Ž .Geoderma 88 1999 13–38

Nonparametric cross-covariance modeling asexemplified by soil heavy metal concentrations

from the Swiss Jura

Tingting Yao )

Department of Geological and EnÕironmental Sciences, Stanford UniÕersity, Stanford, CA 94305,USA

Received 22 October 1997; accepted 23 June 1998

Abstract

Geostatistics is used in soil science to map spatial distributions of soil properties from limitedsamples. Covariance models provide the basic measure of spatial continuity which is used toweight the information available at different sample locations, as in kriging. Traditionally, aclosed-form analytical model is fitted to allow for interpolation of sample covariance values whileensuring the positive definiteness condition. For cokriging where several different properties are

Ž .cross correlated with each other, the simultaneous modeling of several cross covariances is mademore difficult because of the restrictions imposed by the linear coregionalization model. An

Ž .algorithm for automatic joint modeling of multiple cross covariance tables is proposed, buildingon an extension of Bochner’s positive definiteness theorem and eigenvalue correction. Theobjective of this paper is to present the new methodology and demonstrate its application tomodeling large cross-covariance matrix. Such task is not easily done or, more bluntly, seldomdone in the conventional way when the number of coregionalized variables becomes large. Thedata set used for the case study relates to heavy metal pollution in soil. q 1999 Elsevier ScienceB.V. All rights reserved.

Keywords: geostatistics; multivariate; covariance; spectrum; heavy metals

1. Introduction

Geostatistics, or more specifically, kriging estimation or stochastic simula-tion, provides a tool for the evaluation of spatial distribution of soil properties

) Present address: Mobil Technology Company, Reservoir Characterization Group, 13777Midway Road, Dallas, TX 75244, USA. Tel.: q1-972-851-8271; Fax: q1-972-851-8703; E-mail:tingting_yao@email.mobil.com

0016-7061r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0016-7061 98 00074-3

( )T. YaorGeoderma 88 1999 13–3814

Ž .Goovaerts, 1992; Stein, 1993, 1994 . A prior requirement is to quantify thespatial correlation between properties at different locations so that the informa-tion from samples can be weighted into an estimator of the values at unsampledlocations. Covariance models provide the basic measure of such spatial correla-tion. A permissible positive definite model is required to interpolate samplecovariance values towards uninformed distance vectors. The positive definite-ness property ensures existence and uniqueness of solutions to the kriging

Ž .systems Armstrong and Jabin, 1981; Armstrong and Diamond, 1984 . Thetraditional modeling approach considers only positive linear combination of

Žbasic models known to be positive definite Journel and Huijbregts, 1978;.Goovaerts, 1997 . This restriction to linear combinations is sometimes limiting

and the modeling process can be time-demanding and is somewhat subjective. Inthe multivariate case, to apply cokriging, the modeling of a cross-covariance

Ž .matrix between primary and secondary variable s is even more difficult becauseŽ .of further restrictions imposed by the linear model of coregionalization LMC

ŽJournel and Huijbregts, 1978; Myers, 1983; Goulard, 1989; Bourgault andMarcotte, 1991; Myers, 1992; Goovaerts, 1994; Webster et al., 1994; Bourgault

.et al., 1995; Wackernagel, 1994, 1995 . Iterative algorithms to fit a linearŽ .coregionalization model have been proposed by Goulard 1989 and Goulard

Ž .and Voltz 1992 .Ž .The LMC model includes Lq1 permissible basic structures c h which arel

Ž . Ž .Xcombined linearly to model each cross variogram C h . For two variables,kk

that coregionalization model is written:L

XlX XC h s b c h , ; ks1, 2; k s1, 2Ž . Ž .Ýkk kk l

ls0

w Ž .xXFor the matrix of covariances C h to be positive definite, it is sufficientkkŽ .that the coefficient matrix for each of the Lq1 basic structures c h ,l

l lb b11 12lB s , ; ls0, . . . , L, be positive definitel lb b21 22

ˆ lGoulard’s algorithm starts with a set of arbitrary coefficient matrices B . Thecoefficients values within these matrices are then iteratively perturbed so as tominimize a weighted sum of squared differences between sample and modelcovariance values subject to the positive definiteness constraints. Each squareddifference is weighted proportionally to the number of data pairs used in thecalculation of the corresponding sample covariance value. Theoretically, theprocedure may not converge nor is it guaranteed to provide a unique solution.The most critical problem, however, is the prior specification of the number and

Ž .type of the basic structures c h , which remains somewhat arbitrary and is al

major cause of poor fit. With an increasing number of coregionalized variables,even with Goulard’s algorithm, the modeling becomes impractical. In addition,

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the linear coregionalization model is only a sufficient, not necessary, modelensuring positive definiteness of the matrix of cross-covariance functions. TheMarkov coregionalization model is an example of a nonlinear coregionalization

Ž .model Almeida and Journel, 1994 but it only holds under a specific Markov-type screening assumption.

