Designation: D5923 − 96 (Reapproved 2010)Standard Guide forSelection of Kriging Methods in Geostatistical SiteInvestigations1This standard is issued under the fixed designation D5923; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.INTRODUCTIONGeostatistics is a framework for data analysis, estimation, and simulation in media whosemeasurable attributes show erratic spatial variability yet also possess a degree of spatial continuityimparted by the natural and anthropogenic processes operating therein. The soil, rock, and containedfluids encountered in environmental or geotechnical site investigations present such features, and theirsampled attributes are therefore amenable to geostatistical treatment. Kriging methods are geostatis-tical techniques for spatial estimation belonging to the class of least-squares estimators. This guidereviews criteria for selecting a kriging method, offering direction based on a consensus of viewswithout recommending a standard practice to follow in all cases.1. Scope1.1 This guide covers recommendations for selecting appro-priate kriging methods based on study objectives, exploratorydata analysis, and analysis of spatial variation.1.2 This guide considers commonly used forms of kriging,including ordinary kriging, simple kriging, lognormal kriging,universal kriging, and indicator kriging. Multivariate, space-time, and other less-frequently used kriging methods are notdiscussed; however, this is not intended to reflect any judge-ment as to the validity of these methods.1.3 This guide describes conditions for which kriging meth-ods are not appropriate and for which geostatistical simulationsapproaches should be used.1.4 This guide does not discuss non-geostatistical alterna-tives to kriging, such as splines or inverse-distance techniques.1.5 This guide does not discuss the basic principles ofkriging. Introductions to geostatistics and kriging may befound in numerous texts including Refs (1-3).2A review ofkriging methods is given in Ref. (4).1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.1.7 This guide offers an organized collection of informationor a series of options and does not recommend a specificcourse of action. This document cannot replace education orexperience and should be used in conjunction with professionaljudgment. Not all aspects of this guide may be applicable in allcircumstances. This ASTM standard is not intended to repre-sent or replace the standard of care by which the adequacy ofa given professional service must be judged, nor should thisdocument be applied without consideration of a project’s manyunique aspects. The word “Standard” in the title of thisdocument means only that the document has been approvedthrough the ASTM consensus process.2. Referenced Documents2.1 ASTM Standards:3D653 Terminology Relating to Soil, Rock, and ContainedFluidsD5549 Guide for The Contents of Geostatistical Site Inves-tigation Report (Withdrawn 2002)4D5922 Guide for Analysis of Spatial Variation in Geostatis-tical Site InvestigationsD5924 Guide for Selection of Simulation Approaches inGeostatistical Site Investigations1This guide is under the jurisdiction ofASTM Committee D18 on Soil and Rockand is the direct responsibility of Subcommittee D18.01 on Surface and SubsurfaceCharacterization.Current edition approved May 1, 2010. Published September 2010. Originallyapproved in 1996. Last previous edition approved in 2004 as D5923 – 96 (2004).DOI: 10.1520/D5923-96R10.2The boldface numbers in parentheses refer to a list of references at the end ofthe text.3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.4The last approved version of this historical standard is referenced onwww.astm.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 additivity, n—a mathematical property of a regional-ized variable stating that it can be combined linearly in order todefine a similar variable on a larger support.3.1.2 block kriging, n—a form of kriging in which thevariable to be estimated has a rectangular or possibly irregularone-, two-, or three-dimensional support.3.1.3 drift, n—in geostatistics, a systematic spatial variationof the local mean of a variable, usually expressed as apolynomial function of location coordinates.3.1.4 estimation, n—a procedure by which the value of avariable at an unsampled location is predicted using a weightedaverage of sample values from the neighborhood of thatlocation.3.1.5 field, n—in geostatistics, the region of one-, two- orthree-dimensional space within which a regionalized variableis defined.3.1.6 indicator kriging, n—a form of kriging in which alldata are indicator variables.3.1.7 indicator variable, n—a regionalized variable that canhave only two possible values, 0 or 1.3.1.8 