Appendix A Geostatistical Concepts

This appendix gives an overview of the geostatistical framework and introduces some key concepts without which the geostatistical approach cannot be properly understood and used. It also details at which stages of a geostatistical study VARIOWIN should be used. It has been included in order to remind the user of the necessary theoretical background required to understand the construction of a geostatistical model. Since this appendix serves only as a reminder, the reader is strongly advised to browse through the textbooks listed at the end of the appendix. Within the geostatistical framework, a probabilistic model is used: the random function model. Estimations and/or simulations are performed using the properties of this model. The values or set of values produced by the estimation or simulation procedure are considered as a correct approximation of reality. Therefore, distinguishing between what belongs to the real world and what belongs to the probabilistic model is of paramount importance. Figure A.I displays the various steps involved in a geostatistical study. VARIOWIN provides tools to deal with the operations that appear in italic in the figure.

A.I Random Variables, Regionalized Variables, and Random Functions

A is a variable whose values are randomly generated according to some probabilistic mechanism. For instance, the result of casting a die can be considered as a random variable that can take one of six equally probable values. A regionalized variable is a variable distributed in ; it is used to represent natural phenomena. For example, the heavy-metal content of the top layer of soil can be considered as a regionalized variable in the 2D space. A natural phenomenon, or a regionalized variable, usually shows a structured aspect as well as an erratic aspect: 74 VARlOWIN: Software for Spatial Analysis in 2D

Variogram modeling

REAL WORLD PROBABILISTIC MODEL Observations - samples - qualitative information Conceptual tools - random variable Check RF hypothesis - regionalized variable Operations ..... - random function (RF) - descriptive - exploratory data analysis (EDA) Operations - exploratory variography (EV) '- - estimation - simulations

Apply the results obtained within the probabilistic model to reality

Figure A.I Conceptual framework of the geostatistical approach.

• The structured aspect is related to the overall distribution of the natural phenomenon. For instance, in a polluted area, there are zones with, on average, a greater heavy-metal content than others. • The erratic aspect is related to the local behavior of the natural phenomenon. For instance, within a given zone of a polluted area, the heavy-metal content seems to fluctuate randomly.

The formulation of a natural phenomenon must take this double aspect of and structure into account. A consistent and operational formulation is the probabilistic representation provided by random functions. A random function is a set of random variables {Z(x) I location x belongs to the area investigated} whose dependence on each other is specified by some probabilistic mechanism. It expresses the random and structured aspects of a natural phenomenon in the following way: • Locally, the point value z(x) is considered as a random variable. • This point value z(x) is also a random function in the sense that for each pair of points Xi and Xi + b, the corresponding random variables Z(Xi) and Z(Xi + b) are not independent but are related by a correlation expressing the spatial structure of the phenomenon.

The regionalized variable representing the phenomenon (the set of all of its possible values distributed in space) is considered as a particular realization of Appendix A: Geostatistical Concepts 75 the random function constructed on this phenomenon (the set of an infinity of random variables representing the values of the phenomenon at each point in space).

A.2 Moments Considered in Linear Geostatistics

Consider the random function Z(x). For every set of k points in space xl' X2' ... , xk, there is a corresponding set of k random variables

This set of random variables is completely characterized by the k-variable distribution function

Fx x x I> 2"'" k (ZI,Z2, ... ,Zk) = Prob {Z(XI)

The set of all these distribution functions, for all positive integers k and for every possible choice of point Xi, constitutes the spatial law of the random function Z(x). Exhaustive knowledge of the spatial law is not required for most estimation/simulation problems. Moreover, the amount of data is generally insufficient to infer the entire spatial law. In linear geostatistics, only the first two moments of the random function are used. Thus, no distinction will be made between two random functions having the same first- and second-order moments.

A.2.1 First-Order - Mathematical Expectation

Consider a random variable Z(x) at location x. If the distribution function of Z(x) has an expectation, then this expectation is generally a function of X and is written as

E{Z(x)} = m(x).

