fitting negative spatial covariances to geothermal field temperatures in nea kessani (greece

ENVIRONMETRICSEnvironmetrics 2007; 18: 759–773Published online 2 August 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/env.871

Fitting negative spatial covariances to geothermalfield temperatures in Nea Kessani (Greece)

Jorge Mateu1∗,†, Emilio Porcu1, George Christakos2 and Moreno Bevilacqua3

1Department of Mathematics, Universitat Jaume I, Campus Riu Sec, E-12071 Castellon, Spain2Department of Geography, San Diego State University, 5500 Campanile Drive San Diego, CA 92182-4493, USA

3Department of Statistics, University of Padova, Via Cesare Battisti 241/243, I-35121 Padova, Italy

SUMMARY

In this paper, we investigate the problem of negative empirical covariances for temperature data in a three-dimensional geothermal field. We make use of permissibility criteria that allow a linear combination of covariancefunctions to be positive definite even if some of the weights of the linear combination are set to be negative. We thuspropose a generalized sum of spatial covariance structures to model negative spatial covariances. We evaluate theperformance of the proposed model, together with that of other celebrated, and widely used covariance functions.Comparison is done in terms of cross-validation, which allows to evaluate the relative bias, accuracy of estimation,and goodness-of-fit. Copyright © 2007 John Wiley & Sons, Ltd.

key words: composite likelihood; geostatistics; geothermal field; GSP model; negative covariances; permissi-bility

1. INTRODUCTION: THE GEOTHERMAL FIELD OF NEA KESSANI ANDSTATISTICAL-MODELING INTEREST

1.1. Geology of the study area

The geothermal field of Nea Kessani (NE Thrace, Greece) is located within the Ksanthi–Komotinibasin, next to the Aegean Sea and Vistonis Lake, as shown in the geological map in Figure 1 (Pa-pantonopoulos and Modis, 2005). The geothermal area is located near the southwestern margin of theKsanthi–Komotini basin which is a post-orogenic Tertiary sedimentary basin, and covers an area ofabout 1600 km2 between the Rhodope Mountains and the Aegean coast (Figure 2, left). It is mainlyconstituted of clastic sediments, and reaches maximum depth at the foot of the Rhodope chain, and min-imum depth in the vicinity of the coast, where the Nea Kessani geothermal field is located (Thanassoulasand Tsokas, 1990). During the Paleogene, predominantly molassic sediments were deposited over thehighly fractured substratum, which are represented by the metamorphic Paleozoic basement, mainlyoutcrops in the Rhodope Mountains. Local outcroppings of the basement, which is composed of gneiss,

∗Corresponding to: J. Mateu, Department of Mathematics, Universitat Jaume I, Campus Riu Sec, E-12071 Castellon, Spain.†E-mail: [email protected]

Received 20 November 2006Copyright © 2007 John Wiley & Sons, Ltd. Accepted 22 June 2007

760 J. MATEU ET AL.

Figure 1. Geological map of Nea Kessani geothermal area (rectangle) and vicinity

amphibolites with interbedded marbles, micaschists, migmatites, and intruded granites, are located tothe southwest of the study area. The Eocene–Oligocene sequence is mainly consisting of basal brecciasand conglomerates, nummulitic limestones and arkosic sandstones, which make way to flysch forma-tions in the upper part of the sequence. These formations are overlain by Pliocene lacustrine sediments,such as marls, clays, and sands, as well as alluvial Quaternary deposits. During the Tertiary, as a resultof the sinkage of the African plate below the European plate, an andesitic magmatism developed in thebasin, with emplacement of subvolcanic rocks. Small intrusions of such rocks are inter-bedded with theclastic horizons in the study area. Two major tensile fault systems, striking N160◦ and N70◦, developedfrom the Miocene on. The most active is the N160◦ system, which is probably related to the movementsof the North Anatolic Fault.

