Standard Guide for Selection of Kriging Methods in Geostatistical Site Investigations

SIGNIFICANCE AND USE
This guide is intended to encourage consistency and thoroughness in the application of kriging methods to environmental, geotechnical, and hydrogeological site investigations.
This guide may be used to assist those performing a kriging study or as an explanation of procedures for qualified nonparticipants that may be reviewing or auditing the study.
This guide encourages the use of site-specific information for the selection of an appropriate kriging method; however, the quality of data, the sampling density, and site coverage cannot be improved or compensated by any choice of kriging method.
This guide describes conditions for which kriging or particular kriging methods are recommended. However, these methods are not necessarily inappropriate if the stated conditions are not encountered.
This guide should be used in conjunction with Guides D5549, D5922, and D5924.
SCOPE
1.1 This guide covers recommendations for selecting appropriate kriging methods based on study objectives, exploratory data 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 not discussed; however, this is not intended to reflect any judgement as to the validity of these methods.
1.3 This guide describes conditions for which kriging methods are not appropriate and for which geostatistical simulations approaches 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 principles of kriging. Introductions to geostatistics and kriging may be found in numerous texts including Refs (1-3). 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 responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability 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 course of action. 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 replace the standard of care by which the adequacy of a given professional service must be judged, nor should this document be applied without consideration of a project's many unique aspects. The word “Standard” in the title of this document means only that the document has been approved through the ASTM consensus process.

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NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: D5923 − 96 (Reapproved 2010)
Standard Guide for
Selection of Kriging Methods in Geostatistical Site
Investigations
This standard is issued under the fixed designation D5923; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
INTRODUCTION
Geostatistics is a framework for data analysis, estimation, and simulation in media whose
measurable attributes show erratic spatial variability yet also possess a degree of spatial continuity
imparted by the natural and anthropogenic processes operating therein. The soil, rock, and contained
fluids encountered in environmental or geotechnical site investigations present such features, and their
sampled 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 guide
reviews criteria for selecting a kriging method, offering direction based on a consensus of views
without recommending a standard practice to follow in all cases.
1. Scope priate safety and health practices and determine the applica-
bility of regulatory limitations prior to use.
1.1 This guide covers recommendations for selecting appro-
1.7 This guide offers an organized collection of information
priate kriging methods based on study objectives, exploratory
or a series of options and does not recommend a specific
data analysis, and analysis of spatial variation.
course of action. This document cannot replace education or
1.2 This guide considers commonly used forms of kriging,
experienceandshouldbeusedinconjunctionwithprofessional
including ordinary kriging, simple kriging, lognormal kriging,
judgment. Not all aspects of this guide may be applicable in all
universal kriging, and indicator kriging. Multivariate, space-
circumstances. This ASTM standard is not intended to repre-
time, and other less-frequently used kriging methods are not
sent or replace the standard of care by which the adequacy of
discussed; however, this is not intended to reflect any judge-
a given professional service must be judged, nor should this
ment as to the validity of these methods.
document be applied without consideration of a project’s many
1.3 This guide describes conditions for which kriging meth- unique aspects. The word “Standard” in the title of this
document means only that the document has been approved
odsarenotappropriateandforwhichgeostatisticalsimulations
approaches should be used. through the ASTM consensus process.
1.4 This guide does not discuss non-geostatistical alterna-
2. Referenced Documents
tives to kriging, such as splines or inverse-distance techniques.
2.1 ASTM Standards:
1.5 This guide does not discuss the basic principles of
D653 Terminology Relating to Soil, Rock, and Contained
kriging. Introductions to geostatistics and kriging may be
Fluids
found in numerous texts including Refs (1-3). A review of
D5549 Guide for The Contents of Geostatistical Site Inves-
kriging methods is given in Ref. (4).
tigation Report (Withdrawn 2002)
1.6 This standard does not purport to address all of the
D5922 Guide for Analysis of Spatial Variation in Geostatis-
safety concerns, if any, associated with its use. It is the
tical Site Investigations
responsibility of the user of this standard to establish appro-
D5924 Guide for Selection of Simulation Approaches in
Geostatistical Site Investigations
This guide is under the jurisdiction ofASTM Committee D18 on Soil and Rock
and is the direct responsibility of Subcommittee D18.01 on Surface and Subsurface
Characterization. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved May 1, 2010. Published September 2010. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 1996. Last previous edition approved in 2004 as D5923 – 96 (2004). Standards volume information, refer to the standard’s Document Summary page on
DOI: 10.1520/D5923-96R10. the ASTM website.