A.2.2 Second-Order Moments

In geostatistics three second-order moments are considered: 1. the of Z(x), which is the second order moment about the expectation m(x) of the random variable Z(x): 76 V ARIOWIN: Software for Spatial Data Analysis in 2D

Var{Z(x)} = E{[Z(X)- m(x)f}

= E{Z(x)2}- 2· E{Z(x)}. m(x) + m2(x)

= E{Z(x)2}- m2 (x).

As with the expectation m(x), the variance is generally a function of the location x. 2. the . If two random variables Z(Xl) and Z(X2) have , then they also have a covariance, which is a function of the two locations Xl and X2:

C(Xj, X2) = E{[z(xj) - m(xj)J [Z(X2) - m(x2)>>

= E{Z(xj)}. E{Z(X2)}- E{Z(xj)}. m(x2)- E{Z(X2)}· m(xj) +m(xd· m(X2) = E{Z(xj)}. E{Z(X2)}- m(xj). m(x2}

3. the or semivariogram. The variogram is defined as half the variance of the increment [Z(Xj)-Z(X2)}

1 Y(Xj' X2) = 2" Var{Z(xd- Z(X2)}'

2y(xj, X2} = E{[Z(xd- z(x2)f} - [E{Z(xj)- Z(X2)}f,

Y(Xj,X2) = ~E{[Z(Xj)-Z(X2)f} when E{z(xd-z(X2)} = o.

A.3 Ergodicity

The probabilistic framework deals with expected values, which are merely the integrals over all possible realizations of a random process. For an ergodic process, the expectation over all possible realizations is equal to the average over a single unbounded realization. Without the ergodic property, the moments of the random function could not be inferred from the spatial values calculated on the regionalized variable. Appendix A: Geostatistical Concepts 77 AA Hypothesis of Stationarity

In order to infer the spatial law or the first two moments of a random function Z(x), where x is a point in space, many realizations Zl (x), Z2(X), .•• , Zk(X), of Z(x) would be required. However, unlike the casting of a die, it is not possible to obtain a new realization of the natural phenomenon at location x. The reality is unique, and our model parameters must be inferred from this unique "realization". If the phenomenon under study is homogeneous over a certain region, we can consider that its representation as a regionalized variable repeats itself in space. This repetition provides the equivalent of many realizations of the same random function Z(x) and allows a certain amount of . Two experimental values z(Xo) and z(Xo + h) at two different points Xo and Xo + h can thus be considered as two different realizations of the same random function Z(x) (Figure A.2). Note also that such statistical inferences are possible only when the regionalized variable can be considered as ergodic. This approach is not peculiar to geostatistics and random functions; it is more commonly used either to infer the of a random variable from the of data values or to infer the mathematical expectation from the average value of the data. The hypotheses of stationarity are related to various degrees of the spatial homogeneity of the phenomenon under study. The type of stationarity that is assumed will tell what kind of statistical inference is permitted within the

Concept Application (stationarity hypotheSIS)

2nd realization 3rd reallzallon

reality considered as a 1st "realization" 01 the regionalized variable

Figure A.2 Multiple realizations of a regionalized variable and the hypothesis of stationarity. 78 V ARIOWIN: Software for Spatial Data Analysis in 2D probabilistic model.

A.4.1 Strict Stationarity

A random function is said to be stationary, in the strict sense, when its spatial law is invariant under translation:

are identical in law (have the same k-variable distribution law) for any separation vector h. In practice, this very strong hypothesis is rarely assumed; it is usually replaced by the hypothesis of second-order stationarity or by the intrinsic hypothesis.

A.4.2 Second-Order Stationarity

Since linear geostatistics considers only the first two moments of the spatial law, it is sufficient first to assume that these moments exist and then to limit the stationarity assumption to them. A random function Z(x) is said to be second-order stationary if • the mathematical expectation E{Z(x)} exists and does not depend on location x:

E{Z(x)} = m, \fx;

• for each pair of random variables {Z(x), Z(x + h)}, the covariance exists and depends only on the separation vector h:

C(h) = E{Z(x).Z(x+h)}-m2, \fx;

The stationarity of the covariance implies the stationarity of the variance:

Var{z(x)} = E{[Z(x)- m f} = C(O), \fx.

