Kriging and Conditional Geostatistical Simulation Based on Scale ...

Kriging and Conditional Geostatistical SimulationBased on Scale-Invariant Covariance Models

Diploma Thesisby

Rolf SidlerSupervisor: Prof. Dr. Klaus Holliger

INSTITUTE OF GEOPHYSICSDEPARTMENT OF EARTH SCIENCE

October 20, 2003

Copyright Diplomarbeiten Departement Erdwissenschaften, ETH Zurich‘Der/Die Autor/in erklart sich hiermit einverstanden, dass die Diplomarbeit fur pri-vate und Studien-Zwecke verwendet und kopiert werden darf. Hingegen ist eineVervielfaltigung der Diplomarbeit oder die Benutzung derselben zu kommerziellenZwecken ausdrucklich untersagt. Wenn wissenschaftliche Resultate aus der Arbeitverwendet werden, mussen diese wie in allen wissenschaftlichen Arbeiten ublichentsprechend zitiert werden.’

Copyright Diploma Thesis, Department of Earth Sciences, ETH Zurich‘The author hereby agrees that the diploma thesis may be copied and used forprivate and personal scholarly use. However, this is to emphasize that the makingof multiple copies of the thesis or the use of the thesis for commercial purposes isstrictly prohibited. When making use of the results found in this thesis, the normalscholarly methods of citation are to be followed.’

Unterschrift des/der DiplomierendenSignature of Diploma-Student

Contents

1 Introduction 2

2 Theory 42.1 First- and Second-Order Stationarity . . . . . . . . . . . . . . . . . . 42.2 Covariance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 von Karman Covariance Model . . . . . . . . . . . . . . . . . . . . . 72.4 Kriging Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Search Neighborhood . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Basic Types of Kriging . . . . . . . . . . . . . . . . . . . . . . 102.4.3 Simple Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.4 Ordinary Kriging . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Description of the Implementation 143.1 Program Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Overall Structure of the Code . . . . . . . . . . . . . . . . . . . . . . 143.3 Description of the Individual Functions . . . . . . . . . . . . . . . . . 15

3.3.1 vebyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 kriging3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.3 inputmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.4 neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.5 buildbigc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.6 buildsmallc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.7 displacement3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.8 covcalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.9 ordinary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.10 rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Testing of the Implementation 234.1 Comparison with a published example . . . . . . . . . . . . . . . . . 234.2 Structural Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Search Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4.1 Leaving-One-Out Cross-Validation . . . . . . . . . . . . . . . 294.4.2 Jackknife Cross-Validation . . . . . . . . . . . . . . . . . . . . 31

4.5 Sensitivity of Kriging Estimation with Regard to Auto-CovarianceModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3

5 Application of Kriging to Conditional Geostatistical Simulations 405.1 Unconditional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Conditional Simulations of Porosity Distributions in Heterogeneous

Aquifers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.1 Boise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2.2 Kappelen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Conclusions 58

7 Acknowledgments 60

Appendix 61

A Kriging 61

B Boise 64

C Kappelen 70

Abstract

Kriging is a powerful spatial interpolation technique, especially for irregularly spaceddata points, and is widely used throughout the earth and environmental sciences.The estimation at an unsampled location is given as the weighted sum of the cir-cumjacent observed points. The weighting factors depend on a model of spatialcorrelation. Calculation of the weighting factors is done by minimizing the errorvariance of a given or assumed model of the auto-covariance for the data with re-gard to the spatial distribution of the observed data points. As part of this work,I have developed a flexible and user-friendly matlab-program called vebyk (valueestimation by kriging), which performs ordinary kriging and can be easily adaptedto other kriging methods. Extensive tests demonstrate that (i) heavily clustereddata require an adaption of the search neighborhood, (ii) kriging may cause artefactsin anti-persistent media when using the “correct” auto-covariance model and (iii)best performance for kriging scale-invariant media is obtained when using smootherauto-covariance models than those indicated by the observed dataset. Conversely,my results indicate that kriging is relatively insensitive to the absolute value of thecorrelation lengths used in the auto-covariance model as long as the structural as-pect ratio is approximately correct. Finally, this kriging algorithm has been usedas the basis for conditional geostatistical simulations of the porosity distributionin two heterogeneous sedimentary aquifers. The stochastic simulations were con-ditioned by porosity values derived from neutron porosity logs and georadar andseismic crosshole tomography. The results indicate that conditional simulations in“hydrogeophysics” will prove to be similarly useful for quantitatively integratingnumerous datasets of widely differing, resolution, coverage and “hardness” as it hasfound to be in more established fields of reservoir geophysics.

Chapter 1

Introduction

Gathering information about subsurface properties is always a difficult task, as directaccess is generally not possible. Measurements are therefore more sparsely sampledthan desired. Although more samples can not be generated without additional mea-surements, it is possible to take advantage of characteristic properties of a datasetand to estimate values at unsampled locations that fit these characteristics. Themathematical methods to evaluate these characteristics lie in the realm of statistics,but for reliable results in earth science, the characteristics have to fit not only math-ematical but also geological criteria. For this reasons, the methode for estimatingvalues at unsampled locations is therefore referred to as geostatistics.

Basic classical statistic data analysis consists of data posting, computation ofmeans and variances, scatter-plots to investigate the relationship between two vari-ables and histogram analysis. In the early 1950s, these techniques where found to beunsuitable for estimating disseminated ore reserves. D. G. Krige, a South Africangeoscientist, and H. S. Sichel, a statistican, developed a new estimation technique(Krige, 1951). The method was based on statistical properties of the investigatedregion. Matheron (1965) formalized the concepts of the new branch of geosciencethat combines “structure and randomness” and proposed the name geostatistics forthis new scientific discipline. By the early 1970s, kriging has proved to be very usefulin the mining industry. With the arise of high-speed computers, geostatistics spreadto other areas of earth science and has become more popular ever since. Journeland Huijbregts (1978) summarized the state of the art of “linear” geostatistics withan emphasis mostly with regard to practical and operational efficiency. They alsodescribed the then novel technique of conditional simulation that leaves the field of“linear” statistics.

The goal of a conditional simulation is to model a region numerically so thatthe auto-covariance of the model complies with the auto-covariance of the observeddata and the model data coincide with the data at sampled locations. As mostconditional simulations go hand in hand with unconditional simulation, the qualityof which depend strongly on computational capabilities, the technique has onlyrecently seen wider application. Conditional simulation has proved to be essentialfor the planing and control of mining and hydrocarbon recovery processes as well asfor the optimization of ground-water and contaminant flow simulations.

In the course of this diploma thesis I have developed a Matlab implementation of

2

3

the kriging algorithm. A major motivation of this work is that there are few avail-able codes and that these codes are gennerally inflexible and/or poorly documented.Given the rapidly increasing importance of geostatistics in applied and environmen-tal geophysics, the availability of a comprehensible, expandable and well documentedcode was considered to be essential. After outlining the theory in Chapter 2 anddescribing the implementation of the code in Chapter 3, to convey confidence in itsreliability and present various tests of the code to point to issues that have to beconsidered when interpolating with kriging. Conditional simulations for “hhydro-geophysical” field data from Boise (Idaho,USA) and Kappelen (Bern, Switzerland)are presented in Chapter 5. The simulations were obtained by adapting stochas-tic models to measured data trough kriging interpolation. Finally, the Appendicescontains results of kriging with different ν-values and correlation lengths as well asadditional realizations of the presented conditional simulations.

Chapter 2

Theory

The goal of this chapter is to provide a summary of the theory needed for under-standing the implementation of the kriging technique, realised as part of this work.Detailed descriptions of the theory of kriging are given, for example, by Armstrong(1998); Isaaks and Sarivastava (1989); Journel and Huijbregts (1978); Kelkar andPerez (2002) and Kitanidis (1997).

Interpolation with kriging is based on the spatial relationship of random, spa-tial variables. A random variable can take numerical values according to a certainprobability distribution. For instance, the result of casting an unbiased dice can beconsidered as a random variable that can take one of six equally probable values(Journel and Huijbregts, 1978). The spatial relationship is given by the covariancefunction or, equivalently, by the semi-variogram. A thorough understanding of theconcept of covariance is therefore necessary for working with kriging interpolation.For this reason, this topic is covered before the methodological foundations of krigingare outlined.

2.1 First- and Second-Order Stationarity

Geostatistics tries to predict the values of a random spatial variable at unsampledlocations by using values at sampled locations. For reasonable predictions some basicassumptions have to be made. An inherent problem of this approach is that for theunsampled locations no information is available and a verification of the assumptionsis not possible until these missing samples are available. Nevertheless, practicehas proven geostatistics to be a powerful tool for improving data-based subsurfacemodels, for optimizing the recovery and sustainable use of natural resources and forassessing environmental hazards.

A key assumption in geostatistics is that the data are first- and second-orderstationary. First-order stationarity implies that the arithmetic mean within theconsidered region is constant and independent of the size of the region and of thesampling locations. This in turn implies that local means inside a region are constantand correspond to the global mean of the entire region. The mathematical definitionof first-order stationarity is more general:

f [X(~u)] = f [X(~u + ~τ)], (2.1)

4

2.2 Covariance Function 5

where X is a random variable, f [...] is any function of a random variable and ~uand ~u + ~τ are two locations of the random variable. For all practical intents andpurposes, however, the condition of constant arithmetic mean within a certain regionis most important for ensuring first-order stationarity.

Second-order stationarity requires that the relationship between two data valuesdepends only on the distance of the two points but is otherwise independent on theirabsolute position. Mathematically, this is expressed as:

f [X( ~u1), X( ~u1 + ~τ)] = f [X( ~u2), X( ~u2 + ~τ)]. (2.2)

2.2 Covariance Function

The covariance function describes the spatial relationship of the data points as func-tion of their distance vectors. The definition of the covariance function can beexpressed in terms of the expected value. The expected value is the most probablevalue for a random data distribution and is defined as:

E[X(k)] =

∫ ∞

−∞Xp(X) dX = µX , (2.3)

where X(k) is a random variable and p(X) is its probability of occurrence (Ben-dat and Piersol, 2000). The variance of a random variable is given by:

σ2 = V ar(X) = E(X2)− [E(X)]2, (2.4)

where σ is the standard deviation. The cross-covariance function describes therelation between distance and the variance. It is defined as:

C(X, Y ) = E[(X − µX) · (Y − µY )], (2.5)

where µX and µY are the expected values of the two random variables X and Yand therefore constants. A constant factor can be taken out of the expression for theexpected value (see equation 2.3) and the above definition for the cross-covariancefunction can be rewritten as:

C(X, Y ) = E(XY )− E(X)E(Y ). (2.6)

For the so called auto-covariance C(X, X) the value for zero distance, or zero lag,thus corresponds to the variance (equation 2.4). With increasing lag the covariancedecreases depending on the spatial relationship of the dataset. If there is cyclicityin the data set, the auto-covariance will mirror this cyclicity as a function of thelag. It should, however, be noted that, in principle, the presence of cyclicity, does,however violate the assumption of second order stationarity.