2 4
The boldface numbers in parentheses refer to a list of references at the end of The last approved version of this historical standard is referenced on
the text. www.astm.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D5923 − 96 (2010)
3. Terminology 3.1.17 simple kriging, n—a form of kriging for which the
meanoftheestimatedvariableisaknownconstantandthesum
3.1 Definitions of Terms Specific to This Standard:
of sample weights is unconstrained.
3.1.1 additivity, n—a mathematical property of a regional-
3.1.18 simulation, n—in geostatistics, a Monte-Carlo proce-
izedvariablestatingthatitcanbecombinedlinearlyinorderto
dure for generating realizations of fields based on the random
define a similar variable on a larger support.
function model chosen to represent a regionalized variable. In
3.1.2 block kriging, n—a form of kriging in which the
addition to honoring a random function model, the realizations
variable to be estimated has a rectangular or possibly irregular
may also be constrained to honor data values observed at
one-, two-, or three-dimensional support.
sampled locations.
3.1.3 drift, n—in geostatistics, a systematic spatial variation
3.1.19 smoothing effect, n—in geostatistics, the reduction in
of the local mean of a variable, usually expressed as a
spatial variance of estimated values compared to true values.
polynomial function of location coordinates.
3.1.20 spatial average, n—a quantity obtained by averaging
3.1.4 estimation, n—a procedure by which the value of a
a regionalized variable over a finite region of space.
variableatanunsampledlocationispredictedusingaweighted
3.1.21 support, n—in geostatistics, the spatial averaging
average of sample values from the neighborhood of that
region over which a regionalized variable is defined, often
location.
approximated by a point for sample-scale variables.
3.1.5 field, n—in geostatistics, the region of one-, two- or
3.1.22 universal kriging, n—a form of kriging in which
three-dimensional space within which a regionalized variable
additional weighting constraints are introduced in order to
is defined.
account for a drift in the estimated variable.
3.1.6 indicator kriging, n—a form of kriging in which all
3.1.23 variogram, n—a measure of spatial variation defined
data are indicator variables.
asonehalfthevarianceofthedifferencebetweentwovariables
and expressed as a function of the lag; it is also sometimes
3.1.7 indicator variable, n—a regionalized variable that can
referred to as the semi-variogram.
have only two possible values, 0 or 1.
3.2 For definitions of other terms used in this guide, refer to
3.1.8 kriging, n—an estimation method where sample
Terminology D653 and Guides D5549, D5922, and D5924.A
weights are obtained using a linear least-squares optimization
complete glossary of geostatistical terminology is given in Ref
procedure based on a mathematical model of spatial variability
(7).
and where the unknown variable and the available sample
values may have a point or block support.
4. Significance and Use
3.1.9 kriging variance, n—theexpectedvalueofthesquared
4.1 This guide is intended to encourage consistency and
difference between the true value of an unknown variable and
thoroughness in the application of kriging methods to
its kriging estimate, sometimes used as a measure of kriging
environmental, geotechnical, and hydrogeological site investi-
precision.
gations.
3.1.10 lognormal kriging, n—thekrigingoflog-transformed
4.2 This guide may be used to assist those performing a
variables followed by a back-transformation procedure based
kriging study or as an explanation of procedures for qualified
on a lognormal distribution model.
nonparticipants that may be reviewing or auditing the study.
3.1.11 nugget effect, n—the component of spatial variance
4.3 This guide encourages the use of site-specific informa-
unresolved by the sample spacing, including the variance due
tion for the selection of an appropriate kriging method;
to measurement error.
however, the quality of data, the sampling density, and site
3.1.12 ordinary kriging, n—a form of kriging for which the
coverage cannot be improved or compensated by any choice of
mean of the estimated variable is an unknown constant and the
kriging method.
sample weights sum to one.
4.4 This guide describes conditions for which kriging or
3.1.13 point, n—in geostatistics, the location in the field at
particular kriging methods are recommended. However, these
which a regionalized variable is defined. It also commonly
methods are not necessarily inappropriate if the stated condi-
refers to the support of sample-scale variables.
tions are not encountered.
3.1.14 point kriging, n—a form of kriging in which the
4.5 This guide should be used in conjunction with Guides
variable to be estimated has the
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