The stationarity of the covariance also implies the stationarity of the variogram: Appendix A: Geostatistical Concepts 79

2C(h) = 2E{Z(x+h).Z(x)}-2m2

= [ E{Z(X + ht}- m2 ] + [E{Z(x)2}- m2 ]

-[E{Z(X + h)2}_ 2E{Z(x + h). Z(X)} + E{Z(x)2}]

2C(0) - 2y(h), C(h) = C(O)-y(h).

This relation indicates that under the hypothesis of second-order stationarity, the covariance and the variogram are two equivalent tools for characterizing the correlation between two random variables Z(x) and Z(x + h) separated by a vector h. The hypothesis of second-order stationarity assumes the existence of a covariance and, thus, of a finite a priori variance Var{Z(x)} = C(O). Under this condition, the correiogram, p(h) , and the standardized variogram, y s(h) , can be defined and used similarly to the covariance and the variogram:

(h) = C(h) = (0) _ y(h) = p(O) - ys(h). P C(O) P C(O)

While those probabilistic functions are perfectly equivalent, their estimators are not. In practice, once the hypothesis of second-order stationarity has been accepted, one is free to use the probabilistic function that is the easiest to estimate in order to characterize the second moment of the random function Z(x) [Srivastava & Parker, 1989].

Consequently, within the Model program, a 2D nested model of spatial continuity of a second-order stationary, random function can be fitted against anyone of these measures of spatial continuity.

A.4.3 Intrinsic Hypothesis

Many physical phenomena have an infinite capacity for dispersion (i.e., brownian motion). These phenomena do not have an a priori variance or a covariance. However, the variance of their increments, or their variogram, exists and can be defined. As a consequence, the second-order stationarity hypothesis can be slightly reduced when assuming only the existence and stationarity of the variogram. A random function Z(x) is said to be intrinsic if • the mathematical expectation E{Z(x) }exists and does not depend on location x, 80 VARIOWIN: Software for Spatial Data Analysis in 2D

E{Z(x)} = m, "Ix;

• for all separation vector h the increment [Z( x + h) - Z( x)] has a finite variance which does not depend on x:

lvar{Z(x+h)-Z(X)} =: lE{[Z(X+h)-Z(x)f} =: Y(h), "Ix.

The intrinsic hypothesis can also be seen as the limitation of the second• order stationarity to the increments of the Random Function Z(x). Note that second-order stationarity implies intrinsic behavior, but the converse is not true.

When the intrinsic hypothesis is accepted without the second-order stationarity hypothesis, only the variogram exists and can be used to characterize the second moment of the random function Z(x).

Consequently, an experimental variogram that does not reach a plateau indicates that the underlying random function is not second order stationary. Within the Model program, a model of spatial continuity of such a random function may only be fitted against the variogram since the standardized variogram, the covariance, and the do not exist within the probabilistic framework (Figure A.I).

A.S WbyDo We Need to Model ?

In linear geostatistics, values to be estimated or simulated are considered as a realization of a random variable that is a linear combination of known random variables:

n Y 2,AjZ(Xi} i=1

This linear combination of random variables is itself a random variable whose variance is given by

n n

Var{Y} =: 2, 2,Ai AF(Xi -xj } i=1 j=1 Appendix A: Geostatistical Concepts 81

The appearing in this expression must ensure that the variance will always be positive or equal to zero. By definition, such a covariance function ceh) is said to be positive-definite.

Thus, not any function g(h) can be considered as the covariance of a stationary random function: it must be positive-definite.

Assuming second-order stationarity [A.4.2], the variance of Y can be written in terms of the variogram:

n n n n Var{Y} C(O)~>'i2,Aj - 2,2, AiA/Y(Xi -Xj). i=l j=l i=l j=l

When the variance ceO) does not exist, only the intrisic hypothesis [A.4.3] is assumed and the variance of Y is defined for authorized linear combinations, that is, linear combinations of random variables for which the sum of the weights is 0:

In this case, the ceO) term is eliminated and the variance of Y is reduced to

n n Var{Y} = 2, 2, AiA/Y(Xi - x j). i=l j=l

The variogram function must be such that the variance of an authorized linear combination of random variables is positive or zero. By definition, -y(h) is said to be a conditional positive-definite function. Using the property that -y(h) is a conditional positive-definite function, it can be shown that the semivariogram increases more slowly at infinity than the squared magnitude of the separation vector h does:

lim yeh) = 0 when Ihl ~ 00 Ihl2

As a consequence, an experimental variogram [4.5] whose variogram values [4.7.1] increase at least as rapidly as the squared magnitude of the separation vector is incompatible with the intrinsic hypothesis. This behavior most often indicates the presence of a drift, that is, a nonstationary mathematical expectation of the random function associated with the studied phenomenon. 82 V ARIOWIN: Software for Spatial Data Analysis in 2D

This drift can often be subtracted from the data values using either a trend surface or a moving window map. The removal of the drift must be realized outside of VARIOWIN, which can then be used to perform the spatial data analysis of the residuals. All geostatistical estimations and simulations are then performed on the residuals, and the drift is added back afterwards.

A.6 The Multivariate Gaussian Random Function

This section describes one of the most often used geostatistical models as well as how VARIOWIN can be used to check that the properties attributed to the random function within this model are not in contradiction with the data. The multivariate Gaussian model is extremely convenient because of its extreme analytical simplicity. It is also the limit distribution of many analytical theorems globally known as central limit theorems. If the spatial phenomenon is seen as being generated by the addition of several independent sources having similar spatial distributions, then the spatial distribution of the phenomenon can be modeled by a multivariate Gaussian random function:

K Z(x) = LYk(x); k=! where Z(x) is the random function representing the phenomenon and Yk(x) are independent random functions having similar spatial distributions. Natural processes that generate an observed phenomenon are rarely independent from one another and the way they interact is rarely additive. However, multivariate Gaussian models have been successful in a number of applications. This practical success combined with the extremely convenient analytical properties of the Gaussian models makes them the preferred choice for modeling continuous variables. Of course, if the multivariate Gaussian model is determined to be inappropriate, it should not be used. The multivariate Gaussian model consists of the following two hypotheses: • It is possible to transform the original variable into a new variable having a univariate standard normal distribution:

Z(x) =

where Z(x) is the original random variable, Y(x) is univariate standard normally distributed, and

function Z(x) and a standard normal cumulative density function [Verly,1985]. • The multivariate distribution of the random function Y(x) is Gaussian and strictly stationary [A.4.l].

As a consequence, the random function Y(x) is fully determined by two parameters: • its mean, m(x):

E{Y(x)} = m(x) = 0

for all locations x; • its covariance function, C(b):

E{Y(x). Y(x + b)} = c(b)

for all locations x;

Note that since Y(x) is Gaussian, the hypothesis of strict stationarity is equivalent to the hypothesis of second order-stationarity [AA.2]. The property that makes the multi-Gaussian random function such a pleasant model for geostatistical estimation and simulation is the fact that all conditional distributions of the random function Y(x) are also multivariate normal. When dealing with spatial data, this that the local conditional distribution at an unknown point (i.e., the cumulative density function at the unknown point, given the surrounding known values) can be reconstructed with only two parameters: its mean and its variance. • The mean, or conditional expectation, is identical to the simple (SK) estimate of that point value:

n m(xo)+ ~)l.u[Yu - m(xu)]. u=!

The n weights Au are given by the simple kriging system:

n 2:AuC(xu, xii) = C(xo, xu} ~ = 1, ... , n. u=! 84 VARIOWIN: Software for Spatial Data Analysis in 2D

• The variance, or conditional variance, is equal to the simple kriging variance associated with the simple kriging estimate:

n var{Y(xo)I y(xa) = Ya' a = 1, ... ,n} = C(O)- :~)'aC(xo,xa)' a=l

Note that this conditional variance does not depend on the data values but only on the data configuration and the covariance model.

