rL~gL 0AV IS STP 1283 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 Environmental geology-Statistical methods-Congresses 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 Photocopy Rights Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by the AMERICAN SOCIETY FOR TESTING AND MATERIALS for users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the base fee of $2.50 per copy, plus $0.50 per page is paid directly to CCC, 222 Rosewood Dr., Danvers, MA 01923; Phone: (508) 750-8400; Fax: (508) 750-4744 For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged The fee code for users of the Transactional Reporting Service is 0-8031-2414-7/96 $2.50 + 50 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 Describing Spatial Variability Using Geostatistical Analysis-R MOHAN SRI VASTA VA 13 Geostatistical Estimation: Kriging-sHAHROKH ROUHANI 20 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 162 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 200 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 Principal Investigator, Sandia National Laboratories/Spectra Research Institute, MS 1324 P.O Box 5800, Albuquerque, NM 87185-1342 YOUNG ON ANALYSIS OF EXCAVATIONS 267 condition of bars (or their continuity conditions), and the whole truss structure was represented by the global connectivity matrix Then, the independent small size structure representing a block formed by joints can be searched and identified as an independent block matrix system in the global connectivity matrix Each of the independent matrix elemental blocks in the whole system matrix has its own size, shape, and location So, the complete information for a block geometry is known and available from the nodal numbers of the matrix elemental block Element Model The element constructing the whole system of the rock mass is replaced with a truss formed by simple bars connected between two nodes The element can be in any shape or size with different numbers of nodes For simplicity in this paper, a rectangular element with nodal points was used for the rock block calculations Then, the 8-point equal parameter truss element appears like the usual 8-point solid element in the finite element method, but it consists of 28 two-force bars as shown in Figure The volume of this element is the same to that of a solid element and distributed equally on to its nodal points Therefore, the nodal point system and the number of freedoms in the element model were not changed from the finite element model systems The global continuous truss structure for the entire rock mass was developed by constructing this type of truss element on every element in the model The inner nodal points will have 26 bars connecting to adjacent nodes around it FIG 28 bar truss element model Elemental and Global Connectivity Matrices The connectivity of the truss element is an 8x8 matrix since the element has nodal points and only one freedom at each node is needed for the purpose of block calculations Also, the distribution of connection bars at a node is not required to be known exactly Consequently, two indicator numbers are enough to define the continuity condition of the connection bars between any two nodes, such as for no connection or the connection cut by a joint, and for the positive connection, so there is a bar to connect them By applying this indicator system, the connectivity matrix of a truss element can be written simply as follows: 268 K = GEOSTATISTICAL APPLICATIONS 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Compared with the exact stiffness matrix of the finite element method, the freedom of [KJ was reduced from 24 to and the matrix element of [KJ does not express exactly the mechanical behavior of the element However, the connectivity matrix [KJ remains symmetric to follow the principles of mechanics Also, the global connectivity matrix can be assembled from the element connectivity matrix [KJ by following the same procedure of assembling the global stiffness matrix in the finite element method The global connectivity matrix of the entire truss structure in the rock mass, [MJ, is a (nxn) matrix, where n is the number of nodal points in the whole structural matrix Therefore, the information of the continuity or connection between two nodes can be stored at each matrix element in [MJ, because each node has one freedom The [MJ matrix is symmetric and banded along the diagonal direction as in the way of the finite element method Block Calculation Simulated rock joints were introduced into the whole structural matrix one by one (Young and Hoerger 1988a) Whenever one joint plane was introduced, the elements were searched that may be cut