on the geology and obtaining an adequate number of sampling points beyond the limits of the suspected landfill cells to establish background subsurface conductivity levels. Electromagnetic Survey

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Electromagnetic techniques measure terrain conductivity to identify geologic and other subsurface formations. In most environmental EM applications, differing conductivity measurements are interpreted as a change in geologic formations or subsurface conditions (McNeill 1980). The EM instrument operates by generating alternating current loops with a transmitter coil (Tx). A time-varying magnetic field arising from the alternating current induces secondary currents sensed by a receiver coil (Rx) along with the primary field. The EM receiver coil intercepts a portion of the magnetic field from each loop generated by the transmitter coil and results in an output voltage which is also linearly proportional to the terrain conductivity. The resulting reading is in milli-Siemans per meter (mS/m)

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The reading obtained from the EM instruments is a conductivity measurement averaged over a volume of subsurface media. Because the effective depths of penetration are small in comparison to the overall horizontal and vertical dimensions of the site, these readings were interpreted as being representative of a sampling point at the calculated effective depth. The effective depth of penetration by the induced current is directly proportional to the intercoil spacing and depends on the orientation of the instrument. By varying the intercoil spacing, conductivity measurements can be collected at varying depths. Also, operating the instrument in the horizontal dipole mode reduces the effective depth of penetration by approximately one halfthat of the vertical dipole mode. Therefore, the instrument was operated in the horizontal and vertical dipole positions at four different intercoil spacings to obtain readings at eight different depths.

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Theoretically, the total instrument response represents a weighted average of subsurface conductivities with a depth of infinity, but it does have practical limits. Interpretation, or modeling, of geophysical data to determine a reasonable unique solution to the nonunique problem was not performed. Although modeling this data provides a more comprehensive interpretation of the data set, the preliminary nature ofthis research did not warrant the level effort involved with the modeling process. Instead, Hagemeister (1993) calculated effective exploration depths offour intercoil spacings for both the vertical and horizontal dipole modes. These calculations are based on the assumption that 60 percent of the total signal contribution over a volume of subsurface media is associated with a discernible layer. Based on the small diameter and thickness of the support volume, in relation to the overall area of the site and the preliminary nature of the investigation, each instrument reading was assigned to a point located at the centroid of the calculated effective depth of penetration. Table 2 depicts the exploration depths at various intercoil spacings.

Table 2. Exploration Depths Exploration Depth (meter) Intercoil Spacing Horizontal Mode Vertical Mode 3.7 meters 1.0 2.5 10.0 meters 3.5 6.5 20.0 meters 8.0 13.0 26.0 40.0 meters 15.5

GEOSTATISTICAL ANALYSIS Environmental professionals are often confronted with the problem of providing detailed information about a site based on a minimum number of sampling points. Geostatistics provides a means of utilizing spatial continuity for estimating the expected value at unsampled locations. Geostatistics is commonly utilized to describe the spatial continuity of earth science data and aims at understanding and modeling the spatial variability of the data. The geostatistical analytical process for this study consisted of 1) describing and understanding the statistical distribution of the data, 2) modeling the spatial variability of the data, 3) estimating expected values at unsampled locations and 4) computing estimation variance values at the unsampled locations. Univariate Description Univariate description deals with organizing, presenting and summarizing data and provides an effective means of describing the data by identifying outliers and extreme values. The univariate descriptive tools utilized to analyze the conductivity data set were:

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1) histograms, 2) probability plots and 3) summary statistics. Because the data set is three dimensional, a descriptive scatter plot could not effectively be obtained.

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Geo-EAS 1.2.1 Geostatistical Environmental Assessment Software C>400 5 (Englund and Sparks 1988) was utilized to g. prepare histograms and probability plots for ~ 300 both the observed data set and logarithmic transformations of the observed data set. 200 Characteristic of many environmental data sets, the observed data exhibited a large 100 nwnber of low values which offset the mean of the data distribution to the left of the o median. The data were transformed to Conductivity (mmhoslm) prepare lognormal histograms and Figure 1. Histogram probability plots to determine if the data exhibits a lognormal distribution .. The tests for lognormality indicated that the data does not approach this distribution. Therefore, the observed data set was utilized for the analysis. Figure 1 presents a histogram plot for the observed conductivity data set. The summary statistics presented in Table 3 nwnerically describe the location, spread and shape of the observed data distribution. Table 3. Observed Value Summary Statistics Minimum Number 1679 0.5 Mean 23 .8 25th Percentile 10.2 Variance 263.6 Median 22.7 Standard Deviation 16.2 75th Percentile 31.9 Skewness 1.1 Maximum 125.0 Experimental Variogram A variogram is a plot of the variance, or one-half the mean squared difference, of paired data points as a function of the distance between the two points (Deutsch and Journel 1992). An omnidirectional variogram can be developed to obtain a general understanding of the spatial characteristics of the sample data. The omnidirectional variogram does not take into account spatial continuity changes due to directional changes in the data. Therefore, directional variograms are developed to identify these changes, if present. To ensure a more realistic sample variogram, the window for the lag distance did not extend greater than one-half the length or width of the data set. The lag distance between points

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was also restricted such that a minimum of 30 pairs per lag distance were available to increase the confidence of variogram calculations. GSLIB - Geostatistical Software Library and User's Guide (Deutsch and Journe11992) was utilized to develop directional and omnidirectional experimental variograms from the three dimensional conductivity data set. Generally, two sets of directional variograms were developed by restricting the paired data points to be either horizontally coplanar or vertically cocolumnar. Attempts to identify directions of maximum and minimum continuity within a horizontal plane revealed the same structure as the omnidirectional variogram in the all directions. Therefore, isotropic conditions were considered for the horizontal plane. Paired data

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The exponential variogram model was fit to both the horizontal omnidirectional and the vertical variograms (Figure 2) utilizing the parameters presented in Table 4. Although the nugget and sill are identical, the range significantly decreases in the vertical direction. This is characteristic of geometric anisotropy and is commonly encountered in earth SCIence. Table 4. Variogram Model Parameters Parameter Horizontal Model Vertical Model Model Structure Exponential Exponential Range 250 50 Sill 180 180 Nugget Effect 130 130 Cross Validation

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Model variograms were cross validated to compare the sample point values to the estimated values at those locations. It is important to develop a variogram model that would minimize the standard deviation of the estimation error as determined by the cross validation process. A variogram model that produces good results does not necessarily indicate that the estimation at unknown locations will be accurate. However, good results from cross validation will suggest, with more confidence, the effectiveness ofthe selected model.

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Cross validation consists of removing a data point from the data set and calculating an estimated value utilizing the model variogram. Once the estimate af6 is calculated, a comparison can be made between the estimated and observed values at each sampling point by calculating the difference between the two values. The three summary statistics utilized to evaluate the cross validation results are : 1) average kriging error (AKE), 2) mean squared error (MSE) and 3) standardized mean squared error (SMSE) (Woldt 1990). The AKE provides a measure of the degree of bias introduced by the kriging process and should equal 0 if the data is unbiased. The MSE should be less than the variance of the measured values. The SMSE is a measure of the consistency and is satisfied if the SMSE is within the interval 1.0±[2(2/n)1/2]. The results are summarized in Table 5 along with their calculated expected values.