The geothermal field is created by distribution of heat in the geological formations of the groundat some depth due to thermal fluid circulation. This causes an increase of the earth natural geothermalgradient, which is 30◦C/km at average, by several degrees reaching 35◦C/km. The hot reservoir createdby the thermal fluids has an average temperature of 75–80◦C and covers an area of 5 km2. The roof ofthe reservoir is found in a depth of 100–120 m at its south part near Aegean Sea, while its basement inthis area is located at the depth of 450 m. At the north part of the reservoir its roof is located at the depthof 300–350 m and its basement at greater depth than 1 km. A schematic representation of the reservoirand the thermal fluid circulation can be seen in Figure 2 (right) where a N–S vertical section of the areais presented.

1.2. Drill–hole data

In order to explore and study the geothermal anomaly in the area, 25 exploratory boreholes up to a depthof 500 m were drilled during 1980–1991 by IGME (Institute of Geological and Mineral Investigations,Athens, Greece). The first phase of the drilling campaign began in 1980 and produced 11 drill holes

Copyright © 2007 John Wiley & Sons, Ltd. Environmetrics 2007; 18: 759–773DOI: 10.1002/env

NEGATIVE EMPIRICAL COVARIANCES 761

Figure 2. Left: Geological features of Ksanthi–Komotini basin and the geothermal area (rectangle) along with a vertical section:(1) Quaternary and Tertiary formations, (2) Paleozoic basement, and (3) fault. Right: A N–S section of the area, showing a thermalfluid circulation model and the morphology of the thermal field: (1) cover formations, (2) arkosic reservoir with hydrogeologicalboundaries (shaded area), (3) basement, (4) deep geothermal fluids, (5) fluid circulation within the arkosic reservoir, and (6)

thermal springs

in depths varying from 65 to 475 m. The second phase, which started in 1990, produced 14 drill holesto depths varying from 200 to 500 m. The drill–hole data consist of temperature measurements takenin the drill wells by an electrical resistance thermometer with ±0, 1◦C error. Thus, we have a three-dimensional data set formed by the longitude and latitude (measured in meters) of the drill hole togetherwith the depth at which the temperature is measured. In Figure 3 the locations of the drill–hole collars

Figure 3. Drill hole (G’s) locations and vertical cross-sections in area of interest (dashed parallelogram). In sections EF andGH the temperature distribution derived from drill data is presented. (1) Drill collar, (2) thermal spring, (3) roof formations, (4)reservoir, and (5) isotherms in ◦C. No thermal springs appear in the two specific sections presented above, although springs canbe seen in other sections of the area not shown here but sharing the same legend as the present two sections. The reader is referred

to the geophysics reports mentioned in the text for further details and cross-sections


762 J. MATEU ET AL.

Figure 4. Geothermal three-dimensional data set formed by the longitude and latitude (measured in meters) of the drill holetogether with the depth at which the temperature is measured. This geothermal field consists of a total of 162 data distributed in

25 exploratory drills, and the domain of the study is 2250 m × 2250 m × 400 m

are shown, along with two vertical sections showing the temperature distribution in the area. Note thatthis geothermal field consists of a total of 162 data distributed in 25 exploratory drills, and the domainof the study is 2250 m × 2250 m × 400 m. This is more clearly shown in Figure 4.

