Basic Steps in Geostatistics: The Variogram and Kriging

100 Pages·2015·3.11 MB·english

by  Margaret A. Oliver

SPRINGER BRIEFS IN AGRICULTURE
Margaret A. Oliver
Richard Webster
Basic Steps in
Geostatistics: The
Variogram and
Kriging
123
SpringerBriefs in Agriculture
More information about this series at http://www.springer.com/series/10183
Margaret A. Oliver Richard Webster
(cid:129)
Basic Steps in Geostatistics:
The Variogram and Kriging
123
Margaret A.Oliver Richard Webster
SoilResearch Centre RothamstedResearch
UniversityofReading Harpenden
Reading UK
UK
ISSN 2211-808X ISSN 2211-8098 (electronic)
SpringerBriefs inAgriculture
ISBN 978-3-319-15864-8 ISBN 978-3-319-15865-5 (eBook)
DOI 10.1007/978-3-319-15865-5
LibraryofCongressControlNumber:2015932443
SpringerChamHeidelbergNewYorkDordrechtLondon
©TheAuthor(s)2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilarmethodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
Preface
This Brief takes readers, in particular environmental scientists, through the
important steps of a geostatistical analysis. Most properties of the environment,
such as rainfall, plant nutrients in the soil and pollutants in the air, are measured
effectively at points between which there are large gaps. The environment is con-
tinuous, however, and environmental scientists and their clients typically want to
know the values of those properties between the points, in the gaps; they want to
predict in a spatial sense from their data, taking into account the locations of their
observations. Geostatistics comprises a set of tools that enable them to do that
optimally by methods established for properties that appear to vary randomly in
one, two or three dimensions. The variogram is the central tool of geostatistics.
Itenablesscientiststoassesswhethertheirdataarespatiallycorrelatedandtowhat
extent.Withasuitablemodelforittheycancombineitwiththeirdatatopredictby
kriging, which in its simpler forms is one of weighted averaging. Kriging is an
optimal method of prediction in that it provides unbiased estimates with minimum
variance.The techniqueisnowavailable in many statistical packagessothat users
can apply it at the press of a few buttons without any idea of what their experi-
mental variogram is like or whether an appropriate model has been fitted to it.
We warn against the practice.
We take readers through the stages of computing a reliable experimental vari-
ogram from sufficient data and the fitting of suitable mathematical models. These
are the most important stages of a geostatistical analysis. If they are done with
proper care kriging provides the best possible predictions from data.
Chapter1introducesthebackgroundtogeostatisticswithexamplesofthebreadth
of current applications. Almost all statistical analyses of environmental data,
includinggeostatistics,dependonsampledata,andinthischapterweintroducethe
basic concepts of sampling: the specification of variables, the support and suitable
sampling designs. Chapter 2 describes random variables and regionalized variable
theorybriefly;thisisthetheorythatunderpinsgeostatistics.Forreaderswhowishto
knowmoreofthetheorywerecommendfurtherreading.Chapter3explainshowto
compute the experimental variogram from regular and irregular sampling designs,
the factors that affect the reliability of the variogram and how to model the
v
vi Preface
experimental values reliably. Most users of geostatistics are eager to obtain maps
of thevariables that interest them, and Chap. 4 illustrates how todo this optimally
with kriging. The theory of ordinary kriging is described, and we show how the
choiceofmodelforthevariogramaffectsthekrigingweights.Thewaytheweights
are obtained in kriging makes the method different from other interpolators. In
Chap.5wereturntosamplingtomeettheneedsofspatialanalysis.Weconsiderthe
situationwherenothingisknownaboutthescaleorpatternofspatialvariationand
forwhichanestedsurveyandanalysisprovideasolution.Wesuggestotherwaysof
determining the spatial scale to sample for mapping that use variograms from
existingdata,eitherofthevariablesofinterestorofintensiveancillarydata,suchas
those from satellites or aerial photographs. Finally, in Chap. 6 we explore the
difficulties that spatial trend poses for geostatistical analysis and how they can
be overcome by residual maximum likelihood estimation of the variogram and
universal kriging.
Acknowledgments
We thank Dr. A. Vásquez for the data from his survey of trace metals in soil near
MadridforthedataoncadmiumusedforFig.3.7andDr.B.P.Marchantforfitting
the REML variogram (Fig. 6.3) with his own Matlab program.
vii
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background to Geostatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications of Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Mining and Engineering . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Environmental Pollution . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Precision Agriculture (PA) . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 Fisheries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 The Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 The Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Units and Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 Practical Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The Essence of Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Regionalized Variable Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Random Variables and Regionalized Variable Theory. . . . . . . . . 11
2.1.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 The Variogram and Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 The Experimental Variogram. . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Computing the Variogram from Regular Sampling
in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Computing the Variogram from Regular and Irregular
Sampling in Two Dimensions . . . . . . . . . . . . . . . . . . . . 18
3.2 Factors Affecting the Reliability of Experimental Variograms . . . 19
3.2.1 Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Sampling Interval and Spatial Scale . . . . . . . . . . . . . . . . 23
3.2.3 Lag Interval and Bin Width. . . . . . . . . . . . . . . . . . . . . . 24
3.2.4 Statistical Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.5 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.6 Trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ix
x Contents
3.3 Modelling the Variogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Principal Features of the Variogram . . . . . . . . . . . . . . . . 30
3.3.2 Variogram Model Functions . . . . . . . . . . . . . . . . . . . . . 32
3.4 Factors Affecting the Reliability of Variogram Models . . . . . . . . 37
3.4.1 Fitting Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Geostatistical Prediction: Kriging . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Kriging Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.2 Kriging Neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.3 Punctual and Block Kriging for Mapping . . . . . . . . . . . . 57
4.2.4 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.5 Simple Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.6 Lognormal Kriging. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Sampling for the Variogram . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.1 Nested Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Sampling Plans for Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.1 Illustrative Example: Sampling to Map Chromium
in the Swiss Jura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Dealing with Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1 Trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.1 Variogram and Model. . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Illustration from a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99