Table 5. Cross Validation Summary Statistics SMSE MSE AKE <263.6 0.931 to 1.069 Expected Value 0.0 -0.03 131.0 0.89 Cross validation Results

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As depicted in Table 5, the criteria meet the recommended AKE and MSE values and is just outside the range of expected SMSE values. These results are generally considered acceptable. Ordinary krigin~ Ordinary point kriging was selected to estimate expected values at unsampled locations. This method was selected because it is a linear unbiased estimator that attempts to minimize the error variance and generally has the lowest mean absolute error and mean squared error in comparison to other estimation methods (i.e. polygonal, triangulation, local sample mean, and inverse distance squared). _gSLIB (Deutsch and Journel 1992) was utilized to calculate the expected values at 8,000 unsampled locations, or nodes, on a 20-x 20-~ 20 grid from the three dimensional conductivity data set. The nodes were ~paced meters in the north and east horizontal directions and 2 meters in the vertical direction. These spacings were selected based on the anticipated spatial orientation and depths of the pertinent subsurface features.

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Search Nei~hborhood The search neighborhood was established based on the following criteria: 1) selecting the greatest distance on the variogram model that closely fit the experimental variogram, 2) at least 30 pairs were utilized to calculate the experimental variogram at each point and 3) the distance did not exceed half the length ofthe horizontal sampling grid diagonal. The search neighborhood was defined in the horizontal plane at 250 meters. The geometric anisotropy limited the search to within 25 meters in the vertical direction which reflects the region of the variogram with the higher level of confidence. Background Conductivities Generally, the expected values located west and south of the site and outside the property boundaries were considered to represent background subsurface conductivities, or expected values assumed not to be impacted by past site activities. This was established based on the groundwater flow direction. These background values generally ranged from less than 0 mS/m to 12 mS/m in the vadose zone and from 12 mS/m to 24 mS/m below the water table. These ranges of values established the basis of interpretation for identifying hydrogeologic features, landfill cells and potential leachate migrating from the cells.

DISCUSSION The previous sections discussed a methodology that can be utilized to interpret surface based electrical data in an effort to construct reliable maps of suspected subsurface features. Based on the selected cross sections of expected conductivity values presented

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in Figures 3 and 4 and limited knowledge of the site, interpretations of: 1) site specific hydrogeology, 2) horizontal and vertical extents of waste and 3) potential sources for leachate migration were developed Hydro geolo (;:y Station 000 North (Figure 3) depicts a vertical cross section of the background expected subsurface conductivity values for background reference. This station is located upgradient of the site and was utilized as an indication of subsurface conditions not impacted by past landfill operations. information obtained from NDEQ records, the natural subsurface environment adjacent to the boreholes consists of fine to medium sand with a static water table elevation near 450 meters. Therefore, 0 to 12 and 12 to 24 mS/m were determined to represent unsaturated and saturated fine to medium sands, respectively. A variance from these ranges was interpreted as an indication of differing subsurface structures, or impact from the landfill operation. Landfill Cell Identification Based on the nature of the site operations, landfill cells are expected to be located from the ground surface down to an approximate depth of 5 meters. The selected vertical cross sections (Figure 3) depicted expect conductivity values near the surface ranging in excess of 24 mS/m within the 0 to 5 meter BGL depth range. Generally, the landfill cells appear to cover the entire site. Based on the vertical cross section maps (Figure 3), two areas exhibiting conductivity values in excess of24 mS/m were elongated in the north and south directions with the centerlines located near stations 150 East and 300 East. Based on a personal interview with the owner, it appears that these two areas are actually several landfill cells spaced close together. Potential Leachate Mi(;:ration The primary concern with SWDS is the potential leachate contamination associated with the nature of the operation. Leachate is a liquid that consists of refuse moisture and all precipitation that mixes with this moisture as it migrates through the landfill. Leachate migrating from a landfill naturally due to gravity or forced out as a result of the consolidation of refuse may transport contaminants from the refuse to the groundwater environment. The presence of elevated conductivity values within the vadose and directly below the landfill cells was interpreted as potential leachate or possible instrument interference from the overlying waste. These conductivity values ranged from 12 mS/m to 24 mS/m (Figure 3).

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Intemreted Subsurface Environment Based on the information obtained from this study, it appears that leachate may have migrated from the landfill and impacted the groundwater table. The plume appears to be migrating horizontally in the northeast direction with a vertical component. Figure 5 consists of a plan view and cross section illustrating an interpreted leachate plume located relative to the identified waste.

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Computer Software Support All data description and estimation efforts requiring computer software support was completed utilizing an IBM 80386 processor. Software support utilized in this study consisted of Geo-EAS, GSLIB and TecPlot Version 6.0 (TecPlot). Probability and histogram plots describing the 1679 observed conductivity values were prepared utilizing Geo-EAS. Geo-EAS generated on-screen plots within minutes allowing efforts to be focused on the descriptive analyses. The expected values. are presented on cross section contour maps included as Figures 3 and 4. The three dimensional expected value data set was imported into TecPlot. TecPlot utilizes linear interpolation to construct each contour line. Tecplot generated cross section maps based on a three dimensional data set. By fixing one dimension, a cross section at a desired location was generated within minutes.

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GEOSTATISTICAL APPLICATIONS

CONCLUSIONS Geostatistical analysis demonstrated that the data is spatial correlated which allowed for an interpreted subsurface model to be developed based on kriged estimated values. As an alternative to traditional intrusive characterization techniques, surface based electromagnetic surveying techniques proved to be a key non-intrusive, cost-effective element in the refinement of the second phase of the hydrogeologic investigation. Review of kriging error maps can further refine this second phase by focusing on the areas with the largest error. This study demonstrated that this methodology, as a preliminary field screening tool, can provide sufficient information to optimize the placement and minimize the number of permanent groundwater monitoring wells.