1.3. Scientific motivation

The scientific interest on the data lies upon answering the question why should a spatial model ofthe geothermal field be created. The importance of geothermal energy as an energy source is easilyunderstandable, especially during the current trends of oil prices. In addition, geothermal energy is aclean and environmental friendly form of energy. Thus it is of worth to explore deposits of thermalenergy, since it can be effectively used in many applications as heating and or cooling of buildings,power generation, etc. The parameter of interest in exploring a geothermal field is the temperature and itsdistribution in the underground area, since the subsequent exploitation of the deposit is tightly connectedto it. For example, a power generator application can be implemented only if the temperatures of thefield exceed 100◦C, which can lead to the sufficient generation of steam. In the Nea Kessani area thetemperature is known in certain positions due to the drilling campaign, so the aim of our work is to createa three-dimensional model of the geothermal field to describe the temperature in a domain as large aspossible. As in the case of a metal deposit where the ore grade must be known almost everywhere in orderto efficiently plan the exploitation, the temperature distribution in a geothermal field must be also knownin underground space, in order to obtain a complete image of the phenomenon and identify the possibleexploitation sites. In the Nea Kessani case, the drilling campaign produced only a small number of data.These data are insufficient to model the field using deterministic tools like inverse distance squares(IDS). Thus, the application of more sophisticated methods, like Geostatistics, was clearly necessary.In this context, spatiotemporal data are used in the study for purely spatial modeling, presumably basedon temporal dynamics stability evidence. In our study the temporal variation was considered negligible,and thus a spatial three-dimensional model was taken into account. An extension of the model to thespatiotemporal case could also be considered by extending the methodology proposed in this paper.However, we leave this as an open question at this stage.



1.4. Exploratory analysis

Stationarity-isotropy-ergodicity are most often modeling assumptions that are justified on the basis ofthe outcome. On the basis of our study of the physical information underlying the data, we found thatthey obey an isotropic heat transfer equation, that is a Laplace equation. In such an equation the heat con-ductivity is isotropic and the transfer ability for the heat is also the same at all directions. The assumptionof a stationary and isotropic covariance is thus consistent with this physical knowledge. As far as trendanalysis is concerned, and before the calculation of the covariance, a three-dimensional nonparametricmeantrend was calculated and removed from the data. In particular, we used a nonparametric movingaverage method similar to the moving window method that uses a spatial moving window to obtain thespatial moving average based on the data within its specified neighborhood. In this case, the spatial win-dow was a square cubic with 100 m long sides. Also, an exponential smoothing technique was appliedto the calculated moving average to have a smoother mean trend. The detrended data, or residuals, con-stituted the base data to be spatially modeled. The empirical estimation of the spatial covariance of theresiduals was based on a 200 m lag, except for few very short lags at the origin. This resulted in a quitesmooth empirical covariance easy to be fitted by a theoretical model. Note that we chose this lag becauseit was the distance for which a satisfactory understanding of the geophysical phenomenon was assured,and was also consistent with the sampling survey available. This lag distance was calculated as a distancein three dimensions. Once the data was detrended, a QQ plot highlighted a slight left tail, which didnot prevent the residuals from being Gaussian distributed. Indeed, we performed Kolmogorov–Smirnovand Shapiro–Wilks tests, and the results, in terms of p-values, confirmed the graphical results, so thatwe could assume to work with residuals that were stationary and approximately Gaussian.

Calculation of the residuals allowed us to obtain the empirical covariance, which highlighted thepresence of negative values. As shown in the description of the data, we have enough data for reliable(geo)statistical modeling, and thus the presence of negative values in the empirical covariance wasconsidered an intrinsic characteristic of the data. This could be a problem for fitting procedures, as thegreat majority of celebrated models of covariance functions for spatial data only attain positive values.This strong motivation prompted our research. We are looking for spatial covariance models satisfyingtwo main features:

1. They may be negative or oscillate between negative and positive values.2. They would preferably allow for an easy interpretation.

We believe it is reasonable to select a class of models satisfying property 2, and then answer, if possible,the natural question: can we obtain negative covariances starting from a class which is easy-to-implementand interpretable? The answer is yes, and a satisfactory solution for this data set will be exposedthroughout the paper.

1.5. Outline

In view of the above considerations, the paper is organized as follows: Section 2 discusses the necessarybackground, notation, and the proposed methodology; it also provides the main result that will be usedthroughout the paper. In Section 3, we perform data analysis and diagnostics. We briefly present thecomposite likelihood (CL) estimation technique adapted to spatial data, and the fitting strategy for fittingnegative covariances. We compare the estimates obtained with our model with those obtained by usingother celebrated models, and perform cross-validation-based diagnostics as a measure of goodness-of-fit. Section 4 ends the paper with some conclusions and a discussion.


