chapter 1 geostatistics in hydrology: kriging interpolationberga/teaching/stam/stat.pdf ·...

Chapter 1

Geostatistics in Hydrology: Kriginginterpolation

Hydrologic properties, such as rainfall, aquifer characteristics (porosity, hydraulic conductivity,transmissivity, storage coefficient, etc.), effective recharge and so on, are all functions of space(and time) and often display a high spatial variability, also called heterogeneity. This variabilityis not in general random. It is a general rule that these properties display a so called “scaleeffect”, i.e., if we take measurements at two different points the difference in the measuredvalues dicreases as the two points come closer to each other.

It is convenient in certain cases to consider these properties as random functions havinga given spatial structure, or in other word having a given spatial correlation, which can beconveniently described using appropriate statistical quantities. These variables are called “re-gionalized” variables [5].

The study of regionalized variables starts from the ability to interpolate a given field startingfrom a limited number of observation, but preserving the theoretical spatial correlation. This isaccomplished by means of a technique called “kriging” developed by [4, 5] and largely applied inhydrology and other earth science disciplines [3, 6, 7, 1] for the spatial interpolation of variousphysical quantities given a number of spatially distributed measurments.

Although theoreticallly kriging cannot be considered superior to other surface fitting tech-niques [7] and the use of a few arbitrary parameters may lead absurd results, this method iscapable of obtaining “objective” interpolations evaluating at the same time the quality of theresults.

In the next sections we discuss first the statistical hypothesis that are needed to develop thetheory of kriging and then we proceed at describing the method under different assumptionsand finally report a few applications.

1.1 Statistical assumptions

Let Z(~x, ξ) be a random function (RF) simulating our hydrological quantity of interest. Werecall that with this notation we imply that ~x denotes a point in space (two or three dimensional

1

2 CAPTER 1. GEOSTATISTICS IN HYDROLOGY

space), while ξ denotes the state variable in the space of the realizations. In other words:

• Z(~x0, ξ) is a random variable at point ~x0 representing the entire set of realizations of theRF at point ~x0;

• Z(~x, ξ1) is a particular realization of the RF Z;

• Z(~x0, ξ1) is a measurement point.

Given a set of sampled values Z(~xi, ξ1), i = 1, 2, . . . we want to reconstruct Z(~x, ξ1), i.e. apossible realization of Z(~x, ξ).

1.2 Kriging for Weak or Second Order stationary RF

An RF is said to be second order stationary if:

1. Constant mean:

E[Z(~x, ξ)] = m (1.1)

2. the autocovariance (another name for the covariance) is a function of the distance betweenthe reference points ~x1 and ~x2:

cov[~x1, ~x2] = E[(Z(~x1, ξ)−m)(Z(~x2, ξ)−m)] = C(~h) (1.2)

In practice second order stationarity implies that the first two statistical moments (expectedvalue and covariance) be translation invariant. Note that by saying that E[Z(~x, ξ)] = m weimply that effectively the expected value taken on all the possible realizations ξ does not varywith space. However, for a given realization Z(~x, ξ1) is a function of ~x.

Because of (1.1), the covariance (1.2) can be written as:

cov[~x1, ~x2] = E[(Z(~x1, ξ)−m)(Z(~x2, ξ)−m)]

= E[Z(~x1, ξ)Z(~x2, ξ)]−mE[Z(~x2, ξ)]− E[Z(~x1, ξ)]m+m2

= E[Z(~x1, ξ)Z(~x2, ξ)]−m2 (1.3)

Obviously if ~h = 0 we have the definition of variance, also called the “dispersion” variance:

C(0) = var[Z] = σ2Z

For simplicity of notation, from now on, we will denote x = ~x, h = ~h and we will drop thevariable ξ in the RF.