The auto-covariance function can alternatively be calculated through the so-called autocorrelation function (Bendat and Piersol, 2000):

RXX(τ) = E[X(u)X(u + τ)], (2.7)

6 Theory

where u is a location and τ is the lag to this location. The auto-correlationfunction distinguishes itself form the covariance function only for non-zero meanvalues. The relation between the both is:

CXX(τ) = RXX(τ)− µ2X . (2.8)

This is important as the Wiener-Khinchine theorem describes the relation be-tween the autocorrelation and the spectral density function, which is often usedto generate random data with a given auto-covariance function. N.Wiener andA.I.Khinchine proved independently of each other in the USA and in the UdSSR,respectively, that the spectral density function SXX is the Fourier transformed ofthe correlation function (Bendat and Piersol, 2000):

SXX(f) =

∫ ∞

−∞RXX(τ)e−j2πfτ dτ, (2.9)

where f is the frequency, τ the lag and j =√−1. The validity of this theorem

rests on the condition that the integral over the absolute values of the correlationfunction is finite, which is indeed always the case for finite record lengths. Theinverse Fourier transform of the spectral density function therefore yields to thecorrelation function:

RXX(f) =

∫ ∞

−∞SXX(f)ej2πfτ df. (2.10)

An alternative way to describe the spatial relationship of a dataset is the var-iogram, or the semi-variogram (i.e. half the variogram). The semi-variogram iscommonly used in geostatistical data analysis, because, as opposed to the covari-ance function, it can also be calculated if the mean of a dataset is not known. Thesemi-variogram is therefore more convenient to analyze the spatial relationship ofan unknown dataset. It is defined as:

γ(~τ) =1

2V ar[X(~u)−X(~u + ~τ)], (2.11)

where ~u is a location and τ is the lag to this location. For a first- and second-order stationary function, the relation between the auto-covariance function and thesemi-variogram is given by (Kitanidis, 1997):

γ(~τ) = C(0)− C(~τ). (2.12)

As the auto-covariance function starts at the variance of the data and decreaseswith increasing lag, the variogram starts at zero and increases as lag increases (Figure2.1). If the data are stationary, the semi-variogram will reach a sill where γ(~τ)reaches the value of the variance. The distance τ at which the sill is reached is calledrange. When the semi-variogram does not reach a sill and the auto-covariance istherefore not defined, a “pseudo-covariance” can be defined, such as γ(τ) = A−C(τ),where A is a positive value of γ(τ) at a distance τ that lies beyond the regionof interest. This does not affect the calculation of the kriging weights, but the

2.3 von Karman Covariance Model 7

C(0) = s2

semi-variogram

covariance function

lag

Figure 2.1: Shematic illustration between the auto-covariance function and the corre-sponding semi-variogram

error variance, which is a measure of the accuracy of the estimation (Journel andHuijbregts, 1978; Kelkar and Perez, 2002).

The “experimental semi-variogram” is calculated from the given data and there-fore a discrete and often irregularly sampled function. The experimental variogramcan then be approximated through a continuous parameter model. Due to the lim-ited scale of sampled values, it is, however, often not possible to estimate the correctvariogram or auto-covariance function of a region (Western and Bloschl, 1999).

As we shall see, the use of the auto-covariance function instead on the semi-variogram is preferable for the purposes of kriging, because the equations are thenlargely identical for simple and ordinary kriging.

2.3 von Karman Covariance Model

Theodore von Karman introduced a novel family of auto-covariance functions tocharacterize the seemingly chaotic, random velocity fields observed in turbulent me-dia (von Karman, 1948):

C(r) =σ2

2ν−1Γ(ν)(r/a)νKν(r/a), (2.13)

where Γ is the gamma function, and Kν is the modified Bessel function of thesecond kind of order 0 ≤ ν ≤ 1, r is the lag, and σ is the variance. The mediadescribed by this correlation functions are self-affine for 0 ≤ ν ≤ 0.5 and self-similar for 0.5 ≤ ν ≤ 1, at distances considerably shorter than the correlation lengtha (Klimes, 2002). Figure 2.3 shows von Karman correlation functions for differentvalues of ν. Small ν-values characterize rapidely decaying covariance and thus highlyvariable media. The von Karman auto-covariance function with ν = 0.5 correspondsto the well-known exponential auto-covariance function:

8 Theory

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/a

Aut

o−C

ovar

ianc

e

ν =0.1

ν =0.2ν =0.3ν =0.4

ν =0.5

ν =0.7

ν =0.9

ν =1

Figure 2.2: Set of one-dimensional von Karman covariance functions with a correlationlength of a = 1.5, varying ν, and variance σ2 = 1.

C(r) = σ2e(−r/a). (2.14)

2.4 Kriging Interpolation

Kriging is a linear interpolation technique, in which the value X at an unsampledlocation ~u is estimated as:

X( ~u0) =n∑

i=1

λiX(~ui), (2.15)

where X(~ui) are the values at neighboring sampled locations and λi are thekriging weights assigned to these values. The estimated value is therefore a weightedaverage of the surrounding sampled values. The “crux” in kriging is to computethe kriging weights, which depend on the spatial relationship of the data. Thisspatial relationship is quantified by the semi-variogram, or equivalently, by the auto-covariance function, which is inferred from available data and/or constrained bycomplementary or a priori information. Kriging is an unbiased estimator and istherefore referred to by the acronym “BLUE”, which stands for best linear unbiasedestimator.

2.4.1 Search Neighborhood

The search neighborhood defines the sampled points used for estimation of valuesat unsampled locations. Using all sampled values would lead to the most accuratesolution. In practice, there are, however, several reasons not to do so and to choosea smaller search neighborhood. The reasons are the following:

2.4 Kriging Interpolation 9

• Computing the kriging weights involves a matrix inversion. The size of thismatrix increases with the number of points in the search neighborhood. Theincrease in computational cost for inverting the matrix does not scale linearlyto the number of sample points used. For the algorithm considered here,doubling the number of sample points results in an eightfold increase of CPUtime and a fourfold increase in memory.

• Figure 2.1 shows that with increasing distance, the spatial relationship betweendata, as defined by the auto-covariance function, is decreasing. Therefore,distant points are associated with small kriging weights and hence, do notnecessarily improve the estimation.

• Kriging algorithms are based on the assumption of first- and second-orderstationarity. This assumption is not always satisfied for the entire sampledregion. By restricting the search neighborhood, local stationarity is enforcedand the estimation is therefore more representative.

• Estimating a variogram using a fixed number of data points in a limited sam-pling region is quite tricky and succeeds in most cases only for small lags(Western and Bloschl, 1999). For this reason, using points with large lags,may in fact be detrimental.

• If too many points are used, especially clustered sample points, there is apossibility that the matrix to be inverted becomes quasi-singular, thus makingits inversion very problematic.

On the other hand, it is clear that the use too few sampled points cannot providean accurate and representative estimation. So a decision has to be made with regardto the appropriate number of points used for interpolation. In practice, this numberoften lies somewhere between 12 and 32 (Armstrong, 1998; Kelkar and Perez, 2002).

The search neighborhood can have a significant influence on the result, partic-ularly for irregularly sampled data. The size, direction and shape of the searchneighborhood therefore depends on the nature of the dataset considered and has tobe chosen carefully. It has to be taken in account that conditions may change withina region and it may not be sufficient to estimate and test the parameters at a singlelocation. The size of the search neighborhood defines also the size of a region wherethe mean is assumed to be constant. Problems can arise if many sample points areclustered or if only few sample points are close to the point to be estimated. If thenumber of points used for interpolation is constant and a large number of points isclustered in only one direction and sparse, far away points lying in other directionsare neglected, the estimation may lose reliability. If not enough data points arepresent whitin a reasonable distance and the samples come from farther and fartheraway, the validity of stationarity may not be given anymore. Nevertheless, the es-timation often improves if also points outside the range of the covariance functionare considered, rather than when too few points are used for interpolation (Isaaksand Sarivastava, 1989).

10 Theory

2.4.2 Basic Types of Kriging

There are a number of different versions of kriging. The most important ones are:ordinary kriging, simple kriging, cokriging, indicator kriging and universal kriging.In cokriging, the estimation of a variable is not only based on its own auto-covariancefunction, but also on its spatial relationship to another variable. This can be usefulif a variable is sparsely sampled but has a similar spatial relationship as extensivelysampled variable. Indicator kriging not only estimates a value for an unsampledlocation, but also provides information about the uncertainty of the estimation.The data is indicator transformed (Kelkar and Perez, 2002) and for every threshold,interpolation is applied. Therefore indicator kriging is computational intensive asseveral kriging runs are necessary for a single estimation. An other common krigingprocedure is universal kriging, in which the sample data are assumed not to bestationary, but to follow a trend. The most common kriging versions are simplekriging and ordinary kriging, which are discussed in some detail below and havebeen implemented in vebyk .

2.4.3 Simple Kriging

In simple kriging the mean of the kriged region is assumed to be known and constant.As this is not often the case, simple kriging is relatively rarely used. The method issometimes used in very large mines, such as those in South Africa, where the meanof the kriged areas is known because the region has been mined for a long time.Simple kriging estimates an unsampled value X at location ~u, as:

X( ~u0) = λ0 +n∑

i=1

(λiX(~ui)), (2.16)

where λ0 is the regional mean and λi is the kriging weight at location ~ui withthe sampled value X(~ui). By assuming that over a large number of estimations theerrors cancel each other out, we call for the condition of unbiasedness:

E[X(~u0)− X(~u0)] = 0. (2.17)

This implies that the mean error of the estimation is zero. Together with equation(2.16) this yields:

E[X( ~u0)] = λ0 +n∑

i=1

(λiE[X(~ui)]), (2.18)

where the expected values E[X( ~u0)] and E[X(~ui)] could in principle be differentform each other. Enforcing the first-order stationarity, i.e. E[X(~u0)] = E[X(~ui)] =m, the expression for λ0 is:

λ0 = m

(1−

n∑i=1

λi

). (2.19)

The error variance for the differences between the true and the estimated valuesis given by:


σ2E = V ar

[X(~u0)− X(~u0)

]= V ar

[X(~u0)−

n∑i=1

(λi ·X(~ui))− λ0

](2.20)

Going back to the definitions of variance (equation. 2.4) and covariance (equation2.6) it can be shown that:

V ar[X − Y ] = V ar[X] + V ar[Y ]− 2C(X, Y ). (2.21)

Given that λ0 is a constant, it does not influence the variance and equation 2.20can be written as:

σ2E = V ar[X(~u0)] + V ar

[ n∑i=1

(λiX(~ui))

]− 2C

[X(~u0),

n∑i=1

(λiX(~ui))

]= C(~u0, ~u0) +

n∑i=1

n∑j=1

λiλjC(~ui, ~uj)− 2n∑

i=1

λiC(~ui, ~u0).

(2.22)

To find the optimal weighting factors that minimize the error of variances, weset the first derivative with respect to λ to zero:

∂σ2E

∂λi

= 0 = 2n∑

j=1

λjC(~ui, ~uj)− 2C(~ui, ~u0) for i = 1, ..., n (2.23)

Written in matrix form this is:C(~u1, ~u1) · · · C(~u1, ~un)...

C(~un, ~u1) · · · C(~un, ~un)

λ1

...λn

=

C(~u0, ~u1)...

C(~u0, ~un)

. (2.24)

This system of equations is now solved for the kriging weights λi, which thenallows us to estimate data at unsampled locations ~u0 as:

X(~u0) = m

(1−

n∑i=1

λi

)+

n∑i=1

(λiX(~ui)

)(2.25)

2.4.4 Ordinary Kriging

Ordinary kriging is probably the most widely used form of kriging. As opposed tosimple kriging, the mean value of the region has not to be known in advance. Thisis a reasonable assumption because the only way the mean could be predicted is byassuming that the means of the often sparse datasets are representative of the globalmeans of the sampling regions, i.e. that the data obey first-order stationarity. Inpractice, however, the mean is often subject to lateral changes, which is accountedfor by ordinary kriging.