REFERENCES Barlow, P.M., Ryan, B.J., 1985,"An Electromagnetic Method of Delineating GroundWater Contamination, Wood River Junction, Rhode Island," Selected Papers in Hydrologic Sciences, U.S. Geological Survey Water-Supply Paper 2270, pp. 35-49. Cooper, R.M., Istok, J.D., 1988, "Geostatistics Applied to Groundwater Contamination. I: Methodology, "Journal of Environmental Engineering, Vol. 114, No.2., pp. 270-285. Cressie, N.A., 1989, "Geostatistics," American Statistician, Vol. 43, pp. 197-202. Deutsch, C.V., Journel, A.G., 1992, "GSLIB - Geostatistical Software Library and User's Guide," Oxford University Press. Environmental Protection Agency, 1994, "EPA Criteria for Municipal Solid Waste Landfills," The Bureau of National Affairs, Inc., 40 CFR Part 258. Englund, E. and Sparks, A., 1988, "Geo-EAS 1,2,1 User's Guide", EPA Report #60018911008, EPA-EMSL, Las Vegas, Nevada. French, R.B., Williams, T.R., Foster, A.R., 1988, "Geophysical Surveys at a Superfund Site, Western Processing, Washington," Symposium on the Application of Geophysics to Engineering and Environmental Problems, Golden, Colorado, pp. 747-753. Greenhouse, J.P., Slaine, DD, 1986, "Geophysical Modeling and Mapping of Contaminated Groundwater Around Three Waste Disposal Sites in Southern Ontario," Canadian Geotechnical Journal, Vol. 23, pp. 372-384. Hagemeister, M.E., 1993, "Systems Approach to Landfill Hazard Assessment with Geophysics (SALHAG)," Unpublished Masters Thesis, University of Nebraska - Lincoln. 1994, "Handbook of Solid Waste Management," McGraw-Hill Publishing

It

Ii11

KUHN ET AL. ON SOLID WASTE DISPOSAL

177

Isaaks E.H., Srivastava R.M., 1989, "An Introduction to Applied Geostatistics," Oxford University Press, New York. Joumel, A., Huijbregts, C., 1978, "Mining Geostatistics," Academica Press, New York. McNeill J.D., October 1980, "Electromagnetic Terrain Conductivity Measurement at Low Induction Numbers," Geonics Limited Technical Note TN-6. NDEQ, February 1990, "Ground Water Quality Investigation of Five Solid Waste Disposal Sites in Nebraska", Nebraska Department of Environmental Quality. SCS Engineers, December 1991, "Volume 1 - Recommendations to State and Local Governments," Nebraska Solid Waste Mana~ement Plan, Nebraska Department of Environemtnal Quality. Woldt, W.E., 1990, "Ground Water Contamination Control: Detection and Remedial Planning," Ph.D. Dissertation, University of Nebraska - Lincoln.

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Craig H. Benson1 and Salwa M. Rashad 2 ENHANCED SUBSURFACE CHARACTERIZATION FOR PREDICTION OF CONTAMINANT TRANSPORT USING CO-KRIGING REFERENCE: Benson, C. H. and Rashad, S. M., "Enhanced Subsurface Characterization for Prediction of Contaminant Transport Using Co-Kriging," Geostatisticsfor Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Groundwater flow and advective transport were simulated in a heterogeneous synthetic aquifer. These simulations were conducted when the aquifer was fully defined and when it was characterized using a limited amount of hard and soft data (hydraulic conductivity data and soil classifications). Co-kriging was used to combine the data types when estimating the hydraulic conductivity field throughout the aquifer. Results of the flow and transport simulations showed that soil classifications were useful in characterizing the hydraulic conductivity field and reducing errors in statistics describing the plume. KEYWORDS: kriging, co-kriging, ground water, contaminant transport, hydraulic conductivity, soil classifications INTRODUCTION

Simulating flow and contaminant transport is often an essential feature of remediation projects dealing with contaminated groundwater. In recent years, numerous sophisticated groundwater models have been developed to conduct such simulations. The complexity of these models allows one to realistically simulate the fate of contaminants provided properties of the aquifer affecting transport are adequately characterized. Unfortunately, what level of characterization is "adequate" is unknown, especially at sites where the subsurface is heterogeneous. Thus, when limited data are available to describe subsurface conditions, predictions of contaminant transport can be uncertain even when sophisticated models are used. Although many factors affect the fate of groundwater contaminants, the spatial distribution of hydraulic conductivity is the primary factor affecting which pathways are active in transport (Webb and Anderson 1996). To better define these pathways, additional data must be collected and analyzed. The most useful data are hydraulic conductivity measurements. However, "hard" data such as hydraulic conductivity measurements are expensive to obtain, especially if the data are to be collected from a site that is contaminated. It is advantageous, therefore, to investigate the effectiveness of using less expensive "soft" data, such as soil classifications, to reduce uncertainty. Soft data can be readily collected using less expensive exploration techniques such a ground penetrating radar, terrain resistivity surveys, or cone penetrometer soundings. The objective of the project described in this paper was to evaluate how characterizing the subsurface affects predictions of contaminant transport. Simulations of 1Assoc. Prof., Dept. of Civil & Environ. Eng., Univ. of Wisconsin, Madison, WI, 53706, USA. 2Asst. Scientist, Dept. of Civil & Environ. Eng., Univ. of Wisconsin, Madison, WI, 53706, USA.

181

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182

GEOSTATISTICAL APPLICATIONS

groundwater flow and advective transport were conducted in a heterogeneous "synthetic aquifer." The aquifer was characterized using various amounts of hard data (hydraulic conductivities) and soft data (soil classifications). Co-kriging was used to combine the two data types when estimating the hydraulic conductivity field. Similar uses of cokriging have been described by Seo et al. (1990 a,b) and Istok et al. (1993).

SYNTHETIC AQUIFER Characteristics

A "synthetic aquifer" was used in this study because it can be fully-defined; that is, the hydraulic properties throughout the aquifer are defined with certainty. In this particular application, fully-defined means that hydraulic conductivities and soil classifications can be assigned to every cell in the finite-difference grid used in simulating flow and transport in the aquifer. Thus, flow and transport simulations conducted with the "fully-defined" aquifer are representative of its "true" behavior. Comparisons can then be made between results obtained using the fully-defined case and cases where the aquifer has been characterized with a limited amount of sub-surface data. This comparison provides a direct means to evaluate the inherent inaccuracies associated with estimating subsurface conditions from a limited amount of information. A schematic illustration of the aquifer is shown in Fig. 1. It is extremely heterogeneous, as might be encountered in a supra-glacial depositional environment such as those occurring in the upper midwestern United States (Mickelson 1986, Simpkins et al. 1987). Details of the method used to design the aquifer are in Cooper and Benson (1993). Although an attempt was made to create a realistic aquifer, the synthetic aquifer was created without any site-specific data and thus may not be "geologically correct." The reader should keep this limitation in mind when considering the results and conclusions described later.

6 Clay 5 Clayey Sill • Silty Sand 3 Ane Sand

2 Coarse-Medium Sand 1 Clean Gravel

Upstream Boundary

Hydraulic Gradient. 0.01

Downstream

Boundary • FIG. 1 - Synthetic aquifer.