764 J. MATEU ET AL.

2. BACKGROUND AND METHODOLOGY

2.1. Geostatistical background

Througout the paper we shall make reference to real-valued homogeneous zero mean Gaussian randomfields (H/SRF) {Z(x): x ∈ D ⊆ R

d, d ∈ N}. Homogeneity or stationarity in the weak sense means thatthe expected value is constant (in this case, identically equal to zero), and the covariance functioncov (Z(x), Z(y)) = E (Z(x)Z(y)) := C(h), with h = y − x, the separation vector between two pointslying in D, and where C: R

d → R is a positive definite function, that is satisfiesn∑

i=1

aiajC(xi − xj) ≥ 0

for all finite collections of real numbers {ai}ni=1 and points xi ∈ D. With alternative notation, Chris-takos (2000, p. 63) calls this property permissibility. A classical result in Bochner (1933) sets up theequivalence between covariance functions and characteristic functions of nonnegative finite measures.Namely, a continuous function C as defined above is positive definite if and only if it is the Fouriertransform of a nonnegative measure F on R

d , that is

C(h) =∫Rd

eiω′h dF (ω) (1)

If, additionally, F is absolutely continuous with respect to the Lebesgue measure, then (1) can be writtenas

C(h) =∫Rd

eiω′hf (ω) d(ω) (2)

and f : Rd → R+ is called the spectral density of the H/SRF.

A stationary covariance function is called isotropic if

C(h) := C1(h), h ∈ R+ (3)

where h = ‖h‖ and where ‖ · ‖ denotes the Euclidean norm. Thus, C is said to be represented by thepositive definite function C1: R+ → R, that is rotation-translation invariant (or radially symmetric).This is the most popular case in spatial and spatio–temporal statistics (Matern, 1986; Yaglom, 1987;Christakos and Papanicolau, 2000). For the remainder of the paper, we shall omit the subscript inEquation (3) when referring to an isotropic covariance function, whenever no confusion may arise.

A real mapping γ:D ⊆ Rd → R is called an intrinsically stationary variogram (Matheron, 1965) if

it is conditionally negative definite, that is

n∑i=1

n∑i=j

aiajγ(xi − xj) ≤ 0

for all finite collections of real weights ai summing up to zero, and all points xi ∈ D. The restrictionto the isotropic case is analogical to that of covariance functions, that is γ(h) := γ1(h), h = ‖h‖ ∈ R+and γ1: R+ → R conditionally negative definite.



2.2. Fitting negative covariances: methodology

In a recent paper, Gregori et al. (2007) proposed a novel and very general approach to building space-time covariance functions that attain negative values or oscillate between positive and negative ones. Theapproach, called Generalized Sum of the Products, is based on a linear combination of pairwise productsof continuous spatial and temporal covariance functions. Some permissibility criteria are given to allowsome of the weights in the linear combination to be negative. In this paper, we are only involved inspatial analysis, but the results contained therein can be very useful in order to implement the followingidea. Consider an arbitrary natural number n of continuous covariances defined on R

d , and additionallyintegrable on their domain. Thus, we propose the following covariance function

C(h) =n∑

i=1

kiCi(h), (h, u) ∈ Rd × R (4)

whose permissibility is in principle guaranteed whenever the weights ki, i = 1, . . . , n are non-negative.By using Proposition 1 in Gregori et al. (2007) one can show the result proposed subsequently, forwhich we restrict to the case n = 2. We skip the proof as it follows exactly the same arguments of thepreviously cited authors.

Proposition 1. Let Ci be spatial continuous and integrable covariance models, i = 1, 2. For ϑ ∈ R,let us define the function

C(h) = ϑC1(h) + (1 − ϑ)C2(h), h ∈ Rd

Let us denote by fi the Fourier transforms of covariances Ci, assume f2 does not vanish and write

m := infω∈Rd

f1(ω)

f2(ω), M := sup

ω∈Rd

f1(ω)

f2(ω)

Then, C is a valid spatial covariance if and only if

[1 − max(1, M)]−1 ≤ ϑ ≤ [1 − min(1, m)]−1

(where 0−1 = −∞ and (−∞)−1 = 0 in the left hand side, and 0−1 = +∞ in the right hand side).

