Astm stp 1283 1996

REFERENCES Desbarats, A.J., "Modeling Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R.. Rouhani, S.

rL~gL 0AV IS

ASTM Publication Code Number (PCN):

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

CIP

Copyright © 1996 AMERICAN SOCIETY FOR TESTING AND MATERIALS, West Conshohocken,

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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

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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

Describing Spatial Variability Using Geostatistical Analysis-R MOHAN SRI V ASTA V A 13

Modeling Spatial Variability Using Geostatistical Simulation-ALEXANDER J

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

Effective Use of Field Screening Techniques in Environmental Investigations: A

Multivariate Geostatistical Approach-MIcHAEL R WILD AND SHAHROKH

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

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

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 200

Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities

Design at Yucca Mountain, Nevada-MARc V CROMER, CHRISTOPHER A

Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective after the 1994 Northridge, California Earthquake-JAMES R CARR 236

Spatial Patterns Analysis of Field Measured Soil Nitrate-FARIDA S GODERY A, M F

DAHAB, W E WOLDT, AND I BOGARD!

Geostatistical Joint Modeling and Probabilistic Stability Analysis for

Overview Papers

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

4 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 risk-based decision analyses For example, a geostatistically-based, costlrisklbenefit approach to sample optimization has been shown to provide a framework for examining the many trade-offs encountered when juggling the risks associated with remedial investigation, remedial

CROMER ON A TECHNOLOGY TRANSFERRED 5 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

6 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 off aster, 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-

CROMER ON A TECHNOLOGY TRANSFERRED 7

FIGURE 1: EXHAUSTIVE PB DATA

Number of samples: 7700 Number of samples = 0 ppm: 213 (3%) Number of samples> 500 ppm: 1426 (19%)

Minimum: 0 ppm Lower quartile: 120 ppm Median: 261 ppm Upper quartile: 439 ppm Maximum: 1066 ppm Mean: 297 ppm Standard deviation: 218 ppm

100 200 300 400 500 600 700 800 900 1000

Pb (ppm)

FIGURE 3: SAMPLE PB DATA

Minimum: 0 ppm Lower quartile: 239 ppm Median: 449 ppm Upper quartile : 613 ppm Maximum : 1003 ppm

Mean : 431 ppm Standard deviation: 237 ppm

~~~~~~~~~~~

100 200 300 400 500 600 700 800 900 1000

0

Pb (ppm)

J J L.1 l W 4-J-L.4-J.~Ll JJ.dJl!:ILt ill ~ ~ ~ ;

0 20 40 60 80 100 120 140 160 180 200

As (ppm)

12 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-In-Soils 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, Be, Canada V5V 3K8

13

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 phe-nomenon 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 dif-ferent 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

Separation distance (in m)

us-ing the sample lead data set described by

Separation distance (in m) 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 gram are, in an approximate sense, mirror images As the variogram gradually rises and

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

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

3The "semivariogram", which is simply the variogram divided by two, has a sill that is approximately

equal to the variance of the data

16 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 lyzed

ana-80000

!!!

.~ 40000 I'll

Separation distance (in m)

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

SRIVASTAVA ON SPATIAL VARIABILITY 17

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

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

(a) Northwest - Southeast

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 dent In terms of the variogram, the range of correlation often depends on direction

depen-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 cal studies typically involve the calculation of separate variograms and correlograms for different directions Figure 5 shows directional variograms for the sample lead data pre-sented by Cromer The range of correlation shown by the NW-SE variogram (Figure 5a)

Geostatisti-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

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

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 gations, 1996, Draft standard from D18.01.07 Section on Geostatistics

Investi-Cromer, M.V., 1996, "Geostatistics for Environmental and Geotechnical Applications:

A Technology Transfer," Geostatistics for Environmental and Geotechnical cations, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V Cro-mer, A Ivan Johnson, Ed., American Society for Testing and Materials, West Conshohocken, P A

Appli-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 tal and Geotechnical Applications, ASTM STP ma, R Mohan Srivastava, Shah-rokh Rouhani, Marc V Cromer, A Ivan Johnson, Ed., American Society for Test-ing and Materials, West Conshohocken, PA

Environmen-Srivastava, R.M and Parker, H.M., 1988, "Robust measures of spatial continuity,"

Geostatistics, M Armstrong (ed.), Reidel, Dordrecht, p 295-308

Shahrokh Rouhani

GEOSTATISTICAL ESTIMATION: KRIGING

REFERENCE: Rouhani, S., "Geostatistical Estimation: Kriging," Geostatistics for ronmental 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

Envi-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 weighting functions, reveals a number of advantages (Rouhani, 1986) Kriging directly incorporates the model of the spatial variability of data This allows kriging to produce site-specific and variable-specific interpolation schemes Estimation criteria of kriging are based on

distance-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

I 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

f

~

24 GEOSTATISTICAL APPLICATIONS

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

i:} 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

0

U ", '"

o 500 1000

(Blank spaces are not estimated)

-t

o l>

r l> -0

Gi

Z

Gl

I\)

(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 non-bias 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

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

i This resulted in an improved transmissivity map The present STP provides examples of

(Non-linea, Kriging

i ~' The above linear kriging techniques do not require any implicit assumptions about the

~:"underlying 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

I , ""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

~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

~.',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)

:~~

:~i

'l i ~or more information on ~iging, readers are referred to Journel and Huijbregts (1978),

~de Marslly (1986), Isaaks and Snvastava (1989), and ASCE (1990) ASTM Standard D , 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

5549-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

30 GEOSTATISTICAL APPLICATIONS

References

(1) 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

(2) 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

(3) 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

(4) 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

(5) 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

(6) Delhomme, J.P., "Kriging in the hydro sciences , " Advances in Water Resources, 1(5),

251-266, 1978

(7) 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

(8) 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

(9) Isaaks, E H and R M Srivastava, An Introduction to Applied Geostatistics, Oxford

University Press, New York, 561 p., 1989

(10) Joumel, A G and C Huijbregts, Mining Geostatistics, Academic Press, London, 600

p.,1978

(11) 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

(12) Marsily, G de, Quantitative Hydrogeology, Academic Press, Orlando, 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

de-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)

Regard-less 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

estima-tion 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

de-termine 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

repro-duce the histograms and variograms of Pb and As in addition to observed

measure-ments at sampled locations Although the full potential of the simulation method

is truly achieved only in sensitivity or risk analysis studies involving multiple

out-comes 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

variabil-ity is often a sobering warning to planners whereas maps of kriged values are easily

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