The aquifer is discretized into 12,500 cells that comprise a finite-difference grid used in simulating flow and transport. The aquifer is segregated into 25 layers. Each layer contains 20 rows and 25 columns of finite-difference cells. Each cell is 100 cm

BENSON AND RASHAD ON CO-KRIGING

183

long per side. Groundwater flow was induced by applying an average hydraulic gradient of 0.01. Constant head boundary conditions were applied at the upstream and downstream boundaries of the aquifer. No flow boundaries were applied along the remaining surfaces of the aquifer. An important feature of the aquifer is that soil types are layered to create continuous and non-continuous soil lenses. Lenses with high hydraulic conductivity, such as clean gravel and coarse to medium sand, simulate preferential flow paths that might not be detected during a subsurface investigation. Low hydraulic conductivity soils such as clayey silt and clay are layered to create pinches and stagnation points that may cause the flow of groundwater to slow or even stop. These intricacies of the aquifer also might not be detected during a subsurface investigation. Hydraulic Conductivity of Geologic Units

A soil classification was assigned to each geologic unit (i.e., the geologic facies) in the fully-defined synthetic aquifer. The soil classifications used to describe geology of the aquifer are: (1) clean gravel, (2) coarse to medium sand, (3) fine sand, (4) silty sand, (5) clayey silt, and (6) clay. These soil classifications are represented numerically using the integers 1-6. The writers note that the integer ordering of these classifications is arbitrary. Consequently, results somewhat different than those described herein may have been obtained had a different categorical scheme been used. Each cell in a given geologic unit was assigned a single realization from the distribution of hydraulic conductivity corresponding to the unit. Single realizations were generated using Monte Carlo simulation via inversion. In addition, no spatial correlation was assumed to exists within a geologic unit. Thus, the correlation structure inherent in the aquifer is due primarily to the relative location and size of the geologic units. The triangular distribution (Fig. 2) was used to describe spatial variability in hydraulic conductivity for a given soil type. The distribution is defined using an upper bound (Kmax), a lower bound (Kmin), and the peak of the density function (Kp). To select K max , Kmin, and Kp for each soil classification, a chart was developed that summarizes hydraulic conductivities assigned to various soil types in thirteen publications (Fig. 3). The hydraulic conductivities recommended by others were synthesized into a single "composite chart" having the six different soil types that comprise the synthetic aquifer, each with a corresponding range of hydraulic conductivities (Table 1).

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FIG. 2 - Distribution of hydraulic conductivity in a geologic unit.

184

GEOSTATISTICAL APPLICATIONS

Oaa (1985) Bowl .. (1984) Domenico & Schwartz (1990) HoH.&

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Kmin (em/sec) 5 x 10- 1

Kp (em/sec) 5 x 100

I x 10-3 I x 10-4

5 x 10-2 5 x 10- 3

5 x 10- 2

Clayey Silt

5 x 10-5 I x 10-7

5 x 10-4 I x 10-6

5 x 10- 3 5 x 10- 5

Clay

I x 10- 10

I x 10- 8

1 x 10- 6

Coarse - Med. Sand Fine Sand Silty Sand

Kmax (em/sec) 5 x 102 I x 100

Spatial Correlation Structure

The spatial correlation structure inherent in the soil type and hydraulic conductivity fields was characterized by computing directional experimental variograms in three dimensions. A model was then fit to the experimental variograms. A similar approach was also used to characterize the spatial cross-correlation structure between hydraulic conductivity and soil type. Experimental variograms were computed using the program GAM3 from the GSLIB geostatisticallibrary (Deutsch and Joumel 1992). The experimental variograms were computed by: N(h)

1 ~ Y*In K (h) = 2 N(h) £.J [In K(xi + h) - In K(xi) ] 2 1=1

(1)

BENSON AND RASHAD ON CO-KRIGING

185 (2)

In Eqs. 1-2, Y*lnK(h) is the estimated variogram for InKs separated by the vector h, y* s(h) is the estimated variogram for soil classifications (S), N(h) is the number of data pairs separated approximately by the same vector h, and Xi is a generic location in the aquifer. The cross-variogram between InK and S is computed as;

1 Y~nK,S (h) = 2 N(h)

N(h)

L [S(Xj + h) - S(Xj)] [lnK(xj+h)-lnK(xD]

(3)

1=1

The principal axes for soil classification were identified by computing a series of experimental variograms each having a different orientation relative to the traditional Cartesian axes. The analysis showed that mild anisotropy exists in the X- Y plane, with the principal axis oriented 45° counterclockwise from the X-axis. For the vertical direction, the principal axis coincided with the vertical (Z) axis (Benson and Rashad 1994). The principal axes for the hydraulic conductivity field were assumed to correspond to the principal axes for soil type because the hydraulic conductivity field was generated directly from the soil type field. A similar assumption was made regarding the cross-variogram (InK-soil type). Experimental directional variograms for soil type and hydraulic conductivity corresponding to the principal axes are shown in Figs. 4 a & b. The experimental crossvariogram (InK vs. S) is shown in Fig. 4c. For each set of variograms, the range is largest in the Y' direction and smallest in the Z' direction, which is consistent with the size and shape of the geologic units shown in Fig. 1. In contrast, the sill is essentially the same for the Y' and Z' directions, but is smaller in the X' direction. A spherical model with no nugget was found to best represent the experimental variograms. The spherical variogram is described by (Issaks and Srivastava 1989); y(h) = C[ I.S(h/a) -D.S(h/a)3] y(h)

=C

ifh
(4a)

ifh ~ a

(4b)

where C is the sill and a is the range. Table 2 provides a summary of C and a for each variogram. The directional experimental variograms exhibit a mixture of geometric and zonal anisotropies. Geometric anisotropy is characterized by directional variograms that have approximately the same sill, but different range. In contrast, zonal anisotropy corresponds to changes in the sill with direction, while the range remains nearly constant (Issaks and Srivastava 1989). The X'-Z' anisotropy is primarily geometric, whereas the X'- Y' and Z'- Y' anisotropies are primarily zonal.

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FIG_4 - Experimental variograms: (a) soil classifications, (b) hydraulic conductivities, and (c) soil classification-hydraulic conductivity_

BENSON AND RASHAD ON CO-KRIGING

187

TABLE 2 - Sill (C) and range (a) for variograms

X'

Parameter InK - Sill InK - Range (em) Soil Class. - Sill Soil Class. - Range (em) Cross-Sill Cross - Range (em)

25 925 1.5 925 -5.8 925

Direction Y'

32 1840 2.25 1840 -7.7 1840

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A three-level nested structure was used to combine the geometric and zonal anisotropies into a single variogram model. The model has the form: (5)

=

The model YI,j is a spherical function (Eq. 4) having C 1; it provides the basis for the nested structure. The subscript j denotes the variable being described (S, InK, or S-lnK for the cross-variogram). The separation distance hi is an "equivalent" distance:

h 1-

J

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(6)

The weight WI,S corresponds to the smallest sill for soil classification (X' axis: sill = WI,S = 1.5 for S). The models Y2 and Y3 are also spherical models and they are used to ensure that sills corresponding to the Z' and Y' axes are preserved. That is: W2,S = 1.75 - WI,S = 0.25

(7a)

w3,S = 2.25 - (WI,S + w2,S) = 0.50

(7b)

and

The equivalent distances h2 and h3 are:

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and

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A summary of the weights used for the soil classifications, hydraulic conductivity, and cross-variogram models is contained in Table 3. It is important to note that data for hydraulic conductivity and soil classification were available for each cell in the aquifer when computing the cross-variogram. In more realistic cases, both types of data will probably not be available at each location. Problems associated with this disparity can be resolved by using the pseudo-cross variogram (Meyers 1991).