by the joint, and which of the 26 bars within the element to be cut were checked Whenever a bar is cut by a joint, the continuity of the global truss structure will be weakened This weakness was reflected on the global connectivity matrix by modifying its matrix element corresponding to the bar that was cut by the joint Actually, the matrix element was subtracted by one from the original number of 26 Consequently, the global connectivity matrix was modified constantly while all of the joints were introduced and all the connection bars were tested for their continuities The final global connectivity matrix must be singular, and will consist of as many independent substructures as blocks formed by the joint systems It is obvious that when a rock block is isolated by joints from the rock mass, its corresponding small substructure will be separated from the global structure These substructures are independent from each other, i.e no connection exists among them This means that the global connectivity matrix, which has been modified completely after the whole joint was introduced, can be transformed into a block matrix by elemental transformation (Golub and Van Loan 1983) Then, each elemental matrix block in the final block matrix transformed from the global connectivity matrix represents an independent substructure, which is nothing but the rock block isolated by the joints and the block searched for the block calculations In this way the problem of searching blocks in the rock mass was replaced with the problem of identifying the independent matrix blocks on the global connectivity matrix The knowledge of nodal numbers within the independent matrix block is enough to identify the shape, size or volume, and location of a rock block formed by the joint systems The volume of a rock block was calculated simply by counting the nodes within its substructural matrix block and summing up their volumes For this connectivity matrix method, there are no limits on the sizes and shapes of joint planes as well as the geometry of rock blocks YOUNG ON ANALYSIS OF EXCAVATIONS 269 formed by the joint systems, including any random aggregation of elements in the three-dimensional space Key Block Failure Once blocks were identified, key blocks were sorted out by following three steps: Collect the joint and excavation plane geometry associated with a particular block Evaluate kinematic stability of the block with an algorithm based on Shi's theorem (Goodman and Shi 19984) and if the block is kinematically unstable Evaluate mechanical stability with Warburton's algorithm (Warburton 1980) • positional Probability of Failure The probabilistic analysis of key block failures was achieved by the positional probability of failure, which was defined by the number of times the position (or node) was evaluated as being contained in a key block In this way the probabilistic key block analysis can be effectively combined with the stochastic joint system simulation, and the realistic structural stability can be obtained in probabilistic terms One of the interesting aspects of this type of analysis is that it provides for the evaluation of progressive key block failures If it is assumed that a key block displaces into the excavation, the next level of blocks exposed to the excavation surface modified by the initial key block failures become potential key blocks The process may continue over many levels of failure With this type of analysis it is very simple to evaluate the successive levels of key block failure around an excavation surface It has a specific application in mining engineering: cavability analysis for the caving method of mining CASE S~IES A few cases were analyzed to demonstrate the capability of localized discrete cell block models and corresponding improvements achieved in the engineering analysis by the finite element method of probabilistic key block theorem Open Pit Slope Stability An extensive statistical analysis was made on a total of 939 joint survey data taken by the cell mapping technique from an open pit mine When pole vectors were projected on the upper hemisphere, three joint sets were identified and separated by FRACTAN computer code (Shanley and Mahtab 1975) as shown in Figure Characteristic parameters and their statistics on those joint sets were summarized in Table 1, which was computed on Grossman's tangent plane (Grossman 1985) Also, their variogram parameters for the isotropic spherical model were presented in Table (Young and Hoerger 1988b) 270 GEOSTATISTICAL APPLICATIONS TABLE l Joint parameters and their statistics Mean Attitude Spacing Roughness Avg Length (deg ) (m) (m) Joint Set Dip Direction (deg.) Dip (deg.) Mean/ Variance Mean/ Variance Mean/ Variance 92.21 353.05 190.3 74.1 70.5 49.2 0.845/0.582 3.17/7.34 0.545/0.235 TABLE Spherical variogram of joint parameters for East set orientation Spacing Roughness Avg Length Sill Nugget Range (m) 0.145 0.06 250 0.53 0.275 250 6.87 4.00 250 0.23 0.135 250 ~~r r ~ ~ ~~ ~ -, _.