ŽAn automatic covariance modeling method has been proposed Rehman,.1995; Yao and Journel, 1998 based on Bochner’s theorem. The main idea is to

Ž .transform the experimental cross covariance tables into quasi-density spectrumŽ . Ž .tables using fast Fourier transform FFT Sneddon, 1951 . These quasi-density

spectrum tables are then smoothed under constraints of positivity and unit sum.Ž . Ž . Ž .A back-transform FFT yields permissible jointly positive definite cross

Ž .covariance tables. Through this FFT ‘roundtrip’, permissible cross covariancestablesrmaps are obtained without calling for any analytical expression. Thepurpose of this paper is to present the methodology to a soil science audienceand improve the algorithm so that it is applicable to any large multivariate case.The conventional way of fitting the auto and cross-covariances using the linearmodel of coregionalization becomes impractical when the number of variablesincreases beyond three. A case study using a soil contaminant data set with fourcoregionalized variables is developed to illustrate the application of this method-ology.

2. Recall of Bochner’s theorem

2.1. UniÕariate case

Ž . Ž .Bochner’s theorem Bochner, 1949 states that a function C h is positivedefinite and, hence, can be used as a covariance model if and only if it can beexpressed as the Fourier transform of a bounded nondecreasing positive measureŽ .S w :

C h sH d e2p i hPw dS wŽ . Ž .R

where h is the distance vector in the d dimension space R d and w is thefrequency vector in the corresponding frequency domain; u is the vector oflocation coordinates in R d.

Ž . Ž .If the stationary random function RF model Z u has unit variance, i.e.,Ž . � Ž .4 Ž .C 0 sVar Z u s1, then S w can be seen as a cumulative distribution

Ž . Ž . Ž .function cdf , with dS w ss w dw. Bochner’s theorem then requires that theŽ . Ž . Ž .density spectrum s w sdS w rdw be a probability density function pdf , i.e.,

such that:

s w G0, ; w , and: s w dws1 1Ž . Ž . Ž .HdR

Ž . Ž .s w is called the spectral density or density spectrum of RF Z u .

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Therefore, in the single-variable case, Bochner’s theorem maps the positivedefinite constraints on covariance into a much simpler positive constraint on thedensity spectrum. Similarly, applying Bochner’s theorem, we proposed to

Ž .transform the set of experimental covariance values called a table into aquasi-density spectrum table using FFT. Because sampling densities vary fromone lag to another and from one variable to another, the experimental covari-ances are not positive definite, hence the corresponding spectrum values are notall positive, hence the term ‘quasi-density spectrum’. These quasi-density spec-

Ž . Ž Ž ..trum tables s w are smoothed under the previous constraints Eq. 1 ofpositivity and unit sum using a smoothing window in the frequency domainŽ . Ž .Chatfield, 1996; Yao and Journel, 1998 . That is, for each entry j , j of the1 2

Ž .spectrum table, define a symmetric square smoothing window m ,m centered1 2

on this cell, the smoothed value is:

m r2 m r21 21s j , j s s j q l , j q l 2Ž . Ž . Ž .ˆ Ý Ý1 2 1 1 2 2m m1 2 l sym r2 l sym r21 1 2 2

Ž . Ž .The smoothing window m ,m is increased gradually from 1,1 until either1 2Ž .the averaged spectral density value s j , j within this window becomesˆ 1 2

nonnegative or the window has reached a predetermined maximum size: in thelatter case the entry is assigned a value of 0 to ensure nonnegativity. All positivedensity values are then standardized to sum to one. A back-transform FFT of the

Ž .smoothed spectral density table s j , j yields permissible positive definiteˆ 1 2

covariance tables. At no point any analytical modeling is called for, in particularthe resulting model need not be a linear combination of a few basic component

Ž .structures as in the general linear model Journel and Huijbregts, 1978 .Ž . Ž .In Eq. 2 , had we used an asymmetric e.g., elliptical smoothing window,

we may have imparted artefact anisotropy to the resulting smoothed spectrummap. It is preferable, when enough data is available, to let the originalexperimental covariance and spectrum values establish anisotropy if any. Anasymmetric smoothing window could, however, be used to enhance anisotropyif that anisotropy is known to exist from ancillary information.

