Geostatistics in soil science: state-of-the-art and ...mmc2.geofisica.unam.mx/cursos/geoest/Articulos/Geostatistics... · Geoderma 89 1999 1–45 . Geostatistics in soil science:

Ž .Geoderma 89 1999 1–45

Geostatistics in soil science: state-of-the-art andperspectives

P. Goovaerts )

Department of CiÕil and EnÕironmental Engineering, The UniÕersity of Michigan, Ann Arbor,MI 48109-2125, USA

Received 4 November 1997; accepted 6 July 1998

Abstract

This paper presents an overview of the most recent developments in the field of geostatisticsand describes their application to soil science. Geostatistics provides descriptive tools such assemivariograms to characterize the spatial pattern of continuous and categorical soil attributes.

Ž .Various interpolation kriging techniques capitalize on the spatial correlation between observa-tions to predict attribute values at unsampled locations using information related to one or severalattributes. An important contribution of geostatistics is the assessment of the uncertainty aboutunsampled values, which usually takes the form of a map of the probability of exceeding criticalvalues, such as regulatory thresholds in soil pollution or criteria for soil quality. This uncertaintyassessment can be combined with expert knowledge for decision making such as delineation ofcontaminated areas where remedial measures should be taken or areas of good soil quality wherespecific management plans can be developed. Last, stochastic simulation allows one to generate

Ž .several models images of the spatial distribution of soil attribute values, all of which areŽ .consistent with the information available. A given scenario remediation process, land use policy

Žcan be applied to the set of realizations, allowing the uncertainty of the response remediation.efficiency, soil productivity to be assessed. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: geostatistics; spatial interpolation; risk assessment; decision making; stochasticsimulation

1. Introduction

During the last decade, the development of computational resources hasfostered the use of numerical methods to process the large bodies of soil data

) Fax: q1-313-734-2275; E-mail: [email protected]

0016-7061r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0016-7061 98 00078-0

( )P. GooÕaertsrGeoderma 89 1999 1–452

that are collected across the world. A key feature of soil information is that eachobservation relates to a particular location in space and time. Knowledge of anattribute value, say a pollutant concentration, is thus of little interest unlesslocation or time of measurement or both are known and accounted for in theanalysis. Geostatistics provides a set of statistical tools for incorporating spatialand temporal coordinates of observations in data processing.

Until the late 1980s, geostatistics was essentially viewed as a means todescribe spatial patterns by semivariograms and to predict the values of soilattributes at unsampled locations by kriging, e.g., see review papers by Vieira et

Ž . Ž . Ž .al. 1983 , Trangmar et al. 1985 , and Warrick et al. 1986 . New tools haverecently been developed to tackle advanced problems, such as the assessment ofthe uncertainty about soil quality or soil pollutant concentrations, the stochasticsimulation of the spatial distribution of attribute values, and the modeling ofspace–time processes. Because of their publication in a wide variety of journalsand congress proceedings, these new developments are generally barely knownby soil scientists who must also struggle with different sets of notation toestablish links between all these techniques. This paper aims to provide acoherent and understandable overview of the state-of-the-art in soil geostatistics,refer to recent applications of geostatistical algorithms to soil data, and point outchallenges for the future.

I have extracted most of the material in this paper from my recent bookŽ .Goovaerts, 1997a on the application of geostatistics to natural resourcesevaluation. The presentation follows the usual steps of a geostatistical analysis,introducing tools for description of spatial patterns, quantitative modeling ofspatial continuity, spatial prediction and uncertainty assessment. Particular atten-tion is paid to practical issues such as the modeling of sample semivariograms,the choice of an interpolation algorithm that incorporates all the relevantinformation available, or the incorporation of uncertainty assessment in decisionmaking. Some common misunderstandings regarding the modeling of crosssemivariograms, the use of the kriging variance or Gaussian-based algorithmswill also be reviewed. The different concepts will be illustrated using multivari-ate soil data related to heavy metal contamination of an area of the Swiss JuraŽ .Atteia et al., 1994; Webster et al., 1994 , kindly provided by Mr. J.-P. Duboisof the Swiss Federal Institute of Technology.

2. Description of spatial patterns

Analysis of spatial data typically starts with a ‘posting’ of data values. Forexample, Fig. 1 shows the spatial distribution of five stratigraphic classes and ofthe concentrations of two heavy metals recorded, respectively, at 359 and 259locations in a 14.5 km2 area in the Swiss Jura. For both continuous and

( )P. GooÕaertsrGeoderma 89 1999 1–45 3

Fig. 1. Locations of sampling sites superimposed on the geologic map, and concentrations in CdŽ y1.and Ni at 259 of these sites unitssmg kg .

categorical attributes, the spatial distribution of values is not random in thatobservations close to each other on the ground tend to be more alike than thosefurther apart. The presence of such a spatial structure is a prerequisite to theapplication of geostatistics, and its description is a preliminary step towardsspatial prediction or stochastic simulation.

2.1. Continuous attributes

Consider the problem of describing the spatial pattern of a continuousattribute z, say a pollutant concentration such as cadmium or nickel. Theinformation available consists of values of the variable z at n locations u ,a

Ž .z u , as1,2, . . . ,n. Spatial patterns are usually described using the experi-a

Ž .mental semivariogram g h which measures the average dissimilarity betweenˆ


data separated by a vector h. It is computed as half the average squareddifference between the components of data pairs:

Ž .N h1 2g h s z u yz u qh 1Ž . Ž . Ž . Ž .ˆ Ý a a2 N hŽ . as1

Ž .where N h is the number of data pairs within a given class of distance andŽ .direction. Webster and Oliver 1993 showed that reliable estimation of semivar-

iogram values requires at least 150 data, and larger samples are needed toŽ .describe anisotropic direction-dependent variation. This does not mean, how-

ever, that geostatistics cannot be applied to smaller data sets. It is noteworthythat geostatistics has become the reference approach for characterization ofpetroleum reservoirs where the information available typically reduces to a fewwells. Sparse data are often supplemented by expert knowledge or ancillaryinformation originating from a better sampled area which is deemed similar tothe study area; see also latter discussion on semivariogram modeling.

Ž .Fig. 2 top graphs shows the omnidirectional semivariograms computed fromthe heavy metal concentrations displayed in Fig. 1. These graphs point out

Fig. 2. Experimental omnidirectional semivariograms for Cd and Ni: original concentrations andŽ . Ž .indicator transforms using thresholds corresponding to the second- — , fifth- – – and eighth-de-

Ž .cile - - - of the sample histogram. To facilitate the comparison, indicator semivariogram valueswere rescaled by the indicator variance.


distinct spatial behaviors of the two metals: Ni concentrations appear to varymore continuously than Cd concentrations, as illustrated by the smaller nuggeteffect and larger range of its semivariogram. In combination with a goodknowledge about the phenomenon and the study area, such a spatial descriptioncan improve our understanding of the physical underlying mechanisms control-

Žling spatial patterns Oliver and Webster, 1986a; Goovaerts and Webster, 1994;.Webster et al., 1994 . In the present study, the long-range structure of the

semivariogram of Ni concentrations is probably related to the control asserted byrock type, while the short-range structure for cadmium suggests the impact of

Ž .local man-made contamination Goovaerts, 1997a, p. 38 .Description of spatial patterns is too often restricted to the spatial variation

over the full range of attribute values. Spatial patterns may, however, differdepending on whether the attribute value is small, medium, or large. Forexample, in many environmental applications, a few random ‘hot spots’ of largeconcentrations coexist with a background of small values that vary morecontinuously in space. Depending on whether large concentrations are clusteredor scattered in space, our interpretation of the physical processes controllingcontamination and our decision for remediation may change. The characteriza-tion of the spatial distribution of z-values above or below a given threshold

Ž .value z requires a prior coding of each observation z u as an indicatork a

Ž .datum i u ; z , defined as:a k

1 if z u FzŽ .a ki u ; z s 2Ž . Ž .a k ½0 otherwise

Indicator semivariograms can then be computed by substituting indicator dataŽ . Ž . Ž .i u ; z for z-data z u in Eq. 1 :a k a

Ž .N h1 2g h; z s i u ; z y i u qh; z 3Ž . Ž . Ž . Ž .ˆ ÝI k a k a k2 N hŽ . as1

Ž .The indicator variogram value 2g h; z measures how often two z-valuesˆ I k

separated by a vector h are on opposite sides of the threshold value z . In otherkŽ .words, 2g h; z measures the transition frequency between two classes ofˆ I k

Ž .z-values as a function of h. The greater is g h; z , the less connected in spaceˆ I k

are the small or large values.Ž .Fig. 2 bottom graphs shows the omnidirectional indicator semivariograms

computed for the second-, fifth- and eighth-decile of the distributions ofcadmium and nickel concentrations. To facilitate the comparison, all semivari-ograms values were standardized by dividing them by the indicator variance. Forboth metals, indicator semivariograms for small concentrations have smallernugget effect than those for larger concentrations, which suggests that homoge-neous areas of small concentrations coexist within larger zones where large and


medium concentrations are intermingled. Two clusters of small concentrationsare indeed apparent on the location maps of Fig. 1 and correspond to theArgovian rocks.