The hypothesis of multinormality of the random function Y(x) implies that each random variable [Y(x), x being any location] is normally distributed. It also implies that each pair of random variables [Y(u), Y(v), u and v being any location] is binormally distributed, that each triplet of random variables [Y(u), Y(v), Y(w), u, v, and w being any location] is trinormally distributed and so on. Just as the strict stationarity hypothesis, the complete hypothesis cannot be tested. However, the multinormal distribution has many properties that can be used to design various checks: 1. Since any linear combination of a multinormal vector is also normally distributed, a check on binormality consists of verifying that the difference [Y(x) - Y(x + h)] is normally distributed with mean 0 and variance 2')'(h). Checking that all h-scatterplots [4.3] are elliptical clouds which show a greater density of points in the center and that are symmetric around the 45° bisector is a rough equivalent of this check of binormality. 2. For a multinormal random function, the conditional variance is independent of the conditional mean. As a consequence, multinormally distributed data should show no proportional effect, that is, there should not be a linear relationship between the local mean and the local variance. The presence of a proportional effect can easily be checked using moving windows statistics [Murray & Baker, 1991]. 3. The sample indicator variogram can be compared with its theoretical bivariate counterpart (use the BIGAUSS program found in GSLIB [Deutsch & Joumel, 1992]).

These three checks are quite effective. The first two are the easiest to realize. There is a fourth check, which is somewhat less effective, that uses the relationship between the experimental variogram values and the experimental Madogram values [4.7.5]. Appendix A: Geostatistical Concepts 85

Note that, in practice, only the bivariate normality of the random function Y(x) is checked. Checking for trivariate, quadrivariate, K-variate normality would require many K-tuples sharing the same data configuration. This condition is rarely met in practice (unless the data are gridded and numerous), and the standard technique is to accept multivariate normality of the random function Y(x) if its bivariate normality has been backed up by at least one of the four checks.

Further Reading

Chauvet, P., "Processing Data with a Spatial Support: Geostatistics and Its Methods," Cahiers de Geostatistique, Fasc. 4, Centre de Geostatistique, Fontainebleau, France, 1993. Cressie, N. A. C., "Statistics for Spatial Data," John Wiley & Sons, New York, NY, 1991. Cressie, N. A. C., "The Origins of Kriging," Mathematical Geology, Vol. 22, No.3, pp. 239-252, 1990. Deutsch, C. V. & Journel, A. G., "GSLIB Geostatistical Software Library and User's Guide," Oxford University Press, New York, NY, 1992. Isaaks, E. H. & Srivastava, R. M., "An Introduction to Applied Geostatistics," Oxford University Press, New York, NY, 1989. Journel, A. G., "Fundamentals of Geostatistics in Five Lessons," Short Course in Geology, Vol. 8, American Geophysical Union, Washington, DC, 1989. Joumel, A. G. & Huijbregts, C. J., "Mining Geostatistics," Academic Press, London, 1978. Matheron, G., "Estimer et Choisir," Les Cahiers du Centre de Morphologie Mathematique, Fasc. 7, Ecole Nationale Superieure des Mines de Paris, 1978. Murray, M. R. & Baker, D. E., "MWINDOW: An Interactive FORTRAN-77 Program for Calculating Moving-Window Statistics," Computers & Geosciences, Vol. 17, No.3, pp. 423-430, 1991. Myers, D. E., "To Be or Not to Be ... Stationary? That Is the Question," Mathematical Geology, Vol. 21, No 3, pp. 347-362, 1989. Schofield, N., "Ore Reserve Estimation at the Enterprise Gold Mine, Pine Creek, Northern Territory, Australia. Part 2: The Multigaussian Kriging Model," CIM Bulletin, Vol. 81, No. 909, pp. 62-66,1988. Srivastava, R. M. & Parker, H. M., "Robust Measures of Spatial Continuity in Geostatistics," in Geostatistics, M. Armstrong (ed.), Proc. of the Third International Geostatistics Congress, September 5-9, 1988, Avignon, France, D. Reidel, Dordrecht, Holland, pp. 295-308, 1989. Starks, T. H. & Fang, J. H., "The Effect of Drift on the Experimental Semivariogram." Mathematical Geology, Vol. 14, No.4, pp. 309-319,1982. Strang, G., "Linear Algebra and Its Applications," Harcourt Brace Jovanovitch, San Diego, CA, 1976. Third edition, 1988. Verly, G., "Estimation of Spatial Point and Block Distributions: The Multi-Gaussian Model," Ph.D. thesis, Stanford University, Stanford, CA, 1984. Bibliography