2.2. The multiÕariate case

Bochner’s theorem extends to the multivariate case stating that the inversew Ž .xXFourier transform of a jointly positive definite Hermitian matrix s w , ; w,kk

w Ž .xXis a jointly positive definite auto- and cross-covariance matrix C h ; k andkk

kX are the indices of any two variables.Ž . Ž . 2Consider two RFs Z u and Y u in RR . Their autocorrelograms C andZZ

Ž . Ž . Ž .C are always symmetric in the sense that: C h sC yh , C h sY Y ZZ ZZ Y YŽ . Ž . Ž .C yh ; but the cross-correlograms may be asymmetric: C h sC yh ,Y Y ZY YZ

Ž .Goovaerts, 1997, p. 48 . Therefore, the autospectra s and s are real valuesZZ Y Y

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Žand the cross-spectra s and s are complex but conjugate values Bracewell,ZY YZ.1986 .

Their joint spectral density matrix must be positive definite Hermitian writtenas:

s w ,w s w ,wŽ . Ž .ZZ 1 2 Z Y 1 2s w ,w s 3Ž . Ž .1 2 s w ,w s w ,wŽ . Ž .YZ 1 2 Y Y 1 2

with:

s w ,w sa w ,w q ib w ,w ,Ž . Ž . Ž .ZY 1 2 1 2 1 2

s w ,w ss w ,w sa w ,w y ib w ,w ,Ž . Ž . Ž . Ž .YZ 1 2 Z Y 1 2 1 2 1 2

Ž . Ž .where the two real functions a w ,w and b w ,w are even and odd,1 2 1 2

respectively.w Ž .xIf the spectral matrix s w ,w verifies the condition above, then its inverse1 2

Fourier transform

C h C hŽ . Ž .ZZ ZYC h sŽ .

C h C hŽ . Ž .YZ Y Y

is a real positive definite matrix, whereas:

C 0 C h PPP C hŽ . Ž . Ž .ZY ZY 12 Z Y 1n

C h C 0 PPP C hŽ . Ž . Ž .ZY 21 Z Y Z Y 2 nC h sŽ . . . . .ZY . . . .. . . .

C h C h PPP C 0Ž . Ž . Ž .ZY n1 Z Y n2 Z Y

h is the distance vector between ith and jth datum.i j

The joint smoothing of the spectrum maps s , s , s , s uses movingZZ ZY YZ Y YŽ .averages similar to those used for the autospectrum smoothing in Eq. 2 with,

in addition, the following Schwarz’s inequality-type constraints:

s w ,w G0.0, s w ,w G0.0,Ž . Ž .ZZ 1 2 Y Y 1 2

s w ,w s w ,w Fs w ,w s w ,w 4Ž . Ž . Ž . Ž . Ž .ZY 1 2 Y Z 1 2 ZZ 1 2 Y Y 1 2

Ž . Ž . ŽNote that s P and s P are real values since autocorrelograms areZZ Y Y. Ž . Ž .symmetric in space ; s P and s P are complex conjugate values.YZ ZY

The latter constraint is equivalent to:

a2 w ,w qb2 w ,w Fs w ,w s w ,w 5Ž . Ž . Ž . Ž . Ž .1 2 1 2 ZZ 1 2 Y Y 1 2

Ž . Ž . Ž .where a w ,w and ib w ,w are the real and imaginary part of s w ,w .1 2 1 2 Z Y 1 2Ž . Ž .The smoothing window m ,m , similar to that defined for Eq. 2 , is1 2

Ž Ž ..increased gradually until conditions Eq. 4 are met or until the window hasŽ .reached a predetermined maximum size. In the latter case, 1 if the autodensity

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Ž . Ž .spectrum values s w ,w or s w ,w are still negative, they are set to 0;ZZ 1 2 Z Y 1 2Ž . Ž .2 if the inequality constraint 5 is still not respected, a correction factor p is

Ž . Ž .applied to the coefficients a w ,w , b w ,w , such that the equality holds:1 2 1 2

2 2a w ,w Pp q b w ,w Pp ss w ,w Ps w ,wŽ . Ž . Ž . Ž .Ž . Ž .1 2 1 2 ZZ 1 2 Y Y 1 2

that is:

s w ,w Ps w ,wŽ . Ž .ZZ 1 2 Y Y 1 2ps 6Ž .2 2) a w ,w qb w ,wŽ . Ž .ž /1 2 1 2

A straightforward Fourier back-transform of the tables s , s , s , sZZ Y Y ZY YZŽ . w Ž .xwill then provide a jointly positive definite cross covariance matrix C h .

