A state-space model approach to optimum spatial sampling design

Environmental and Ecological Statistics 5, 29±44 (1998)

A state-space model approach to optimum spatial sampling design based on entropy M.C. BUESO, J.M. ANGULO and F.J. ALONSO Departamento de EstadõÂstica e I.O. Universidad de Granada, Campus de Fuentenueva s/n, E-18071 Granada, Spain Received February 1996. Revised July 1997 We consider the spatial sampling design problem for a random field X. This random field is in general assumed not to be directly observable, but sample information from a related variable Y is available. Our purpose in this paper is to present a state-space model approach to network design based on Shannon's definition of entropy, and describe its main points with regard to some of the most common practical problems in spatial sampling design. For applications, an adaptation of Ko et al.'s (1995) algorithm for maximum entropy sampling in this context is provided. We illustrate the methodology using piezometric data from the VeÂlez aquifer (MaÂlaga, Spain). Keywords: Shannon's entropy, spatial sampling, state-space model 1352-8505  1998 Chapman & Hall

1. Introduction Spatial sampling design is a common problem in many fields of application (geology, geophysics, agriculture, etc.), with spatial dependence playing a crucial role. Of particular significance is the Gaussian case, for which, without considering the effect of the deterministic trend defined by the mean, the stochastic spatial dependence is determined by the covariance structure, which can be derived from a specific model or represented by an empirical function. This problem can be formulated differently depending upon the situation and, of course, on the purpose. The design problem is to find a set of sampling locations (optimum under some specific criterion) either observing X or some related variable (random field) Y , with or without assuming any restrictions, and with or without considering any prior sample or model information. It would be a difficult, if not impossible, task to give a full answer to this general problem and all its many possible derivations. Different approaches have been introduced in the literature (see, for example, De Gruijter and Ter Braak, 1990). A geostatistical approach is used by Bras and RodrõÂguez-Iturbe (1976), BoÂgardi et al. (1985), Aspie and Barnes (1990), Samper and Carrera (1990, Chapter 19), Trujillo-Ventura and Ellis (1991), Haas (1992), and Journel (1994), among others. Cressie summarises the main aspects of the general geostatistical approach (1991, Sections 4.6.2 and 5.6.1). 1352-8505  1998 Chapman & Hall

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Bueso, Angulo, Alonso

A more random-field focused formulation can be found in Christakos (1992, Chapter 10). Using an information theory approach, a point of view that we adopt in this paper, Caselton and Hussian (1980) propose the choice of a network which maximizes the entropy of the random variables at gauged sites. Along the same lines, Caselton and Zidek (1984) consider the problem in a Bayesian framework, formulating it as a decision problem. Their optimal choice maximizes the information in the random variables at gauged sites on the random variables at ungauged sites. Caselton et al. (1991) assume that the random vector depends on a parameter with a prior distribution and their purpose is also to reduce uncertainty about this parameter. Thus, they select stations to be observed that minimize the residual uncertainty. Wu and Zidek (1992) study the problem of reducing a network considering clusters, applying for each cluster the method proposed by Caselton et al. (1991). Guttorp et al. (1993) examine the complementary problem of extending an existing network; their optimal network reduces the uncertainty about future observations and model parameters. In the Gaussian case, Ko et al. (1995) provide an upper bound for the entropy and develop an exact algorithm based on this bound for solving the design problem. In this paper we consider the idea of using the Shannon entropy (Shannon, 1948) to sampling network design on a state-space model framework. The method is applicable to discrete or continuous parameter spatial processes, and is described in detail for the Gaussian case, where the special distribution properties lead to a particularly convenient mathematical treatment. In section 2 we present the general formulation of the problem, analyse the problems of extending and reducing a pre-existing network, and provide an adaptation to a state-space-model framework of Ko, Lee and Queyranne's (1995) algorithm for maximum entropy sampling. An application to piezometric data from the VeÂlez aquifer (MaÂlaga, Spain) is presented in section 3, where the non-observable process of interest is assumed to satisfy a Laplace stochastic partial differential equation. This model has been considered by several authors (Whittle, 1954; Jones, 1989; Angulo et al., 1994, among others).

2. Fundamentals and the method Below we describe in detail the main formal aspects of the procedure proposed for sampling network design based on entropy.