Armstrong, M., "Common Problems Seen in Variograms," Mathematical Geology, Vol. 16, No.3, pp. 305-313, 1984. Barnes, R. J., 'The Variogram Sill and the Sample Variance," Mathematical Geology, Vol. 23, No.4, pp. 673-678,1991. Bradley, R. & Haslett, J., "Interactive Graphics for the Exploratory Analysis of Spatial Data - The Interactive Variogram Cloud," in Proc. of the Second CODATA Conference on Geomathematics and Geostatistics, P. A. Dowd & J. J. Royer (eds.), Sci. de la Terre, Ser.Inf, Nancy, No. 31, pp. 373-386,1992. Carr, J. R., Bailey, R. E. & Deng, E. D., "Use of Indicator Variograms for an Enhanced ," Mathematical Geology, Vol. 17, No.8, pp. 797-811,1985. Chauvet, P., "Processing Data with a Spatial Support: Geostatistics and Its Methods," Cahiers de Geostatistique, Fasc. 4, Centre de Geostatistique, Fontainebleau, France, 1993. Chauvet, P., "The Variogram Cloud," Internal Note N-725, Centre de Geostatistique, Fontainebleau, France, 1982. Cressie, N. A. C., "Statistics for Spatial Data," John Wiley & Sons, New York, NY, 1991. Cressie, N. A. c., "The Origins of Kriging," Mathematical Geology, Vol. 22, No.3, pp. 239-252, 1990. David, M., "Geostatistical Ore Reserve Estimation," Elsevier, New York, 1977. Deutsch, C. V. & Journel, A. G., "GSLIB Geostatistical Software Library and User's Guide," Oxford University Press, New York, NY, 1992. Englund, E. & Sparks, A., "Geo-EAS 1.2.1 User's Guide," US-EPA Report #600/8- 91/008, EPA-EMSL, Las Vegas, NV, 1991. Froidevaux, R., "Geostatistical Toolbox Primer, version 1.30," FSS International, Chemin de Drize 10, 1256 Troinex, Switzerland, 1990. Golden Software Inc., "SURFER for Windows User's Guide," Golden Software Inc., Golden, CO, 1994. Goulard, M. & Voltz, M., "Linear Coregionalization Model: Tools for Estimation and Choice of Cross-Variogram Matrix," Mathematical Geology, Vol. 24, No.3, pp. 269-286, 1992. Isaaks, E. H. & Srivastava, R. M., "An Introduction to Applied Geostatistics," Oxford University Press, New York, NY, 1989. Isaaks, E. H. & Srivastava, R. M., "Spatial Continuity Measures for Probabilistic and Deterministic Geostatistics," Mathematical Geology, Vol. 20, No.4, pp. 313-341, 1988. Journel, A. G., "Fundamentals of Geostatistics in Five Lessons," Short Course in Geology, Vol. 8, American Geophysical Union, Washington, DC, 1989. Journel, A. G. & Huijbregts, C. J., "Mining Geostatistics," Academic Press, London, 1978. Bibliography 87