12 Theory

The mathematics of ordinary kriging is quite similar to that of simple kriging.Adopting the starting equation (equation 2.16) for an unknown mean by assumingE[X( ~u0)] = E[X(~ui)] = m( ~u0) yields:

λ0 = m(~u0)

(1−

n∑i=1

λi

). (2.26)

The value of the regional mean λ0 is not known, but it can be forced to zero,assuming the local mean m(~u0) to be the mean of the dataset. This results in thefollowing condition for unbiasedness:

n∑i=1

λi = 1 (2.27)

The minimization of the variance is quite similar to that in simple kriging. Theerror variance expressed in terms of auto-covariance (equation 2.5) is the same as inequation (2.22). For minimizing the error variance, the condition defined in equation(2.27) has to be taken in account. This can be done using the Lagrange multipliermethod (Luenberger, 1984), which defines the function F as:

F = σ2E + µ

( n∑i=1

λi − 1

)= C(~u0, ~u0) +

n∑i=1

n∑j=1

λiλjC(~ui, ~uj)− 2n∑

i=1

λiC(~ui, ~u0) + 2µ

( n∑i=1

λi − 1

),

(2.28)

where µ is the Lagrange parameter. Minimization the error variance is nowachieved by differentiating F with respect to λi and µ and setting the resultingequations to zero:

∂F

∂λi

= 2n∑

j=1

λjC(~ui, ~uj) + 2µ− 2C(~ui, ~u0) = 0 for i = 1,...,n (2.29)

and

∂F

∂µ=

n∑i=1

λi − 1 = 0. (2.30)

The kriging weights λi and µ can now be calculated by solving the two aboveequations simultaneously. To this end, these equations are written in matrix form:

C(~u1, ~u1) · · · C(~u1, ~un) 1...

...C(~un, ~u1) · · · C(~un, ~un) 1

1 · · · 1 0

λ1...

λn

µ

=

C(~u0, ~u1)

...C(~u0, ~un)

1

. (2.31)

The corresponding estimation at the unsampled location ~u0 is then given as:


X( ~u0) =n∑

i=1

λiX(~ui). (2.32)

As already mentioned above, a mayor benefit of ordinary kriging is that it doesnot require the data to be strictly first-order stationary. The mean can therefore belaterally variable and only considered to be constant within the search neighborhoodsof the points that we aim to estimate. The data are thus allowed to have a larger-scale trend, which is indeed often the case.

Chapter 3

Description of the Implementation

This section documents the Matlab implementation of the kriging interpolation de-veloped in this thesis. The code consists of a main function and several subfunc-tions (Figure 3.1). This classical modular approach facilitates the understanding ofthe code and allows later users to adapt it to their requirements by changing onlyindividual functions. The code of the functions is straightforward and largely self-explanatory. Only reference functions were used and hence no Matlab toolboxes areneeded to run the current implementation. The program was tested using MatlabVersion 6.1.

3.1 Program Options

The program is designed to interpolate values on a regular two-dimensional gridusing ordinary kriging. By slightly modifying the code, it is also possible to useit for simple kriging (see section 3.3.9). The grid of estimated or “kriged” valuesis rectangular and spans the range of coordinates in the dataset of sampled values.The spacing between estimated points for x - and y-axes can be specified individuallyand the sampled values do not have to follow a spatial order. For the search neigh-borhood, the number of points used for interpolation can be specified. Arbitrarilyoriented spatial anisotropy of the covariance function can be accounted for. Thecurrent implementation is based on the von Karman family of covariance function.The correlation length a and the ν-value of the von Karman function can be spec-ified. Finally, “handles” are provided to switch on a wait bar or a cross validationmode for comparison of the interpolated value at a sampled location.

3.2 Overall Structure of the Code

The implementation of the code is structured hierarchically, as shown in Figure 3.1.The main function vebyk (value estimation by kriging) manages the data in/outputand administrates the different steps before and after the actual kriging routine. Theactual interpolation for unsampled points is performed by the kriging3 functionthat gets support from other functions.

14

3.3 Description of the Individual Functions 15

vebyk

inputmatrix rotationneighborhoodkriging3

buildbigc3buildsmallc3ordinary

covcalc displacement3

Figure 3.1: Hierarchical structure of functions used in the implementation of the krigingalgorithm.

3.3 Description of the Individual Functions

3.3.1 vebyk

vebyk is the main function and therefore has to be given all the information used.Table 3.1 describes the parameters which have to be specified. In order to keepthe code simple, vebyk does not check or echo input parameters. The calculation ofdistances and relative positions are based on two-dimensional Cartesian coordinates.The sampled values for input do not have to lie on a specific grid, but can beanywhere in the interpolated region. The first thing the function does, is to calculatethe coordinates of the grid, for which values should be estimated. This is donefor a domain of rectangular shape defined by minimum and maximum x - and y-coordinates of the sampled values. Spacing between points is given by the inputparameter dgrid. The coordinate grid is created using the function inputmatrix.The next step is the rotation of the coordinates for cases in which the anisotropyaxes are not parallel to the axes of the coordinate system. This is described in moredetail in sections 3.3.10 and 4.2. Then a for loop is started, which repeats thenext steps for every point for which a value has to be estimated. The estimationis performed by first calling up the neighborhood function (see section 3.3.4) forevaluating the points used for kriging. The relative coordinates of these points aregiven to the kriging3 function, which calculates the kriging weights (see section3.3.2). The weights are summed up with the corresponding values to to establishthe estimation. After this has been done for every point in the grid, any previouscoordinate transformation is reversed by the rotation function.

3.3.2 kriging3

This function calculates the kriging weights for a specific point. The system ofequations for the kriging weights is built up by calling buildbigc (section 3.3.5)

16 Description of the Implementation

Output Input

outputoutput grid with es-timated values

coordcoordinates andvalues of sampledpoints

error-variance

errorvariance at es-timated locations

dgriddistance betweengrid points

pointsnumber of pointsused for interpola-tion

anisotropyproportion of aniso-tropy

alphaangle between co-ordinate and aniso-tropy axes

nuparameter of thevon Karman covari-ance function

rangedistance of spatialcorrelation

crossvhandle to switchon cross validationmode

verbosehandle to switch onweight bar

Table 3.1: Parameters for vebyk.

Output Input

lambda kriging weights positionrelative coordinatesof sampled points

error-variance

errorvariance at es-timated locations



rangehorizontal correla-tion length

Table 3.2: Parameters for kriging3.

and buildsmallc (section 3.3.6). Using the input parameters shown in Table 3.5,the first function constructs the matrix C containing all covariances among the sam-pled points, while the second function returns a vector ~c containing the covariancesbetween all sampled point and the point to estimate. This would already sufficefor simple kriging. The function ordinary expands the matrix and the vector forordinary kriging as described in section 2.4.4. For changing the interpolation rou-tine to run with simple kriging, the line calling up the ordinary function can be


commented out. Solving the system of equations is done by inverting the matrixC and multiplying with ~c. The error variance is calculated to provide informationabout the reliability of the estimated value. Input and output parameters of thefunction are shown in Table 3.2.

3.3.3 inputmatrix

Output Input

inputn×3 matrix of coor-dinates and values

matrixrectangular matrixcontaining only val-ues

dxdistance betweenvalues in x direc-tion

dydistance betweenvalues in y direc-tion

Table 3.3: Parameters for inputmatrix.

The function inputmatrix generates a matrix with three columns containingthe x- and y-coordinates in the first two columns and the corresponding parametervalues in the third column. Each row in the matrix thus specifies one data point.The input is for this function an array of values defined on a evenly spaced grid.The spaces between the sampled points in x- and y-direction are given by the inputparameters of the function (Table 3.3). The function is used in vebyk to create thegrid of estimation points, but can also be used to convert a rectangular input matrixwith equally spaced values to the format required by vebyk.

3.3.4 neighborhood

neighborhood is the function that chooses the points used for the estimation process.As mentioned in section 2.4.1, the search neighborhood has considerable influenceon the result of the kriging interpolation. Therefore, the search neighborhood shouldbe adapted to the dataset used. The neighborhood function defines a simple searchneighborhood by choosing a defined number of sampled points lying next to thepoint to be estimated. The points are figured out by giving respect to the distancebetween sampled and estimated point only. Anisotropy is taken in account as thesearch neighborhood becomes quickly too small in one direction if there is significantstructural anisotropy. As kriging is a smooth estimator, a too small search neigh-borhood may result in artefacts in the form of high frequent oscillations (see section4.2 and Figure 4.5). neighborhood calculates the distances between sampled pointsand the location on which the estimation is made. Therefore the x- and y-axesare scaled in accordance with the structural anisotropy. Then it takes the specifiednumber of points with the smallest lags and calculates the relative coordinates. Ifthe cross-validation handle is on, the first point is skipped in order not to use the


Output Input

valuessampled values ofthe points used forkriging

xx-coordinate of thepoint to estimate

positionrelative coor-dinates of thesampled points

yy-coordinate of thepoint to estimate

pointsnumber of pointsused for interpola-tion

coord

coordinates andvalues of all avail-able sampledpoints

crossvhandle to switchon cross validationmode


Table 3.4: Parameters for neighborhood.

sampled data at a sampled location (see section 4.4). Alternatively, neighborhood2,which is the implementation of a quadrant search (Isaaks and Sarivastava, 1989),can be used instead of neighborhood. This is useful if the sampled points are clus-tered. Section 4.3 shows an example of artefacts caused by non adequate searchneighborhood.

3.3.5 buildbigc

Output Input

Cmatrix with covari-ances between sam-pled points

positionrelative coordinatesof sampled points




Table 3.5: Parameters for buildbigc.

This is a subfunction of kriging3, which calculates the matrix containing thecovariance values among the sampled points (see equation 2.24). displacement3 cal-


culates a matrix containing the distances between the sampled points. Therefore, thex- and y-axis are scaled according to the structural anisotropy. The displacement3function uses the relative coordinates of the search neighborhood, which are pro-vided by the neighborhood function. The lag values of the matrix are then replacedwith the corresponding covariance values by the function covcalc. As C is a sym-metric matrix, only one half has to be explicitly calculated. Parameters used andcalculated by buildbigc are shown in Table 3.5

3.3.6 buildsmallc

Output Input

c

matrix with covari-ances between sam-pled points and es-timated location





Table 3.6: Parameters for buildsmallc.

buildsmallc generates a vector containing the covariance values for the lagsbetween the data points used for interpolation and the point which is subject tointerpolation. In equation 2.24 this is the vector on the right. The functionality ofbuildsmallc is the same as in buildbigc with the difference that only a few pointshave to be calculated and the computation of the lags is straightforward by usingthe relative coordinates. Table 3.6 shows that the input parameters are the same asfor buildbigc.

3.3.7 displacement3

Output Input

dist

distances betweensampled pointswith respect to theanisotropy



Table 3.7: Parameters for displacement3.


The displacement3 function is called by buildbigc for calculating the lagsamong the sampled points used for interpolation. The first row of the resultingmatrix contains the lags between the first point and all other points. The secondrow contains the lags between the second point and all other points and so on. Thediagonal of the matrix are the lags between identical points and hence uniformlyzero. As the lag is the same from point A to B as from B to A only one half ofthe matrix has to be calculated explicitly. Anisotropy is taken in account in thecalculation of the lags by multiplying the axis with the shorter correlation lengthwith the anisotropy factor (correlation length in x-direction divided by correlationlength in y-direction). Table 3.7 shows the input and output parameters.

3.3.8 covcalc

Output Input

covariancevalue of the covari-ance function at acertain lag

lagargument of the co-variance function


ahorizontal correla-tion length

Table 3.8: Parameters for covcalc.