-pi loe.t i ona ~L- ~ ~ ~ ~ ~ ~ 'UL 1"' no • EAST (METERS) FIG Contours of poles projected on a stereonet ·,et "- FIG Discrete cell block model model for an open pit mine The entire mine was subdivided into cell-blocks as shown in Figure to build a discrete cell-block model Then geostatistical operations were performed to define the spatial variability of every joint parameter by their variograms and to characterize the joint systems within each cell-block by estimating those parameters by kriging techniques First, the local deterministic cell-block model of three joint sets was generated for the mine by using OK The average values of joint parameters were obtained for every cell-block, which is equivalent to the characterization of joint systems within a small unit block In other words, the average local joint system properties were inferred and characterized from the global sample data Then the key block theorems (Goodman and Shi 1984) were applied to study the local slope stability in terms of the maximum safe slope angles within each local cell-block area The local joint parameters kriged previously were used as input for this local slope analysis The localized slope stability was then compared with the slope stability obtained from the global average values of the joint parameters The maximum safe slope angles based on local input showed YOUNG ON ANALYSIS OF EXCAVATIONS 271 significant deviations from those based on global averages as input To illustrate these results graphically, the localized maximum safe slope angles were plotted along the various pit slope dip directions as shown in Figure These significant local deviations would have a major effect on the overall behavior of the mine slope (Hoerger and Young 1987) • For an open pit already designed using global averages as design inputs, geostatistics can identify areas whose slopes could be steepened as well as local high risk areas deserving increased monitoring; for a new mine, using only the limited information available during the development stage, kriging can create a block model of joint orientations which could be used to design not only the final pit slope, but also to design the intermediate slopes " ,· , · · 15 - - glob.l IVg o loca l esti ma te 80 )0 60 ~ a 40 ooO!O~ zo 10 j 00 ~ ~ ~Pf( ""J J 00000 o~ 00000 · 20 180 ZZ5 Z70 315 360 Probabill lyof C.:.llIolU , (I) Pit Slop Oip Di r.ct i on (dog) FIG Safe slope angles using global averages and local estimates as input FIG Histograms of PF for cell-blocks Secondly, the local stochastic joint model was developed for the mine by using IK as described before (Young 1987b) In this case the full probabilistic distribution of joint parameters are available for each local block and the probabilistic stability analysis can be performed on every local block to achieve the localized probability of slope stability Then, the probability of failure (PF) based on the localized probabilistic model of joint systems [PF (IK)) was compared with PF (sample), which was calculated similarly from the global sample distributions, by constructing a histogram of failure probability for local blocks (Figure 6) PF (sample) yielded a marginal PF of 50\ to every cell block, which could be expected from the symmetric distribution of the global sample data (Young and Hoerger 1988b) Therefore, it could be said that the PF (sample) did not improve the stability analysis over the deterministic method used currently PF (IK) draws a distinctively different histogram, which spreads over a wide range of PF between 25\ and 80\ Only 17\ of a total of 36 cellblocks showed the marginal 50\ of PF by the PF (IK) Fifty percent of the 36 blocks had a higher PF than the marginal PF and the remaining 33% showed a PF lower than the marginal PF This clearly indicates that the local variation or spatial variability of joints plays a significant role in the slope stability, and the local probabilistic approach should be applied to achieve an effective slope analysis Also, local risk assessments on the regional pit slopes can be achieved from the PF (IK) analysis at any time period of the mine life It is an important improvement in slope design in general and should be exercised routinely in field projects 272 GEOSTATISTICAL APPLICATIONS "¥ ; r -.,. - , - , , -, -, !L- '- . . -'- -'- -' "01 EAST ,METERSI " IIUt "" '1f1 FIG Spatial distribution of PF (IK) in the pit The spatial distribution of PF (IK) plotted in Figure indicates cell-blocks of higher and lower PF's than the marg i nal 50% Regional zones of higher and lower PF's were formed and zones were scattered throughout the pit, randomly, but it was noticed that block PF's were not scattered randomly in the pit, showing that the analysis can identify local stability trends in the mine A Subway Tunnel A metropolitan subway tunnel was studied to illustrate the difference between traditional key block theorem and the positional probability of key block failures by the finite element approach for the block failure The joint systems and their statistical details were published by cording and Mahar (1974) An unit length of the tunnel was isolated based on the discrete cell-block model The joint systems within this cell-block were modeled and simulated for the stability analysis as done for pit slopes When the frequency of positional block failure was projected along the unit length of tunnel, the cumulative probability of positional failure can be plotted around the tunnel as shown in Figure Compared with the worst case type of analysis by the traditional key block theorem, the positional probability analysis shows clearly the size and frequency of key block occurrences in this projection : ",""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 1"" "fI'"'''''''''''''''''''''''''''''''''''' ",""""",,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, '1#'''''''''''''''''''''''.' '-.11""'"'''''''''''''' i"lliI""''''''II,,,,,,,,,''OEEctJ'-'II'''''''''''''''''' """""",,,,,,,,,,-o71,,-,,,,,,,,,,,,,,"""" iN IIII."""'U,,,,,'''' 10' E92''''''''''',,,'''''' 6-"-,."""""", 'II' -z ,",,"'11 ',',2 " , •••• " ••" ' - C ,,,, -""'2211223 ""UU''' """-'-" ' '-''' '2Jl lJKI -,-."""" 11- - f l ' " ",,,,,,,,-11223 #"""'''''-'23' n""",,'''-'37 mm:m:m;~ 31'5' "'" , "',,, , " " " 51-'''''''' 82""'''''' n-,II""" 1- ~;mmm: Z~~::;;=m:m::""""",,,,,,,,,,,,,,,m:::m::: a ) HaxiDnDI reJlO••bl are b) Positional probabiliti.s of failur FIG Comparison of key block failures (a) with the positional probability of block failures (b) for a tunnel YOUNG ON ANALYSIS OF EXCAVATIONS 273 The positional probability of key block failure carries important features that can be simply implemented in the geotechnical design of excavations For the design of a roof bolting system, the anchor should be located in the area where the positional probability is zero or low The positional probability is the probability of the bolt not being anchored effectively to a stable portion of the rock mass Also, the parts of the excavation requiring the most support can be identified easily from this The other measures of key block stat"istics included here are: The distribution (or histogram) of key block sizes and its mean and standard deviation The total volume of key blocks which summarizes the suspectability to key block failure The frequency of different sizes of key blocks Block Size Distributions The matrix approach for block calculations is general; general in block size, shape and location Therefore, block calculations can be completed within a discrete element Results can be presented in a histogram which shows the frequency of block size distribution (Figure 9) In this case the circular disk model was applied in the joint simulation It should be a part of site characterization for risk assessments in geotechnical and geohydrological engineering " ~ ~ ~ ~ ~ ~ block volume ~ ~ ~ - = FIG The frequency distribution of block sizes formed by three joint sets CONCLUSIONS The most important conclusion, in general, which could be drawn from this work is that the localized probabilistic stability analysis for geotechnical structures can be made at the early stages of engineering design and construction, when only sparse sample data is available It leads to the optimum design of geotechnical structures, optimum in their relative locations and orientations with other peripheral structures, and their shapes and sizes This is achievable through the geostatistical model of characteristic parameters of rock masses Then, it can be said that this is an ideal model of joint systems for many engineering analyses in both rock mechanics and geohydrology As comparing PF (lK) and PF (sample), the local probabilistic analysis of pit slopes is more powerful to draw a detailed and realistic picture of slope stability conditions The local variation of joint orientations played an important role in the slope stability and should 274 GEOSTATISTICAL APPLICATIONS be included in slope design and construction Also a localized fullscale