2.2. Categorical attributes

Many soil variables such as texture or water table classes take only a limitednumber of states which might be ordered or not. Spatial patterns of suchcategorical variables can also be described using geostatistics. Let S be acategorical attribute with K possible states s , ks1,2, . . . , K. The K states arek

exhaustive and mutually exclusive in the sense that one and only one state sk

occurs at each location u . The pattern of variation of a category s can bea kŽ .characterized by semivariograms of type 3 defined on an indicator coding of

the presence or absence of that category:

1 if s u ssŽ .a ki u ; s s 4Ž . Ž .a k ½0 otherwise

Ž .The indicator variogram value 2g h; s measures how often two locations aˆ I kŽ .Xvector h apart belong to different categories s /s . The smaller is 2g h; s ,ˆk k I k

the more connected is category s . The ranges and shapes of the directionalk

indicator semivariograms reflect the geometric patterns of s .k

Fig. 3 shows the indicator semivariograms of two stratigraphic classes of Fig.1 computed in four directions with an angular tolerance of 22.58. For bothclasses the indicator semivariogram value equals zero at the first lag, whichmeans that any two data locations less than 100 m apart belong to the same

Ž .formation. The longer SW–NE range larger dashed line reflects the corre-sponding preferential orientation of these two lithologic formations.

Fig. 3. Experimental indicator semivariograms of Argovian and Sequanian rocks computed in fourŽ .directions —: 22.58, – –: 67.58, – – –: 112.58, . . . : 157.58; angular tolerances22.58 .


Fig. 4. Experimental omnidirectional cross semivariogram between Cd and Ni.

2.3. BiÕariate description

Soil information is generally multivariate, and geostatistics is increasinglyused to investigate how the correlation between two soil properties varies inspace. A measure of the joint variation of two continuous attributes z and z isi j

the experimental cross semivariogram:

Ž .N h1g h s z u yz u qh P z u yz u qh 5Ž . Ž . Ž . Ž . Ž . Ž .ˆ Ýi j i a i a j a j a2 N hŽ . as1

In other words, one looks at the joint variation of gradients of z - and z -valuesi j

from one location to another a vector h away. If both attributes are positivelyŽ .related, an increase decrease in z from u to u qh tends to be associatedi a a

Ž .with an increase decrease in z , and so the cross semivariogram value isj

positive as for the pair cadmium–nickel in Fig. 4.Ž .Cross semivariograms can also be computed from indicator values of type 2

in order to characterize the spatial connection of small or large values of twoŽ .soil properties Goovaerts, 1997a, p. 52 . For example, a similarity in the spatial

distribution of large values for two heavy metals could indicate the existence ofcommon sources of contamination. Another application of indicator cross semi-variograms is the study of the spatial architecture of categories such as soil types

Ž .in Goovaerts 1994a .

3. Modeling spatial variation

Description of spatial patterns is rarely a goal per se. Rather, one generallywants to capitalize on the existence of spatial dependence to predict soilproperties at unsampled locations. A key step between description and predic-tion is the modeling of the spatial distribution of attribute values. Most of


geostatistics is based on the concept of random function, whereby the set ofunknown values is regarded as a set of spatially dependent random variables.

Ž .Each datum z u is then viewed as a particular realization of a randoma

Ž .variable Z u . An important characteristic of the random function is itsa

semivariogram which is modeled from the experimental values.

3.1. UniÕariate case

Consider the problem of modeling the semivariogram of a single attribute zŽ .from the set of experimental values g h computed for a finite number of lags,ˆ k

h , ks1, . . . , K , in distance and direction. Most soil scientists are now awarek

that semivariograms can be modeled using only functions that are conditionallyŽnegative definite, such as the exponential or spherical models McBratney and

.Webster, 1986 . Fewer know that the Gaussian semivariogram model is gener-ally unrealistic and leads to unstable kriging systems and artifacts in the

Ž .estimated maps Wackernagel, 1995, pp. 109–111 . In many situations, two ormore permissible models must be combined to fit the shape of the experimentalsemivariogram, as illustrated for cadmium in Fig. 5. Such nested model iswritten as:

Ll lg h s b g h with b G0 6Ž . Ž . Ž .Ý l

ls0

where bl is the positive sill or slope of the corresponding basic semivariogramŽ .model g h .l

The way in which these permissible models are chosen and their parametersŽ . Ž .range, sill are estimated is still controversial Goovaerts, 1997a, pp. 97–107 .Several methods have been proposed, ranging from full black-box procedureswhich involve an automatic choice and fitting of the model to visual approachesin which the model is selected so that the fit looks satisfactory graphically. An

Fig. 5. Experimental omnidirectional semivariogram for Cd, and the nested model fitted whichincludes a nugget effect and two spherical models with ranges of 200 m and 1.3 km.


Ž .intermediate approach consists of an automatic least-squares estimation ofŽ .parameters of models chosen by the user semi-automatic procedure .

Black-box procedures should be avoided because they cannot take intoaccount ancillary information that is critical when sparse or preferential sam-pling makes the experimental semivariogram unreliable. Differences betweensemi-automatic and visual procedures reside in the criterion used to assess thegoodness of the fit. The user might feel more comfortable if the choice of aparticular model can be based on statistical criteria such as the sum of squares ofdifferences between experimental and model semivariogram values. Using suchcriteria amounts to reducing semivariogram modeling to an exercise in fitting acurve to experimental values, which I think is too restrictive. The objective ofthe structural analysis is to build a permissible model that captures the majorspatial features of the attribute under study. Although experimental semivari-ogram values play an important role in this process, ancillary information suchas provided by physical knowledge of the area and phenomenon may be of greatinterest. For example, strong prior qualitative information may lead one to adoptan anisotropic model even if data sparsity prevents seeing anisotropy from theexperimental semivariograms.

In all situations one should avoid overfitting semivariogram models. Themore complicated model usually does not lead to more accurate estimates.

3.2. BiÕariate case

Modeling the coregionalization between two variables z and z requires thei jŽ . Ž .modeling of the two direct semivariograms g h and g h plus the crossii j j

Ž .semivariogram g h . A difficulty lies in the fact that the three models must bei j

built together. More precisely, one must ensure that the matrix of semivariogramŽ . w Ž .xmodels G h s g h is conditionally negative semi-definite for all lags h,i j

which requires the following inequality to be satisfied for all lags h:

< <g h F g h g h ;h 7Ž . Ž . Ž . Ž .(i j ii j j

Ž .The apparent complexity of condition 7 has discouraged some practitionersfrom modeling cross semivariograms and so using multivariate interpolation. In

Ž . Ž .fact, condition 7 is easily checked as long as the three cross semivariogramsŽ .are modeled as linear combinations of the same set of basic models g h :l

Llg h s b g h ; i , j 8Ž . Ž . Ž .Ýi j i j l

ls0

or, using matrix notation,L

G h s B g h 9Ž . Ž . Ž .Ý l lls0

w l xwhere B s b is referred to as a coregionalization matrix. Sufficient condi-l i jŽ .tions for the so-called linear model of coregionalization LMC to be permissible


Ž . Ž . Ž .are: 1 the functions g h are permissible semivariogram models, and 2 eachl

coregionalization matrix is positive semi-definite, which implies that the coeffi-cients bl satisfy the following constraints:i j

b l G0, bl G0 ; l 10Ž .ii j j

l l l< <b F b b ; l 11Ž .(i j ii j j

In practice, the modeling is done in two steps:1. both direct semivariograms are first modeled as linear combinations of

Ž .selected basic structures g h ,l

2. the same basic structures are then fitted to the cross semivariogram under theŽ .constraint 11 .

Ž .This approach was used to fit visually the following model to the crosssemivariograms of Cd and Ni displayed in Fig. 6:

g h s 0.3 g h q0.3 Sph hr200 m q0.26 Sph hr1.3 kmŽ . Ž . Ž . Ž .Cd 0

g h s 11 g h q71 Sph hr1.3 kmŽ . Ž . Ž .Ni 0

g h s 0.6 g h q3.8 Sph hr1.3 kmŽ . Ž . Ž .Cd – Ni 0

12Ž .The requirement that all semivariograms must share the same set of basicstructures might seem a severe limitation of the linear model of coregionaliza-

Fig. 6. Experimental omnidirectional direct and cross semivariograms for Cd and Ni, and thelinear model of coregionalization fitted.


tion. Variables that are well cross-correlated are however likely to show similarpatterns of spatial variability. In addition, there is no need for the direct andcross semivariograms to include all the basic structures; for example, the Nisemivariogram and the cross semivariogram Cd–Ni do not include the short-

Ž .range 200 m spherical structure. The reader is warned against the use ofŽ .alternative models e.g., Myers, 1982; Zhang et al., 1992 that are more flexible

Ž .but do not provide easy way to check their permissibility Goovaerts, 1994b . Itis unfortunate that such models have been mainly used in soil science.

3.3. MultiÕariate case

Ž .For more than two variables N )2 , checking the permissibility of theÕ

linear model of coregionalization becomes cumbersome because, for eachŽ .structure g h , the N =N matrix of coregionalization B must be positivel Õ Õ l

Ž .semi-definite. Fortunately, Goulard 1989 has developed an iterative procedurethat fits the linear model of coregionalization directly under the constraint ofpositive semi-definiteness of all matrices B . Most applications of this innova-l

Žtive fitting technique have been in the field of soil science Goulard and Voltz,.1992; Goovaerts, 1992; Voltz and Goulard, 1994; Webster et al., 1994 .