1.2 SECOND ORDER STATIONARY RF 3

Case with m and C(~h) known. If E[Z(x)] = m and C(h) are known, then we can define anew variable Y (x) with zero mean:

Y (x) = Z(x)−mE[Y (x)] = 0

Given the observed values:x1 x2 . . . xnY1 Y2 . . . Yn

with Yi = Y (xi) being the observation at point xi, we look for a linear estimator Y ∗(x0) ofY (x0) at point x0 using the observed values. The form of the estimator is:

Y ∗(x0) =n∑

i=1

λiYi (1.4)

Note that the estimator (1.4) is a realization of the RF

Y (x0, ξ) =n∑

i=1

λiY (xi, ξ)

The weights λi are calculated by imposing that the statistical error

ε(x0) = Y (x0)− Y ∗(x0)

has zero expecetd value and minimum variance:

E[ε(x0)] = 0

var[ε(x0)] = E[(Y (x0)− Y ∗(x0))2] = minimum

Substituting eq. (1.4) in the expression of the variance we have:

E[(Y (x0)− Y ∗(x0))2] = E[(n∑

i=1

λiYi − Y0)2]

= E[(n∑

i=1

λiYi − Y0)(n∑

i=1

λiYi − Y0)]

= E[(n∑

i=1

λiYi)(n∑

i=1

λiYi)]− 2E[n∑

i=1

λiYiY0] + E[Y 20 ]

=n∑

i=1

n∑

j=1

λiλjE[YiYj]− 2n∑

i=1

λiE[YiY0] + E[Y 20 ]

but, since m = 0E[YiYj] = C(xi − xj) +m2 = C(xi − xj)


and

E[Y 20 ] = C(0) = var[Y ]

is the dispersion variance of Y . Then:

E[(Y (x0)− Y ∗(x0))2] =n∑

i=1

n∑

j=1

λiλjC(xi − xj)− 2n∑

i=1

λiC(xi − x0)

+ C(0)

The minimum is found by setting to zero the first partial derivatives:

∂

∂λi

(E[(Y (x0)− Y ∗(x0))2]

)= 2

n∑

j=1

λjC(xi − xj)− 2C(xi − x0) = 0

j = 1, . . . , n

This yields a linear system of equations:

Cλ = b (1.5)

where matrix C is given by:

C =

C(0) C(x1 − x2) . . . C(x1 − xn)· · ·· ·

C(xn − x1) C(0)

and the right hand side vector b is given by:

b =

C(x1 − x0)···

C(xn − x0)

Matrix C is the spatial covariance matrix and does not depend upon x0. It can be shownthat if all the xj’s are distinct then C is positive definite, and thus the linear system (1.5)can be solved with either direct or iterative methods. Once the solution vector λ is obtained,equation (1.4) yields the estimation of our regionalized variable at point x0. Thus the calculatedvalue for λ is actually function of the estimation point x0. If we want to change the estimationpoint x0, for example if we need to obtain a spatial distribution of our regionalized variable,we need to solve the linear system (1.5) for different values of x0. In this case it is convenientto factorize matrix C using Cholesky decomposition and then proceed to the solution for thedifferent right hand side vectors.

1.3. KRIGING WITH THE INTRINSIC HYPOTHESIS 5

Evaluation of the estimation variance The estimation variance is defined as the varianceof the error:

ε = Y ∗0 − Y0

Hence:var[Y ∗0 − Y0] = E[(Y ∗0 − Y0)2]− E[(Y ∗0 − Y0)]2

but:E[(Y ∗0 − Y0)] = E[Y ∗0 ]− E[Y0] =

∑

i

λiE[Yi]− E[Y0] = 0 (E[Y ] = m = 0)

and since ∑

j

λjC(xi − xj) = C(xi − x0)

we finally obtain:

var[Y ∗0 − Y0] = E[(Y ∗0 − Y0)2]

=∑

i

∑

j

λiλjC(xi − xj)− 2∑

i

λiC(xi − x0) + C(0)

= −∑

i

λiC(xi − x0) + C(0)

= var[Y ]−∑

i

λiC(xi − x0)

which, since∑

i λiC(xi − x0) > 0, shows that the estimation variance of Y0 is smaller than thedispersion variance of Y (the real variance of the RF). In statistical terms, we can interpretthis result by saying that since we have observed Y at some points xi then the uncertainty onY decreases.