kriging, n—an estimation method where sampleweights are obtained using a linear least-squares optimizationprocedure based on a mathematical model of spatial variabilityand where the unknown variable and the available samplevalues may have a point or block support.3.1.9 kriging variance, n—the expected value of the squareddifference between the true value of an unknown variable andits kriging estimate, sometimes used as a measure of krigingprecision.3.1.10 lognormal kriging, n—the kriging of log-transformedvariables followed by a back-transformation procedure basedon a lognormal distribution model.3.1.11 nugget effect, n—the component of spatial varianceunresolved by the sample spacing, including the variance dueto measurement error.3.1.12 ordinary kriging, n—a form of kriging for which themean of the estimated variable is an unknown constant and thesample weights sum to one.3.1.13 point, n—in geostatistics, the location in the field atwhich a regionalized variable is defined. It also commonlyrefers to the support of sample-scale variables.3.1.14 point kriging, n—a form of kriging in which thevariable to be estimated has the same support as the sampledata.3.1.15 regionalized variable, n—a measured quantity or anumerical attribute characterizing a spatially variable phenom-enon at a location in the field.3.1.16 search neighborhood, n—the region within whichsamples are considered for inclusion in the kriging estimationprocess.3.1.17 simple kriging, n—a form of kriging for which themean of the estimated variable is a known constant and the sumof sample weights is unconstrained.3.1.18 simulation, n—in geostatistics, a Monte-Carlo proce-dure for generating realizations of fields based on the randomfunction model chosen to represent a regionalized variable. Inaddition to honoring a random function model, the realizationsmay also be constrained to honor data values observed atsampled locations.3.1.19 smoothing effect, n—in geostatistics, the reduction inspatial variance of estimated values compared to true values.3.1.20 spatial average, n—a quantity obtained by averaginga regionalized variable over a finite region of space.3.1.21 support, n—in geostatistics, the spatial averagingregion over which a regionalized variable is defined, oftenapproximated by a point for sample-scale variables.3.1.22 universal kriging, n—a form of kriging in whichadditional weighting constraints are introduced in order toaccount for a drift in the estimated variable.3.1.23 variogram, n—a measure of spatial variation definedas one half the variance of the difference between two variablesand expressed as a function of the lag; it is also sometimesreferred to as the semi-variogram.3.2 For definitions of other terms used in this guide, refer toTerminology D653 and Guides D5549, D5922, and D5924.Acomplete glossary of geostatistical terminology is given in Ref(7).4. Significance and Use4.1 This guide is intended to encourage consistency andthoroughness in the application of kriging methods toenvironmental, geotechnical, and hydrogeological site investi-gations.4.2 This guide may be used to assist those performing akriging study or as an explanation of procedures for qualifiednonparticipants that may be reviewing or auditing the study.4.3 This guide encourages the use of site-specific informa-tion for the selection of an appropriate kriging method;however, the quality of data, the sampling density, and sitecoverage cannot be improved or compensated by any choice ofkriging method.4.4 This guide describes conditions for which kriging orparticular kriging methods are recommended. However, thesemethods are not necessarily inappropriate if the stated condi-tions are not encountered.4.5 This guide should be used in conjunction with GuidesD5549, D5922, and D5924.5. Selection of Kriging Methods5.1 The following subsections describe conditions for whichvarious kriging methods are appropriate. Each section corre-sponds to a step in a geostatistical site investigation where adecision concerning the most appropriate form of kriging mayhave to be made. Ordinary kriging is the most common form ofD5923 − 96 (2010)2kriging and is the conventional default unless any of thefollowing conditions makes another method more appropriate.5.2 Study Objectives—A common objective of geostatisticalsite investigations is to produce a two- or three-dimensionalspatial representation of a regionalized variable field from a setof measured values at different locations. Such spatial repre-sentations are referred to here as maps. Estimation approaches,including all forms of kriging, yield maps that exhibit asmoothing effect, whereas simulation approaches yield mapsthat preserve the spatial variability of the regionalized