4. Spatial prediction

The main application of geostatistics to soil science has been the estimationand mapping of soil attributes in unsampled areas. Kriging is a generic nameadopted by the geostatisticians for a family of generalized least-squares regres-sion algorithms. The practitioner often gets confused in the face of the palette ofkriging methods available: simple, ordinary, universal or with a trend, cokriging,kriging with an external drift . . . . This section presents a brief description of themain methods and provides references to soil applications. A detailed presenta-tion of the mathematics can be found in textbooks such as Isaaks and SrivastavaŽ . Ž .1989, pp. 278–337 and Goovaerts 1997a, pp. 125–258 .

4.1. UniÕariate kriging

Consider the problem of estimating the value of a continuous attribute z at� Ž . 4any unsampled location u using only z-data z u , as1, . . . ,n . All kriginga

)Ž .estimators are but variants of the basic linear regression estimator Z u definedas:

Ž .n u)Z u ym u s l u Z u ym u 13Ž . Ž . Ž . Ž . Ž . Ž .Ý a a a

as1


Ž . Ž .where l u is the weight assigned to datum z u interpreted as a realizationa a

Ž . Ž .of the random variable Z u and located within a given neighborhood W ua

Ž .centered on u. The n u weights are chosen so as to minimize the estimation or2Ž . � )Ž . Ž .4error variance s u sVar Z u yZ u under the constraint of unbiasednessE

of the estimator. These weights are obtained by solving a system of linearequations which is known as ‘kriging system.’

Differences between kriging variants reside in the model considered for theŽ . Ž .trend m u in expression 13 .

Ž . Ž . Ž .1 Simple kriging SK considers the mean m u known and constantthroughout the study area AA:

m u sm , known ; ugAAŽ .

Ž . Ž .2 Ordinary kriging OK accounts for local fluctuations of the mean byŽ .limiting the domain of stationarity of the mean to the local neighborhood W u :

m uX sconstant but unknown ;uX gW uŽ . Ž .

Unlike simple kriging, the mean here is deemed unknown.Ž . Ž .3 Kriging with a trend model KT , also known as ‘universal kriging,’

Ž X.considers that the unknown local mean m u smoothly varies within each localŽ .neighborhood W u , and the trend is modeled as a linear combination of

Ž .functions f u of the coordinates:k

KX X Xm u s a u f u ,Ž . Ž . Ž .Ý k k

ks0

with a uX fa constant but unknown ; uX gW uŽ . Ž .k k

Ž X.The coefficients a u are unknown and deemed constant within each localkŽ . Ž X.neighborhood W u . By convention, f u s1, hence ordinary kriging is but a0

particular case of KT with Ks0.These differences are illustrated in the one-dimensional example of Fig. 7

where Cd concentration is estimated every 50 m using at each location the fiveclosest data. The middle graph shows the mean implicitly used by each krigingvariant: global mean for SK, constant mean within local search neighborhoodsfor OK, local linear function of the x-coordinates for KT. For the latter two

Ž .Fig. 7. Impact of the kriging algorithm on the estimation of the trend middle graph and of CdŽ .concentration bottom graph along a transect. The vertical dashed lines delineate the segments

that are estimated using the same five Cd concentrations. For example, the first segment, 1–2.1km, includes all estimates that are based on the data at locations u to u .1 5


algorithms, the parameters of the trend model are constant within each segmentwhere the same five neighboring data are used, and these are delineated by


vertical dashed lines. The corresponding estimates of Cd concentration aredisplayed at the bottom of the figure. Simple kriging is inappropriate because itdoes not account for the increase in Cd concentration along the transect.Ordinary kriging with local search neighborhoods provides similar results tokriging with a trend model while being easier to implement. Large differencesbetween the two estimators arise only beyond location u and are due to the10

arbitrary extrapolation of the trend model evaluated from the last five data. Notethat in this example the linear trend model yields negative estimates of concen-tration around 7 km. Such aberrant results show that the blind use of complextrend models is risky and may yield worse estimates than the straightforwardordinary kriging.

4.2. Accounting for exhaustiÕe secondary information

When measurements are sparse or poorly correlated in space, the estimationof the primary attribute of interest is generally improved by accounting forsecondary information originating from other related categorical or continuousattributes. This secondary information is said to be exhaustively sampled when itis available at all primary data locations and at all nodes of the estimation grid,e.g., a soil map or an elevation model. In this case, secondary data can beincorporated using three variants of the aforementioned univariate methods:kriging within strata, simple kriging with varying local means, and kriging withan external drift.

Where major scales of spatial variation are related to changes in land use, soiltype or lithology, secondary categorical information such as land use, soil or

Žgeologic maps can be used to stratify the study area e.g., see Stein et al., 1988;.Voltz and Webster, 1990; Van Meirvenne et al., 1994 . Kriging is then

performed within each stratum using a semivariogram model and data specific tothat stratum. When data sparsity prevents computing reliable semivariogramestimates within each stratum, experimental values can be combined into asingle pooled within-stratum semivariogram. Fig. 8 shows a stratification of thetransect of Fig. 7 based on geology. Within-stratum semivariograms computedfrom the entire study area indicate that Cd concentration varies more continu-

Žously within the stratum AA . Kriging within strata yields estimates bottom1.graph, solid line that suddenly change at the strata boundaries, as opposed to

Ž .ordinary kriging without stratification dashed line .A second approach is simple kriging where the global mean is replaced by

local means derived from a calibration of the secondary information. Forexample, the calibration of the geologic map of Fig. 1 yields an average Cd

Ž .concentration for each rock type, see Fig. 9 left top graph . Residual data canbe computed by subtracting from each Cd measurement the local mean of theprevailing rock type. Residual values are then estimated at unsampled locations


Fig. 8. Kriging within strata. The transect is first split into two strata AA and AA , according to1 2Ž . Žgeology top graph . Within each stratum, the Cd semivariogram is inferred and modeled middle

.graph , and Cd concentrations are estimated using ordinary kriging and stratum-specific dataŽ .bottom graph, solid line . Vertical arrows depict discontinuities at the strata boundaries. Thedashed line represents the OK estimate without stratification.



Ž .using simple kriging and the semivariogram of residuals, see Fig. 9 third row .The final estimates of Cd concentration are obtained by adding the local meansto the residual estimates. Unlike kriging within strata, data across geologicboundaries are used in the estimation, which attenuates discontinuities at

Ž .boundaries. A similar approach was used by Bierkens 1997 to incorporate soilmap information in the mapping of heavy metal concentrations. To account forthe accuracy of the soil map information in mapping, Heuvelink and BierkensŽ .1992 proposed a weighted average of soil map predictions and kriged esti-mates where more weight is given to the information with the smallest predic-tion error variance. A case study showed this heuristic method to produce amore accurate map of mean water table than either kriging or soil mapprediction as long as the soil map accuracy is correctly estimated and pointobservations are scarce.

If the secondary attribute is continuous, the local means in simple kriging canŽ .be derived by regression procedures. For example, Odeh et al. 1997 used a

Ž .combination of multiple linear regression and ordinary instead of simplekriging to account for landform attributes derived from a digital elevation modelin the prediction of percent topsoil organic carbon.

Kriging with an external drift is but a variant of kriging with a trend modelŽ .where the trend m u is modeled as a linear function of a smoothly varying

Ž . Ž .secondary external variable y u instead of a function of the spatial coordi-nates:

m u sa u qa u y uŽ . Ž . Ž . Ž .0 1

Ž . Ž . K Ž . Ž .as opposed to: m u sa u qÝ a u f u for kriging with a trend model.0 ks1 k k

Besides the difficult inference of the residual semivariogram, this methodrequires that the relation between primary trend and secondary variable is linearand makes physical sense. One of the few applications of the method to soil

Ž .science is a paper by Bourennane et al. 1996 where the slope gradient is usedas external drift for predicting the thickness of a pedological horizon. Gotway

Ž .and Hartford 1996 used a similar approach to account for corn yield measure-ments and soil nitrate concentration data in the prediction of the amount of

Ž .nitrogen left in the soil after harvest. In Heuvelink 1996 , the external drifttakes the form of a polygonal map of mean highest water table derived from acalibration of a soil map.

Fig. 9. Simple kriging with varying local means. The trend component at each location isestimated by the average Cd concentration for the rock type prevailing at that location. Residuals

Ž .are interpolated by simple kriging, and the results third row are added to the trend estimates toŽ .yield the Cd estimates bottom graph .