It is important to remark the difference between estimation and dispersion variance. Thelatter is representative of the variation interval of the RF Y within the interpolation domain,while the estimation variance represents the residual uncertainty in the estimation of the real-ization Y ∗0 of Z when n observations are available. The dispersion variance is a constant, whilethe estimation variance varies from point to point and is zero at the observation points.

Our original variable was Z = Y +m and its estimate is thus:

Z∗0 = m+∑

i

λi(Zi −m)

var[Z∗0 − Z0] = var[Z0]−∑

i

λiC(xi − x0)

1.3 Kriging with the intrinsic hypothesis

The hypothesis of second order stationarity of the RF is not always satisfied, for example C(0)increases with the distance, violating hypothesis (1.2). In this case the “Intrinsic hypothesis”must be used, in which we assume that the first order increments

δ = Y (x+ h)− Y (x)


(h)

γ

High spatial correlationLow spatial correlation

C(h

)

hh

range

range

sill

Figure 1.1: Behavior of the covariance as a function of distance (left) and the correspondingvariogram (right).

are second order RF:

E[Y (x+ h)− Y (x)] = m(h) = 0 (1.6)

var[Y (x+ h)− Y (x)] = 2γ(h) (1.7)

where the function γ(h) is called the variogram. If the mean m(h) is not zero an obviouschange of variable is required. The variogram is defined as the mean quadratic increment ofY (x) (divided by 2) for any two points xi and xj separated by a distance h:

γ(h) =1

2var[Y (x+ h)− Y (x)] =

1

2E[(Y (x+ h)− Y (x))2] (1.8)

and is related to the covariance function by:

γ(h) =1

2E[(Y (x+ h)− Y (x))2]

=1

2E[Y 2(x+ h)]− E[Y (x+ h)Y (x)] +

1

2E[Y 2(x)]

= C(0)− C(h)

The intrinsic hypothesis requires a finite value for the mean of Y (x) but not for its variance.In fact, hypothesis (1.2), as changed into (1.3), implies (1.8), but not viceversa.

The covariance C(h) has a decreasing behavior as shown in Fig. 1.1. When C(h) is knownthen the variogram can be directly calculated. When C(0) is finite, the variogram γ(h) isbounded asymptotically by this value. The value of h at which the asymptot can be consideredachieved is called the “range“, while C(0) is called the “sill” (see Fig. 1.1).


1.3.1 The variogram

The variogram is usually calculated from the experimental observations, and describes thespatial structure of the RF. It can be shown [5] that given x1, . . . , xn are n points belonging tothe domain of interpolation, for all the coefficients η1, . . . , ηn ∈ R satisfing:

n∑

i=1

ηi = 0

then

−n∑

i=1

n∑

j=0

ηiηjγ(xi − xj) ≥ 0

i.e. −γ(h) is a positive semidefinite function, and

lim|h|→∞

γ(h)

h2= 0

i.e. γ(h) is tends to infinity slowlier than |h|2 as |h| → ∞.In principle there γ(h) could assume different behaviors also with the direction of vec-

tor h (“anisotropy”), but this is in general not easily verifiable due to the limited numberof data points usually available for hydrologic variables. If the experimental variogram dis-plays anisotropy, then the intrinsic hypothesis is not verified and one has to use the so called“universal” kriging [2, 3].

The most commonly used isotropic variograms are shown in Fig. 1.2 and are of the form:

1. polinomial variogram:γ(h) = ωhα 0 < α < 2

2. exponential variogram:γ(h) = ω

[1− e−αh

]

3. gaussian variogram:

γ(h) = ω[1− e−(αh)2

]

4. spherical variogram:

γ(h) =

{12ω[

3hα−(hα

)3]

h ≤ α

ω h > α

where ω and α are real constants.The variogram is estimated from the available observations in the following manner. The

data points are subdivided into a prefixed number of classes based on the distances betweenthe measurement locations. For each pair i and j of points and for each class calculate:

1. the number M of points that fall within the class;


1 2 3 4 5

2

4

6

8

10

α=1.5

α=1.0

α=0.5

1 2 3 4 5

0.2

0.4

0.6

0.8

10.95

α 3α

α 3

1 2 3 4

0.2

0.4

0.6

0.8

10.95

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

2α/3 α

Figure 1.2: Behavior of the most commmonly used variograms: polinomial (top left); exponen-tial (top right); gaussian (bottom left); spherical (bottom right)


2. the average distance of the class;

3. the half of the mean quadratic increment

1

2

∑(Yi − Yj)2/M

In general the pairs are not uniformly distributed among the different classes as usually there aremore pairs for the smaller distances. Thus the experimental variogram will be less meaningfulas h increases. A best fit procedure together with visual inspection is then used to select themost appropriate variogram and evaluate its optimal parameters.

Remark 1. An experimental variogram that is not bounded above (for example the polinomialvariogram with α ≥ 1) implies an infinite variance, and thus the covariance does not exist. Onlythe intrinsic hypothesis is acceptable. If the variogram achieves a “sill” then the phenomenonhas a finite variance and the covariance exists.

Remark 2. For every variogram, γ(0) = 0. However sometimes the data may display ajump at the origin. This apparent discontinuity is called the “nugget effect” and can be due tomeasurement errors or to microregionalization effects that are not evidenced at the scale of theactual data points. If the nugget effect is present the variogram needs to be changed into:

γ(h) = δ + γ0(h)

where δ is the jump at the origin and γ0(h) the variogram without the jump. In Fig. 1.3 wereport a variogram with the nugget effect together with some other special cases that may beencountered.

Remark 3. When we look for an estimator of the type

Y ∗(x0) =n∑

i=1

λiYi

from the intrinsic hypothesis, i.e. eq. (1.6), we see that our random variable Y (x) must have aconstant mean m, so that:

E[Y ∗(x0)] = E[n∑

i=1

λiYi] = m

and, since E[Y (xi)] = E[Yi] = m, we readily obtain:

n∑

i=1

λi = 1


γ(h) γ(h)

γ(h)γ(h)

γ(h)

δ δmeasured values

nugget effect

h

h

h

h

h

random structure(no correlation)

nested structure periodic structure

hole effect

Figure 1.3: Example of typical spatial correletion structures that may be encountered in ana-lyzing measured data

1.4. REMARKS ABOUT KRIGING 11

The condition of minimum variance becomes now a constrained minimization problem whichcan be solve by introducing a lagrange multiplier µ. The system (1.5) must be rewritten as:

0 γ(x1 − x2) . . . γ(x1 − xn) 1...

...γ(xn − x1) 1

1 1 . . . 1 0

λ1...λnµ

=

γ(x1 − x0)...

γ(xn − x0)1

1.4 Remarks about Kriging

a. Kriging is a BLUE (Best Linear Unbiased Estimator) interpolator. In other word it is aLinear estimator that matches the correct expected value of the population (Unbiased)and that minimizes the variance of the observations (Best).

b. Kriging is an exact interpolator if no errors are present. In fact, if we set x0 = xi in (1.5)we obtain immediately λi = 1, λj = 0, j = 1, . . . , n, j 6= i.

c. If we assume that the the error ε is Gaussian, then we can associate to the estimate Y ∗(x0)a confidence interval. For example, the 95% confidence interval is ±2σ0 where:

σ0 =√

var[Y ∗(x0)− Y (X0)]

Then the kriging estimator (1.4) becomes:

Y ∗(x0) =n∑

i=1

λiYi ± 2σ0

d. The solution of the linear system does not depend on the observed value but only on xiand x0.

e. A map of the estimated regionalized variable, and possibly its confidence intervals, canbe obtained be defining a grid and solving the linear system for each point in the grid.