The strength of Bochner’s theorem lies in the decomposition of a difficultproblem, that of ensuring positive definiteness of a large covariance matrixw Ž .xC h for many lag distance vectors h and many covariates, into many lesserproblems each involving only one lag distance. Indeed, for K coregionalizedvariables and n lag distances, checking positive definiteness involves the

Ž .potentially very large nK=nK covariance matrix:

C h C h PPP C hŽ . Ž . Ž .11 12 1K

C h C h PPP C hŽ . Ž . Ž .21 22 2 KC h s 7Ž . Ž .. . . .. . . .. . . .

C h C h PPP C hŽ . Ž . Ž .K1 K 2 K K

whereas:

C 0 C h PPP C hŽ . Ž . Ž .k k k k 12 k k 1n1 2 1 2 1 2

C h C 0 PPP C hŽ . Ž . Ž .k k 21 k k k k 2 n1 2 1 2 1 2C h sŽ . . . . .k k1 2 . . . .. . . .

C h C h PPP C 0Ž . Ž . Ž .k k n1 k k n2 k k1 2 1 2 1 2

Bochner’s theorem reduces that problem to ensuring positive definiteness ofŽ .many smaller K=K matrices, one per lag w:

s w s w PPP s wŽ . Ž . Ž .11 12 1K

s w s w PPP s wŽ . Ž . Ž .21 22 2 K8Ž .. . . .. . . .. . . .

s w s w PPP s wŽ . Ž . Ž .K1 K 2 K K

However, if the number K of coregionalized variables is greater than 2, theŽ Ž .. Ž .constraints Eq. 4 become increasingly complex, e.g., 1 for two secondary

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variables Y and Y and a primary variable Z, i.e., for a total of three covariates1 2

with spectrum matrix:

s w s w s wŽ . Ž . Ž .ZZ ZY ZY1 2

s w s w s wŽ . Ž . Ž . ,ZY Y Y Y Y1 1 1 1 2

s w s w s wŽ . Ž . Ž .ZY Y Y Y Y2 1 2 2 2

Ž Ž ..the constraint Eq. 5 calls for the following two determinants to be nonnega-tive:

s w s w ys2 w G0,Ž . Ž . Ž .ZZ Y Y ZY1 1 1

and:

s w s w s w q2 s w s w s w ys w s2Ž . Ž . Ž . Ž . Ž . Ž . Ž .ZZ Y Y Y Y ZY ZY Y Y ZZ Y Y1 1 2 2 1 2 1 2 1 2

ys w s2 w ys w s2 w G0Ž . Ž . Ž . Ž .Y Y ZY Y y ZY1 1 2 2 2 1

Ž .2 for four covariates with spectrum matrix:

s w s w s w s wŽ . Ž . Ž . Ž .ZZ ZY ZY ZY1 2 3

s w s w s w s wŽ . Ž . Ž . Ž .ZY Y Y Y Y Y Y1 1 1 1 2 1 3

s w s w s w s wŽ . Ž . Ž . Ž .ZY Y Y Y Y Y Y2 1 2 2 2 2 3

s w s w s w s wŽ . Ž . Ž . Ž .ZY Y Y Y Y Y Y3 1 3 2 3 3 3

an additional constraint on the 4=4 determinant is written:

s w s w s w s w q2 s w s w s w s wŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž .ZZ Y Y Y y Y Y ZY Y Y Y y Y Z1 1 2 2 3 3 1 1 2 2 3 3

2 2 2 2 2qs w s w ys w s w ys w s w s wŽ . Ž . Ž . Ž . Ž . Ž . Ž .ZY Y Y Y Y ZY ZZ Y Y Y Y2 1 3 1 2 3 2 2 1 3

2 2 2ys w s w ys w s w s w G0Ž . Ž . Ž . Ž . Ž .ZY Y y Y y Y Y ZY1 2 3 1 1 3 3 2

Ž .Therefore, in presence of many coregionalized variables K large , rather thanchecking a large number of determinants, we propose ensuring positive definite-ness through eigenvalue correction.

3. Eigenvalues correction

Ž .In cases where there are many secondary variables K large , the correctionŽ .proposed as in Eq. 6 is difficult to implement. The eigenvalue correction

Ž .initially proposed by Wackernagel 1995 can be expanded to ensure positivedefiniteness.