4.3. Cokriging

When the secondary information is not exhaustively sampled, the estimationŽ Ž ..can be done using a multivariate extension of the kriging estimator Eq. 13

which is referred to as cokriging:Ž .N n uÕ i

)Z u ym u s l u Z u ym u 14Ž . Ž . Ž . Ž . Ž . Ž .Ý Ý1 1 a i a i ai i iis1a s1i

Ž . Ž . Ž .where l u is the weight assigned to the primary datum z u and l u ,a 1 a ai i i

Ž .i)1, is the weight assigned to the secondary datum z u . Like kriging, threei a i

cokriging variants can be distinguished according to the models adopted for thetrends of primary and secondary variables: simple, ordinary, and universal orwith trend models. To be complete, one must also mention a variant of ordinary

Žcokriging, called standardized ordinary cokriging Isaaks and Srivastava, 1989,.p. 416 , which uses a single unbiasedness constraint that calls for all primary

and secondary data weights to sum to one. To be unbiased though, the methodrequires a prior rescaling of all secondary variables to the primary mean.Because it does not call for the secondary data weights to sum to zero, ordinarycokriging with a single unbiasedness constraint gives a larger weight to thesecondary information while reducing the occurrence of negative weightsŽ .Goovaerts, 1998 . Numerous examples of cokriging can be found in the soil

Ž . Ž .literature, e.g., McBratney and Webster 1983 , Yates and Warrick 1987 , SteinŽ . Ž .et al. 1988 , Leenaers et al. 1990 . Most of these studies, however, do not

check the permissibility of the coregionalization model which is used inprediction; recall previous remark on the linear model of coregionalization.

Ž .Cokriging is much more demanding than kriging in that N N q1 r2 directÕ Õ

and cross semivariograms must be inferred and jointly modeled, and a largeŽ .cokriging system must be solved. As emphasized by Wackernagel 1994 on the

occasion of Pedometrics’92, the additional modeling and computational effortimplied by cokriging is not worth doing when primary and secondary variables

Ž .are recorded at the same locations isotopic case and the direct semivariogramŽ .g h of the primary variable is proportional to the cross semivariograms with11

Ž .the secondary variables, g h , i)1. More generally, practice has shown that1i

cokriging improves over kriging only when the primary variable is undersam-pled with regard to the secondary variables and those secondary data are well

Žcorrelated with the primary value to be estimated Journel and Huijbregts, 1978,.p. 326; Goovaerts, 1998 .

Cokriging can also be used to incorporate secondary information that isŽexhaustively sampled. Several studies Asli and Marcotte, 1995; Goovaerts,

.1998 have shown that secondary data that are close or even co-located with theestimated location tend to screen the influence of further away secondary data.Thus, in the presence of highly redundant secondary information, one gains littleby retaining more than one secondary datum in cokriging. Besides the screening


effect, the use of redundant information can make the cokriging system unstable.Cokriging using the single secondary datum collocated with the location being

Ž .estimated is referred to as collocated cokriging Almeida and Journel, 1994 .Whereas kriging with an external drift uses the secondary exhaustive informa-tion only to inform on the shape of the trend of the primary variable, cokrigingexploits more fully the secondary information by directly incorporating thevalues of the secondary variable and measuring the degree of spatial associationwith the primary variable through the cross semivariogram. Moreover, in somesituations, it is more realistic to use the secondary information as a full covariate

Ž .than just an indicator of the trend shape Gotway and Hartford, 1996 . Forexample, it is important to account for the actual values and magnitude of localfluctuations of hydrologic parameters, such as specific capacity, in the predic-tion of conductivity.

Another application of cokriging is the combination of different measure-ments of the same attribute. For example, a few precise laboratory measure-ments of clay content can be supplemented by more numerous field datacollected using cheaper measurement devices. Measurement errors are likely tobe larger for field data, and their semivariogram is likely to have a largerrelative nugget effect. To account for such a difference in the patterns of spatialcontinuity, precise laboratory measurements and less precise field data areweighted differently through cokriging. Note that the secondary information cantake the form of constraint intervals indicating that the primary attribute isvalued between specific bounds, e.g., reference colors on a colorimetric paper to

Ž .evaluate acidity levels. Secondary data are thus coded into indicators of type 2Žprior to their incorporation in the cokriging estimator Goovaerts, 1997a, pp.

.241–244 .

4.4. Smoothing effect and kriging Õariance

Estimation by kriging is best in the least-squares sense because the local error� )Ž . Ž .4variance Var Z u yZ u is minimum. A shortcoming of the least-squares

criterion, however, is that the local variation of z-values is smoothed: estimatedvalues are typically much less variable than actual values, which is expressed byan overestimation of small values while large values are underestimated. An-other drawback of the estimation is that the smoothing depends on the local dataconfiguration; it is small close to the data locations and increases as the locationbeing estimated gets farther away from sampled locations. This uneven smooth-ing yields kriged maps that artificially appear more variable in densely sampledareas than in sparsely sampled areas. For all these reasons, interpolated mapsshould not be used for applications sensitive to presence of extreme values andtheir patterns of continuity, typically soil pollution data and physical propertiesŽ .permeability, porosity that control solute transport in soil. A better alternativeis to use simulated maps which reproduce the spatial variability modeled fromthe data, see later section.


Kriging provides not only a least-squares estimate of the attribute but also theattached error variance. The so-called kriging variance is unfortunately oftenmisused as a measure of reliability of the kriging estimate, as reminded by

Ž .several authors Journel, 1993; Armstrong, 1994 . By doing so, one assumes thatthe variance of the errors is independent of the actual data values and dependsonly on the data configuration, a situation referred to as ‘homoscedasticity.’ Inthe example of Fig. 10, the kriging variance is similar at locations uX and uX

1 2

with similar data configurations, although the potential for error is expected tobe greater at location uX , which is surrounded by a very large value and a small2

one, compared with location uX , which is surrounded by two consistently small1

Cd values. Homoscedasticity is rarely met in practice because the local varianceŽ .of data usually changes across the study area nonstationarity . For example, the

Fig. 10. Ordinary kriging estimates of Cd concentration with the associated kriging variance.


sills of the within-stratum semivariograms of Fig. 8 indicated that the varianceof Cd concentrations is larger within the stratum AA . In such a case, one might1

consider the possibility of subdividing the study area into more homogenouszones which are then treated as different statistical populations. Such subdivi-sion, however, is conditioned by our ability to delineate the different zones inthe field, and the availability of enough data to infer statistics such as semivari-ograms for each zone. Stationarity is a property of the random function modelthat is needed for statistical inference, and in most situations the ‘stationaritydecision’ is taken by the user simply because he cannot afford to split data intosmaller subsets! An interesting approach consists of rescaling locally a relativesemivariogram computed from all the data so that its sill equals the variance of

Žthe data within the kriging search neighborhood Isaaks and Srivastava, 1989,.pp. 516–524 . Whereas this local rescaling does not change the kriging weights

and so the estimated value, it might provide kriging variances that better informon the actual estimation errors.

Another way to correct for the lack of homoscedasticity is to transform thedata to stabilize the variance, typically when the sample histogram is asymmet-

Žric and the local variance of data is related to their local mean proportional.effect . A frequent transform consists of taking the logarithm of strictly positive

Ž . Ž .measurements, y u s ln z u . The problem lies in the back-transform of thea a)Ž .estimated value y u to retrieve the estimate of the original variable z at u. For

example, the unbiased back-transform of the simple lognormal kriging estimateis:

) ) 2z u sexp y u qs u r2 15Ž . Ž . Ž . Ž .SK

2 Ž .where s u is the simple lognormal kriging variance. Because of the exponen-SK

tiation of both kriging estimate and variance, final results are very sensitive tothe semivariogram model which controls the kriging variance!

4.5. Factorial kriging

Semivariogram models used in earth science are frequently nested in thatseveral structures with different ranges or slopes are combined to fit the

Ž .experimental curves. Recall expression 6 for the nested model:L

lg h s b g h 16Ž . Ž . Ž .Ý lls0

Ž . Ž .The random function Z u with a nested semivariogram g h can be interpretedŽ . lŽ .as the sum of Lq1 independent random functions Z u , each with zero mean

l Ž .and semivariogram b g h :l

LlZ u s Z u qm u 17Ž . Ž . Ž . Ž .Ý

ls0


Ž .where the trend component m u is assumed locally constant as in the practiceŽ .of ordinary kriging. According to Burrough 1983 , the variation of soil proper-

ties appears to be consistent with the hypothesis of the nested model, andexamples of sources of variation affecting soil at different spatial scales areearthworms, geology, relief, to which one can add tree-throw and man’s

Ž .divisions into farms and fields Oliver and Webster, 1986b . Once severalspatial scales have been identified on the semivariograms, the corresponding

lŽ .spatial components Z u can be estimated and mapped using a variant ofŽkriging, known as factorial kriging or kriging analysis Matheron, 1982;

.Goovaerts, 1992 .Ž .Several authors Goovaerts, 1994c; Webster et al., 1994 have used factorial

kriging to separate local variation in soil properties due to field-to-field differ-ences or local sources of pollution from regional variation related to differentsoil types or geological classes. In these studies, the maps of spatial componentsserved mainly as descriptive tools to improve our understanding of the sources

Ž .of spatial variation. More recently, Bourgault et al. 1995 proposed to use themap of the regional component of soil electromagnetic response as secondaryinformation in the cokriging of electrical conductivity. They capitalized on thefact that these two soil properties were better correlated at a regional scaleŽ . Ž .rs0.63 than at a local scale rs0.30 . Such scale-dependent relations are

Ž .frequent in soil science, e.g., see Goulard and Voltz 1992 , Goovaerts andŽ . Ž .Webster 1994 , Dobermann et al. 1995, 1997 . Many soil properties of the soil

are controlled by the same physical processes which operate at different spatialscales and influence these properties in different ways. Accounting for thespatial scale in the study of correlations may enhance a relation betweenvariables that is otherwise blurred in an approach where all different sources ofvariation are mixed, leading to a better understanding of the physical underlyingmechanisms controlling spatial patterns.