This function evaluates the used parametric model of the auto-covariance func-tion. The von Karman family of auto-covariance functions (see section 2.3), which iscurrently used in the covcalc routine has a wide range of shapes that are controlledby the parameter ν and the correlation length a. These parameters are input di-rectly into vebyk. This covariance model can fit a wide variety of practically relevantauto-covariance functions. Therefore, a wide range of datasets can be interpolatedwithout changing the covcalc function. At zero lag, the auto-covariance functionis normalized to the value of one (assuming that there is no nugget effect) as thenumerical evaluation of the von Karman function at lag zero tends to be problem-atic, particularly for small ν-values. The parameters of the function are describedin Table 3.8.

3.3.9 ordinary

Ordinary kriging does not assume the mean of a region to be given, but insteadforces the kriging weights to sum up to one (see section 2.4.4). The correspondingeffect on the calculation of the kriging weights is relatively small. The equation forcalculating the Lagrange parameter has to be added. This is done by expanding thematrix C by one row and one column of ones and the enlargement of the vector ~cby one element of value one. As the Lagrange parameter is not to be included insummation of the kriging weight, the value at the bottom right corner of the matrix


Output Input

Cmatrix with covari-ances between sam-pled points

CC added a line anda column with onesfor ordinary kriging

c

matrix with covari-ances between sam-pled points and es-timated location

cc added a one foreordinary kriging

Table 3.9: Parameters for ordinary.

is set to zero. Input and output are described in Table 3.9. If the implementationhas to be changed to run with simple kriging, the function ordinary has to becommented out and the line where the estimation is performed using the krigingweights must be changed to conform with equation (2.25).

3.3.10 rotation

Output Input

rotcoord

coordinates rotatedaround the originat the angle givenby alpha

coordcoordinates andvalues of thesampled points

alphaangle between theaxes and the aniso-tropy

Table 3.10: Parameters for rotation.

In cases where the axes of the anisotropy ellipsoid do not coincide with the axesof the coordinate system, this can be specified in vebyk through the correspond-ing rotation angle. One possibility to take this into account would be to performto corresponding adjustments for every estimation. This would take place in thedisplacement function, where also the anisotropy parallel to the axis is considered.The relative coordinates would have to be rotated by the specific angle. An eas-ier way to account for the angle of anisotropy is to rotate the coordinates of allsampled points and the grid of estimated values in order to have the anisotropy inthe direction of the axes. This is equivalent to an eigenvalue transformation. Nowthe interpolation can be made as if the axes of the coordinate system coincide withthose of the anisotropy ellipsoid. Afterwards, the coordinates are rotated back tothe places they belong. The direction of the anisotropy is expected to be the same inthe whole region and therefore it does not matter which point is the center of the ro-tation. Here the rotation is done around the origin of the coordinate system becausethe equations are simplest for this case. The corresponding rotation of Cartesiancoordinates is given by (Papula, 1994):


u = y sin(ϕ) + x cos(ϕ),v = y cos(ϕ) − x sin(ϕ),

(3.1)

where x and y are the primary coordinates, u and v the corresponding rotatedcoordinates and ϕ is the rotation angle. The angle ϕ has to be measured as shownin Fig. 3.2. The anisotropy ratio corresponds to ratio of lengths of the axis of theanisotropiy ellipsoid (i.e., to the correlation lengths in u- and v-directions)(Figure3.2):

kany =au

av

. (3.2)

Input parameters for this function are the rotation angle in radians, and thematrix containing the coordinates to be rotated. The first two rows of the matrixare the x- and y coordinates and the third column contains the data values.

jx

y

au

av

v u

Figure 3.2: If the direction of the axes of the anisotropy ellipsoid differ from the directionof the axes of the coordinate system, the angle φ is measured as shown above. au and av

denote the correlation lengths in the u- and v-directions.

Chapter 4

Testing of the Implementation

4.1 Comparison with a published example

P1 P2 P0 P3 P4

Figure 4.1: Locations of the four points P1, P2, P3, P4 with observed values and the pointP0, where the estimation is performed. After Armstrong (1998, pp. 105 ff)

PointsKriging weightsobtained by Arm-strong (1998)

Kriging weights ob-tained by vebyk

P1, P4 -0.47 -0.475P2, P3 0.547 0.5475

Table 4.1: Comparison of the results published by Armstrong (1998) with those obtainedby vebyk.

To verify the accuracy of the implemented kriging algorithm, we first compareits output with that of a simple published example (Armstrong, 1998, pp. 105ff).The spatial arrangement of the example is shown in Figure 4.1. The four sampledpoints are equally spaced on a line with 1 m intervals and the point to be estimatedlies in the center. The used covariance function, is given by C(τ) = 1 − |τ |1.5.Unfortunately, I could not find any suitable published examples for which the vonKarman auto-covariance model could be used. It should, however, be noted thatthe definition of the model auto-covariance function only corresponds to one line inmy code that is essentially independent of the core of the actual kriging algorithm.Table 4.1 shows a comparison of the results published by Armstrong (1998) withthose obtained with my program. The slight differences may be caused by differentequation solvers or may simply represent a rounding effect, as only three digits aregiven by Armstrong (1998, pp. 105ff). Overall, this comparison indicates that thecore of the algorithm works correctly.

23

24 Testing of the Implementation

4.2 Structural Anisotropy

Structural anisotropy implies that the decay of the auto-covariance function dependson the direction in which it is measured or modeled. Anisotropy is generally assumedto be elliptical in nature, i.e., there are two perpendicular main axes along whichthe auto-covariance function exhibits different correlation lengths (Figure 3.2). Theanisotropy factor denotes the ratio between the maximum and minimum correlationlength (equation 3.2). The effects of structural anisotropy can be accounted forthrough a suitable coordinate transform. As the coordinate axes are rotated andcompressed, the ellipse is transformed into a circle and kriging can then be performedas if there were no anisotropy.

In the presence of anisotropy, care has to be taken with regard to choice of thesearch neighborhood. The kriging weights decrease in the direction of the long axisof the anisotropy ellipsoid (Figure 3.2). If the search neighborhood is symmetricaround the estimated point it tends to be too small in the direction of the longanisotropy axis and unnecessarily large in the other direction. This has the effectthat kriging weights at the border of the search neighborhood become larger thankriging weights in the immediate vicinity of the estimated point (Fig. 4.2). This is

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Figure 4.2: Kriging weights for too small a search neighborhood. A total of 16 pointswhere used for the interpolation. The estimated point is in the center of the region. Thekriging weights exist actually only on locations where a sampled point is available. Thispicture is obtained by interpolating a regular grid of zeros with one sampled point witha value of one in the middle of the region. The estimation grid is sampled at half thedistance of the input dataset and the estimation grid is slightly shifted to avoid that thesampled data points dominate the output of the estimation. The kriging weights dependon the spatial relationship amongst each other. This implies that if sampled data pointswhere not on a regular grid, the shape of the kriging weight function would be somewhatdifferent.

4.2 Structural Anisotropy 25

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Figure 4.3: Same as Figure 4.2 but for 36 points. The kriging weights on the border of thesearch neighborhood have become notably smaller. The computational efforts have risennintefold compared to those for Fig. 4.2.

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Figure 4.4: Same as Figures 4.2 and 4.3, but for an anisotropic search neighborhood usinga total of 16 points. Larger kriging weights on the border of the search neighborhood haveessentially disappeared.

clearly an artefact as, according to the auto-covariance model, the spatial relation-ship decreases monotonously with increasing distance. If the search neighborhoodis enlarged this effect decreases, as can be seen in Fig. 4.3. It can also be seen that


in the direction of the short anisotropy axis far too many points are used, whichunnecessarily increases the computational effort. It is therefore most appropriate toadapt the shape of search neighborhood to that of the anisotropy ellipsoid. Figure4.4 shows the corresponding results.

In summary, if the search neighborhood is too small, the kriging weights are spa-tially cropped. As illustrated by Figure 4.5, this results in high-frequent oscillationsin the output. Kriged regions are expected to be smooth. But as the search neigh-borhood gets too small, the surface becomes more “edgy” and finally high-frequentoscillations appear. This effect is not very obvious on a two-dimensional plot of alarger kriged region, but it can be clearly discerned when taking profiles across theinterpolated surface (Figure 4.5).

4.3 Search Neighborhood

The use of a simple search neighborhood is a suitable practice if the sampled valuesare evenly or randomly spread. As kriging accounts for slight to moderate cluster-ing of the data, it is usually not necessary to use an elaborate search neighborhood.Nevertheless, some heavily clustered datasets may cause artefacts in the interpola-tion. A corresponding example is the kriging interpolation of two neutron porosity

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Figure 4.5: High-frequent oscillation caused by too small a neighborhood. The graph showsa slice through an interpolated region. The abscissa shows the number of the interpolatedpoint, whereas the ordinate gives the estimated value at this point. Every fourth pointis a sampled point and is green marked in the illustration. The red curve is obtainedwhen using a too small search neighborhood. For the blue curve a search neighborhood ofadequate size was used.

4.3 Search Neighborhood 27

logs. This implies that a large area without any observed data has to be inter-polated based on densely sampled, but inherently one-dimensional borehole logs.Using a simple search neighborhood causes the interpolation to use data of of theone borehole only closest to the point to be estimated. Figure 4.6 shows the resultingartefacts in the middle of the interpolated region, where the two unrelated halvesmeet. The quadrant search method (Isaaks and Sarivastava, 1989), where points ofall four quadrants are considered, is an effective tool to avoid this kind of artefacts.This is illustrated in Figure 4.7.

Figure 4.6: The use of a simple search neighborhood can cause artefacts if the sampled datapoints are heavily clustered. This figure shows a kriging interpolation of two neighboringborehole logs. As only the nearest points are considered for interpolation, the right andleft halves of the interpolated region are independent of each other. The obvious artefactsin the middle clearly demonstrate that a simple search neighborhood fails in this example.


Figure 4.7: Kriging interpolation of the same borehole data as in Figure 4.6. This timea quadrant search neighborhood was used. Considering data of all four quadrants is astraightforward and efficient way to avoid artefacts in heavily clustered data.

4.4 Cross-Validation

Cross-validation allows to assess how well kriging works for a given dataset. This isachieved by performing the interpolation at locations where observations are avail-able and assessing the discrepancies between the observed and estimated values. Inparticular, cross-validation is commonly used to test the practical validity of theused model for the auto-covariance function (Kelkar and Perez, 2002).

In this study, cross-validation is primarily used to check the implementation ofthe kriging algorithm, as the auto-covariance function of the sampled dataset issupposed to known in advance. There are several methods for cross-validation. Thetechniques used in this study are referred as “leaving-one-out” and “jackknifing”.

As indicated by its name, the leaving-one-out cross-validation technique esti-mates a value for a sampled point by leaving out the observed point at which theestimation is made. A comparison of the resulting discrepancies allows to assess the

4.4 Cross-Validation 29

accuracy of the used auto-covariance model. If all the information of the sampleddata is used to model the spatial relationship, as this is the case for the leaving-one-out cross-validation, the estimated value is not strictly independent from thesampled value.

For the more rigorous jackknifing test, the estimated value is independent fromthe sampled value. This is achieved by first dropping part of the sampled data andthen estimating them through kriging. Jackknifing is, however, only applicable if asufficient amount of sampled points is available.

4.4.1 Leaving-One-Out Cross-Validation

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Figure 4.8: Crossplot of true versus estimated values. As the absolute error is not in-creasing with increasing data values, the estimation is conditionally unbiased. This is animplicit characteristic of kriging.