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A nonpositive definite covariance matrix is characterized by the presence ofŽ .one or more negative eigenvalues. Wackernagel 1995 has proposed to set these

negative eigenvalues to zero and defines the corrected covariance matrix as:

TC̃ h s A B A 9Ž . Ž .

w x w xwhere: A is the matrix of eigenvectors arranged in columns and B is thediagonal matrix of eigenvalues with all negative values set to 0.

In the following, we will develop as an example the 3=3 matrix case:

C h C h C hŽ . Ž . Ž .11 12 13

C h C h C hŽ . Ž . Ž .21 22 23

C h C h C hŽ . Ž . Ž .31 32 33

l 0 01 T

0 l 0s 2z z z z z z1 2 3 1 2 30 0 l3

where l , l , l are eigenvalues in increasing order and z , z , z are the1 2 3 1 2 3

corresponding eigenvectors. If l -0 and l -0, which means that the matrix1 2Ž .C h is not positive definite, then the corrected positive definite matrix is built

as:

˜ ˜ ˜C h C h C hŽ . Ž . Ž .11 12 13

˜ ˜ ˜C h C h C hŽ . Ž . Ž .21 22 23

˜ ˜ ˜C h C h C hŽ . Ž . Ž .31 32 33

T0 0 00 0 0s

z z z z z z1 2 3 1 2 30 0 l3

˜w Ž .xThe covariance matrix C h is now positive definite.˜w Ž .xOne concern relates to how much the corrected result C h departs from the

w Ž .xoriginal matrix C h and, does it still reflect the original spatial structures.Ž .Also, calculating eigenvalues and eigenvectors of a large matrix nK=nK

w Ž .xmatrix C h can be cpu-intensive.Ž .Rather than starting from one single large nK=nK covariance matrix, such

Ž . Ž .as Eq. 7 , we propose to start from n small K=K spectrum matrices s w asŽ .in Eq. 8 , whose eigenvalues are easier to calculate. For each K=K spectrumw Ž .x Ž .matrix s j , j at frequency cell j , j , . . . , with j s1, . . . ,n, j s1, . . . ,n,1 2 1 2 1 2

we first check whether all the eigenvalues, l , is1, . . . , K , are positive or not.i

If all eigenvalues are positive, go to the next spectrum matrix at another cell

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Ž X X . Ž . Ž .j , j ; otherwise, define a smoothing window m ,m around cell j , j and1 2 1 2 1 2

substitute the original frequency values with the window averages for both theauto and cross-density spectra, i.e.,

m r2 m r21 21s j , j s s j q l , j q l ,Ž . Ž .ˆ Ý Ýk k 1 2 1 1 2 21 2 m m1 2 l sym r2 l sym r21 1 2 2

k s1, . . . , K , k s1, . . . , K 10Ž .1 2

w Ž .xThen, check the new eigenvalues of the smoothed spectrum matrix s j , j . Ifˆ 1 2Ž .some eigenvalues are still negative, increase the smoothing window m ,m1 2

until the constraint of positive eigenvalues are met. If the window becomes toolarge while there are still some negative eigenvalues, as a last resort, shift alleigenvalues by the absolute value of the most negative one l , i.e.,1

< < Xl sl q l , is2, . . . , K the l s are in increasing orderŽ .i 1 1 i

< <It is preferable to shift the eigenvalues with a constant value l rather than1

resetting the negative ones to 0, because the first option keeps unchanged therank order between auto- and cross-correlation values. More precisely, all the

< <spectral densities are shifted by the same value l , i.e.,1

X X Xs s PPP s11 12 1KX X Xs s PPP s21 22 2 K s. . . . z z PPP z. . . . 1 2 K. . . .X X Xs s PPP sK1 K 2 K K

=

< <l q l 0 PPP 01 1

< <0 l q l PPP 02 1. . . .. . . .. . . .

< <0 0 PPP l q lK 1

=z z PPP z1 2 K

< <s q l s PPP s11 1 12 1K

< <s s q l PPP s21 22 1 2 Ks . . . .. . . .. . . .

< <s s PPP s q lK1 K 2 K K 1

A case study using a published multivariate data set follows to demonstrateimplementation of this algorithm.

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4. The Jura data set case study

The Jura data set relates to soil pollution by heavy metal and was collected byŽ .the Swiss Federal Institute of Technology Goovaerts, 1997 . It includes 359

Ž .sample locations scattered in space see top of Fig. 1 . Concentrations of sevenŽ .heavy metals cadmium, cobalt, chromium, copper, nickel, lead, and zinc were

measured at each sample location. A detailed description of the sampling, fieldŽ .and laboratory procedures is given in the paper of Atteia et al. 1994 and

Ž .Webster et al. 1994 . The most important feature of this reasonably large dataset is that several attributes are significantly cross correlated, which allows

Žtesting the proposed algorithm. The colocated linear correlation matrix at lag.hs0 is given in Table 1.