ŽMultivariate factorial kriging, also called factorial kriging analysis Matheron,.1982; Wackernagel, 1988, 1995, pp. 160–165 , allows one to analyse relations

between variables at the spatial scales detected and modeled from experimentalsemivariograms. Like factorial kriging, which is based on the linear model of

Ž Ž ..regionalization Eq. 16 , multivariate factorial kriging is based on the specificlinear model of coregionalization fitted to the experimental direct and crosssemivariograms:

Llg h s b g h ; i , jŽ . Ž .Ýi j i j l

ls0

Ž .Under that particular model, each random function Z u can be interpreted asilŽ .the sum of independent random functions Z u :i

LlZ u s Z u qm u 18Ž . Ž . Ž . Ž .Ýi i i

ls0


Characterizing the linear relation between the two variables Z and Z at ai j

particular scale l amounts to computing the covariance between the twolŽ . lŽ .corresponding spatial components Z u and Z u , which is but the contributioni j

Ž . Ž . Ž .sill, slope of the structure g h to the cross semivariogram g h :l i j

Cov Zl u ,Zl u sbl 19Ž . Ž . Ž .½ 5i j i j

A standardized measure of the spatial relation is the structural correlationcoefficient defined as:

bli jl l lr sCorr Z u ,Z u s 20Ž . Ž . Ž .½ 5i j i j l lb Pb( ii j j

Ž . Ž .Application of expression 20 to the linear model of coregionalization 12yields the following structural correlation coefficients for the pair Cd–Ni: 0.33,

Ž . Ž . Ž .0.0, and 0.88 at the micro- nugget effect , local 200 m and regional 1.3 kmscales, respectively. The linear correlation between the original concentrationswhich embody all three different scales is 0.49. Taking into account the spatialscale in the correlation analysis reveals a strong correlation of the two metals atthe regional scale that matches the scale of the stratigraphy, suggesting that thesource of these metals is geochemical.

Whereas multivariate analysis is usually done without regard to spatialposition of observations, multivariate factorial kriging accounts for the regional-

Ž .ized nature of variables by analyzing the Lq1 matrices of structural correla-tion coefficients separately. Each correlation structure is thus distinguished byfiltering the structures belonging to other scales of spatial variation. Factorialkriging has been increasingly used these last five years in soil science. One mustbe aware that the decomposition into spatial components, and hence anysubsequent interpretation, depends on the somewhat arbitrary decision of whether

Ž .to include a particular component in the linear model of co regionalization.Therefore, during the structural analysis, it is crucial to take into account anyphysical information related to the phenomenon and the study area.

5. Modeling of local uncertainty

There is necessarily some uncertainty about the value of the attribute z at anunsampled location u. In geostatistics, the usual approach for modeling local

)Ž .uncertainty consists of computing a kriging estimate z u and the associatederror variance, which are then combined to derive a Gaussian-type confidence

Ž . Žinterval Isaaks and Srivastava, 1989, pp. 517–519 , see Fig. 11 right top


Fig. 11. Different models of uncertainty about the Cd concentration at the unsampled location u:95% confidence interval derived from the ordinary kriging estimate and the associated kriging

Ž .variance, and local distributions of probability ccdf established using either a multi-Gaussian orŽ .an indicator approach bottom graphs .

.graph . A more rigorous approach is to model the uncertainty about theŽ .unknown z u before and independently of the choice of a particular estimate

for that unknown. The interpretation of the unknown as a realization of aŽ .random variable Z u leads one to model the uncertainty at u through the

Ž .conditional cumulative distribution function ccdf of that variable:

< <F u; z n sProb Z u Fz n 21� 4Ž . Ž . Ž . Ž .Ž .

<Ž .where the notation ‘ n ’ expresses conditioning to the local information, say,Ž . Ž .n u neighboring data z u . The ccdf fully models the uncertainty at u since ita

gives the probability that the unknown is no greater than any given threshold z,Ž .see Fig. 11 bottom graphs . This section presents the main algorithms for


modeling these ccdfs, with an emphasis on indicator methods. The use of ccdfmodels in spatial prediction and decision making is discussed.

5.1. The parametric approach

Determination of ccdfs is straightforward if an analytical model defined by afew parameters can be adopted for such distributions. Disjunctive krigingŽ . ŽMatheron, 1976; Webster and Oliver, 1989 and multi-Gaussian kriging Verly,

.1983 are two parametric approaches capitalizing on the congenial properties ofthe Gaussian model. The theory underlying disjunctive kriging is rather complexand requires the use of orthogonal polynomial expansions with their relatedconvergence problems. The second approach is easier to implement in that,under the multi-Gaussian model, the mean and variance of the Gaussian ccdf atan unsampled location are but the simple kriging estimate and variance, see Fig.

Ž .11 left bottom graph .The congeniality of the parametric approach is balanced by strong assump-

tions about the normality of the two-point cdf for disjunctive kriging and of themulti-point cdf for multi-Gaussian kriging. The normality of the one-point

Ž .distribution of data sample histogram can be easily checked and, if required,data can be normalized using a transform which amounts at replacing theoriginal values by the corresponding quantiles in the standard normal distribu-

Ž .tion Deutsch and Journel, 1998, p. 141 .The normality of the one-point cdf is a necessary but not sufficient condition

to ensure that a random function is bivariate, and a fortiori multivariate,Gaussian. A common misunderstanding in the soil literature about disjunctivekriging is that the normal score transform ensures the normality of the two-pointdistribution. In fact, such a univariate transform has no impact on the bivariate

Žproperties of the random function Deutsch and Journel, 1998, p. 143; Rivoirard,. Ž .1994, p. 46 . Deutsch and Journel 1998, p. 142 provide a way to check the

normality of the two-point distribution from the shape of indicator semivari-Žograms computed for a series of threshold values, see also Goovaerts 1997a, p.

.271 . Unfortunately, such checks are rarely performed, and the adoption of aGaussian-based approach is essentially a subjective decision too often driven bythe simplicity of the corresponding algorithms.

Multi-Gaussian models have their own merits but they should not be adoptedif critical features of the data are not reproduced. In particular, the Gaussianmodel does not allow for any significant spatial correlation of very large or verysmall values, a property known as destructuration effect. Connected strings ofsmall or large values are very important features in particular for soil pollution,and so the Gaussian model should never be considered when the structural

Ž .analysis indicator semivariograms or qualitative information indicates that verylarge or very small values are spatially correlated. Even in the absence ofinformation about the connectivity of extreme values, the Gaussian model is not


conservative in the sense that it leads one to understate the potential for hazard,Ž .as shown by Gomez-Hernandez 1997 for mass transport in an aquifer section´ ´

with heterogeneous hydraulic conductivity.

5.2. The nonparametric approach

Unlike the Gaussian-based techniques, the nonparametric algorithms do notassume any particular shape or analytical expression for the conditional distribu-

Ž <Ž ..tions. Instead, the value of the function F u; z n is determined for a series ofK threshold values z discretizing the range of variation of z:k

< <F u; z n sProb Z u Fz n ks1, . . . , K 22� 4Ž . Ž . Ž . Ž .Ž .k k

The resolution of the discrete ccdf is then increased by interpolation within eachŽ xclass z , z and extrapolation beyond the two extreme threshold values zk kq1 1

Ž .and z . For example, Fig. 11 right bottom graph shows the model fitted to theK

nine ccdf values estimated at location u.Ž .Nonparametric geostatistical estimation of ccdf values Journel, 1983 isŽ Ž ..based on the interpretation of the conditional probability Eq. 22 as theŽ .conditional expectation of an indicator random variable I u; z given thek

Ž .information n :

< <F u; z n sE I u; z n 23� 4Ž . Ž . Ž . Ž .Ž .k k

Ž . Ž .with I u; z s1 if Z u Fz and zero otherwise. Ccdf values can thus bek kŽ .estimated by least-squares kriging interpolation of indicator transforms of data.

Practical implementation of the indicator approach involves the following stepsŽ .Goovaerts, 1997a, pp. 284–328 .

Ž . Ž .1 Code each observation z u into a vector of K indicator values:a

1 if z u FzŽ .a ki u ; z s ks1, . . . , K 24Ž . Ž .a k ½0 otherwise

The set of K threshold values is typically chosen such that the range of z-valuesŽ .is split into Kq1 classes of approximately equal frequency, e.g., the nine

deciles of the sample cumulative distribution.Ž .2 For each threshold z , compute the experimental indicator semivariogramk

Ž Ž .. Ž Ž ..Eq. 3 , and model it using a linear combination Eq. 6 of permissiblesemivariogram models.