variable.5.2.1 If mapped values of the regionalized variable arerequired to provide a least-squares estimate of actual values atunsampled points, then a kriging method is appropriate.5.2.2 If mapped values of the regionalized variable are topreserve the spatial variability of values at unsampled points,then simulation rather than kriging should be used.NOTE 1—Preservation of in-situ spatial variability is important ifmapped values of the regionalized variable are to be entered in anumerical model of a dynamic process and therefore simulation shouldgenerally be used. For example, mapped values of transmissivity to beentered in a numerical model of groundwater flow should not be generatedby kriging since this may produce spurious flow patterns (6, 7). However,if the numerical process model is insensitive to spatial variations of theregionalized variable, then kriging methods may also be used.5.2.3 If an objective of the study is to generate multiplepossible outcomes of a regionalized variable field for thepurpose of risk analyses or sensitivity studies, then krigingmethods are inappropriate and simulation approaches arerecommended.5.2.4 If an objective of the study is to estimate probabilitydistributions for regionalized variables over an entire field, asrequired for calculating site-wide compliance probabilities,then kriging methods are inappropriate and simulation ap-proaches are recommended.5.2.5 If an objective of the study is to provide the best linearunbiased estimate of a regionalized variable at unsampledlocations, and the mean is assumed constant but unknown, thenordinary kriging is the appropriate estimation method.5.2.6 If an objective of the study is to provide the best linearunbiased estimate of a regionalized variable at unsampledlocations, and the mean is presumed known, then simplekriging is the appropriate estimation method.NOTE 2—However, knowledge of the mean is an assumption seldomjustified unless the mean can be confidently represented by some priordeterministic model. The model for the mean is then used to remove trendsin the original data leaving the residuals with a mean of zero.5.2.7 If an objective of the study is to quantify uncertaintyusing the kriging variance and data are adequately representedby a Gaussian distribution, then ordinary or simple kriging areappropriate estimation methods.5.2.8 If an objective of the study is to quantify uncertaintyusing the kriging variance and log-transformed data are ad-equately represented by a Gaussian distribution, then ordinaryor simple lognormal kriging are appropriate estimation meth-ods.5.2.9 If an objective of the study is to quantify uncertaintyand data are not adequately represented by a Gaussiandistribution, then the use of kriging variances is notappropriate, and indicator kriging is the preferred estimationmethod.5.3 Choice of Regionalized Variable—The choice of region-alized variable made at the beginning of a geostatistical siteinvestigation may affect the selection of an appropriate krigingmethod.5.3.1 If the regionalized variable is binary or categorical,then indicator kriging is an appropriate estimation method.5.3.2 If the regionalized variable has the same support as thesample data, then point forms of kriging are appropriate.5.3.3 If the regionalized variable is additive and has a blocksupport, and the data have a point support, then block forms ofkriging are appropriate.NOTE 3—However, if indicator or log-transformed regionalized vari-ables are considered, then estimated block values should be interpretedwith caution.5.4 Exploratory Data Analysis—Exploratory data analysisduring a geostatistical site investigation often reveals featuresof the data probability distribution function that affect theselection of an appropriate kriging method.5.4.1 If log-transformed data are approximately Gaussian,then lognormal kriging may be an appropriate estimationmethod.5.4.2 If the data include extreme values that cannot betreated as spatial outliers or separate populations, then indica-tor kriging is an appropriate estimation method.NOTE 4—However, if the data are skewed and this skewness is causedby only a few outliers or clustered sampling, then ordinary krigingremains an appropriate estimation method.5.5 Analysis of Spatial Variation—Analysis of spatial varia-tion during a geostatistical site investigation often revealsfeatures of the data spatial variability structure that affect theselection of an appropriate kriging method.5.5.1 If the analysis of log-transformed data reveals abetter-defined spatial variation structure than an analysis of theoriginal data, then lognormal kriging may be an appropriateestimation method.5.5.2 If calculated indicator variograms for different th