From the above table, we selected the best correlated four variables: Cd, CrNi, Zn. To test the algorithm, we keep the first 259 samples as data and estimatethe remaining 100 sample values for cross-validation. The corresponding loca-tion maps are shown in Fig. 1.

Fig. 1. Location maps of all 359 cadmium samples, the 259 samples to be used as data, and 100samples to be used for cross-validation.

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Table 1Correlation table

Element Cd Cr Ni Zn Cu Pb Co

Cd 1.00Cr 0.61 1.00Ni 0.49 0.69 1.00Zn 0.67 0.67 0.63 1.00Cu 0.12 0.21 0.23 0.57 1.00Pb 0.22 0.30 0.31 0.59 0.78 1.00Co 0.25 0.45 0.59 0.75 0.22 0.19 1.00

4.1. Smoothing of multiÕariate correlogram tables

Using the 259 data kept as samples, we calculated the experimental correlo-Ž .gram maps for the four coregionalized variables Deutsch and Journel, 1997 .

Ž .Fig. 2. Experimental autocorrelogram maps calculated from 259 samples of cadmium C11 ,Ž . Ž . Ž .chromium C22 , nickel C33 and zinc C44 .

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The four autocorrelograms are shown in Fig. 2, and the six cross-correlogramsare shown in Fig. 3. We then perform a preliminary smoothing to dampen large

Ž .fluctuations and fill-in missing entries if any due to data sparsity. TheŽ .smoothing detail can be referred to Yao and Journel 1998 . This preliminary

smoothing results in the correlograms shown in Figs. 4 and 5.Next, we carry out a second smoothing in the frequency domain to ensure

positive definiteness through eigenvalue correction. The original and somesmoothed spectral density maps s , s , s , s , s and s are shown in Figs.11 22 33 44 12 34

6 and 7.A final back-transform provides the licit covariance maps shown in Figs. 8

and 9. Compared to the original experimental covariances of Figs. 2 and 3, thesmoothed final covariance tables still reflect the large scale structures andanisotropy, such as the most continuous direction being NW for variable 1

Ž . ŽFig. 3. Experimental cross-correlogram maps between four heavy metals. C12 Cd, Cr , C13 Cd,.Ni , etc.

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Fig. 4. Experimental autocorrelogram maps after primary interpolation.

Ž . Ž .cadmium and NE for variable 3 nickel . Note that the proposed method doesnot assume symmetry of the correlogram tables: more precisely, the autocorrelograms are symmetric per definition, while the cross correlograms may not

X Ž . X Ž .be: c h /c yh .kk kk

Note that all axes in Figs. 2 and 9 are in grid unit, 250 m=250 m for bothsampling and estimation grids.

For a more detailed comparison, some views of the final smoothed vs. theexperimental correlograms are given in Fig. 10. The final correlogram values areall slightly higher than the experimental values, which is acceptable because,due to sparse sampling, experimental correlograms typically underestimate theactual correlation.

4.2. Cross-Õalidation using the smoothed coÕariance tables

Using the positive definite covariance tables established above, we can buildup any kriging system by simply reading from these tables. Such kriging wasused to reestimate the test 100 values from the 259 data retained as samples. In

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Fig. 5. Experimental cross-correlogram maps after primary interpolation.

the following study, we will perform both ordinary kriging using only thecadmium data, and standardized ordinary cokriging using cadmium data and thecorrelated nickel, cadmium and zinc data.

4.2.1. Kriging cadmium using only 259 cadmium dataBy reading from the smoothed correlogram map of cadmium, we can build up

the ordinary kriging system to estimate the cadmium values at the 100 cross-validation locations. Alternatively, we have built the same kriging system using

Ž .the analytical variogram model proposed by Goovaerts 1997 :

h hg h s0.3q0.3Sph q0.26SphŽ . ž / ž /200 m 1300 m

The cross-validation results, using Goovaerts’ analytical variogram model andour proposed correlogram map, are shown in Fig. 11. The estimation maps are

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Fig. 6. Original and smoothed-spectrum density maps of four autocorrelogram maps.

very similar and the correlation coefficients between estimated and true valuesdo not show significant difference whether one uses the traditional analyticalmodel or our proposed covariance table. The scatterplot between the two sets of

Ž .estimates shows a very high correlation see bottom of Fig. 11 . Thus, using theeasier spectral approach to modeling does not detract from the LMC results.