188

GEOSTATISTICAL APPLICATIONS

TABLE 3 - Summary of variogram weights.

Variable Soil classification, S Hydraulic Conductivity, InK Cross-Variogram InK vs. S

wI 1.5 25.0

w2 0.25 4.0

w3 0.5 3.0

-5.75

-0.75

-1.2

FLOW AND ADVECTIVE TRANSPORT MODELS MODFLOW

The three-dimensional finite difference program MODFLOW was used to simulate steady-state saturated flow in the aquifer. MODFLOW uses a block-centered finite difference scheme to solve the groundwater flow equation. A detailed description of MODFLOW can be found in MacDonald and Harbaugh (1988). MODFLOW was modified for use in this study by adding subprograms and changing the existing data collection and storage procedures. A detailed description of these modifications can be found in Benson and Rashad (1994). For each field of hydraulic conductivity that was simulated, MODFLOW was used to compute the total heads at each node and the total flow rate (Q) emanating from cells at the downstream end. The hydraulic head field is used by the advective transport model for simulating contaminant transport. PATH3D

The program PATH3D (Zheng 1988) was used to simulate advective contaminant transport. PATH3D is a general particle-tracking program for calculating groundwater paths and travel times in steady-state or transient, two- or three-dimensional flow fields. A detailed description of PATH3D can be found in Zheng (1988). Changes in PATH3D were required before it could be used in this study. These changes included modifying the algorithm for time step adjustment and modifying the post-processor to describe characteristics of the plume. Details of these changes can be found in Benson and Rashad (1994). Transport was initiated by releasing contaminant fluid particles along a vertical profile located at X =0, Y = 1000 cm, and Z =0 to 2500 cm. Eight particles were placed in each cell along this profile (200 total particles in aquifer). RESULTS AND ANALYSIS Ground water flow and contaminant transport simulations were conducted for three different conditions: (1) fully-defined aquifer, (2) partially defined aquifer using only hydraulic conductivity data, and (3) partially-defined aquifer using hydraulic conductivity and soil classification data. In the partially-defined cases, a co-kriging program (based on the program COKB3D, Deutsch and Journel 1992) was used to

BENSON AND RASHAD ON CO-KRIGING

189

estimate hydraulic conductivity for each finite-difference cell in the aquifer using a liner co-regionalization model for the variograms (see previous sections). In this application, co-kriging was only used to estimate the primary variable, hydraulic conductivity. In addition, point kriging was used instead of block kriging because it was more easily implemented and the cells used to discretize the aquifer were small. Nevertheless, a small error was introduced by point kriging. The variogram models previously discussed were used to describe the spatial correlation structure. Input for the co-kriging program included subsurface information consisting of profiles of hydraulic conductivity or soil classifications. A description of the co-kriging implementation can be found in Benson and Rashad (1994). Estimating the Hydraulic Conductivity Field

Initial Comparison: Kriging vs. Co-Kriging An initial comparison was made between the estimated hydraulic conductivity field and the fully-defined hydraulic conductivity field along a transect (single row of cells; X=50 cm, Z=50 cm, Y=O to 2500 cm) through the aquifer. In one case, the hydraulic conductivity field was estimated with kriging using only hydraulic conductivity data; co-kriging using hydraulic conductivity and soil classification data was used for the other case. u

The hydraulic conductivity fields estimated using kriging and co-kriging are shown in Fig. 5. Hydraulic conductivities measured along three vertical profiles were used as input when only kriging was conducted. The resulting estimated InK field is a smooth, nearly linear interpolation between the profiles at which measurements were made. The estimated InK field is very different from the "true" hydraulic conductivity field obtained from the fully-defined synthetic aquifer. That is , the irregular spatial variations in InK are not preserved.

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500

1000

1500

2000

2500

Y(cm)

FIG. 5 - Estimated hydraulic conductivities along a transect. Co-kriging was conducted using three profiles of hydraulic conductivity (primary variable) and 11 profiles of soil classifications (secondary variable). Figure 5 shows that

190

GEOSTATISTICAL APPLICATIONS

addition of the secondary variable greatly improves the estimated hydraulic conductivity field. The estimated InKs along the transect more closely resemble the "true" InKs. However: even with co-kriging, the estimated field is smoother then the "true" field. Selecting Profiles -- Various exploration schemes (i.e., collections of hydraulic conductivity and soil classification profiles) were selected to evaluate the relative effectiveness of different types of subsurface data in estimating the accuracy of the hydraulic conductivity field. The initial exploration scheme consisted of five profiles of hydraulic conductivity and five profiles of soil classifications. Subsequent schemes incorporating more data were constructed by adding more profiles of either hydraulic conductivities or soil classifications. A consistent method was needed to select locations for additional profiles. The writers chose to select subsequent profiles at locations where the co-kriging variance is largest. These locations have the greatest uncertainty in the estimated hydraulic conductivity. The writers note, however, that locations where the co-kriging variance is largest are not necessarily the critical locations where uncertainty in hydraulic conductivity has the greatest impact on contaminant transport. However, these critical locations cannot be identified a priori, because under normal circumstances the detailed characteristics of the aquifer are unknown. The aforementioned methodology was used to select 14 exploration schemes. The first seven schemes consist of five profiles of hydraulic conductivity (NK=5) and a varying number of soil classification profiles (NS=O, 5, 9, 15,22,32, 125). The second set of seven schemes was similar, except ten hydraulic conductivity profiles (NK=lO) were used. The layout of each exploration scheme is contained in Benson and Rashad (1994). Precision of the Hydraulic Conductivity Field -- In this section, two statistics are used to characterize the precision of the estimated field. These statistics are: the maximum co-kriging variance and the mean co-kriging variance. The mean co-kriging variance (cr~) is used to quantify the global estimation error. The mean co-kriging variance is computed as: H

.... 2

_~ ~ 2 N c 1= .1..1 cr cki'

uci ,

(9)

where cr~k,i is the co-kriging variance at the ith cell in the finite-difference grid and Nc is the total number of grid points (Nc = 12,500). In each case, cr c2k,I· is the variance of the primary variable (hydraulic conductivity) being estimated, as described in Issaks in Srivastava (1989, p. 404). The maximum co-kriging variance is shown in Fig. 6. The maximum co-kriging variance is essentially the same for the schemes using five and ten profiles of hydraulic conductivity; the co-kriging variance is only slightly larger for the scheme using five profiles, More importantly, however, addition of soil classification profiles results in a significant reduction in the maximum co-kriging variance. The mean co-kriging variance is shown in Fig. 7 as a function of the number of soil classification profiles. In this case, the error is significantly larger when five hydraulic conductivity profiles are used instead of ten profiles. A significant difference is