risk assessment can be made from PF (IK) and the pit design and operation can be optimized progressively Geostatistics contributed significant improvements into the modeling of rock joint systems (or site characterizations) for goetechnical structural analysis The spatial variability was fully incorporated in this modeling and the characteristic model parameters were localized The non-parametric approach by IK simplified the modeling of three-dimensional probability distributions of pole vectors projects on the reference sphere Otherwise, the local probability distribution of poles will never be achievable from the sparse sample data available at the early stages of engineering design The mathematical probability density function (pdf) for directional data projected on a sphere is little known and a non-parametric approach has been desirable for a long time [13] Even though the mathematical pdf is available, the sample data available at the early stages will never be enough to describe the local statistical distribution Geostatistics is general and applicable to any characteristic parameters of physical properties for geotechnical materials such as strength values, elastic or plastic constants and flow parameters Therefore, local probabilistic models of these parameters are readily available from geostatistics and corresponding geotechnical analysis can be made in terms of the probability as seen here Considering the dispersion of these geotechnical parameters around their mean values as well as their spatial variations, the local probability analysis is a natural choice for various geotechnical fields in the future The stochastic analysis based on the global sample data distribution did not improve the overall picture of slope stability conditions over the deterministic analysis by using sample mean attitudes, although it yielded a probability of slope failure over a full range of slope angles The deterministic approach based on block theorems treats joint orientations as constant and requires "engineering judgement" to qualitatively incorporate the quantitatively ignored factors of joint sizes and spacings, and their variabilities Because of the fixed joint orientations and assumptions of infinite size of joint, the maximum removable area approach of the deterministic key block analysis provides an upper bound to the key block size identified When the probabilistic analysis is coupled with the localized stochastic model of joint systems in geological formations, a Significant amount of engineering judgement required to optimize the size, shape and orientation of an excavation could be replaced by quantitative solutions The positional probability of key block failure carries important features; the parts of excavation requiring the most support, probability of roof bolts not being anchored in stable zones, distributions of key block volumes, and key block sizes and frequency In-situ block size distribution should be a part of geotechnical and geohydrological site characterizations It is a pertinent parameter to be included in key block failures, and it is related directly to the transmissibility of fluid flow through the fractured rock as in the granular materials A further study is deserved for this hydrological application of the block size distribution REFERENCES Baecher, G B., Lanney, N A and Einstein, H H., 1977, "statistical Description of Rock Properties and Sampling," Proceedings of the Eighteenth Symposium on Rock Mechanics, Golden, Colorado Chiles, J P., 1988, "Fractal and Geostatistical Methods for Modeling of a Fracture Network," Mathematical Geology, Vol 20, pp 631-654 YOUNG ON ANALYSIS OF EXCAVATIONS 275 Cording, E J and Mahar, J W., 1974, "The Effects of Natural Geologic Discontinuities on Behavior of Rock in Tunnels," Proceedings of 1974 Rapid Excayation and Tunneling Conference, San Francisco, CA, pp 107138 Golub, G H and VanLoan, C F., 1983, "Matrix Computations," John Hopkins University Press, Baltimore, MD Goodman, R E and Shi, G H., 1984, "Block Theory and Its Application to Rock Mechanics," Prentice-Hall, Englewood Cliffs Grossman, N F., 1985, "The Bivariate Normal Distribution on the Tangent Plane at the Mean Attitude," proceedings of International Symposium on Fundamentals of Rock Joints," Bjorkliden, Sweden, pp 3-11 Hoerger, S F and Young, D S 1987, "Predicting Local Rock Mass Behavior Using Geostatistics," Proceedings of Twenty-eighth U.S Symposium on Rock Mechanics, Tucson, AZ, pp 99-106 Journel, A G and Huijbregts, Ch J., 1978, "Mining Geostatistics," Academic Press, London Lemmer, C., 1984, "Estimating Local Recoverable Reserves via Indicator Kriging," Proceedings of