1.5 Kriging with uncertainties

We now assume that the observations Yi are affected by measurement errors εi, and that:

1. the errors εi have zero mean:

E[εi] = 0 i = 1, . . . , n

2. the errors are uncorrelated:

cov[εi, εj] = 0 i 6= j


3. the errors are not correlated with the RF:

cov[εi, Yi] = 0

4. the variance σ2i of the errors is a known quantity and can vary from point to point.

The new coefficient matrix C of the linear system (1.5) is changed by adding to the maindiagonal the quantity −σ2

i :

C := C −

σ2

1 · 00 · 00 · σ2

n

and everything proceeds as in the standard case.

1.6 Validation of the interpolation model

The chosen model (in practice the variogram) can be validated by interpolating observed values.If n observations Y (xi), i = 1, . . . , n are available, the validation process proceeds as follows:

For each j, j = 1, . . . , n:

• discard point (xj, Y (xj));

• estimate the Y ∗(xj) by solving the kriging system having set x0 = xj and using theremaining points xi, i 6= j for the interpolation;

• evaluate the estimation error εj = Y ∗j − Yj,

The model can be considered theoretically valid if the error distribution is approximatelygaussian with zero mean and unit variance (N(0, 1)), i.e. satisfies the following:

1. there is no bias:1

n

n∑

i=1

εi ≈ 0

2. the estimation variance σi is coherent with the error standard deviation:

1

n

n∑

i=1

(Y ∗i − Yiσi

)2

≈ 1

One can also look at the behavior of the interpolation error at each point looking at the meansquare error of the vector ε:

Q =

√√√√1

n

n∑

i=1

ε2i

1.7. COMPUTATIONAL ASPECTS 13

The uncertainties connected to the choice of the theoretical variogram from the experimentaldata can be minimized by anaylizing the validation test. In fact, among all the possible vari-ograms γ(h), that close to the origin display a slope compatible with the observations and givesrise to a theoretically coherent model, one can choose the variogram with the smallest value ofQ.

1.7 Computational aspects

In the validation phase n linear systems of dimension n − 1 need to be solved. The systemmatrices are obtained by dropping one row and one column of the complete kriging matrix.This can be efficiently accomplished by means of intersections of n − 1-dimensional lines withappropriate coordinate n-dimensional planes.

Note that the kriging matrix C is symmetric, and thus its eigenvalues µi are real. However,since

n∑

i=1

µi = Tr(C) =n∑

i=1

cii = 0

where Tr(C) is the trace of matrix C, it follows that some of the eigenvalues must be negativeand thus C is not positive definite. For this reason, the solution of the linear systems is usuallyobtained by means of direct methods, such as Gaussian elimination or Choleski decomposition.Full Pivoting is often necessary to maintain stability of the algorithm.

1.8 Kriging with moving neighborhoods

Generally the experimental variogram is most accurate for small values of h, with uncertaintiesgrowing rapidly when h is large. The influence of this problem may be decreased by usingmoving neighborhoods. With this variant, only the points that lie within a prefixed radiusR from point x0 are considered, provided that we are left with an adequate number of data(Fig. 1.4). The radius R is selected so that the lag h will remain within the range of maximumcertainty for γ(h).

This approach leads also to high saving in the computational cost of the procedure be-cause each linear system is now much smaller (Gaussian elimination has a computational costproportional to n3, where n is the number of equations).

1.9 Detrending

In certain cases the observations display a definite trend that needs to be taken into account.This is the case, for example, when one needs to interpolate piezometric heads observed fromwells in an aquifer system where a regional gradient is present. To remove the trend from thedata it is possible to work with residuals (= measurements - trend) that have a constant mean.However this detrending procedure may be dangerous as it may introduce a bias in the results.For this reason it is important that the trend be recognized not only from the raw data but


x0

R

Figure 1.4: Example of interpolation with moving neighborhood

also from the physical behavior of the system from which the data come. If the trend cannotbe removed from the observations by simple subtraction, the universal kriging approach [2, 3]can be used.