BENSON AND RASHAD ON CO-KRIGING

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191

192

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GEOSTATISTICAL APPLICATIONS

expected, because the mean co-kriging variance represents a global measure of uncertainty whereas the maximum co-kriging variance is a local measure of uncertainty. Adding more hydraulic conductivity profiles will have a significant effect on the maximum co-kriging variance only if the profiles are located near or directly at the location where the maximum co-kriging variance exists because the co-kriging variance is a point measure. In contrast, because the mean co-kriging variance is a global measure of uncertainty, it will be reduced by adding more profiles, regardless of their location. Figure 7 also shows that exploration schemes employing more soil classification profiles with fewer hydraulic conductivity profiles can be as effective in reducing uncertainty as schemes that simply use more hydraulic conductivity profiles. For example, the scheme consisting of five hydraulic conductivity profiles and five soil classification profiles has similar mean co-kriging variance as the scheme using ten hydraulic conductivity profiles and no soil classification profiles. Furthermore, the scheme using more soil classification profiles and fewer hydraulic conductivity profiles is likely to be less expensive. Thus, a similar reduction in uncertainty can be obtained at less cost. Total Flow

One means to evaluate how well the aquifer is characterized is to compare the total flow rate across the compliance surface for the fully-defined condition with the total flow rate when the aquifer is characterized using a limited amount of subsurface data. For the synthetic aquifer, the compliance surface was defined as the downstream boundary (Fig. 1). If the flow rates are not nearly equal, then the aquifer is not adequately characterized. If the flow rate is too high, low conductivity regions blocking flow have been missed. In contrast, a flow rate that is too low is indicative of missing preferential pathways (Fogg 1986). Figure 8 shows total flow rate when the aquifer is characterized with 5 or 10 profiles of hydraulic conductivity and a variable number of soil classification profiles. When no profiles of soil classifications are used (kriging only), the flow rate is one-third to one-half the total flow rate. Apparently, the sampling program has inadequately defined the preferential pathways controlling true total flow. However, when more soil classification profiles are added, the flow rate begins to rise and then becomes equal (i.e., > 10 profiles) to the flow rate for the fully-defined condition. Two other characteristics of Fig. 8 are notable. First, similar flow rates were obtained when five or ten profiles of hydraulic conductivity (but no soil classifications) were used to characterize the aquifer. Apparently, neither set of measurements is of sufficient extent to capture the key features controlling flow. Second, the aquifer was better characterized (in terms of total flow rate) using five hydraulic conductivity profiles and 15 soil classification profiles then 10 hydraulic conductivity profiles and 15 soil classification profiles. This indicates that focusing on collecting a greater quantity of index measurements (i.e., soil classifications) may be more useful in characterization than collecting a fewer number of more precise measurements (i.e., hydraulic conductivities). In this case, hydraulic conductivity inferred from a soil classification had a precision of two to three orders of magnitude, whereas the hydraulic conductivity "measurements" were exact. Thus, in this case, simply defining the existence of critical flow paths apparently is more important than precisely defining their hydraulic conductivity.

BENSON AND RASHAD ON CO-KRIGING

193

120 100

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10 100 FullyNo. Soil Classification Profiles Defined

FIG. 8 - Total Flow Rate Through the Synthetic Aquifer Trajectory of the Plume - Centroid

Trajectory of the plume can be characterized by the coordinates (X, Y, Z) of its centroid. Trajectories for several different explorations schemes are shown in Fig. 9. In each case, the trajectory is recorded for only four years. For longer times, a portion of the plume has passed the downstream edge of the aquifer. Consequently, the statistics used to describe the plume (centroid and variance) are ambiguous. At early times, the trajectory of the centroid does not depend greatly on the exploration scheme. However, as the plume evolves, different trajectories of the centroid are obtained. In particular, the plume moves more slowly in the down-gradient (Xdirection) when the aquifer is characterized with a limited amount of subsurface data. Apparently, the preferential pathways controlling down-gradient movement were inadequately characterized. Addition of soil classification data did not consistently improve the trajectory in the X-direction. Adding five soil classification profiles improved the trajectory significantly, but the worst trajectory was obtained when 22 soil classification profiles were used. Adding even more soil classification profiles (NS = 32 or 125) improved the trajectory only slightly. This is particularly discouraging because 125 soil classification profiles corresponds to sampling 25% of the entire aquifer. The cause of this discrepancy is inadequate representation of subsurface anomalies that affect movement of the plume. At approximately 0.2 years, the centroid of the plume moves dramatically as the particles flow around a low conductivity region. The Y-coordinate increases and the Z-coordinate decreases (i.e., the plume moves upward and towards the rear face of the aquifer). None of the exploration schemes provided enough information to adequately characterize this movement. However, adding soil classification profiles did improve the prediction. When only hydraulic conductivity profiles were used (kriging only), the plume moved downward and to the front which is the exact opposite behavior occurring in the fully-defined case. Adding soil classification profiles did prevent the centroid from moving in the opposite direction

194

GEOSTATISTICAL APPLICATIONS 900 - NK =10, Ns=5

800 700

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- NK =10, Ns=22 -- -. - -- NK =10, Ns=32 - NK =10, N s=125

600

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0

300

/: '

~ -~

/-

200

.--- ,.,0:

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- .-

>--0

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-- - M- --

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-

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-

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1150 1200

Q)E

-u 0:1-

1250

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OC u.,

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1450 1500

L -_ _ _ _ _ _ _ _

0.01

~

_______

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FIG, 9 - Centroid of plume: (a) X-coordinate, (b) Y-coordinate, and (c) Z-coordinate

BENSON AND RASHAD ON CO-KRIGING

195

and, in the case where 125 profiles were used, did result in a subsurface where the plume moved upward and to the rear of the aquifer. Unfortunately, the degree of plume movement existing when 125 soil classification profiles were used is still too small to simulate the fully-defined condition. For brevity, graphs of trajectory of the centroid are not shown for the exploration schemes where 10 hydraulic conductivity profiles were used. These graphs can be found in Benson and Rashad (1994). Smaller errors in the predicted trajectory occurred when ten hydraulic conductivity profiles were used in the exploration scheme. In this case, addition of soil classification profiles also had a smaller impact on the predicted down-gradient movement of the plume. However, adding soil classification profiles did improve the Y and Z-coordinates of the centroid. When only ten hydraulic conductivity profiles were used (kriging only), the plume moved in the opposite direction as was observed in Fig. 9 (NK=5, NS=O). But, when soil classification profiles were added, the plume moved in the correct direction. Nevertheless, the movement occurred more slowly than in the fully-defined case, which caused the down-gradient movement of the plume in the estimated aquifers to lag behind the down-gradient movement of the plume in the fullydefined aquifer. Spreading - Variance of the Plume