Ž .3 At each unsampled location u, the following should be done.Ø Estimate each of the K ccdf values as a linear combination of neighboring

indicator data by kriging as for continuous attributes; for example, the ordinaryindicator kriging estimator is:

) )

<F u; z n s I u; zŽ . Ž .Ž .k k OKO IK

Ž .n u 25Ž .s l u; z I u ; zŽ . Ž .Ý a k a k

as1


w Ž <Ž ..x)Ø Correct the estimated probabilities F u; z n that do not meet thek

following constraints:)

<F u; z n g 0,1 ;z 26Ž . Ž .Ž .k k

) )

< <X XF u; z n F F u; z n ;z )z 27Ž . Ž . Ž .Ž . Ž .k k k k

Ø Interpolate or extrapolate ccdf values to build a continuous model for theconditional cdf, which allows one to retrieve the probability of being no greaterthan any threshold z, in addition to the K original thresholds z .k

The indicator approach appears much more demanding than the multi-Gaus-sian approach both in terms of semivariogram modeling and computer require-ments. This additional complexity is balanced by the possibility of modelingspatial correlation patterns specific to different classes of attribute valuesthrough indicator semivariograms. In particular, the connectivity of extremevalues can be accounted for. Note that in many applications the objective is notto model the whole ccdf but rather to assess the probability of exceeding a

Žparticular value, say a regulatory threshold in soil pollution Leonte and. Ž .Schofield, 1996 or a critical value for soil quality Smith et al., 1993 . In this

case, the indicator approach is as straightforward as the parametric approachsince a single indicator semivariogram and kriging system need to be consid-ered.

Ž .Fig. 12 first two rows shows the different steps of the indicator approach toŽmap the probability of contamination by cadmium in the region regulatory

y1.threshold z s0.8 mg kg . A usual criticism of the indicator approach is thatc

the indicator coding amounts to discarding much of the information in the data.In the example of Fig. 12, all concentrations ranging from 0.81 mg kgy1 to 5.2mg kgy1 yield the same unit indicator value, while all concentrations no greaterthan 0.8 kgy1 are translated into zero indicator values. In theory, this loss ofinformation can be compensated by accounting for indicator values defined atdifferent thresholds that is using indicator cokriging instead of kriging. Practicehas shown, however, that indicator cokriging improves little over indicator

Ž .kriging Goovaerts, 1994d because cumulative indicator data carry substantialinformation from one threshold to the next one, and all indicator values are

Ž .available at each sampled location isotopic or equally-sampled case . AnŽalternative to indicator cokriging is probability kriging Journel, 1984; Goovaerts,

.1997a, p. 301 where ccdf values are estimated as a linear combination ofneighboring indicator and uniform transforms of the data:

) )

<F u; z n s I u; zŽ . Ž .Ž .k k PKPK

Ž . Ž .n u n u

s l u; z I u ; z q n u; z X uŽ . Ž . Ž . Ž .Ý Ýa k a k a k a

as1 as1

28Ž .


Fig. 12. Estimation of the probability of contamination by cadmium using the indicator approach.The original concentrations are transformed into indicators of exceedence of the regulatorythreshold 0.8 mg kgy1 that are then kriged using an indicator semivariogram. The bottomprobability map accounts for additional information in the form of soft probabilities derived froma calibration of the geologic map of Fig. 1.

Ž . Ž . Ž .The standardized rank x u of observation z u is computed as r u rn,a a a

Ž . w xwhere r u g 1,n is the rank of the datum in the sample cumulative distribu-a

w xtion. The data ranks valued in 0,1 allow one to discriminate Cd concentrations


with similar indicator transforms 0 or 1, and so corrects for the loss of resolutioncaused by the use of a single threshold in indicator kriging.

A limitation of the indicator approach is the a posteriori correction ofŽ . Ž .estimated probabilities that do not meet constraints 26 and 27 , although

practice has shown that order relation deviations are generally of small magni-Ž .tude Goovaerts, 1994d . A more elegant solution would consist of implement-

ing these constraints directly into the indicator kriging system, as proposed byŽ .de Gruijter et al. 1997 for categorical variables. A potential pitfall is the

interpolation or extrapolation of the corrected probabilities to derive a continu-ous ccdf model. Characteristics of the ccdf such as the mean or variance mayoverly depend on the modeling of the upper and lower tails of the distributionŽ .Goovaerts, 1997a, p. 338 . A linear model is usually adopted for interpolation

Ž xwithin each class z , z , whereas power or hyperbolic models are used fork kq1Žextrapolation beyond the two extreme threshold values z and z Deutsch and1 K

.Journel, 1998, pp. 135–138 . The choice of these models is fully arbitrary, and IŽprefer to capitalize on the higher level of discretization of the cdf i.e., the

.cumulative histogram to improve the within-class resolution of the ccdfŽ .Goovaerts, 1997a, p. 327 . For example, the resolution of the discrete ccdf of

Ž .Fig. 11 right bottom graph has been increased by performing a linear interpola-tion between tabulated bounds provided by the histogram of 259 Cd concentra-tions. An alternative to the piecewise interpolationrextrapolation of the ccdfmodel consists of fitting a continuous parametric model to the set of estimatedprobabilities. In all cases, the impact of extrapolation models can be reduced by

Žselecting more threshold values within the two tails of the distribution Deutsch.and Lewis, 1992; Chu, 1996 .

The major advantage of the indicator approach is its ability to incorporate softŽinformation of various types e.g., soil map or qualitative field observations such

.as the smell or color of contaminated soil in addition to direct measurements onthe attribute of interest. The only requirement is that each soft datum must becoded into a vector of K cumulative probabilities of the type:

<Prob Z u Fz specific local information at u ks1, . . . , K 29� 4Ž . Ž .k

Ž . Ž Ž ..Goovaerts and Journel 1995 showed how probabilities Eq. 29 can bederived from the calibration of a soil map and combined with precise measure-ments of metal concentration to map the risk of deficiency of these metals in the

Ž .soil. Similar applications in soil science are presented in Colin et al. 1996 ,Ž . Ž . Ž .Goovaerts et al. 1997 , and Journel 1997 . Fig. 12 bottom graphs illustrates

the use of the geologic map as soft information in mapping the risk ofcontamination by cadmium. Probabilities of contamination derived from acalibration of geology are used as local means in the simple kriging of indicatordata; recall similar approach for continuous attributes in Fig. 9. Taking thegeology into consideration produces a more detailed map and enhances the


contrast between Argovian rocks, on which the soil contains little cadmium, andother rocks with larger probabilities of exceeding the regulatory threshold.

Another interesting feature of the indicator approach is its ability to accountfor a secondary continuous variable, say another metal content, that is nonlin-early related to the primary variable. The idea consists of discretizing the two

Žvariables using two series of K thresholds e.g., deciles of the sample cumula-.tive distributions , then combining primary and secondary indicator data at each

threshold using a cokriging algorithm similar to the one introduced for continu-Ž .ous variables Zhu and Journel, 1993; Goovaerts, 1997a, p. 308 .

5.3. Using the model of local uncertainty

Once the uncertainty about an unsampled value has been modeled usingeither parametric or nonparametric approaches, an estimate for that unknowncan be retrieved from the ccdf, say the mean or the median of the conditional

Ž .distribution. For example, the map in Fig. 13 left top graph depicts the mean ofthe conditional cdfs of Cd concentration modeled using an indicator approach.

Fig. 13. Ordinary kriging estimates of cadmium concentration and the corresponding estimationŽ . Ž .variances right column . Left maps show the mean E-type estimate and variance of ccdfs

modeled using an indicator approach.


This map of conditional means, also known as E-type estimates, looks similar toŽthe map of Cd concentrations estimated using ordinary kriging Fig. 13, right

.top graph . The advantage of the indicator approach over ordinary kriging is thatit provides a measure of uncertainty that accounts for the local data, whereas thekriging variance depends only on the data configuration and semivariogrammodel; recall previous discussion and Fig. 10. These differences are clear on thetwo maps at the bottom of Fig. 13 which depict the variance of the conditionalcdfs and the kriging variance, respectively. The conditional variance is larger inthe high-valued parts of the study area where the Cd measurements fluctuate themost and so the largest uncertainty is intuitively expected. The uncertainty issmaller on Argovian rocks where Cd concentrations are consistently small. Incontrast, the kriging variance map indicates greater uncertainty in the extremewest corner of the study area where data are sparse, whereas the uncertainty issmallest near data locations. Elsewhere the kriging variance is about the samewhatever the surrounding data values. As mentioned previously, several meth-ods could be used to correct for the lack of stationarity of the variance, such asdata transformation, stratification of the study area, or local rescaling of arelative semivariogram.

Mapping metal concentrations or other soil properties is often a preliminarystep towards decision making, such as the delineation of polluted areas or theidentification of zones that are suitable for crop growth. For soil pollution, astraightforward approach consists of declaring contaminated all locations wherethe pollutant concentration estimate exceeds the regulatory threshold. Similarly,a farmer may decide to grow a given crop wherever the estimated value of the

Ž .limiting factor e.g., depth to parent material, soil acidity exceeds someŽ .threshold. Such an approach has two drawbacks: 1 the estimation error is

ignored: a contaminated location can be declared safe on the basis of a wrongestimate of pollutant concentration which is slightly less than the regulatory

Ž .threshold, 2 the decision rule requires crip thresholds such as provided byenvironmental protection agencies for soil pollution. For land evaluation, how-ever, the use of crisp thresholds is generally inappropriate in that the cropresponse is continuous from a slight reduction in crop performance through to

Ž .crop failure Burrough, 1989 .For environmental applications where crisp thresholds exist the uncertainty

about the predicted soil attribute can be accounted for by estimating andmapping the probability of exceeding those thresholds. A common question isthen above which level of risk should we decide to clean a polluted area or

Ž .develop specific land use policies Goovaerts, 1997b . When the probability isvery large or very small, the risk-based decision is quite straightforward.Decision making is much more difficult for locations with intermediate probabil-

w xities, say in the interval 0.3,0.7 . Depending on the resources available, comple-mentary investigation could be made to reduce the uncertainty at these locations,which amounts to decreasing or increasing the intermediate probabilities. Never-


theless, the choice of a probability threshold is mainly subjective: a givenprobability of contamination may be unacceptable for residential areas, yettolerable for industrial yards. One must also keep in mind that probabilities arebut estimates and so are subject to errors as the estimates of attribute values.