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Ž .Fig. 7. Original and smoothed cross-spectrum density maps real and imaginary .

4.2.2. Cokriging cadmium using coregionalized dataSince there are three other coregionalized variables correlated with cadmium,

we can perform cokriging to utilize the information provided by these secondaryŽ .variables. Theoretically, for four coregionalized variables, 10 analytical cross

variogram models would be required to build up the cokriging system. These 10

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Fig. 8. Smoothed jointly positive definite autocorrelogram maps.

Ž .variogram models must satisfy a sufficient condition the linear coregionaliza-Ž .tion model Journel and Huijbregts, 1978, p. 317 to ensure that the cokriging

matrix is positive definite and the estimation variance is nonnegative. Suchmodeling although possible is tedious. Typically in practice only the two bestcorrelated variables are retained to reduce the modeling burden. In this applica-tion, we retain nickel as a covariate for the cokriging of cadmium, utilizing the

Ž .analytical linear model of coregionalization LMC proposed by GoovaertsŽ .1997 :

h hg h s0.3 q0.3Sph q0.26SphŽ . Ž . Ž .Cd_Cd 200 m 1300 m

hg h s0.6 q3.8SphŽ . 11Ž . Ž .Cd_Ni 1300 m

hg h s11 q71SphŽ . Ž .Ni_Ni 1300 m

For our proposed approach, all 10 covariance maps involved by the fourcovariates are generated automatically with no analytical model involved. The10 covariance maps were smoothed simultaneously to satisfy the positive

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Fig. 9. Smoothed jointly positive definite cross-correlogram maps.

definiteness condition of all cokriging systems. Therefore, all four variableswere used together in the cokriging system. Cokriging estimation was thus

Ž Ž ..performed using both the analytical model Eq. 11 limited to two variablesonly, and the proposed smoothed correlogram maps using all four covariates.The limitation of the analytical model to only two variables corresponds to theusual practice of LMC outside academic environments.

The ordinary cokriging constraints call for the primary data weights to sumup to 1 and secondary data weights to sum up to 0. There are two drawbacks

Ž .associated to these constraints: a some of the secondary data weights areŽ .negative, thereby increasing the risk of getting unacceptable estimates; and b

the secondary data weights tend to be small, thus reducing the influence of thesecondary information.

To reduce the occurrence of negative weights and avoid limiting artificiallythe impact of secondary data, a standardized ordinary cokriging is applied

()

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31Ž . Ž .Fig. 10. Cross-views of final smoothed solid line vs. experimental dash line correlograms of C11 and C12.

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Fig. 11. Comparing kriging estimates of cadmium using the analytical variogram model andreading from the automatically derived correlogram table, retaining 259 samples.

Ž .instead Isaaks and Srivastiva, 1989, p. 403 . The secondary variables are nowrescaled to have the same mean as the primary variable so that the previousconstraints for ‘unbiasedness’ are reduced to only one single constraint, that allthe weights applied on both primary and secondary data sum up to 1.

Each of the 259 sample locations is sampled for all four variables. Thus, if weretain all 259 cadmium samples the impact of cokriging is erased, because allsecondary data are collocated with the cadmium primary data hence the covari-

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ate information is screened out. To emphasize the worth of using that covariateŽ .information, we adopted the following sampling strategy: 1 retain only 20

Ž .cadmium data from the original 259 samples; 2 retain only 20 nickel data fromŽ .the original 259 samples colocated with the 20 cadmium samples; 3 keep all

the 259 samples for both chromium and zinc.The corresponding data location maps are shown in Fig. 12. Then, we

perform the following estimations.Ž .a Krige cadmium at the 100 test locations retaining only 20 cadmium

samples and using both the traditional analytical variogram model and thesmoothed correlogram map.

The results are shown in Fig. 13. Due to the small number of sampled values,Ž .the estimation result is poor in terms of correlation less than 0.2 between

estimates and true values. The estimates using the traditional variogram modelŽand our proposed approach do not show any significant difference see the

.scatterplot at the bottom of Fig. 13 . Again, using the easier spectral approach tomodeling does not detract from the LMC results.

Fig. 12. Location maps of the 20 chromium and nickel data, and 259 chromium and zinc data.

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Fig. 13. Comparing kriging estimates of cadmium using the analytical variogram model andreading from the automatically derived correlogram table, retaining only 20 samples.