Spreading of the plume is characterized by the variance (or second central moment); a larger variance corresponds to a greater amount of spreading. Evolution of the variance of the plume is shown for various exploration schemes in Fig. 10. Some general features of Fig. 10 are noteworthy of mention. First, the variance is larger in the X and Z-directions. The Z-variance is large because the particles are uniformly distributed along a vertical profile (i.e., Z-direction) at the onset of the flow and transport simulation. A large X-variance occurs because down-gradient movement of the plume occurs in the X-direction and thus the X-variance corresponds to longitudinal spreading of the plume. Accordingly, the Y-variance is much smaller because it corresponds to lateral spreading orthogonal to the average hydraulic gradient, which is generally smaller than spreading in the longitudinal direction. The X-variance increases with time. At short times, the variance is small because little spreading of the plume has occurred. However, as the plume moves down-gradient, the variance increases as more spreading occurs. Furthermore, the ability to capture the true amount of spreading depends on the amount of subsurface information used in characterization (Fig. 10). When less information is used (e.g., kriging only, NK=5, NS=O), the variance is smallest and when more information is used (i.e., by adding soil classification profiles) the variance increases. This is expected, because a smoother subsurface containing fewer heterogeneities exists when less data are used in characterization. However, adding more soil classification profiles does not consistently improve the X-variance. In fact, the X-variance for NS=5 is closer to the X-variance in the fully-defined case than the schemes having NS=22, 32, and 125. Similar behavior was noted for the Y-variance. However, adding more soil classification profiles had a more consistent effect on the Z-variance. Adding more soil classification profiles consistently resulted in a Z-variance that was closer to the Zvariance in the fully-defined case.

196

GEOSTATISTICAL APPLICATIONS

~ - - NK=5'NS=5~

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-

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>

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'

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-

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X 200

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=5~

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o ~---------~--------

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~

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NK=5, Ns 500 I 0.01

=: 25

I

I I 1

0.1

4

Time (years)

FIG. 10 - Variance of plume: (a) X-variance. (b) Y-variance. and (c) Z-variance

BENSON AND RASHAD ON CO-KRIGING

197

When ten hydraulic conductivity profiles were used for characterization, the Xvariance in the estimated aquifers was similar to the X-variance in the fully-defined case regardless of the number of soil classification profiles used in characterization (Benson and Rashad 1994). Apparently, the ten hydraulic conductivity profiles used for characterization resulted in a sufficiently heterogeneous subsurface such that spreading in the down-gradient direction was preserved. In contrast, spreading in the Y and Z-directions for the fully-defined case was distinctly different from spreading that occurred in these directions when ten hydraulic conductivity profiles were used for characterization (Benson and Rashad 1994). Adding five soil classification profiles resulted in a V-variance that was closer to the V-variance in the fully-defined case. However, as even more soil classification profiles were added (NS=22, 32, 125), the V-variance became much different than was observed in the fullydefined case. Apparently, the heterogeneities causing spreading in the V-direction were inadequately represented when the subsurface was characterized with additional soil classification profiles. Kriging with only ten soil classification profiles resulted in a Z-variance that differed greatly from the Z-variance in the fully-defined case. For larger times, the Zvariance was much smaller than the Z-variance for the fully-defined condition. However, when soil classifications were added, the Z-variance more closely resembled the Zvariance for the fully-defined case. Thus, using soil classifications apparently resulted in heterogeneities that were similar to those controlling spreading in the Z-direction in the fully-defined aquifer. SUMMARY AND CONCLUSIONS The objective of this study was to illustrate how predictions of contaminant transport differ as the quantity and type of information used to characterize the subsurface changes. Groundwater flow and advective contaminant transport were simulated through a heterogeneous synthetic aquifer that was fully defined. The aquifer was highly heterogeneous, as might be encountered in supraglacial sediments, such as those found in the upper mid-western United States. Additional flow and transport simulations were conducted using versions of the aquifer that were characterized using a limited amount of subsurface data. Comparisons were then made between the true movement of the plume (in the fully defined aquifer) and movement of the plume in versions of the aquifer that were characterized with limited subsurface data. Results of the flow and transport simulations show that soil classifications can be used to augment or replace more costly hydraulic conductivity measurements while maintaining similar accuracy in terms of total flow through the aquifer. However, the geologic details that govern transport through the synthetic aquifer apparently were never sufficiently characterized. Bulk movement of the plume (i.e., the centroid) and spreading (i.e., variance) of the plume were never simulated accurately, regardless of the amount of subsurface data (hard or soft) that were used for characterization.

198

GEOSTATISTICAL APPLICATIONS

ACKNOWLEDGMENT

,,

The study described in this paper was sponsored by the U.S. Dept. of Energy (DOE), Environmental Restoration and Waste Management Young Faculty Award Program. This program is administered by Oak Ridge Associate Universities (ORAU). Neither DOE or ORAU have reviewed this paper, and no endorsement should be implied. REFERENCES

Benson, C. and S. Rashad (1994), "Using Co-Kriging to Enhance Hydrogeologic Characterization," Environmental Geotechnics Report No. 94-1, Dept. of Civil and Environmental Engineering, University of Wisconsin, Madison, WI. Bowles, J. (1984), Physical and Geotechnical Properties of Soils, 2nd Edition, McGrawHill, New York. Cooper, S. and C. Benson (1993), "An Evaluation of How Subsurface Characterization U sing Soil Classifications Affects Predictions of Contaminant Transport," Environmental Geotechnics Report No. 93-1, Dept. of Civil and Environmental Engineering, University of Wisconsin, Madison, WI. Das, B. (1985), Principles of Geotechnical Engineering, PWS-Kent Publishing, Boston. Deutsch, C. and A. Joumel (1992), GSLIB Geostatistical Software Library and User's Guide Book, Oxford University Press, New York. Domenico, P. and F. Schwartz (1990), Physical and Chemical Hydrogeology, John Wiley, New York. Fogg, G. (1986), "Groundwater Flow and Sand Body Interconnectedness in a Thick, Multiple-Aquifer system," Water Resources Research, 22(5), 679-694. Holtz, R. and W. Kovacs (1981), An Introduction to Geotechnical Engineering, PrenticeHall, Englewood Cliffs, N1. Hough, B. (1969), Basic Soils Engineering, 2nd Edition, Ronald Press Co., New York. Isaaks, E. and R. Srivastava (1989), Applied Geostatistics, Oxford Univ. Press, New York. Istok, J., Smyth, J., and Flint, A. (1993), "Multivariate Geostatistical Analysis of GroundWater Contamination: A Case History," Ground Water, 31(3), 63-73. Lee, I., White, W., and Ingles, O. (1983), Geotechnical Engineering, Pitman Co., Boston. McCarthy, D. (1982), Essentials of Soil Mechanics and Foundations, Basic Geotechnics, Reston Publishing, Reston, VA. MacDonald, M. and A. Harbaugh (1988), "A Modular Three-Dimensional Finite Difference Ground-Water Flow Model," Techniques of Water-Resources Investigations of the United States Geological Survey, USGS, Reston, VA. Means, R. and 1. Parcher (1963), Physical Properties of Soils, Merrill Books, Columbus.