Note that C1 and C2 only need be continuous and integrable, as the condition is imposed on theassociated Fourier transform. Thus, the argument is valid not only for positive-valued covariancesbut also for possibly oscillating ones. The advantage of the result above is that it allows for buildingcovariance functions that, thanks to the negative weights, attain negative values or may oscillate betweenpositive and negative ones. The procedure is very simple: one must chose two parametric covariancefunctions that admit, possibly, an explicit closed form for the associated spectral density. Then, therange of ϑ can be easily computed.

In this paper we shall be involved in linear combinations of Whittle–Matern type covariance functions(see Matern, 1986), that is

C(h) = σ2{ϑCM(h; α1, ν1) + (1 − ϑ)CM(h; α2, ν2)} (5)


766 J. MATEU ET AL.

Table 1. Results of inf’s and sup’s needed in Proposition 1 for the particular case Matern vs. Matern correlationfunctions, respectively, in terms of their parameters values. Here d denotes the spatial dimension, and we assume

σ21 = σ2

2 = σ2

Matern1/Matern2

Parameters mM1,M2 MM1,M2

ν1 < ν2�(ν1+ d

2

)�(ν2+ d

2

)2ν1−ν2

(α2α1

)d

+∞α2

2α2

1≤ ν2+ d

2

ν1+ d2

ν1 < ν2�(ν1+ d

2

)�(ν2+ d

2

)(ν1+ d

2

)ν2+ d2

(ν2+ d

2

)ν1+ d2

×(

α22−α2

12(ν2−ν1)

)ν2−ν1 α2ν11

α2ν22

+∞α2

2α2

1>

ν2+ d2

ν1+ d2

ν1 = ν2 and α2 ≥ α1

(α1α2

)2ν1(

α2α1

)d

ν1 = ν2 and α2 < α1

(α2α1

)d (α1α2

)2ν1

ν1 > ν2 0�(ν1+ d

2

)�(ν2+ d

2

)(ν1+ d

2

)ν2+ d2

(ν2+ d

2

)ν1+ d2

×(

α22−α2

12(ν2−ν1)

)ν2−ν1 α2ν11

α2ν22

α22

α21

<ν2+ d

2

ν1+ d2

ν1 > ν2 0�(ν1+ d

2

)�(ν2+ d

2

)2ν1−ν2

(α2α1

)d

α22

α21

≥ ν2+ d2

ν1+ d2

with h = ‖h‖, h ∈ Rd , and where we use the Whittle–Matern correlation functions t �→ CM(t; αi, νi) =

(αit)νi Kνi (αit), i = 1, 2, with Kν the modified Bessel function of the second kind and order ν

(Abramowitz and Stegun, 1965). The nonnegative parameters αi represent the scale of the spatialdependence, while νi > 0 governs the level of smoothness of the associated Gaussian random field. Theparameter σ2 is the variance of the Gaussian H/SRF represented by the function (5). Thus, we are assum-ing that the random field under study is a linear combination of two random fields with a Matern-typecorrelation structure, and that additionally have the same variance. Ergo, they can be distinguished onlythrough the scale (αi) and smoothing (νi) parameters characterizing their spatial dependence. This isalso desirable by the fact that the range of permissibility for ϑ negative is a function of these parameters.In particular, one can appreciate this fact in Table 1, where the permissibility conditions for model (5)are resumed. For the remainder of the paper, we shall denote the model in Equation (5) with the acronymGSM, by meaning Generalized Sum of Matern models.

3. DATA ANALYSIS AND DIAGNOSTICS

3.1. Performing estimation through composite likelihood

The term CL was coined by Lindsay (1988). It is a general method of estimation applied in several dis-ciplines in the last years, such as Geostatistics (Curriero and Lele, 1999), point processes (Guan, 2006),



binary spatial data (Heagerty and Lele, 1998), survival analysis (Parner, 2001), and state-space model-ing (Varin and Vidoni, 2007) among others. The aim in using CL is basically to reduce computationalcomplexity and to offer feasible inference tools when working in highly structured models.