Matheron, G., "Estimer et Choisir," Les Cahiers du Centre de Morphologie Mathematique, Fasc. 7, Ecole Nationale Superieure des Mines de Paris, 1978. Microsoft Corporation, "Microsoft Windows 3.1 - Guide to Programming", Microsoft Press, Redmont, W A, 1992. Murray, M. R. & Baker, D. E., "MWINDOW: An Interactive FORTRAN-77 Program for Calculating Moving-Window Statistics," Computers & Geosciences, Vol. 17, No. 3,pp. 423-430, 1991. Myers, D. E., "Pseudo-Cross Variograms, Positive Definiteness and Cokriging," Mathematical Geology, Vol. 23, pp. 805-816, 1991. Myers, D. E., "To Be or Not to Be ... Stationary? That Is the Question," Mathematical Geology, Vol. 21, No 3, pp. 347-362,1989. Pannatier, Y., "V ARIOWIN: Logiciel pour l'analyse spatiale de donnees en 2D - Etude geologique et geostatistique du gite de phosphates de Taiba (Senega!)," Ph.D. thesis, University of Lausanne, Lausanne, Switzerland, 1995. Pannatier, Y., "MS-WINDOWS Programs for Exploratory Variography and Variogram Modeling in 2D," in Statistics of Spatial Processes; Theory and Applications, Capasso,V., Girone, G. & Posa, D. (eds.), Istituto per Ricerche de Mathematica Applicata (IRMA), Bari, Italy, pp. 165-170, 1994. Posa, D., "Conditioning of the Stationary Kriging Matrices for Some Well-Known Covariance Models," Mathematical Geology, Vol. 21, No.7, pp. 755-765,1988. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P., "Numerical Recipes in C - The Art of Scientific Computing - Second Edition," Cambridge University Press, Cambridge, 1992. Rendu, J., "Kriging for Ore Valuation and Mine Planning," Engineering and Mining Journal, Vol. 181, No.1, pp. 114-120, 1980. Schofield, N., "Ore Reserve Estimation at the Enterprise Gold Mine, Pine Creek, Northern Territory, Australia. Part I: Structural and Variogram Analysis," CIM Bulletin, Vol. 81, No. 909, pp. 56-61,1988. Schofield, N., "Ore Reserve Estimation at the Enterprise Gold Mine, Pine Creek, Northern Territory, Australia. Part 2: The Multigaussian Kriging Model," CIM Bulletin, Vol. 81, No. 909, pp. 62-66,1988. Srivastava, R. M. & Parker, H. M., "Robust Measures of Spatial Continuity in Geostatistics," in Geostatistics, M. Annstrong (ed.), Proc. of the Third International Geostatistics Congress, September 5-9, 1988, Avignon, France, D. Reidel, Dordrecht, Holland, pp. 295-308, 1989. Starks, T. H. & Fang, 1. H., "The Effect of Drift on the Experimental Semivariogram." Mathematical Geology, Vol. 14, No.4, pp. 309-319,1982. Strang, G., "Linear Algebra and Its Applications," Harcourt Brace 10vanovitch, San Diego, CA, 1976. Third edition, 1988. Verly, G., "Estimation of Spatial Point and Block Distributions: The Multi-Gaussian Model," Ph.D. thesis, Stanford University, Stanford, CA, 1984. Index

