1、Designation: D5923 96 (Reapproved 2010)D5923 18Standard 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 rev
2、ision, 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
3、 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 theirsamp
4、led 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 viewswith
5、out recommending a standard practice to follow in all cases.1. Scope1.1 This guide covers recommendations for selecting appropriate kriging methods based on study objectives, exploratory dataanalysis, and analysis of spatial variation.1.2 This guide considers commonly used forms of kriging, includin
6、g ordinary kriging, simple kriging, lognormal kriging,universal kriging, and indicator kriging. Multivariate, space-time, and other less-frequently used kriging methods are not discussed;however, this is not intended to reflect any judgement as to the validity of these methods.1.3 This guide describ
7、es conditions for which kriging methods are not appropriate and for which geostatistical simulationsapproaches should be used.1.4 This guide does not discuss non-geostatistical alternatives to kriging, such as splines or inverse-distance techniques.1.5 This guide does not discuss the basic principle
8、s of kriging. Introductions to geostatistics and kriging may be found innumerous texts including Refs (1-3).2 A review of kriging methods is given in Ref. (4).1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibilityof the use
9、r of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine theapplicability of regulatory limitations prior to use.1.7 This guide offers an organized collection of information or a series of options and does not recommend a specific courseof ac
10、tion. This document cannot replace education or experience and should be used in conjunction with professional judgment.Not all aspects of this guide may be applicable in all circumstances. This ASTM standard is not intended to represent or replacethe standard of care by which the adequacy of a give
11、n professional service must be judged, nor should this document be appliedwithout consideration of a projects many unique aspects. The word “Standard” in the title of this document means only that thedocument has been approved through the ASTM consensus process.1.8 This international standard was de
12、veloped in accordance with internationally recognized principles on standardizationestablished in the Decision on Principles for the Development of International Standards, Guides and Recommendations issuedby the World Trade Organization Technical Barriers to Trade (TBT) Committee.1 This guide is un
13、der the jurisdiction of ASTM Committee D18 on Soil and Rock and is the direct responsibility of Subcommittee D18.01 on Surface and SubsurfaceCharacterization.Current edition approved May 1, 2010Dec. 15, 2018. Published September 2010December 2018. Originally approved in 1996. Last previous edition a
14、pproved in 20042010as D5923 96 (2004). 96(2010). DOI: 10.1520/D5923-96R10.10.1520/D5923-18.2 The boldface numbers in parentheses refer to a list of references at the end of the text.This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of wh
15、at changes have been made to the previous version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered th
16、e official document.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States12. Referenced Documents2.1 ASTM Standards:3D653 Terminology Relating to Soil, Rock, and Contained FluidsD5549 Guide for The Contents of Geostatistical Site Investigat
17、ion Report (Withdrawn 2002)4D5922 Guide for Analysis, Interpretation, and Modeling of Spatial Variation in Geostatistical Site InvestigationsD5924 Guide for Selection of Simulation Approaches in Geostatistical Site Investigations3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1
18、additivity, na mathematical property of a regionalized variable stating that it can be combined linearly in order to definea similar variable on a larger support.3.1.2 block kriging, na form of kriging in which the variable to be estimated has a rectangular or possibly irregular one-, two-,or three-
19、dimensional support.3.1.3 drift, nin geostatistics, a systematic spatial variation of the local mean of a variable, usually expressed as a polynomialfunction of location coordinates.3.1.4 estimation, na procedure by which the value of a variable at an unsampled location is predicted using a weighted
20、 averageof sample values from the neighborhood of that location.3.1.5 field, nin geostatistics, the region of one-, two- or three-dimensional space within which a regionalized variable isdefined.3.1.6 indicator kriging, na form of kriging in which all data are indicator variables.3.1.7 indicator var
21、iable, na regionalized variable that can have only two possible values, 0 or 1.3.1.8 kriging, nan estimation method where sample weights are obtained using a linear least-squares optimization procedurebased on a mathematical model of spatial variability and where the unknown variable and the availab
22、le sample values may havea point or block support.3.1.9 kriging variance, nthe expected value of the squared difference between the true value of an unknown variable and itskriging estimate, sometimes used as a measure of kriging precision.3.1.10 lognormal kriging, nthe kriging of log-transformed va
23、riables followed by a back-transformation procedure based ona lognormal distribution model.3.1.11 nugget effect, nthe component of spatial variance unresolved by the sample spacing, including the variance due tomeasurement error.3.1.12 ordinary kriging, na form of kriging for which the mean of the e
24、stimated variable is an unknown constant and thesample weights sum to one.3.1.13 point, nin geostatistics, the location in the field at which a regionalized variable is defined. It also commonly refersto the support of sample-scale variables.3.1.14 point kriging, na form of kriging in which the vari
25、able to be estimated has the same support as the sample data.3.1.15 regionalized variable, na measured quantity or a numerical attribute characterizing a spatially variable phenomenonat a location in the field.3.1.16 search neighborhood, nthe region within which samples are considered for inclusion
26、in the kriging estimation process.3.1.17 simple kriging, na form of kriging for which the mean of the estimated variable is a known constant and the sum ofsample weights is unconstrained.3.1.18 simulation, nin geostatistics, a Monte-Carlo procedure for generating realizations of fields based on the
27、randomfunction model chosen to represent a regionalized variable. In addition to honoring a random function model, the realizations mayalso be constrained to honor data values observed at sampled locations.3.1.19 smoothing effect, nin geostatistics, the reduction in spatial variance of estimated val