Geostatistics for Natural Resources Characterization (ed by G Verlys), D Reidel, Dordrecht, pp 349-364 Miller, S M., 1979, "Geostatistical Analysis for Evaluating Spatial Dependence in Fracture Set Characteristics," Proceedings of the Sixteenth Symposium on Application of Computers and Operations Research in the Mineral Industry, Tucson, AZ, pp 537-545 Shanley, R J and Mahtab, M A., 1975, "FRACTAN: A Computer Code for Analysis of Clusters Defined on Unit Hemisphere," U.S Bureau of Mines, IC 8671, Washington, DC Warburton, P M., 1980, "Stereological Interpretation of Joint Trace Data: Influence of Joint Shape and Implications for Geological surveys," International Journal of Rock Mechanics & Mining Science, Vol 17, pp 305-316 Young, D S., 1987a, "Random Vectors and Spatial AnalysiS by Geostatistics for Geotechnical Applications," Mathematical Geology, Vol 19, pp 467-479 Young, D 5., 1987b, "Indicator Kriging for Unit Vectors; Rock Joint Orientations," Mathematical Geology, Vol 19, pp 481-502 Young, D S and Hoerger, S F., 1988a, "Non-Parametric Approach for Localized Stochastic Model of Rock Joint Systems," Geostatistical Sensitivity and Uncertainty Methods for Ground-Water Flow and Radionuclide Transport Modeling (ed by B Buxton), Battelle Press, Columbus, OH, pp 361-385 Young, D S and Hoerger, S F., 1988b, "Geostatistics Applications to Rock Mechanics," Proceedings of Twenty-ninth U.S Symposium on Rock Mechanics, Balkema, Brookfield, pp 271-282 Author Index N Benson, C H., 181 Bogardi, L., 248 Buxton, B E., 51 Nicoletis, S., 69 C p Carr, J R, 236 Colin, P., 69 Cromer, M V., 3, 218 Pate, A D., 51 Patinha, P J., 146 Pereira, M J., 146 D R Dagdelen, K., 117 Dahab, M F., 248 Desbarats, A J., 32 Rashad, S M., 181 Rautman, C A, 218 Rouhani, S., 20, 88 F s Froidevaux, R, 69 Schofield, N., 133 Schulte, D D., 162 Soares, A 0., 146 Srivastava, R M., 13 G Garcia, M., 69 Goderya, F S., 248 T J Turner, A K., 117 Johnson, R L., 102 Jones, D D., 162 W K Wells, D E., 51 Wild, M R, 88 Woldt, W E., 162,248 Kannengieser, A J., 200 Kuhn, G N., 162 L y Leonte, D., 133 Young, D S., 262 z M Miller, S M., 200 Zelinski, W P., 218 277 Subject Index A Electromagnetics, 162 Estimation procedures, 20 Annealing, 69 Arsenic,13 ASTM standards, 13, 32 F Flow model, 51 Flow simulation, 181 B H Bayesian analysis, 102 Bivariate distribution, 69 Block failure, 262 Block value estimation, 20 Histogram, 32 Hotspots, 133 Hydraulic conductivity, 181,200 Hydraulic head analysis, 51 C I Conceptual interpretation, Conditional probability, 133 Conductivity, 162 Contamination copper, 146 delineation, 88, 102 lead, 13, 133 metal, 13, 51, 133, 146 plume mapping, 162 site mappmg, 69 soil, 133 stationarity, assessment with, 117 subsurface, 162 transport, 181 uranium, 51 Contouring, 133 Copper, 146 Core data, 218 Correlogram, 13 D Indicator simulation, 218 Infiltrometer, 200 Interpolation techniques, 20 Inverse-distance weighting, 51 Irrigation practices, 248 J Joint model, 262 K Kriging,20,32,162,200,262 cokriging, 88, 181 indicator, 69, 102, 117, 133, 236,262 lognormal, 51 L Leachate, 162 Lead, 13, 133 Design analysis, Direct measurement, M Mapping, 20, 32,51,200 plume, 162, 181 probability, 69 Markov-Bayes simulation, 200 Matrix approach, 262 E Earthquakes, 236 Electncal resistivity, 69 279 280 GEOSTATISTICS Mercalli intensity, 236 Metals, heavy, 13, 133 Mine effluent, 146 Modeling flow, 51 histogram, 32 joint, 262 numerical,218 spatial variability measures, 13 stationary, 117 stochastic, 117, 146 transport, 102 variogram, 13,32,218,236, 248 water quality, 146 Multivariate approach, 88, 200 N Nitrate, 248 Numerical methods, Numerical model, 218 p Pit slope stability, 262 Polycyclic aromatic hydrocarbons, 69 Probabilistic stability analysis, 262 Site remediation strategies, 32 Soil classifications, 181 Soil contamination, 133 Soil gas measurements, 88 Soil nitrate, 248 Space-time series, 146 Spatial patterns analysis, 248 Spatial simulation, 200 Spatial temporal analysis, 51 Spatial variation, 13, 20, 248 modeling, 32 Stability analysis, 262 Stationarity, second order, 117 Stationary model, 117 Statistical methods, Stochastic images, 200 Stochastic modeling, 117, 146 Structural stability, 262 Subway tunnels, 262 T Transport simulation, 102, 181 u Uncertainty, 102 characterization, 69 measures, kriging, 51 Uranium, 51 R Repository system, 218 Rock quality designation, 218 s Sampling program adaptive, 102 planning, 13 strategy, 102 Second order stationarity, 117 Seismic hazard, 236 Site characterization, 102 v Variogram, 13,32,218,236 semivariograms, 248 Vario~raphies, 88 Volatile organic screening, 88 w Water quality modeling, 146 ISBN 0-8031-2414-7