Ž .b Cokrige cadmium retaining the 20 cadmium data and the 20 nickel data,again using the traditional analytical coregionalization model and the smoothed

Žcorrelogram maps three maps: two autocorrelogram maps and one cross-corre-.logram map . Since nickel is sampled at the same locations as cadmium, this

secondary information does not help much in improving the estimation results.ŽThe correlation coefficients between estimates and true values are still low see

.middle row of Fig. 14 .

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ŽFig. 14. Comparing cokriging estimates of cadmium using the analytical variogram models one. Žsecondary variable and reading from the automatically derived correlogram tables one then three.secondary variables .

Ž .c Cokrige cadmium retaining the 20 cadmium data, the 20 nickel data, andthe 259 samples of both chromium and zinc, using all smoothed correlogram

Ž .maps 10 maps: four autocorrelogram maps and six cross-correlogram maps .For this case, we applied only our proposed algorithm since all 10 correlo-

gram maps involved can be obtained automatically. The estimation result isshown at the bottom row of Fig. 14. Because the chromium and zinc samples

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provide valuable additional information at those locations where cadmium is notsampled, their utilization improves the estimation results significantly. Thecorrelation coefficient between estimates and true sampled values is increased to0.3. 1

Ž .This last result case c only proves that it is important to use all K covariatesno matter how large is K. Admittedly, one could have fitted an analytical LMCmodel using Goulard’s algorithm and the results would have been similar tothose obtained by our approach, but fitting a LMC as soon as K)2 is tediousand rarely done in real practice.

5. Resolution of the covariance map

One implementation issue of the proposed covariance map concept is associ-ated to resolution. If data are clustered, then the required covariance values forsmall lag distances between two close-by data may not be available from coarsecovariance maps. Because of the lack of resolution of such coarse maps, twodifferent but close-by data locations may end up being seen as colocated in thekriging matrix, resulting in a singular matrix. One solution is to relocate the

Ž .samples to the closest grid node one per node before searching for data, butthen, some information from the clustered data set may be lost. Another solutionis to calculate the covariance maps on a coarse grid, FFT them into thefrequency domain to get the density spectrum maps. These coarse density mapsare then appropriately interpolated to retrieve the higher resolution needed. Forexample, an initial coarse 32=32 covariance map is calculated, with unit lagdistance 2.0. The corresponding 32=32 density spectrum map is then expandedto 64=64 by interpolating four values within each cell of the original coarsegrid. After that interpolation, the density spectra maps can be smoothed underappropriate constraints and, last, they are backtransformed to provide licitcovariance maps at the required 64=64 resolution with unit lag distance 1.0.

6. Conclusions

A major limitation to joint utilization of multiple covariates is the difficultŽ .modeling of their coregionalization cross-covariance matrix . To avoid check-

ing each cokriging matrix for positive definiteness, the traditional approachconsists of fitting a positive definite analytical model of linear coregionalization.But such task becomes quickly very tedious: with four covariates only, one need

1 One explanation for these overall low correlation coefficients in cross-validation is the highnugget effect of cadmium, which accounts for 35% of the total variance.

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Ž .to analytically model jointly at least 10 and up to 16 cross covariances. Thelimitation of using linear combination of basic covariance structures may alsocall for possible drastic approximation of the experimental sample covariancecurves.

We improve the nonparametric, nonanalytical approach initially proposed byŽ . Ž .Yao and Journel 1998 , an approach that can be easily automatized: all cross

Ž .covariance tables are interpolated then perturbed to ensure joint positivedefiniteness. The covariance values required by any cokriging system are thensimply read from such permissible tables. The perturbation to ensure positive

Ž .definiteness is easily done in the frequency domain after FFT .A case study using a multivariate pollution data set demonstrates the possibil-

Ž .ity of simultaneous modeling of cross correlogram maps of many coregional-ized variables, without resorting to any analytical variogram model. The final

Ž .results precision of estimates do not suffer from the automatic modelingproposed. Such automatic modeling may prove especially important in environ-mental and soil sciences where many data types typically coexist and the studyof coregionalization is required.

The main point of this paper has been to prove that the spectral modelingapproach proposed, while being much easier to implement than the LMCapproach, yields results that are no less accurate. This should incite practitionersto actually use all available data without fear of a demanding modeling task.

Acknowledgements

This work has been supported by the Stanford Center for Reservoir Forecast-Ž .ing SCRF , Stanford University. I would like to thank Prof. Andre Journel for

his constant attention on this research. Also, I acknowledge the many usefulcomments of the reviewers.

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