Roughly speaking, CL is obtained by adding together marginal or conditional valid likelihood objects.For the sake of clarity, let us introduce some formalism and notation. Let f (y, θ) be a parametric modelwith y ∈ R

d, θ ∈ � ⊆ Rq, d, q ∈ N. If we consider a set of events (Ai: Ai ⊆ F), i ∈ I ⊆ N, and where

� is a σ-algebra, then a CL function is defined as

LCL(θ, y) =∏i∈I

f (y, θ ∈ Ai), Ai ∈ �.

Following Varin and Vidoni’s (2005) taxonomy, CL methods are of two kind: (a) subsetting methods,based on likelihood of marginal events, and (b) omission methods, based on omitting components tothe full likelihood.

For instance, Besag (1974), Cox (1975), Vecchia (1988) and Stein (2004) belong to the latter, whileHeagerty and Lele (1998), Curriero and Lele (1999) and Varin and Vidoni, (2005) belong to the former.

Since each of the components is a valid likelihood, the resulting estimating function dlCL(θ)dθ

= 0,where lCL(θ) is the logarithm of LCL(θ), is unbiased and, under regularity conditions (Heyde, 1997),the CL estimator θCL is consistent and normally distributed with mean θ and variance V (θ) =H(θ)−1J(θ)H(θ)−1, where J(θ) = V

(dlCL(θ)

dθ

)and H(θ) = E

(d2lCL(θ)dθdθ′

). Thus, standard errors estimation

is performed through efficient estimation of the matrices H(θ) and J(θ). The former does not imply diffi-culties, while the latter needs particular attention (see Varin and Vidoni, 2005, and the references therein).

In order to adapt the CL approach to the spatial setting, we shall follow the procedure in Currieroand Lele (1999). Under the hypothesis of intrinsic stationarity, they assume Gaussian distribution forthe differences

Ui,j = Z(xi) − Z(xj) ∼ N(0, 2γ(xi − xj; θ)), ∀i, j (6)

where γ(·; θ) is a parametric variogram model with θ ∈ � an unknown parameter vector.Let us call L(i, j; θ) the analytical expression of the Gaussian density associated with Ui,j , i �= j =

1, . . . , n and n denoting the spatial sample dimension. Thus, one easily gets that, for every (i, j), thenegative log-likelihood, namely −l(·, ·; θ) := −logL(·, ·; θ), is, up to additive constants,

−l(i, j; θ) ∝ log(γ(xi − xj; θ)

)+U2

i,j

2γ(xi − xj; θ)

that have to be summed up in order to get an estimate for θ ∈ �, namely

CL(θ) =n∑

i=1

n∑j≥i

li,j(θ) (7)

The expression in Equation (7) is a log-CL because each of the components li,j(θ) is a legitimatenegative log-likelihood. Estimates can be obtained by minimizing CL(θ) with respect to θ ∈ �. We notethat the method does not request any matrix inversion though it involves a large number of sums. Theorder of computations is O(n2). The associated estimating function can be easily computed through


768 J. MATEU ET AL.

some algebra manipulation

n∑i=1

n∑j≥i

[(γ ′(xi − xj; θ)

γ(xi − xj; θ)

(1 −

U2i,j

2γ(xi − xj; θ)

))]= 0 (8)

which is an unbiased estimating function, under the intrinsic stationarity hypothesis.The theoretical properties of CL in the spatial setting have been studied in Curriero and Lele (1999). It

can be shown that the estimating function (8) is unbiased, estimation does not request any distributionalassumptions as least squares methods do, and it has considerable computational gains with respect tomaximum likelihood. In a recent paper, Bevilacqua et al. (2007) show, by working in the spatio–temporalsetting, that CL is robust under miss-specification of the model and it works better than least squaresmethods when working on irregular grids.