For testing the implementation a 20× 20 stochastic dataset with a known auto-covariance function was used. By using the leaving-one-out cross-validation methodthe estimated values for the sample points were calculated. As proposed in Kelkarand Perez (2002), a crossplot between true and estimated values (Figure 4.8), acrossplot of errors versus true values (Figure 4.9) and a colorplot of the errors weregenerated.

Figure 4.8 shows the crossplot between true and estimated values. The pointsare equally spread around the 45◦-line showing that the estimate is conditionallyunbiased, that is, the errors are independent of the amplitude of the observed val-ues. This demonstrates that implemented kriging algorithm is indeed unbiased asrequired by the methodological foundations. In Figure 4.9 the corresponding es-timation errors are plotted against the true values. The errors do not show any


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Figure 4.9: Crossplot of true value versus the corresponding estimation error. The evendistribution of the points around the zero line indicates homoscedasticity of the errorvariance.

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systematic variation in magnitude with increasing sample values, which implies thatthe estimation shows homoscedastic behavior. That is, the variance around the re-


gression line is the same for all estimated values. The colorplot of the error valuesshown in Figure 4.10 should reveal any systematic spatial relationships present inthe error values. When using the correct auto-covariance function, the kriging errorsof the estimation should not be spatially related. Figure 4.10 indicates that this isindeed the case.

4.4.2 Jackknife Cross-Validation

For jackknifing, a stochastic dataset with 200 × 200 values and a known exponen-tial auto-covariance function was generated (Figure 4.11). Then an equally spaced“observed” dataset for the interpolation was extracted from this initial model by dec-imating it fourfold (Figure 4.12). This observed data then was used for estimationof the skipped data points. Figure 4.13 shows the result of the kriging interpolation.For comparison the interpolation of the same input dataset was also performed usinga spline interpolation (Figure 4.14), a very common interpolation technique, wherea polynomial is fitted to the sampled data points (de Boor, 1978). Compared tothe spline interpolation, kriging provides a sharper image that is much more closelyrelated to the initial model (Figures 4.11 and 4.14). The computational efforts forkriging interpolation are, however, much larger than those for spline interpolation.

The differences between spline and kriging interpolation increase with increas-ing roughness and complexity of the initial dataset. Figure 4.15 shows a stochasticmodel characterized by a von Karman auto-covariance function with ν = 0. Themodel estimated by kriging (Figure 4.16) is again much more closely related to theoriginal model than the spline interpolation (Figure 4.17). A conspicuous feature of

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Figure 4.11: Stochastic model characterized by von Karman auto-covariance function withσ2 = 1, ν = 0.5, horizontal correlation length a = 100 and anisotropy kany = 10.


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Figure 4.13: Kriging estimation of the input model shown in Figure 4.12. The samplinginterval is the same as in Figure 4.11.

the kriged image is, however, that the sampled/observed points “stand out” fromthe interpolated background. This effect is also observed if the input dataset is nota regularly spaced grid, but randomly chosen as for the input dataset of Figure 4.18.I found this observation to be universally characteristic for input data with ν-values


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Figure 4.14: Spline interpolation of input model shown in Figure 4.12. The samplinginterval is the same as in Figure 4.11.

smaller than 0.5. These “freckles” arise from the high variability of neighboringsampled data points. As the interpolated space is regarded as stationary, the sam-pled values are dispersed around a local mean. Kriging interpolation predicts theexpected value for an estimated point based on the sampled points in the searchneighborhood. It is therefore a prediction of a random value in consideration of itssurrounding values. This is only possible, if the predicted value is not completelyrandom. The stochastic models with auto-covariance functions of the von Karmanfamily are so called fractional Gaussian noise (fGn). fGn has the property, that itsspectral density function scales with:

P ∝ fβ, (4.1)

where β is related to ν as:

β = −(2ν + 1). (4.2)

The short-term predictability for scale-invariant stochastic data depends on theν-value. Bandwidth ranges between good, linear predictability for β ≥ 3 to entirelyunpredictable for β < 1 (Hergarten, 2003). For β < 1 the best value prediction isthe expected value of the dataset, i.e., the global mean. But as the actual medium isin fact of very rough and complex, most sampled data points have values that differsignificantly from the global mean. Kriging of such a medium therefore creates asmooth interpolated surface close to the mean of the used search neighborhoods fromwhich the observed points “stand out” (4.16 and 4.18). Scale-invariant stochasticdata with 0 ≤ ν < 0.5 are refered as “anti-persistent”. This means that the gradientis likely to turn negative if it has been positive before and vice versa. Controversely,


for 0.5 < ν < 1 scale-invariant stochastic data are “persistent” and the gradients areexpected to remain positive if it was positive before and vice versa (Hergarten, 2003).The behavior of kriging interpolation, as of most other interpolation procedures, isinherently persistent in nature and hence the interpolation may create artefacts ifthe data to be interpolated are in fact anti-persistent in nature.

−3

−2

−1

0

1

2

3

x [points]

y [p

oint

s]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure 4.15: Stochastic model characterized by von Karman covariance function withσ2 = 1, ν = 0, horizontal correlation length a = 100 and anisotropy kany = 10.


−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x [points]

y [p

oint

s]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure 4.16: Kriging estimation of a dataset generated by taking every fourth point of themodel in Figure 4.15.

−3

−2

−1

0

1

2

x [points]

y [p

oint

s]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure 4.17: Spline interpolation for the model shown in Figure 4.15.


−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x [points]

y [p

oint

s]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure 4.18: Kriging estimation of a dataset generated by taking 2500 random points fromthe model shown in Figure 4.15.

4.5 Sensitivity of Kriging Estimation with Regard to Auto-Covariance Model 37

4.5 Sensitivity of Kriging Estimation with Regard

to Auto-Covariance Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

ν−value of input model

best

fitti

ng ν

−va

lue

Figure 4.19: Relationship between the ν-value of the auto-correlation function of the initialmodel and the ν-value of the model auto-correlation function that leads to the smallestsum of kriging errors (equation 4.3). Stars denote the explicitly calculated values. Thesolid line is a quadratic interpolation for ν ≤ 0.7 and a corresponding extrapolation forν > 0.7.

The kriging results depend on the search neighborhood and the chosen modelfor the auto-covariance function. As discussed in section 2.2, the auto-covariancefunction often cannot be determined accurately from the available data. Therefore,it is important to know how stable and robust the kriging estimation behaves withregard to the used auto-covariance model.

To this end, eleven stochastic models with known auto-covariance functions werecreated. The auto-covariance models are all of the von-Karman-type (σ2 = 1, a =100, kany = 10) and differ only in terms of their ν-values. The ν-values are increasedin steps of 0.1 from ν = 0 for a very rough and complex model to ν = 1 for avery smooth model. Every fourth value in x- and y- axis direction was taken forthe input dataset. The dataset was then kriged using von Karman auto-covariancefunctions with wide range of ν-values and anisotropy ratios keeping the horizontalcorrelation length fixed at the correct value. The sum of absolute error betweenthe estimated and sampled values is taken as an indicator for the quality of theinterpolation. Figure 4.19 shows the best fitting ν-values against the true ν-valuesof the stochastic models. The resulting “trade-off maps” for input models with ν-values 0, 0.5 and 0.7 are shown in Figures 4.20, 4.21 and 4.22. The minimal errorvalues are marked with a red cross.

It is noticeable that the ν-values that lead to the lowest error values are all


considerably larger than the true ν-value of the auto-covariance function of the inputmodel. This difference increases with increasing roughness character of a medium.At least in the range 0 ≤ ν ≤ 0.7 the ν-values of the input model νtrue are relatedto the best-fitting ν-values νbest as:

νbest = −0.5 ν2true + 1.1 νtrue + 0.55. (4.3)

This reflects the attitude of kriging not to preserve the auto-covariance function ofthe initial dataset, but to calculate the expected value at interpolated locations. Theauto-covariance function of a kriged dataset, therefore, will always be smoother thanthat of used auto-covariance model. This also finds its expression in the fact thatsum of the absolute errors in the trade-off maps shown Figures 4.20, 4.21 and4.22decreases marked with increasing ν-values. Appendix A shows kriging interpolationof the model presented in Figure 4.11 with different ν-values and the “trade-off”maps for cross-validation with different correlation lengths.

540

560

580

600

620

640

Anisotropy

ν

2 4 6 8 10 12 14 16 18 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.20: Sum of absolute kriging errors as a function of the ν-value and the anisotropyratio kany. The input dataset has a von Karman covariance function with ν = 0, ahorizontal correlation length ax = 100 and an anisotropy ratio kany = ax

ay= 10. The

subscripts denote the axis direction of the correlation length.

4.5 Sensitivity of Kriging Estimation with Regard to Auto-Covariance Model 39

100

110

120

130

140

150

Anisotropy

ν

2 4 6 8 10 12 14 16 18 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure 4.21: Sum of absolute kriging errors as a function of the ν-value and the anisotropyratio kany. The input dataset has a von Karman covariance function with ν = 0.5, ahorizontal correlation length ax = 100 and an anisotropy ratio kany = ax

ay= 10. The

subscripts denote the axis direction of the correlation length.

45

50

55

60

65

70

75

80

85

Anisotropy

ν

2 4 6 8 10 12 14 16 18 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure 4.22: Sum of absolute kriging errors as a function of the ν-value and the anisotropyratio kany. The input dataset has a von Karman covariance function with ν = 0.7, acorrelation length ax = 100 and an anisotropy ratio kany = ax

ay= 10. The subscripts

denote the axis direction of the correlation length.

Chapter 5

Application of Kriging toConditional GeostatisticalSimulations

Conditional simulation is a complementary geostatistical tool to interpolation. Asopposed to interpolation, which has the objective to provide estimates that are sta-tistically as close as possible to the unknown true value at any particular location,the goal of conditional simulation is to represent the inherent spatial variability of amedium. Unlike kriging interpolation, conditional simulation therefore reproducesthe second-order statistical attributes (i.e., mean value and auto-covariance function)of the considered dataset. The error variance of the simulated values is, however, notminimized and a simulated value is not necessarily the best estimation, in a statisti-cal sense, for a specific point. Conditional simulation is therefore not adequate, forexample, to estimate total reserves of an ore deposit, but it can adequately illustratethe inherent variability of the ore concentration in the deposit. Many conditionalsimulations allow to draw probability maps of a region. The mean of many condi-tional simulation maps correspond to a map of the expected values, which is similarto a kriging interpolation. This is required, for example, for optimization of miningand hydrocarbon recovery processes as well as for detailed ground water flow andcontaminant transport simulations (Hardy and Beier, 1994).

A conditioned simulation is one of an infinite number of possible realizations ofthe space between sampled data points. The realization is constrained by the meanvalue and the auto-covariance function of the observed dataset and is conditionedby the sampled values. There are many different approaches to conditional simula-tion (Kelkar and Perez, 2002). A simple and effective way is to use unconditionalstochastic simulation and kriging interpolation (Goff and ”Jennings, 1999; Journeland Huijbregts, 1978) as following:

1. Compute unconditional simulation Xu(~u).

2. Sample XU(~u) at locations ~ui (i ∈ 1,2,...,N) with available observed data andcompute differences: ∆X(~ui) = X(~ui)−XU(~ui).

3. Perform kriging interpolation: ∆X(~ui) → ∆XI(~u).

40

5.1 Unconditional Simulation 41

4. Add kriged values to unconditioned simulation: XC = XU(~u) + ∆XI(~u).

The space between the locations with observed data is smoothly interpolatedthrough kriging, as can be seen in Figure 4.7, which shows the state of the conditionalsimulation after step three. The unconditional model is then superimposed on tothis smooth kriged data structure.