BENSON AND RASHAD ON CO-KRIGING

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Meyers, D. (1991), "Pseudo-Cross Variograms, Positive Definiteness and Co-Kriging," Mathematical Geology, 23, 805-816. Mickelson, D. (1986), "Glacial and Related Deposits of Langlade County, Wisconsin," Information Circular 52, Wisc. Geologic and Natural History Survey, Madison, WI. Mitchell, J. (1976), Fundamentals of Soil Behavior, John Wiley and Sons, New York. Scott, C. (1980), An Introduction to Soil Mechanics and Foundations, 3rd Edition, Applied Science Publishers, London. Simpkins, W., McCartney, M., and D. Mickelson (1987), "Pleistocene Geology of Forest County, Wisconsin," Information Circular 61, Wisconsin Geologic and Natural History Survey, Madison, WI. Smith, G. (1978), Elements of Soil Mechanics for Civil and Mining Engineers, 4th Ed., Granada Publishing, London. Seo, D-J, Krajewski, W., and Bowles, D. (1990a), "Stochastic Interpolation of Rainfall Data from Rain Gages and Radar Using Co-Kriging," Water Resources Research, 26(3),469-477. Seo, D-J, Krajewski, W., Azimi-Zonooz, A., and Bowles, D. (1990b), "Stochastic Interpolation of Rainfall Data from Rain Gages and Radar Using Co-Kriging. Results," Water Resources Research, 26(5), 915-924. Sowers, G. and G. Sowers (1970), Introductory Soil Mechanics and Foundations, 3rd Ed., McMillan Co., New York. Webb, E. and Anderson, M. (1996), "Simulation of Preferential Flow in ThreeDimensional, Heterogeneous Conductivity Fields with Realistic Internal Architecture," Water Resources Research, 31(3), 63-73. Whitlow, R. (1983), Basic Soil Mechanics, Construction Press, New York. Zheng, C. (1988), "PATH3D, A Groundwater Path and Travel Time Simulator, User's Manual," S. S. Papadopulos and Associates, Inc., Rockville, MD.

Stanley M. Miller 1 and Anja J. Kannengieser 2

GEOSTATISTICAL CHARACTERIZATION OP HYDRAULIC CONDUCTIVITY USING PIELD

UNSATURATED INPILTROMETER

DATA

REFERENCE: Miller, S. M., and Kannengteser, A. J., "Geostatistical Characterization of Unsaturated Hydraulic Conductivity Using Field Infiltrometer Data," Geostatlstlos for Environmental and Geoteohnioal Applioations, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

ABSTRACT: Estimation of water infiltration and retention in surficial soils is a critical aspect of many geotechnical and environmental site evaluations. The recent development of field-usable tension infiltrometers now allows insitu measurements of unsaturated hydraulic conductivity (K u ), thus avoiding some uncertainties associated with remolded soil samples tested in the laboratory. Several different geostatistical "mapping" methods can be used to spatially characterize Ku , including ordinary and indicator kriging, as well as spatial simulations that provide realizations (stochastic images) of Ku that exhibit more natural variability than do the smoothed spatial estimations of kriging. Multivariate procedures, such as cokriging and Markov-Bayes simulation, can incorporate information from a secondary attribute (e.g., particle size information) to enhance the spatial characterization of an undersampled Ku field. These geostatistical procedures are demonstrated and compared for a case study at a 700 sq. meter site comprised of coarse soil material. Results indicate that percent-by-weight fractions can be used effectively to enhance insitu spatial characterization of Ku' KEY WORDS: unsaturated hydraulic conductivity, particle size, characterization, geostatistics, kriging, spatial simulation.

site

An important physical property to be measured when investigating water infiltration through surficial soils is the insitu unsaturated hydraulic conductivity (K u ). The recent development of field-usable tension infiltrometers now provides the capability to measure insitu Ku , thus avoiding some of the uncertainties associated with remolded soil specimens tested in the laboratory (e.g., loss of insitu soil structure). Even though field measurements of unsaturated hydraulic conductivity exhibit spatial variability, enough spatial dependence typically 10ept. of Geology and Geol. Engrg., Univ. of Idaho, Moscow, 10 20ept . of Mathematics and Statistics, Univ. of Idaho, Moscow, 10

200

83844 83844

MILLER AND KANNENGIESER ON CONDUCTIVITY

201

is observed to warrant a geostatistical investigation to characterize Ku across the study site. Because of a fairly rapid sampling time, as many as 20 to 30 Ku field measurements can be obtained in two days, a much faster and more time-efficient procedure than that associated with laboratory testing of remolded specimens. This provides an adequate number of data for many geostatistical assessments, and the data base can be supplemented by additional data on other physical properties related to Ku (particle-size distribution attributes or insitu density) . Several different geostatistical "mapping" methods can be used to spatially characterize Ku. Univariate procedures include: 1) ordinary kriging, which provides a smoothed map of Ku estimates at unsampled locations across the site, 2) indicator kriging, which provides local estimates of conditional probability distributions of Ku at specified grid locations across the site, and 3) gaussian-based simulations, which provide spatial realizations (stochastic images) of Ku that exhibit more natural variability than do the smoothed spatial estimations of kriging. Multivariate procedures, such as cokriging and Markov-Bayes simulation can incorporate spatial information from a secondary attribute (e.g., the median particle size) to enhance the spatial characterization of an undersampled Ku field. To demonstrate and evaluate these various spatial characterization methods, a case study at a 700 sq. meter site was conducted. The coarse-grained soil material, as represented by 20 sampling locations across the site, had a median particle size of approximately 7 mm and averaged less than 6% by weight fines (i.e., finer than a No. 200 sieve). Insitu Ku values at a 5-cm tension were obtained at each of the 20 sites to provide a minimally sized sample for geostatistical studies. Background information, analytical procedures, and results of the study are presented below.

TENSION

INPILTROMETER

For nearly a hundred years, soil scientists have been describing the flow of water through unsaturated materials. Estimating the amount and rate of such water flow requires knowledge of the Ku/moisture content relationship or the Ku/soil tension (water potential) relationship. The most commonly used method to define these relationships relies on laboratory measurements obtained by pressure desorption of a saturated core of soil material, which leads to the construction of a moisture characteristic curve of moisture content vs. soil tension (Klute 1986). However, there are three problems associated with such testing: 1) the time required to setup samples and then test over a wide range of soil tensions; 2) the cost of field sampling, remolding specimens, and monitoring the laboratory tests, which may take several weeks; and 3) potentially unrealistic results due to the remolding of specimens, which destroys