3.2. Fitting negative covariances to Nea Kessani data

In this section we illustrate the basic procedure for fitting negative covariances by using the previouslydescribed methodology. The aim of this section is twofold. On the one hand, we want to illustrate thatthe methodology we propose is useful for fitting data that exhibit negative empirical covariances. Onthe other hand, we need to show that it is necessary, for this dataset, to use this methodology. For thelatter purpose, the modus operandi is necessarily comparative, so that the reader must be convinced thata covariance model that attains negative values has, for these data, a somehow better performance withrespect to other (strictly positive) covariance models that one can easily find in the classic literature. Asfor the comparative performance of other covariance functions fitted to the same data, we shall evaluatesome measures of error for the associated best linear unbiased predictor (BLUP). This will be explainedin detail subsequently.

As our procedure is necessarily comparative, we have selected two alternative models that attainonly positive values:

1. The Generalized Cauchy model (GC) introduced by Gneiting and Schlather (2004), who study itsproperties in terms of decoupling of the local and global behavior for the associated Gaussian randomfield

C(h) = σ2 (1 + ahα)−β (9)

where a > 0 is the scaling parameter, h = ‖h‖ and h ∈ Rd . Necessary and sufficient conditions for

the permissibility of this model are α ∈ (0, 2] and β positive.2. The Gaussian model (Gau), with equation

C(h) = σ2e−ah2(10)

with positive scaling parameter a.Our selection was oriented towards these models because of their wide use in the geostatistical

literature. In particular, a wide number of applications involving the use of Gaussian models canbe found in the literature. The Generalized Cauchy model allows for identifying separately theHaussdorff dimension and the Hurst effect, which is desirable for those interested in modelingphysical and geological phenomena with a geostatistical approach.



Table 2. Summary of the estimates obtained by fitting the models GC, Gau, and GSM

Model Parameters

GC C(h) = 52.48(1 + 207079h1.54)−337.2791

Gau C(h) = 52.40734e−3178h2

GSM C(h) = 50.67{2.22CM(h; 162, 1.12) − 1.22CM(h; 211.44, 1.12)}

As far as model (5) is concerned, we had a practical problem with the estimation of the parameterϑ. In particular, the problem was finding the bounds for this parameter in order to preserve thepermissibility of Equation (5), as ϑ := ϑ(α1, α2, ν1, ν2). An empirical useful procedure, followed inthis paper that overcomes this problem is the following:

(i) We fix, for computational convenience, ν1 = ν2 = ν. The number of parameters makes com-putation difficult, above all because of the small number of available data. Thus, we assumethe same level of smoothness for the Matern covariances in Equation (5);

(ii) Estimate through CL the parameter vector � := (α1, α2, ν, σ2)′ ∈ R

4+, that is findArgmax

�

CL1(�; ϑ) := CL2(ϑ; �);

(iii) Find the corresponding bounds for the possible estimate of ϑ by using Equation (5) and Table 1;(iv) Maximize, with respect to ϑ, the function CL2(ϑ; �).

This procedure is consistent to that of profile likelihood. Estimation results are reported in Table 2together with those related to GC and Gau models, whilst Figure 5 illustrates the fittings for the threeproposed models.

One can notice that estimates for the variance are very similar for the three models, but very differentin terms of range. This is probably due to the fact that we assume the H/SRF to be obtained through aweighted sum of independent H/SRF with different ranges. This procedure is sometimes used in orderto model zonal anisotropy, but in this case we assume the covariances in the linear combination inEquation (5) to be functions of the same argument. Subsequently we show that this procedure is useful,easy to implement and very effective in terms of goodness-of-fit performance.