5.1 Unconditional Simulation

An unconditional simulation is a second-order stationary realization of a randomvariable with given mean value and auto-covariance function. Although such real-izations are possible with any auto-covariance function, the band-limited “fractal”von Karman model (see section 2.3) has been applied with particular success in manyfields (Goff and Jordan, 1988; Holliger, 1996; Holliger and Levander, 1993) and willalso be used here. There exist different possibilities to obtain a unconditional sim-ulation (Journel and Huijbregts, 1978). A straightforward and efficient way to per-form a unconditioned simulation is the spectrum method (Christakos, 1992). Withthe increasing availability of powerful desktop computers, such realizations usingtwo- (Goff and ”Jennings, 1999) and three-dimensional (Chemingui, 2001) discreteFourier transforms have become very attractive. Unconditional stochastic simula-tions of this type are performed by taking the inverse discrete Fourier transformation(IFFT) of the amplitude spectrum A of considered stochastic process:

XU(~u) = IFFT [A(~k)ei2πϕ(~k)], (5.1)

where ~k is the wavenumber. The phase value ϕ is a uniformly distributed randomnumber sampled between 0 and 1.

As discussed in section 2.2 the spectral density function is the Fourier transformof the correlation function of a variable. The spectral density function is thus alsothe square of the amplitude spectrum. Therefore, a dataset with an auto-covariancefunction C(~r) can be generated by computing the spectral density function of theauto-covariance function and taking the square root to obtain the amplitude spec-trum in equation 5.1. For the von Karman family of auto-covariance functions, theamplitude spectrum can be calculated analytically. The Fourier transform of thevon Karman auto-covariance function (equation 2.3) and therefore also the spectraldensity function for E-dimensional space is given by (Holliger, 1996):

Phh(~k) =σ2

h(2√

πa)EΓ(ν + E/2)

Γ(ν)(1 + ~k2a2)ν+E/2, (5.2)

where a is the correlation length. For the two-dimensional case (E = 2) thisyields:

Phh−2D(~k) =4πaνσ2

h(1 + (~ka)2

)ν+1 . (5.3)

42 Application of Kriging to Conditional Geostatistical Simulations

As the final variance can be adjusted by simple scaling, the variance in equation5.3 can be chosen as σ2

h = 14πaν

. The amplitude spectrum of the unconditionedsimulation with a von Karman covariance function then yields:

|A(~k)| =√(

1 + (~ka)2)−(ν+1)

(5.4)

An unconditional simulation can now realized following equation 5.1. The datasethas to be normalized with regard to mean, amplitude and variance after discreteinverse Fourier transformation. Examples of unconditional simulations based onvon auto-covariance functions are shown in Figure 4.11 and Figure 4.15.

5.2 Conditional Simulations of Porosity Distribu-

tions in Heterogeneous Aquifers

In hydrology, detailed knowledge of the spatial porosity distribution is pre requisitefor constraining the permeability structure, which then allows to simulate the flowand transport properties of an aquifer. Porosity data are obtained from boreholelogs, which have a high resolution in the vertical dimension but generally a very lowlateral coverage. This results in a correspondingly vague interpretation of the regionbetween the boreholes. Various geophysical methods, such as resistivity sounding,crosshole georadar and seismic tomography, can be used to improve our knowledgeof the lateral porosity distribution. However, due to their limited resolution, thesemethods tend to produce an overly smooth picture of the subsurface. As flow andtransport simulations are strongly influenced by the local variability of porosity data(Hassan, 1998), conditional simulations based on borehole logs and geophysical mea-surements are critical for improving such simulations. I have produced conditionalsimulations for two different sites. At both sites there were porosity borehole logsand crosshole georadar tomography data available. In addition to this, crossholeseismic data were available at one site.

To perform a conditional simulation with these geophysical data, the followingsteps were necessary:

1. Estimate ν-value, vertical correlation length, and variance from porosity logs.

2. Convert tomographic data to porosity and estimate structural aspect ratio andits orientation.

3. “Arbitrarily” fix vertical or horizontal correlation lengths so that structure isscale-invariant at considered model range.

4. Subsample spatial porosity structure to resolution of tomographic data.

5. Scale porosity field derived from tomographic data to match variance of poros-ity logs.

6. Perform conditional simulation of spatial porosity structure at the desired gridspacing.

5.2 Conditional Simulations of Porosity Distributions in Heterogeneous Aquifers 43

7. Repeat previous step several times for different unconditional realizations toobtain an estimate of the bandwidth of the variability.

5.2.1 Boise

Based on crosshole georadar tomography and porosity logs of two nearby boreholesat the Boise Hydrogeophysical Research Site (BHRS) a conditional simulation of theporosity structure has been performed. BHRS is a testing ground for geophysicaland hydrological methods in an unconfined alluvial aquifer near Boise, Idaho. Thecrosshole georadar tomography data and the porosity logs were provided by Tronickeet al. (2003).

To obtain a conditional simulation of the tomographic data and the porositylogs, the following parameters were defined:

ν-value To construct an unconditional simulation, it is critical to estimate the vari-ability of the simulated medium. This can be done on the basis of sampledvalues or by using a priori information. The use of an a priori ν-value in caseof porosity simulations seems feasible as many different analyses of porosityhave shown the spatial distribution to behave uniformly as fractional-Gaussian-noise (fGn) with a ν-value close to zero. A straightforward method to estimate

75

80

85

90

95

0 5 10

2

4

6

8

10

12

14

16

18

Distance [m]

Dep

th [m

]

o = Source x = Receiver

C5 C6

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10

2

4

6

8

10

12

14

16

18

Distance [m]

Dep

th [m

]

o = Source x = Receiver

C5 C6

Figure 5.1: Crosshole georadar tomography between boreholes C5 and C6 on Boise Hy-drogeophysical Research Site (BHRS). Left: velocity in m/µs, right: attenuation in 1/m.After Tronicke et al. (2003)


0.1 0.2 0.3 0.4 0.5

4

6

8

10

12

14

16

18

Porosity

Dep

th [m

]C5

0.1 0.2 0.3 0.4 0.5

4

6

8

10

12

14

16

18

Porosity

Dep

th [m

]

C6

Figure 5.2: Porosity logs for boreholes C5 and C6 at BHRS. Sample interval of both logsis 0.06m. After Tronicke et al. (2003).

the ν-value of a stochastic time-series is to analyze the behavior of the spectraldensity function. The analysis is done here for the densely sampled porositylogs rather than for the sparsely sampled, smoothed tomographic data. Thespectral density function P of a scale-invariant sequence scales with angularfrequency ω as (Hardy and Beier, 1994; Holliger, 1996):

P ∝ ωβ, (5.5)

where β lies is between −1 and −3. If the logarithm of the spectral densityis plotted against the logarithm of the frequency, β denotes the slope of thelinearly decaying spectral density:

ln(P ) ∝ β · ln(ω). (5.6)

Figure 5.3 shows a double-logarithmic plot of the spectral density of one ofBHRS porosity logs. A linear regression of the slope indicates that β is around-0.97. The relationship between the slope of spectral density and the ν-valueis (Hardy and Beier, 1994; Holliger, 1996):

β = −(2ν + 1). (5.7)

This results in a ν-value of -0.015, which is approximated as ν = 0.


−5 −4 −3 −2 −1 0 1 2 3 4 5−7

−6

−5

−4

−3

−2

−1

0

1

log (angular frequency)

2 *

log

(am

plitu

de)

Figure 5.3: Double-logarithmic plot of spectral density of the porosity log in borehole C5.A linear regression of the slope suggests β to be around -0.97. The medium is thereforeexpected to be of rough and complex with a ν-value of close to zero.

Correlation length and structural aspect ratio The estimation of the correla-tion length cannot be done very accurately for different reasons. The limitationof available data can cause scaling effects (Western and Bloschl, 1999) and thevicinity of the boreholes may be disturbed during the drilling process or bewashed out by drilling fluids. It is also possible that the backfill of the casingis measured instead of the in situ porosity of the sediments. The analysisof the auto-covariance function of boreholes results in correlation lengths of2.4 m, which is about 1/10 of the measurement scale and hence points to ascaling effect as discussed by Gelhar (1993) and Western and Bloschl (1999).An other way to estimate the correlation structure is to plot the tomographydata and estimate the horizontal and vertical size of evident structures. Sucha fit-by-eye yields a dominant scale of about 2 m in vertical and 10 m in thehorizontal direction and therefore an anisotropy ratio kany of 5. To achieve adataset that is self-affine at scales in the order of the investigated region, thecorrelation length were defined ten times larger with a = 100 m in horizontaldirection and a = 20 m in vertical direction.

Conversion of georadar velocities to porosity Many different conversions ofgeoradar velocities to porosity exist. Tronicke et al. (2003) found the con-version based on a two-component mixing model (Wharton et al., 1980) forwater-saturated media to provide good results for this site. Porosity Φ there-fore is a function of the relative permittivity of matrix εm

r , water εwr and the

measured relative permittivity εr:

Φ =

√εr −

√εmr√

εwr −

√εmr

. (5.8)

The values used for εmr and εw

r are 4.6 and 80, respectively. The measured


relative permittivity can be calculated from the velocity tomogram using thehigh-frequency approximation:

ε =1

µv2, (5.9)

where v is the electromagnetic velocity and µ is the magnetic permeability. Thelatter can be assumed to be equal of to the magnetic permeability of vacuum(µ0 = 4π · 10−7V sA−1m−1), as rocks and soils are generally non-magnetic.

Subsampling spatial porosity structure Although the tomographic data givesthe impression of a very densly sampled image of the subsurface (Figure 5.1),the actual resolution is much sparser. The limitation of the resolution dependssystematically on the wavelength λ and the propagation distance L. Thesmallest feature to be recovered is expected to be of the order of (Williamsonand Worthington, 1993):

rmin ∼√

Lλ. (5.10)

In this case, the two boreholes are about 10 meters apart. The tomographicdata was recorded with a georadar system with nominal center frequency at250 MHz. The nominal recorded center frequency, however, is around 100Hz and the bandwidth of the signal is about two octaves (50Hz-200Hz). Thecrucial wavelength for tomography resolution seems to be the shortest recordedwavelength, which is in this case approximately half a meter. The resolutionalso depends on the propagation distance and is therefore poorest in the centerof the tomographic image with a value of somewhat more than one meter. Onthe other hand, ray covering is much better in the center of the tomographythan it is near the boreholes. As a result of the inversion process, changesin the tomographic image are inherently smooth and a regular spacing of onemeter for subsampling the tomography data for the conditional simulationseems adequate. Figure 5.4 shows the locations of sampled points on whichthe unconditional simulation has to be adapted.

Scale tomography data to variance of porosity logs The assumption of second-order stationarity implies a constant variance over the entire experimental re-gion. It is likely that the variance of the tomographic data has been reduceddue to the damping and smoothing constraints used in the inversion process.In contrast, the porosity values measured in the boreholes are assumed to rep-resent in situ porosity values and for this reason also the correct variance.To perform the conditional simulation, the variance of porosity values derivedfrom the crosshole georadar tomography has therefore been scaled to matchthe variance of the porosity logs.