28、ues compared to true values.3.1.20 spatial average, na quantity obtained by averaging a regionalized variable over a finite region of space.3.1.21 support, nin geostatistics, the spatial averaging region over which a regionalized variable is defined, oftenapproximated by a point for sample-scale var
29、iables.3.1.22 universal kriging, na form of kriging in which additional weighting constraints are introduced in order to account fora drift in the estimated variable.3 For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer Service at serviceastm.org. For Annual Bo
30、ok of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.4 The last approved version of this historical standard is referenced on www.astm.org.D5923 1823.1.23 variogram, na measure of spatial variation defined as one half the variance of the difference
31、 between two variables andexpressed as a function of the lag; it is also sometimes referred to as the semi-variogram.3.1 For definitions of other terms used in this guide, refer to Terminology D653 and Guides D5549, D5922, and D5924. Acomplete glossary of geostatistical terminology is given in Ref (
32、5).Definitions:3.1.1 For definitions of common technical terms used in this standard, refer to Terminology D653.3.2 Definitions of Terms Specific to This Standard:3.2.1 additivity, na mathematical property of a regionalized variable stating that it can be combined linearly in order to definea simila
33、r variable on a larger support.3.2.2 block kriging, na form of kriging in which the variable to be estimated has a rectangular or possibly irregular one-, two-,or three-dimensional support.3.2.3 drift, nin geostatistics, a systematic spatial variation of the local mean of a variable, usually express
34、ed as a polynomialfunction of location coordinates.3.2.4 estimation, nin geostatistics, a procedure by which the value of a variable at an unsampled location is predicted usinga weighted average of sample values from the neighborhood of that location.3.2.5 field, nin geostatistics, the region of one
35、-, two- or three-dimensional space within which a regionalized variable isdefined.3.2.6 indicator kriging, na form of kriging in which all data are indicator variables.3.2.7 indicator variable, na regionalized variable that can have only two possible values, 0 or 1.3.2.8 kriging, nan estimation meth
36、od where sample weights are obtained using a linear least-squares optimization procedurebased on a mathematical model of spatial variability and where the unknown variable and the available sample values may havea point or block support.3.2.9 kriging variance, nthe expected value of the squared diff
37、erence between the true value of an unknown variable and itskriging estimate, sometimes used as a measure of kriging precision.3.2.10 lognormal kriging, nthe kriging of log-transformed variables followed by a back-transformation procedure based ona lognormal distribution model.3.2.11 nugget effect,
38、nthe component of spatial variance unresolved by the sample spacing, including the variance due tomeasurement error.3.2.12 ordinary kriging, na form of kriging for which the mean of the estimated variable is an unknown constant and thesample weights sum to one.3.2.13 point, nin geostatistics, the lo
39、cation in the field at which a regionalized variable is defined. It also commonly refersto the support of sample-scale variables.3.2.14 point kriging, na form of kriging in which the variable to be estimated has the same support as the sample data.3.2.15 regionalized variable, na measured quantity o
40、r a numerical attribute characterizing a spatially variable phenomenonat a location in the field.3.2.16 search neighborhood, nthe region within which samples are considered for inclusion in the kriging estimation process.3.2.17 simple kriging, na form of kriging for which the mean of the estimated v
41、ariable is a known constant and the sum ofsample weights is unconstrained.3.2.18 simulation, nin geostatistics, a Monte-Carlo procedure for generating realizations of fields based on the randomfunction model chosen to represent a regionalized variable. In addition to honoring a random function model
42、, the realizations mayalso be constrained to honor data values observed at sampled locations.3.2.19 smoothing effect, nin geostatistics, the reduction in spatial variance of estimated values compared to true values.3.2.20 spatial average, na quantity obtained by averaging a regionalized variable ove
43、r a finite region of space.3.2.21 support, nin geostatistics, the spatial averaging region over which a regionalized variable is defined, oftenapproximated by a point for sample-scale variables.3.2.22 universal kriging, na form of kriging in which additional weighting constraints are introduced in o
44、rder to account fora drift in the estimated variable.3.2.23 variogram, na measure of spatial variation defined as one half the variance of the difference between two variables andexpressed as a function of the lag; it is also sometimes referred to as the semi-variogram.4. Significance and Use4.1 Thi
45、s guide is intended to encourage consistency and thoroughness in the application of kriging methods to environmental,geotechnical, and hydrogeological site investigations.D5923 1834.2 This guide may be used to assist those performing a kriging study or as an explanation of procedures for qualifiedno
46、nparticipants that may be reviewing or auditing the study.4.3 This guide encourages the use of site-specific information for the selection of an appropriate kriging method; however, thequality of data, the sampling density, and site coverage cannot be improved or compensated by any choice of kriging
47、 method.4.4 This guide describes conditions for which kriging or particular kriging methods are recommended. However, these methodsare not necessarily inappropriate if the stated conditions are not encountered.4.5 This guide should be used in conjunction with Guides D5549, D5922, and D5924.5. Select
48、ion of Kriging Methods5.1 The following subsections describe conditions for which various kriging methods are appropriate. Each section correspondsto a step in a geostatistical site investigation where a decision concerning the most appropriate form of kriging may have to bemade. Ordinary kriging is
49、 the most common form of kriging and is the conventional default unless any of the following conditionsmakes another method more appropriate.5.2 Study ObjectivesAcommon objective of geostatistical site investigations is to produce a two- or three-dimensional spatialrepresentation of a regionalized variable field from a set of measured values at different locations. Such spatial representations arereferred to here as maps. Estimation approaches, including all forms of kriging, yield maps that exhibit a smoothing effect, whereassimul