Figure 5. Empirical and fitted spatial covariances, with continuous line for GSM, dashed for GC and dotted for Gau


770 J. MATEU ET AL.

Table 3. Cross-validation results for the models GC, Gau, and GSM

Model GC Gau GSM

CRV1 −0.02981434 −0.08080574 −0.06923005CRV2 0.9030204 2.547413 1.23477CRV3 4.005985 5.598049 3.525365

Figure 5 illustrates the empirical covariance together with the three fitted models. It can be appreciatedthat there is substantially no graphical difference between Gau and GC models, as both decay very fastto zero at low spatial lags. The GSM model attains negative values and then goes to zero. Actually, itwould have been desirable, for having better diagnostic results, to have a model with lower negativevalues, but it was not possible, owing to the already mentioned constraints on the bound. It is also worthremarking that implementation and estimation of GSM model are very handy and that the model allowsfor an easy physical interpretation of the phenomena under study.

3.3. Diagnostics

In order to compare the performances of the proposed models in terms of fitting, we followed theclassical cross-validation procedure, whose details can be found in Christakos (1992). In particular, weuse the cross-validation statistics CRV1, CRV2, and CRV3 proposed by Huang and Cressie (1996) andCarroll and Cressie (1997), whose expressions can be found therein. Roughly speaking, to every ofthe n points and corresponding realizations, we associate the simple kriging predictions (we assumethe mean is known and constant, as the data was detrended) obtained by using, for every point xi inthe sampling design, the remaining (n − 1) points. Note that for these individual predictions we keptfixed the covariance function obtained with the whole set of n data. Then, for every point one cancalculate the prediction error and the quadratic prediction, that is, respectively, the difference and thesquared difference between realization and prediction. Thus, CRV1 checks the relative unbiasedness ofthe prediction error and should be close to zero, while CRV2 checks the accuracy of standard deviationof the prediction error and should be close to one. We particularly focus on CRV3, as recommended byHuang and Cressie (1996), as it is a measure of goodness of prediction.

Figure 6. Left: Kernel density estimate for original residuals (continuous line) and GSM (dotted). Right: Observed versuspredicted residuals, using GSM



Figure 7. Simple kriging predictions of Nea Kessani temperatures at several depths

Cross-validation results are reported in Table 3. It is quite evident that, for the index CRV3, whichis the most important, GSM model outperforms GC and Gau, whilst the GC is slightly better in CRV1,which measures the relative unbiasedness. In Figure 6 (right) we show the plot of observed residualsversus predicted ones, obtained with the GSM model, as well as the plot of the nonparametric kernelestimate of the density of the original residuals together with that of the predicted residuals obtainedusing the GSM model (Figure 6, left).

We also mapped the kriging values at some fixed depth levels. In Figure 7, one can appreciate thatthere is a general trend resulting in lower temperatures when sampling at low depth. This is consistentwith the information we have concerning the temperature of the geothermal field.

4. CONCLUSIONS AND DISCUSSION

In this paper, we have shown a valid approach to fit negative empirical covariances to a H/SRF. Theproposed method is easy to implement and allows to check directly the permissibility conditions in termsof bounds for the parameter ϑ that induces negative values on the corresponding covariance model. We


772 J. MATEU ET AL.

showed that under the presence of negative covariances, the proposed model provides better goodness-of-fit and prediction results than other covariance models that are classically used in the geostatisticalframework.

A future topic of research could be the following: find some permissibility criteria in order toensure negative weights on a linear combination of covariance functions that are additionally compactlysupported. This is very challenging, as the arguments that can be used to show Proposition 1 of thispaper are in general difficult to verify for compactly supported covariance functions, as they rarelyadmit a closed form for the associated Fourier transform. Note that the Fourier transform associated toa covariance function with support on the unit ball of R

d is analytic and computation of m and M as inProposition 1 is often infeasible.

ACKNOWLEDGMENTS

The authors thank the guest editor and two anonymous referees whose thorough revision allowed to improve con-siderably an earlier version of the manuscript. The authors are also indebted to Carlo Gaetan for useful discussionsand suggestions during the preparation of the manuscript. This research has been partially supported by the SpanishMinistry of Science and Education (grant no. MTM2004-06231).

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fitting negative spatial covariances to geothermal field temperatures in nea kessani (greece

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