The conditional simulation was performed for the porosity logs only (Figure 5.5)and for the porosity logs in conjunction with the porosity values derived from thecrosshole georadar tomography (Figure 5.6). On both Figures, a zone of very high


Figure 5.4: Unconditional simulation between boreholes C5 and C6. Locations for sampleddata are marked with a circle for the porosity logs and crosses for tomography data.

porosity above four meters depth is visible. The origin of this feature are the veryhigh values in the porosity log of borehole C5. As the porosity values are much higherthan the surrounding values and prevail only over a short depth range, there is afair possibility that the measurement has been disturbed. At a depth of six meters,the log of borehole C5 predicts again high porosity values, which are not present inthe tomography data. It is therefore again most likely to be a local disturbance ofthe porosity log. Conversely, the high porosity zones visible in borehole C6 at 11and 13 meters seem to be real features. If only the borehole logs are used for theconditional simulations, it seems that two independent high porosity zones connectthe two boreholes at 11 m and 15 m depth, whereas the tomographic informationindicates that these two peeks rather belong to one big zone of high average porosity.Further realizations of the conditional simulations can be found in Appendix C.


Figure 5.5: Conditional simulation use porosity logs only.


Figure 5.6: Conditional simulation using both porosity logs and crosshole georadar tomog-raphy data. The same realization of the unconditional simulation was used as in Figure5.5.


5.2.2 Kappelen

Velocity [m/ns]

0.068

0.07

0.072

0.074

0.076

0.078

0.08

0.082

0 5 10 15 20 25 30

4

6

8

10

12

14

Distance [m]

Dep

th [m

]

o = Transmitter x = Receiver

Attenuation [1/m]

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 25 30

4

6

8

10

12

14

Distance [m]

Dep

th [m

]


(b)

Figure 5.7: Crosshole georadar tomography between boreholes K4, K3, K3 and K8 at thehydrogeological test site in Kappelen. After Tronicke et al. (2002).

Velocity [m/ms]

2.1

2.2

2.3

2.4

2.5

0 5 10 15 20 25 30

5

10

15

Distance [m]

Dep

th [m

]


Figure 5.8: Crosshole seismic tomography between boreholes K4, K3, K3 and K8 at thehydrogeological test site in Kappelen. Courtesy H. Paasche (unpublished data).

The second field data set for a conditional simulation of aquifer porosity structureis from the hydrogeological test site in Kappelen, Canton Berne, Switzerland, wherethe Centre d’Hydrogeologie de l’Universite de Neuchatel (CHYN) has drilled 16boreholes. The test site and several pump and tracer tests are described by Probstand Zojer (2001). Neutron porosity logs used for conditioning were digitized fromHacini (2002). Tronicke et al. (2003) provided the georadar crosshole tomography,from which porosity values for conditioning data between the boreholes was derived.The corresponding seismic tomography data have not jet been published, but were


0.1 0.2 0.3

2

4

6

8

10

12

14

Porosity

K2

0.1 0.2 0.3

2

4

6

8

10

12

14

Porosity

K3

0.1 0.2 0.3

2

4

6

8

10

12

14

Porosity

Dep

th [m

]

K4

0.1 0.2 0.3

2

4

6

8

10

12

14

Porosity

K8

Figure 5.9: Neutron porosity logs of boreholes K4, K3, K3 and K8 of Kappelen hydroge-ological test site. Logs are irregularly spaced due to manual digitalization. After Hacini(2002).

kindly provided by H. Paasche. The conditional simulation was obtained on a sectionbetween four boreholes lying on a straight line (K4, K3, K2 and K8).

The parameters for the conditional simulation were compiled in the same wayas for BHRS (see section 5.2.1). For the spatial variability, again a ν-value of 0 wasestimated. The horizontal correlation length was assumed to be 200 m with an aspectratio kany = 5. For the georadar crosshole tomography, the spatial resolution wasestimated to one meter. The variance of the porosity values derived from tomographywas adjusted to that of the porosity logs, which are assumed to represent in situporosity values. The conversion of georadar tomography velocities to porosity wasperformed with the two-component mixture model of Wharton et al. (1980) as wellas with Topp’s equation (Topp et al., 1980):

Φ = −5.3× 10−2 + 2.92× 10−2εr − 5.5× 10−4ε2r + 4.3× 10−6ε3

r. (5.11)

The results obtained with the two conversion methods are quite similar. WithTopp’s equation the resulting porosity values are on average 2% larger than those ob-tained with the two-component mixture model. Somewhat surprisingly, the porosityvalues derived from georadar tomography velocity are, however, significantly higherthan the values of the neutron porosity logs: the mean porosity value derived fromthe georadar data is 29%, whereas the mean porosity from the logs is only 16%.It should be noted that the average porosity estimated from the crosshole georadar


data is rather in line with the expected porosity of an unconsolidated alluvial aquiferthan that obtained from the porosity logs (Schon, 1996).

For the conversion of seismic tomography velocities to porosity values, the em-pirical time-average-equation was used (Wyllie et al., 1958):

Φ =∆t−∆tm∆tf −∆tm

, (5.12)

where ∆t = 1/Vp is the slowness of the P-wave, Vf = 1/∆tm is the velocityof the rock matrix and Vf = 1/∆tf is the velocity of the fluid that fills the porespace. Based on specialized seismic velocity tables (Schon, 1996), the velocitieswere assumed to be Vf = 1400 ms−1 for the fluid and Vm = 5200 ms−1 for the rockmatrix. This results in porosity values with an average of 46%, which is very likelyto be a to high value. Wyllie’s equation is, however, known to yield to high porosityvalues for uncompacted formations (Schon, 1996, p. 233). A correction factor can beadopted to take the effects of compaction or pressure and temperature in account.Unfortunately, for Kappelen test site these informations are not available by presentand an adaption of the porosity values to the neutron log or the georadar crossholetomography data does not seem to be an adequate solution.

Figure 5.10 shows a conditional simulation constrained by the neutron logs withporosity values that are likely to be too small and porosity values derived from theseismic crosshole tomography that are likely to be too high. The upper corner fre-quency of the seismic data was about 800 Hz and the average velocity was about2400 ms−1, which indicates that subsampling 2 m is adequate. It is obvious, that thetwo datasets do not agree, as high porosity pillows are embedded between the gen-erally low values surrounding the boreholes. This result indicates that two datasetsare entirely inconsistent. We also see that the more densely sampled region aroundthe boreholes has only a local effect and does not unduly influence sparser sampledregions.

Figure 5.11 shows a conditional simulation for the neutron logs only. The probedsurface seems to be quite homogeneous so that only few structures can be discerned.The most obvious feature is a zone of higher porosity which is most obvious inFigure 5.12 where the georadar crosshole tomography porosity values are also usedas conditioning data. At borehole K8 it is about four meters thick and its topis located at a depth of about eight meters. Toward borehole K4 its thickness isreduced to approximately two meters and its top is located at a depth of about fivemeters. Figure 5.14 shows a conditional simulation of seismic crosshole tomographydata only. The high porosity structure can also be seen here, even though it is notas pronounced as in the georadar data. In exchange a high porosity zone betweenboreholes K4 and K3 with top at about ten meters and a thickness of about fourmeters is very distinct from the seismic crosshole tomography data and can also beseen in the georadar data is not present in the neutron logs. Overall, the datasets arenot consistent and more information is needed before a consistent interpretation ofthe porosity structure of this aquifer can be obtained. Appendix C shows additionalrealizations of conditional simulations.


Figure 5.10: Conditional simulation between boreholes K4, K3, K2 and K8 conditionedby porosity data from neutron logs and crosshole seismic tomography.


Figure 5.11: Conditional simulation between boreholes K4, K3, K2 and K8 conditionedby borehole logs only.


Figure 5.12: Conditional simulation between boreholes K4, K3, K2 and K8 conditionedby porosity data from neutron logs and crosshole georadar tomography.


Figure 5.13: Conditional simulation between boreholes K4, K3, K2 and K8 conditionedby porosity data from crosshole georadar tomography only.


Figure 5.14: Conditional simulation between boreholes K4, K3, K2 and K8 conditionedby porosity data from crosshole seismic tomography only.

Chapter 6

Conclusions

A program to perform ordinary kriging in two dimensions has been developed. Krig-ing interpolation cannot be used in the same way as standard interpolation proce-dures, but has to be adapted to the considered dataset, at least in terms of structuralanisotropy (section 4.2), auto-covariance function (sections 2.2 and 4.5) and searchneighborhood (section 2.4.1 and 4.3). To achieve optimal results, the user is ex-pected to understand the code and the theory behind it and be able to adapt it tohis purposes. The objective in developing this program, therefore, has been focusedon well structured, understandable and easy expandable code. A documentation ofthe program, called vebyk, is given in Chapter 3.

The implementation was tested on stochastic models characterized by auto-covariance functions of the von Karman type. Somewhat surprisingly, optimal resultwere obtained by using larger ν-values for the interpolation than for the generationof the stochastic models. The reason for this is that the auto-covariance function ofthe experimental dataset is not conserved in the interpolation and the interpolatedmedium is always considerably smoother than the input dataset (section 4.5).

Kriging of so-called anti-persistent models with a auto-covariance functions hav-ing ν-values smaller than 0.5 produced artefacts that become more pronounced withdecreasing ν-values. The predictability of such stochastic models decreases with de-creasing ν-values. For power spectral exponents corresponding to β < 1 the stochas-tic models become entirely unpredictable and the best forecast for an unknown valueis the expected value of the model. Therefore, estimated values calculated by krig-ing progressively approach the expected value of the search neighborhood as theν-values for the auto-covariance function of the stochastic model decrease. Thesampled values in contrast tend to be different from the expected value of the searchneighborhood as the roughness of the medium increases with decreasing ν-values.This systematic discrepancy between sampled and estimated values give rises to theobserved artefacts.

In second part of this work, these implemented kriging algorithm was applied toperform conditional simulations with geophysical data of the hydrological test sitesof Boise, Idaho, USA and Kapplen, Bern,Switzerland. The impacts of ν-values, thecorrelation lengths of the simulated models and subsampling of the spatial structureto resolution were estimated. Conversion of tomographic velocities to porosity valueswas performed with Wharton’s two-component mixture model and Topp’s equation

58

59

for crosshole georadar data and with Wyllie’s time-average-equation for crossholeseismic data. Unconditional simulations were generated by the spectrum methodand were adapted to the conditioning data through kriging interpolation. Whereasthe conversion of the georadar crosshole tomographic data produced good results forBHRS and presumably in Kappelen aswell, the conversion of the seismic velocitiesyielded too high porosity values, due to unknown effects of compression, pressureand temperature. This is a well known effect of Wyllie’s equation for consolidatedsediments. A calibration to the neutron porosity logs seemed also not to be ade-quate. The conditional simulations show that secondary or “soft” data as georadaror seismic crosshole tomography can be successfully incorporated in an aquifer sim-ulation if the data are consistent with primary information, such neutron porositylogs as this was the case for BHRS and to some part also in Kappelen site.

Conditional simulation not only allows to simulate facies properties or to inte-grate secondary data, but also opens the possibility to assess the uncertainties of themodeled aquifer structure. A conditioned simulation is not a deterministic methodas kriging and offers many different stochastic solutions. Therefore it is possible toestablish a probability distribution rather than a single deterministic estimate byassuming that different realizations characterize unbiased and adequate values in a“space of uncertainty”. This also allows to predict minimal and maximal deviationsfrom a “best estimation” as provided by deterministic estimation technique, such askriging. Stochastic porosity models, conditioned by “hydrogeophysical” data thusallow for a better understanding of connectivity between porous and non-porouszones in aquifer and for improving flow simulations in comparison to smooth deter-ministic model.