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Experimental variogram

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Experimental variogram

using 10 classes

The shape is similar: robustness with respect todifferent class subdivisions

0 1 2 3 4 5 6 7 8 9 10

Lag Distance (km)

0

100

200

300

400

500

600

700

800

900

1000

1100

Var

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a (m

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0 1 2 3 4 5 6 7 8 9 10Lag Distance (km)

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100

200

300

400

500

600

700

800

900

1000

1100

Vari

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a(m

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116180

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First try

Linear variogram γ = h

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

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0 1 2 3 4 5 6 7 8 90

1

2

3

4

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0.03 -0.53 0.25 -0.53 0.00 0.00

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0.000.07-0.630.21-0.550.00

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Estimationvariance

interpolation error ε

0 1 2 3 4 5 6 7 8 90

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2

3

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00.10.20.30.40.50.60.70.80.911.11.2

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Kriging reconstruction is robust evenwhen the

variogram is not accurate

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0

100

200

300

400

500

600

700

800

900

1000

1100

Var io

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(m2)

nugget effect = 41.3

Second try

Linear variogram with nugget effect

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0 1 2 3 4 5 6 7 8 90

1

2

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Elevation

Estimationvariance

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0 1 2 3 4 5 6 7 8 9 10Lag Distance (km)

0

100

200

300

400

500

600

700

800

900

Var i

ogra

m( m

2 )

polinomial variogram: ω = 136= 0.5

Third try

Variogram without nugget effect and with large slope at the origin

1.10E

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0 1 2 3 4 5 6 7 8 90

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2

3

2

1

25E-005

3E-007

-1

-7E-007

-3

2E-006

0.2 -0.80.5

-0.08

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

Elevation

Estimationvariance

ε

Estimation variance is large far away fromobservation points

22C

AP

TE

R1.

GE

OST

AT

IST

ICS

INH

YD

RO

LO

GY

0 1 2 3 4 5 6 7 8 9 10Lag Distance (km)

0

100

200

300

400

500

600

700

800

900

1000

1100

Var io

gram

(m2)

= 2.49

Fourth try

Gaussian variogram: small slope at the origin

ω = 317

1.10E

XA

MP

LE

OF

AP

PL

ICA

TIO

N23

0.01 -0.07 0.02 -0.042E-005 8E-008

7E-005

-3E-005

3E-010-0.009-0.20.02-0.03-5E-009

-0.001

0.004

0.04

0.02

-0.03

-0.03

-0.02

-0.03

0.03

0.03

-0.01

0.1

-0.02

0.006

-0.03

0.040.05 0.08

0.03

-0.02

0.04

0.02

-0.02-0.0001

-8E-005

-0.02

-2E-005

-0.08

5E-005

-0.02 -0.070.02

-0.009

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

0123456789101112

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

Elevation

Estimationvariance

ε

small and smooth estimation variance; small errors

Bibliography

[1] de Marsily, G. (1984). Spatial variability of properties in porous media: A stochasticapproach. In J. Bear and M. Corapcioglu, editors, Fundemantals of Transport Phenomenain Porous Media, pages 721–769, Dordrecht. Martinus Nijhoff Publ.

[2] Delhomme, J. P. (1976). Applications de la theorie des varaibles regionalisees dans lessciences de l’eau. Ph.D. thesis, Ecoles des Mines, Fontainebleaus (France).

[3] Delhomme, J. P. (1978). Kriging in the hydrosciences. Adv. Water Resources, 1, 251–266.

[4] Matheron, G. (1969). Le krigeage universel. Technical Report 1, Paris School of Mines.Cah. Cent. Morphol. Math., Fontainbleau.

[5] Matheron, G. (1971). The theory of regionalized variables and its applications. TechnicalReport 5, Paris School of Mines. Cah. Cent. Morphol. Math., Fontainbleau.

[6] Volpi, G. and Gambolati, G. (1978). On the use of a main trend for the kriging techniquein hydrology. Adv. Water Resources, 1, 345–349.

[7] Volpi, G., Gambolati, G., Carbognin, L., Gatto, P., and Mozzi, G. (1979). Groundwatercontour mapping in venice by stochastic interpolators. 2. results. Water Resour. Res., 15,291–297.

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