The practical use of probability maps in decision making faces the difficultchoice of crisp thresholds for attribute values and probability thresholds. Insteadof the abstract notion of risk, decision makers usually prefer dealing with afinancial assessment of the different options. For soil pollution, they would liketo know the financial costs that might result from a wrong decision, i.e.,declaring safe a contaminated location or cleaning a safe location. The idea is togo beyond a mere assessment of the risk and provide decision makers with a set

Žof alternative solutions and the corresponding potential costs Srivastava, 1987;.Journel, 1987; Goovaerts and Journel, 1995; Goovaerts et al., 1997 . This

requires, however, that decision makers provide scientists with the impact orcost functions specific to the problem at hand. Such an approach supports the

Ž .assertion of Bouma 1997 that researchers must interact with stakeholdersŽ .farmers, planners and politicians to analyze problems and propose cost-effec-tive solutions.

Ž .Fig. 14 left top graph shows an example of cost functions for soil contami-Ž y1.nation by cadmium regulatory threshold z s0.8 mg kg . Consider first thec

decision of taking no remedial measure at a location u. If this location isŽ .actually safe, z u Fz , then the decision is correct and there is no cost.c

ŽOtherwise, the cost of misclassifying a contaminated location potential ill.health, legal liability is modeled as proportional to the actual contamination

w Ž . xz u yz :c

0 if z u FzŽ . cL z u s 30Ž . Ž .Ž .1 ½ z u yz otherwiseŽ . c

The second cost function expresses the potential consequences of taking thedecision of cleaning a location. If this location is actually safe, there is undueapplication of remedial measures and the corresponding cost is here modeled asa constant value of 2.5:

0 if z u )zŽ . cL z u s 31Ž . Ž .Ž .2 ½2.5 otherwise

The actual cost attached to either type of decision cannot be computed becauseŽ .the actual concentration z u of pollutant is unknown. A ccdf model allows one

to account for the uncertainty about the unknown value and determine theexpected cost for the two alternatives:

q`

< <w u sE L Z u n s L z u d F u; z n is1,2 32Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž . Ž .Hi i iy`


Fig. 14. Classification of locations as contaminated by cadmium on the basis that the resultingŽ .expected cost unnecessary cleaning is smaller than the cost associated with wrongly classifying a

Ž .location as safe potential ill health . Expected costs are computed using ccdf models and costfunctions specific to each type of decision.

This integral is, in practice, approximated by the discrete sum:Kq1

< <w u , L z F u; z n yF u; z n is1,2 33Ž . Ž . Ž . Ž .Ž . Ž . Ž .Ýi i k k ky1ks1

where z , ks1, . . . , K , are K threshold values discretizing the range ofkŽ xvariation of z-values, and z is the mean of the class z , z , e.g.,k ky1 k

Ž . Ž .z s z qz r2. Fig. 14 bottom graphs shows the two maps of expectedk ky1 k

costs. At each location the health cost and the remediation cost are compared,and the decision of cleaning the location is taken if one can expect the health

Ž .cost to be larger than the cost of unnecessary remediation. Rautman 1997presented a more sophisticated risk-based economic-decision model to evaluatethe performance of different technologies for measuring uranium activity ofin-situ soils.

The concept of a cost function in soil pollution is analogous to that of aŽ .membership function introduced by Burrough 1989 for land evaluation prob-

lems. The membership function is here used to express the suitability of a soil


for a given purpose, say a given crop A, as a function of the value of a soilproperty. For example, the following function illustrates the impact of soilacidity on the growth of crop A:

°1 if z u F5Ž .~0 if z u )7Ž .L z u sŽ .Ž . ¢7yz u r2 if 5-z u F7Ž . Ž .Ž .

The membership ranges from 1 to 0 and reflects the possibility that soil is tooacid for crop A. This gradual response of the crop to soil pH is more realisticthan crop failure below a crisp pH threshold. The actual membership cannot becomputed since the actual pH value at u is unknown. However, the expectedmembership can be computed from the ccdf model using the same expression as

Ž Ž ..for the expected cost Eq. 33 . The concept of expected membership wasŽ .introduced by Lark and Bolam 1997 under the name of ‘weighted member-

ship.’ This approach allows one to combine two types of uncertainty: theuncertainty about the value of a soil property at an unsampled location and the

Ž .imprecision fuzziness of the impact of that soil property on land suitability.Note that one still faces the difficult problem of choosing a membershipthreshold for decision making, which could be overcome by converting member-ships into economic impacts.

5.4. Categorical attributes

Unlike continuous variables, categorical attributes such as texture or watertable classes cannot be estimated as a mere linear combination of neighboringobservations. In many situations, the unsampled location is simply allocated tothe same category as the nearest observation, i.e., in the same Thiessen polygonor Dirichlet tile. Such an approach has two serious weaknesses: it ignores spatialcorrelation and transition probabilities between categories, and it provides nomeasure of the reliability of the prediction.

Qualitative information can be handled by indicator algorithms as long as it iscoded into indicator values, say 1 if the category is present and 0 otherwise;

Ž .recall expression 4 . Soft indicators, i.e., probabilities valued between zero andone, can also be considered to account for the uncertainty in the classification of

Ž .sampled locations fuzzy classification . Then, indicator kriging is used toestimate the probability for each state s of the attribute s to occur at thek

unsampled location u:< <p u; s n sProb S u ss n ks1, . . . , K 34� 4Ž . Ž . Ž . Ž .Ž .k k

Secondary information such as provided by differences in lithology, landform ordrainage can also be incorporated in the estimation of conditional probabilities.

The prediction process amounts to allocating u to a single category on theŽ Ž ..basis of the set of conditional probabilities Eq. 34 . Bierkens and Burrough

Ž .1993a,b proposed to use as predictor the category with the largest probability


of occurrence which defines the ‘map purity.’ Such a criterion typically leadsone to allocate most of the locations to the most frequent categories since theprobability of occurrence is likely to be larger if the corresponding globalproportion is large. Conversely, the less frequent categories tend to be underrep-resented. This approach is thus inadequate if one aims at reproducing sampleproportions that are deemed representative of the entire area. One solutionconsists of preferentially allocating locations to the category with the largestprobability of occurrence under the constraint of reproduction of global propor-

Ž .tions Soares, 1992 . An application to the mapping of land uses is given inŽ .Goovaerts 1997a, p. 357 .

6. Stochastic simulation

Ž .As illustrated by the map of Fig. 15 left column , kriging tends to smooth outlocal detail of the spatial variation of the soil attribute. The variance of ordinarykriging estimates is much smaller than the sample variance s 2 s0.83, and theêxperimental semivariogram has a much smaller relative nugget effect than thesemivariogram model, which indicates the underestimation of the short-rangevariability of Cd values. Unlike kriging, stochastic simulation does not aim atminimizing a local error variance but focuses on the reproduction of statisticssuch as the sample histogram or the semivariogram model in addition to the

Žhonoring of data values. In the example of Fig. 15, the same information data,.semivariogram model is used by the kriging and simulation approaches, but the

simulated map looks more ‘realistic’ than the map of statistically ‘best’ esti-mates because it reproduces the spatial variability modeled from the sampleinformation. Stochastic simulation is thus increasingly preferred to kriging forapplications where the spatial variation of the measured field must be preserved,

Žsuch as the delineation of contaminated areas Desbarats, 1996; Goovaerts,. Ž1997c or the modeling of solute transport in the vadose zone Vanderborght et

.al., 1997 .

6.1. Modeling the spatial uncertainty

One may generate many realizations that all honor the same data and matchreasonably well the same statistics. For example, Fig. 16 shows three realiza-tions of the spatial distribution of Cd values that all honor the 259 measurementsof Cd concentration displayed in Fig. 1 and reproduce approximately thesemivariogram model. The three images are consistent with the sample informa-tion, and their differences provide a measure of spatial uncertainty. Featuressuch as zones of large values are deemed certain if seen on most of the

Žrealizations, and their probability of occurrence i.e., probability that a given.threshold is jointly exceeded at a series of locations can be computed as long as


Fig. 15. Ordinary kriging estimates and simulated values of Cd concentration over the study area.Bottom graphs show the corresponding histograms and standardized experimental and modelŽ .solid line semivariograms. The smoothing effect of kriging leads to underestimation of theshort-range variability of Cd values.

the realizations are equiprobable. Such joint probabilities cannot be derived fromthe ccdfs introduced in the previous section: each ccdf is specific to a singlelocation and thus provides only a measure of local uncertainty.