Chapter 7

Acknowledgments

I am in very grateful to my parents for all their efforts.I thank Klaus Holliger for comments and references that guided me through this

work and for the patience to correct my English.I also thank Jens Tronicke for the crosshole georadar data of the BHRS site and

Hendrick Paasche for providing me the seismic and georadar crosshole tomographydata for Kappelen site.

60

Appendix A

Kriging

Kriging of a stochastic model with a von Karman auto-covariance function withvariance σ = 1, ν = 0.5, horizontal correlation length ax = 100 m and an anisotropyfactor kany = 10. The model is shown in Figure 4.11. Figures A.1 through A.3show kriging interpolations assuming every fourth point to be an observation withdifferent ν-values for the von Karman auto-covariance function.

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x [point #]

y [p

oint

#]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure A.1: Result of kriging interpolation for an auto-covariance function with ν = 0.2.The other parameters for the auto-covariance function are the same as those of the inputmodel.

61

62 Kriging

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x [point #]

y [p

oint

#]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure A.2: Result of kriging interpolation for an auto-covariance function with the sameparameters as for the auto-covariance function of the input model.

−1.5

−1

−0.5

0

0.5

1

1.5

x [point #]

y [p

oint

#]

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

Figure A.3: Result of kriging interpolation for an auto-covariance function with ν = 0.9,which turned out to offer best results.

63

In addition to cross-validation in section 4.5, the stochastic model with the vonKarman auto-covariance function with ν = 0.5 and a horizontal correlation lengthax = 100 m has been cross-validated with von Karman functions that have differentcorrelation lengths. Figures A.4 through A.6 show the sum of absolute error, andthe “x” denotes the location of the global minimum.

100

110

120

130

140

150

160

Anisotropy

ν

5 10 15 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure A.4: Cross-validation with correla-tion length ax = 20m.

100

110

120

130

140

150

160

Anisotropy

ν

5 10 15 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2


100

110

120

130

140

150

160

Anisotropy

ν

5 10 15 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2


Appendix B

Boise

Appendix B presents different realizations of the conditional simulations for BHRS.The kriged version of the observed data is shown as well, and represents the datasetbefore the unconditional realization is added. The data values of this stochasticinterpolation are of qualitative character only, as the unconditional realization atthe observed points is added and the mean of the region is subtracted. The kriginginterpolation between logs only is shown in Figure 4.7. The “seed” is a calibrationfactor for the Matlab random number generator, which allows to regenerate the same“random” dataset. This is useful for comparison of conditional simulations withdifferent input datasets. The seed for conditional simulations used in section 5.2.2was 41. Figures B.4 through B.6 show impact of the magnitude of the correlationlengths on the conditional simulations. Here the horizontal and vertical correlationlengths are 10 m, 2 m, respectively, which is about an order of magnitude smaller.

64

65

Figure B.1: Conditional simulation for BHRS. Only logs were used as conditioning data.Random number seed = 41.

Figure B.2: Conditional simulation for BHRS. Only logs were used as observed data.Random number seed = 1203.

66 Boise

Figure B.3: Conditional simulation for BHRS. Only logs were used as observed data.Random number seed = 1275.

Figure B.4: Conditional simulation for BHRS. Only logs were used as observed data.The correlation length were chosen an order of magnitude smaller than for the simulationsshown in Figures B.1 to B.3 (ax = 10 m, ay = 2 m). Random number seed = 41. Compareto Figure B.1.

67

Figure B.5: Conditional simulation for BHRS. Only logs were used as observed data. Thecorrelation length were chosen an order of magnitude smaller than for the simulationsshown in Figures B.1 to B.3 (ax = 10 m, ay = 2 m). Random number seed = 1203.Compare to Figure B.2.

Figure B.6: Conditional simulation for BHRS. Only logs were used as observed data. Thecorrelation length were chosen an order of magnitude smaller than for the simulationsshown in Figures B.1 to B.3 (ax = 10 m, ay = 2 m). Random number seed = 1275.Compare to Figure B.3.

68 Boise

Figure B.7: Kriging interpolation of the observed datas, consisting of log and georadartomography data.

Figure B.8: Conditional simulation for BHRS site. Log and georadar tomography datawas used as conditioning datas. Random number seed = 41.

69



Appendix C

Kappelen

Appendix C presents different realizations of the conditional simulations for Kap-pelen dataset. The kriged version of the observed data is shown as well, which rep-resents the dataset before the unconditional realization is added. The data valuesof this interpolation are of qualitative character only, as the unconditional simu-lation at the observed points is added and the mean of the region is subtracted.The “seed” is a calibration factor for the Matlab random number generator, whichallows to generate the same “random” dataset again. This is useful for comparisonof conditional simulations with different input datasets. The seed for conditionalsimulations used in section 5.2.2 was 54.

Figure C.1: Kriging interpolation using logs only.

70

71

Figure C.2: Conditional simulation for Kappelen site. Only logs were used as conditioningdata. Random number seed = 43.

Figure C.3: Conditional simulation for Kappelen site. Only logs were used as conditioningdata. Random number seed = 1203

Figure C.4: Conditional simulation for Kappelen site. Only logs were used as conditioningdata. Random number seed = 1275

72 Kappelen

Figure C.5: Kriging interoplation of the observed dataset, consisting of log and georadartomography data.

Figure C.6: Conditional simulation for Kappelen site. Log and georadar tomography datawere used as conditioning constraints. Random number seed = 43

Figure C.7: Conditional simulation for Kappelen site. Log and georadar tomography datawere used as conditioning constraints. Random number seed = 1203

73

Figure C.8: Conditional simulation for Kappelen site. Log and georadar tomography datawas used as conditional constraints. Random number seed = 1275

Figure C.9: Kriging interoplation of the observed dataset consisting of log and crosshole,seismic tomography data. The two datasets do obviously not coincide with each other.

Figure C.10: Kriging interoplation of the observed dataset consisting of crosshole, seismictomography data only.

74 Kappelen

Figure C.11: Conditional simulation for Kappelen site. Seismic tomography data only wasused as conditioning constraints. Random number seed = 43.

Figure C.12: Conditional simulation for Kappelen site. Seismic tomography data was usedonly as conditioning constraints. Random number seed = 1203.

Figure C.13: Conditional simulation for Kappelen site. Seismic tomography data was usedonly as conditioning constraints. Random number seed = 1275.

List of Figures

2.1 Covariance and semi-variogram . . . . . . . . . . . . . . . . . . . . . 72.2 von Karman covariance functions . . . . . . . . . . . . . . . . . . . . 8

3.1 Hierarchical structure of vebyk subfunctions . . . . . . . . . . . . . . 153.2 Anisotropy ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Example arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Kriging weights for too small a search neighborhood . . . . . . . . . . 244.3 Kriging weights for a larger search neighborhood . . . . . . . . . . . . 254.4 Kriging weights for an anisotropic search neighborhood . . . . . . . . 254.5 High-frequent oscillations caused by too small a search neighborhood 264.6 Artefacts generated by a simple search neighborhood . . . . . . . . . 274.7 Kriging with a quadrant search neighborhood . . . . . . . . . . . . . 284.8 Crossplot of true vs. estimated value . . . . . . . . . . . . . . . . . . 294.9 Crossplot of true value vs. corresponding estimation error . . . . . . . 304.10 Spatial distribution of errors in cross-validation . . . . . . . . . . . . 304.11 Stochastic model with ν = 0.5 . . . . . . . . . . . . . . . . . . . . . . 314.12 Subsampled input model for jackknifing . . . . . . . . . . . . . . . . . 324.13 Kriging estimation of subsampled input model . . . . . . . . . . . . . 324.14 Spline interpolation of model with ν = 0.5 . . . . . . . . . . . . . . . 334.15 Stochastic model with ν = 0 . . . . . . . . . . . . . . . . . . . . . . . 344.16 Kriging estimation of regular spaced input model . . . . . . . . . . . 354.17 Spline interpolation of model with ν = 0 . . . . . . . . . . . . . . . . 354.18 Kriging estimation of random spaced input model . . . . . . . . . . . 364.19 Best fitting ν-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.20 Sum of absolute error for ν = 0 . . . . . . . . . . . . . . . . . . . . . 384.21 Sum of absolute error for ν = 0.5 . . . . . . . . . . . . . . . . . . . . 394.22 Sum of absolute error for ν = 0.7 . . . . . . . . . . . . . . . . . . . . 39

5.1 Tomography of Boise, ID . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Boise porosity logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Spectral density of porosity logs . . . . . . . . . . . . . . . . . . . . . 455.4 Locations of sampled data for Boise . . . . . . . . . . . . . . . . . . . 475.5 Conditional simulation for Boise with borehole data only . . . . . . . 485.6 Conditional simulation for Boise with boreholes and georadar porosity

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.7 Georadar tomography of Kappelen . . . . . . . . . . . . . . . . . . . 50

75

76 LIST OF FIGURES

5.8 Seismic tomography of Kappelen . . . . . . . . . . . . . . . . . . . . 505.9 Porosity logs of Kappelen . . . . . . . . . . . . . . . . . . . . . . . . 515.10 Conditional simulation for Kappelen with borehole and seismic poros-

ity data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.11 Conditional simulation for Kappelen with borehole data only . . . . . 545.12 Conditional simulation for Kappelen with borehole and georadar poros-

ity data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.13 Conditional simulation for Kappelen with georadar porosity data only 565.14 Conditional simulation for Kappelen with seismic porosity data only . 57

A.1 Kriging with ν = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.2 Kriging with ν = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.3 Kriging with ν = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.4 Kriging with different correlation length . . . . . . . . . . . . . . . . 63A.5 Kriging with different correlation length . . . . . . . . . . . . . . . . 63A.6 Kriging with different correlation length . . . . . . . . . . . . . . . . 63

B.1 Conditional simulation for BHRS logs . . . . . . . . . . . . . . . . . . 65B.2 Conditional simulation for BHRS logs . . . . . . . . . . . . . . . . . . 65B.3 Conditional simulation for BHRS logs . . . . . . . . . . . . . . . . . . 66B.4 Conditional simulation for BHRS logs with short correlation length . 66B.5 Conditional simulation for BHRS logs with short correlation length . 67B.6 Conditional simulation for BHRS logs with short correlation length . 67B.7 Kriged observed dataset of log and georadar data at BHRS . . . . . . 68B.8 Conditional simulation for BHRS log and georadar data . . . . . . . . 68B.9 Conditional simulation for BHRS log and georadar data . . . . . . . . 69B.10 Conditional simulation for BHRS log and georadar data . . . . . . . . 69

C.1 Kriged dataset of log only at Kappelen . . . . . . . . . . . . . . . . . 70C.2 Conditional simulation for Kappelen logs . . . . . . . . . . . . . . . . 71C.3 Conditional simulation for Kappelen logs . . . . . . . . . . . . . . . . 71C.4 Conditional simulation for Kappelen logs . . . . . . . . . . . . . . . . 71C.5 Kriged observed dataset of log and georadar data at Kappelen . . . . 72C.6 Conditional simulation for Kappelen log and georadar data . . . . . . 72C.7 Conditional simulation for Kappelen log and georadar data . . . . . . 72C.8 Conditional simulation for Kappelen log and georadar data . . . . . . 73C.9 Kriged observed dataset of log and seismic data at Kappelen . . . . . 73C.10 Kriged observed dataset at Kappelen of seismic data only . . . . . . . 73C.11 Conditional simulation for Kappelen with seismic data only . . . . . . 74C.12 Conditional simulation for Kappelen with seismic data only . . . . . . 74C.13 Conditional simulation for Kappelen with seismic data only . . . . . . 74

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