acknowledgments, 5 madogram, 44, see also angular tolerance, 35, 36 madogram Anis. parameter, 55 semivariogram, see cross anisotropy, 53-55, see also variogram geometric anisotropy, zonal standardized variogram, 40, see anisotropy and variogram also standardized variogram surface variogram, 38-39, see also Append, 19,64,66,68,70 variogram A priori CovJCorr. », 52, 53 variogram cloud, see variogram authorized linear combination, 81 cloud variogram surface, see basic model, 47 variogram surface BIGAUSS,84 variography,29 bitmap (copy as), 3 covariance, 41, 76, 78,81 Ctrl + left click, 18,32,37,52 Calculate Directional Variogram... , 14, Data 34 Limits ... , 29 Model. .. , 51, 52 Map!, 12,29 Variogram Cloud ... , 18,36 data sets, 4 Variogram Surface•.• , 13,33 Dir. parameter, 54 central limits theorems, 82 directional conditional positive definite cross variogram, see directional function, 81 variogram conditional variance, 84 variogram, 14-17, 19,21,22, coregionalization, 46, see also 27,28,34-36 of direction angle, 34, 35, 36 coregionalization direct variography, 29 correlogram, 42-43, 79 discontinuity, see nugget effect cross model correlogram, 43, see also drift, 81-82 correlogram duplicate data points, 9, 10,26 covariance, 41-42, see also covariance ergodicity,76 h-scatterplot, see h-scatterplot erratic aspect, 74 Estimator, 52 Index 89 european academic software award, Graph, 13, 14 2 Axis, 15, 35, 56 expectation, 75, 78, 79 CovJCorr., 34, 35 experimetal variogram, see Palette, 33 directional variogram Records, 12 exploratory variography, see Grid Vari02D with PCF Open... , 23 exponential model, 49-50 Grid Display, 4, 33, 57 GSLIB, 61, 71, 84 File Open••• , 51 history, 2 Open Data File ... , 7, 24 homogeneity, see stationarity Save as•.. , 13, 19,22,53,57, h-scatterplot, 15-17,27,28,29-33 64,66,68,70 File (type) identical coordinates, see duplicate data (.dat), 7, 27, 61-62 data points grid (.grd), 23, 33, 46, 57, 63 IGF, see indicative model (.mod), 53, 70-72 indicative goodness of fit, 53, 56 pair comparison (.pcf), 10, 11, inertia, 38 27,37,62 installation, 4 variogram (.var), 19,20,35, internet, 3 46,51,66-68 intrinsic hypothesis, 47, 51, 79-80 variogram cloud (.cld), 37, 68- 70 Keep selected pairs on map, 15 variogram surface (.vs), 34, 64- 66 lag Function keys number of, 34 FI-FS, see Options I spacing, 34, 35, 37 Estimator tolerance, 34,35 F6, see Graph I Records left click, 18,29,32,37,52 FS, see -Graph I CovJCorr. linear F9, see Options I Unmask geostatistics, 75, 75-76 Pairs! model of coregionalization, 57- FlO, see Data I Map! 59 variogram, 51 Gaussian model, 50-51 local conditional distribution, 83 , 36 logarithmic transform, 31 general relative variogram, 39 Geo-EAS, 4, 24, 61 madogram, 44 geometric anisotropy, 53, see also matrix instability, 51 anisotropy maximum geostatistical framework, 73, 74 bandwidth, 35, 36 Geostatistical Toolbox, 24 distance, 36 90 Index measures of spatial continuity. 37- PCF. see pair comparison file 44 plot window. 22. 52. 53. 56-57 memory limitation. 25 positive definite metafile (copy as). 3 condition. 46. 80-82. minimum distance. 36 matrix. see linear model of missing values. 61. 63 coregionalization mixture of populations. 32. 33 variogram model. see basic ~odel.4.20-22.46-60 model modeling window. 22. 52. 52-55 power model. 51. 55 moments. preferences. see V ARIOWIN.INI first-order. see expectation Prevar2D. 4. 7-10. 24-26 second-order. 75-76 probabilistic model. see moving windows. 84 geostatistical framework multi-Gaussian hypothesis. 33. 44. probability density function. 27 82-85 proportional effect. 84 multiple -document interface. 27. 52 random function. 74 nested model. 23.46 variable. 73 nested structure. 47. 53-55 . 47. see also geometric nonergodic. 37 anisotropy correlogram. see correlogram Range parameter. 55 covariance. see covariance rank transform. 31 normal score transform. 31. 82 reduced vector. 53 notation. 3 regionalized variable. 73 nugget effect. 47-48. 51. 53 requirements. 4 right click. see Options I omnidirectional Numerical Results! cross covariance. 35 robustness. 32 variogram. 35 Run!. 8. 24 variogram cloud. 36 Options sample Estimator. 13. 14.37 characteristics. 12 Numerical Results!, 16.30. map. 12. 18.27.28 34.35 second-order stationarity. 48. 49. Unmask Pairs!. 32 50.78-79.83 semivariogram. see variogram pair Settings•••• 29 extracting from PCF. 37 Settings identification. 15. 18.32.37 Limits•••• 26 masking. 15. 17.32 XV-Coordinates•.•• 8. 24; 26 structure Pair2D. 25. 62 sill. 47. see also zonal anisotropy pair comparison file. 24. see simple kriging. 83 Prevar2D smoothed variogram. 34 Index 91 spatial law, 75 spherical model, 48-49 standardized variogram, 39-40, 79 stationarity, 33, 77-80 statistical inference, 77 strict stationarity, 78, 83 structured aspect, 74 subset, 26, 29 SURFER, 63 swapping space, 24 tolerance, 30, 31 transition model, 47

UNINST ALL.LOG, 5 Uninstall utility, 4,5 variance, 75, 78,80,81 variogram, 37-38, 76, 78, 79, 80 cloud, 18,28,36-37 modeling, see Model surface, 13, 14,23,27,28,33- 34,57 VARIOWIN version 1.0, 2 version 2.0, 2 version 2.1, 2 version 2.2, 1, 3 VARIOWIN.INI, 4, 5 Vario2D with PCF, 4,11-19,27- 45

Windows Tile, 14 Windows 95, 5 Windows NT, 5 WIN.lNI,5 world wide web, 2

zonal anisotropy, 55 see also anisotropy