In many situations, decision making concerns areas or blocks that are muchlarger than the measurement support, such as the delineation of 1-ha remediationunits from soil core measurements. Provided the soil variable averages linearly,


Fig. 16. Three realizations of the spatial distribution of Cd values over the study area and thecorresponding standardized semivariograms. Differences between realizations provide a model forthe uncertainty about the distribution in space of Cd values.

the average value over the large support can be estimated by block kriging or,indirectly, by the arithmetical mean of kriging estimates at a series of locationsdiscretizing this support. This linear averaging, however, does not apply toprobability values: the probability that the average attribute value exceeds a

Ž .given threshold block probability is not equal to the linear average of localprobabilities of exceedence defined at a series of discretizing locations! Such anupscaling can be done using disjunctive kriging under the stringent assumption

Ž .of bivariate normality Webster, 1991 . A nonparametric alternative consists ofapproximating numerically the block probability by generating many simulatedblock values as linear averages of simulated point values, and counting the

Žproportion of block values that exceed the critical threshold Isaaks, 1990;.Kyriakidis, 1997 . The major advantage of the simulation-based approach is that

it provides a nonparametric measure of the uncertainty attached to the predictionŽ .of a single block or multiple spatially dependent blocks Goovaerts, 1999 .


Stochastic simulation can also be used to upscale properties like permeabilitythat do not average linearly in space.

Note that the use of block probability may lead to over-optimistic decisions inthat large local probabilities of contamination or large local pollutant concentra-tions are smoothed out through the averaging. One would then prefer dealingwith local probabilities of contamination and, for example, decide to clean theblock if the probability threshold is exceeded for a given proportion of locationswithin the block. Such a proportion can be derived using block-indicator krigingŽ .Goovaerts, 1997a, p. 306 . Other decision making criteria for soil contamina-

Ž . Ž .tion are discussed in Goovaerts 1997b and Kyriakidis 1997 .The impact of a given scenario, such as a particular remediation process or

land use policy, can be investigated from a simulated map that reproducesaspects of the pattern of spatial dependence or other statistics deemed conse-

Žquential for the problem at hand e.g., connectivity of large values, spatial.correlation with secondary attributes . Moreover, having many equiprobable

realizations allows one to assess the uncertainty about the consequences of thisparticular scenario. Consider the problem of assessing the economic impact ofdeclaring the study area safe with respect to Cd. For each realization, this globalcost is obtained as the sum of local costs computed from each simulated value

Ž Ž ..using the cost function Eq. 30 . The three realizations of Fig. 16 yield,respectively, a cost of 1058, 1001, and 825. These differences reflect theuncertainty about the economic impact of taking no remedial measure whichresults from our imperfect knowledge of the spatial distribution of Cd values inthe area. The same operation can be repeated for 100 realizations, and theresulting histogram of costs allows a probabilistic assessment of financialconsequences, e.g., there is here a 75% probability that the cost will be larger

Ž .than 930 Fig. 17 .

Fig. 17. The distribution of costs resulting from a wrong decision to declare the study area safeŽwith respect to Cd. This distribution is obtained by applying the cost function of Fig. 14 solid

.line to 100 realizations of the spatial distribution of Cd values.


6.2. Simulation algorithms

Like estimation, simulation can be accomplished using a growing variety ofŽtechniques which differ in the underlying random function model multi-Gaus-

.sian or nonparametric , the amount and type of information that can beŽ . Ž .accounted for, and the computer requirements. Goovaerts 1997a pp. 376–424

provide a detailed description of the main simulation algorithms, and these areŽavailable in the public domain geostatistical software library GSLIB Deutsch

.and Journel, 1998 . The key concept of each class of algorithms is brieflyoutlined below.

Consider the simulation of the continuous attribute z at N grid nodes uXj

� Ž . 4conditional to the data z u , as1, . . . ,n . Sequential simulation amounts toa

Ž X <Ž .. � Ž X . <Ž .4modeling the ccdf F u ; z n sProb Z u Fz n , then sampling it at eachj j

of the N nodes visited along a random sequence. To ensure reproduction of thesemivariogram model, each ccdf is made conditional not only to the original ndata but also to all values simulated at previously visited locations. Two majorclasses of sequential simulation algorithms can be distinguished, depending onwhether the series of conditional cdfs are modeled using the multi-Gaussian orthe indicator formalism. The probability field approach also requires the sam-pling of N successive ccdfs. However, unlike in the sequential approach, allccdfs are conditioned only to the original n data. Reproduction of the semivari-ogram model is here approximated by imposing an autocorrelation pattern on theprobability values used for sampling these ccdfs. Unlike previous simulationalgorithms, in simulated annealing the creation of a stochastic image is formu-lated as an optimization problem without any reference to a random function

Ž .model. The basic idea is to perturb an initial seed image gradually so as tomatch target constraints, such as reproduction of the semivariogram model.

Stochastic simulation has been seldom used in soil science so far unlike inmining or petroleum. Most applications relate to soil pollution and involve

Ž .diverse algorithms such as sequential Gaussian simulation Desbarats, 1996 ,Žsequential indicator simulation Goovaerts, 1997c; Garcia and Froidevaux,

. Ž .1997 , probability field Bourgault et al., 1995; Kyriakidis, 1997 . SimulatedŽ .annealing was used by Sterk and Stein 1997 to generate unsmoothed maps of

Ž .wind-blown sediment transport. Bierkens and Burrough 1993a,b used thecategorical formalism of sequential indicator simulation to generate maps ofwater table classes which were then used to assess land suitability for pastures.

7. Conclusions and perspectives

The growing interest of soil scientists in geostatistics arises because theyincreasingly realise that quantitative spatial prediction must incorporate thespatial correlation among observations. Also, geostatistics offers an increasingly


wide palette of techniques well suited to the diversity of problems and informa-tion soil scientists have to deal with. The recent developments of data acquisi-tion and computational resources have provided the geostatistician with large

Ž .amounts of information of different types continuous, categorical , which canbe stored and managed in Geographic Information Systems, and which they canprocess rapidly. Multivariate approaches, such as factorial kriging analysis, canbe used to investigate how the correlation between variables changes as afunction of the spatial scale, and to improve our understanding of scale-depen-dent physical processes. Multivariate geostatistical interpolation also allows one

Žto supplement a few expensive measurements of the attribute of interest e.g.,.metal concentrations by more abundant data on correlated attributes that areŽ .cheaper to determine e.g., pH or elevation . In particular, indicator geostatistics

enables secondary categorical information such as provided by land use or soilmap to be accounted for in the prediction of continuous variables. There is stillresearch to be done on the incorporation of variables measured on differentsupports, in particular the combination of field data with remote sensinginformation. Another avenue of research is the three-dimensional spatial model-ing of soil processes since the development of measurement techniques allowsthe collection of a larger amount of data in both the horizontal and verticaldirections.

There is necessarily uncertainty about the attribute value at an unsampledlocation, and its assessment is critical for some applications. The uncertainty canbe assessed from the kriging variance using a Gaussian-type confidence intervalcentered on the kriging estimate. A more promising approach is to assess firstthe uncertainty about the unknown, then deduce an estimate that is optimal insome appropriate sense. This can be achieved using an indicator approach thatprovides not only an estimate but also the probability of exceeding criticalvalues, such as regulatory thresholds in soil pollution or criteria for soil quality.The last five years have witnessed a growing use of geostatistics to createcolorful probability maps, yet the practical use of these maps for decisionmaking, such as delineation of contaminated areas or areas of good soil quality,has received little attention. We should now combine probabilistic modelsprovided by geostatisticians with expert knowledge in order to assess thefinancial impacts of the different options available as is being done in mining. Inthe future, more research should be devoted to this important topic, in particularthe incorporation of the uncertainty related to several correlated attributes suchas multiple criteria for land suitability.

ŽAnother way to model uncertainty is to generate numerous images realiza-.tions that all honor the data and reproduce aspects of the patterns of spatial

dependence or other statistics deemed consequential for the problem at hand. AŽ .given scenario remediation process, land use policy can be applied to the set of

Žrealizations, allowing the uncertainty of the response remediation efficiency,.soil productivity to be assessed. Stochastic simulation is one of the most active


areas of research in geostatistics, and one can expect an increasing use of thesetechniques to model soil spatial uncertainty.

The modeling of the space–time variability is also becoming one of the majoravenues of research is environmental geostatistics. The interested reader shouldrefer to the following papers for a brief overview of the techniques currentlyavailable and references to the few space–time studies that have been conducted

Ž . Ž .so far in soil science: Goovaerts and Sonnet 1993 , Papritz and Fluhler 1994 ,¨Ž .Heuvelink et al. 1997 .

Finally, geostatistics was originally developed to solve practical problems,namely, evaluating recoverable reserves in mining and its growth bears witnessto its utility and success. We should do well to keep this trademark ofpracticality in soil science and to foster the increasing and successful applicationof geostatistics to soil-related issues.

Acknowledgements

This work was partly done while the author was with the Unite Biometrie,´ Úniversite Catholique de Louvain, Belgium. The author thanks Mr. J.-P. Duboisóf the Swiss Federal Institute of Technology at Lausanne for the data and the

Ž .National Fund for Scientific Research Belgium for its financial support. Datacan be downloaded from http:rrwww-personal.engin.umich.edur;goovaertr.

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