2.1 General formulation In this section we formulate the spatial sampling design problem. Assume that the variable of interest X is not directly observable, but that information on X is obtained by sampling a variable Y related to X, the dependence relationship between both variables being known. A simple case of practical interest appears when Y is obtained by adding an observation error to the variable X. In general, however, X and Y may represent different physical magnitudes. Let us also consider that Y is potentially observable on , and we are interested in knowledge of X on a (possibly) different set . For practical purposes, we assume discrete sampling and finite and . Let S be the subset (to be determined) of the locations where Y

Optimum spatial sampling design

31

is actually to be observed, and let S 0 be its complement in , S 0 ÿ S. Denote by X the vector of the random variables Xsi , for all si 2 , and denote by YS the vector of sampled random variables (si 2 S). The Shannon entropy of X is defined as HX EX ÿ log fX ; where fX is the density (or probability mass) function of X . The conditional entropy of X given YS yS } is defined as HX jYS yS EX jYS yS ÿ log fX jYS yS : Thus, the mean conditional entropy of X given YS is defined as HX jYS EYS HX jYS yS EX ;YS ÿ log fX jYS ; where fX jYS is the conditional density (or probability mass) function of X given YS . (EZ denotes the expectation with respect to the distribution of Z.) The amount of information on X in YS is given by IYS ; X HX ÿ HX jYS : Demanding optimum knowledge of X on set forces us to find max IYS ; X ; S

1

or, equivalently, since HX is fixed, min HX jYS ; S

2

where S stands for a set of restrictions such as imposing a maximum cardinality for S or sampling cost. Note that HX jYS HX ; YS ÿ HYS ;

3

where both terms on the right-hand side depend on S. If YS and X are jointly Gaussian for any admissible S (meeting the restrictions), we have HX jYS

N 1 j CX [YS j ; 1 log2 log 2 2 j CYS j

4

where N is the cardinality of , CX [YS is the covariance matrix of the joint vector X0 ; Y0S 0 , and CYS is the covariance matrix of vector YS . The parameters involved in the model and the latter matrices are estimated from historical data, the estimates obtained then being assumed to be the true values. Note that the quotient of determinants in the last term of expression (4) is equal to j CX jYS j, where CX jYS is the conditional covariance matrix of vector X given YS (which does not depend on actual observed values, but on the locations of the sampled random variables in YS ), and j j denotes the matrix determinant. Then problem (2) is reduced to finding min log j CX jYS j; S

5

or, equivalently, since the logarithm is an increasing function, min j CX jYS j : S

6


32

Note that this criterion, as well as certain other related criteria (in the simple case where X and Y are equal and sets and S correspond to the non-observed sites and the observed sites, respectively), has been used by other authors for spatial sampling applications (e.g. Mardia and Goodall, 1993). Our purpose is to give a formal definition and justification of these criteria from an entropy point of view and in a state-space framework. The mutual information between two Gaussian processes can also be expressed in terms of the canonical correlation coefficients, an approach considered by Caselton and Zidek (1984). The computation of CX jYS in (5) can be difficult, for it involves the inversion of a matrix that may be of large dimension. The matrix inversion problem can be approached using an orthogonalization procedure (e.g. a Cholesky decomposition). In practice, search algorithms are commonly considered to find best suboptimal solutions which give, in general, good approximations to optimal solutions, or may be used as a starting point for the application of the optimum entropy exact sampling design algorithm (see Ko et al., 1995, and sections 2.3 and 3 below).

2.2 Adaptation to some pre-existing sampling network redesign problems In this section we consider certain problems common in practice that are related to the extension or reduction of an assumed pre-existing observation network. Again, a variety of particular problems may be obtained from this general idea. In the paragraphs below we describe two simple important cases: adding to and deleting from the network a certain number of sampling locations. Both problems are solved using an adaptation of the approach introduced in section 2.1. We assume the same elements defined at the beginning of section 2.1. 2.2.1 Extending a pre-existing network Let us assume that the pre-existing network is composed of m sites in a set S , and that we want to add some new sampling locations, this set being denoted by Sa . Sa denotes the class of all the subsets of ÿ S to be considered as potential candidates for the extension. For example, a common choice for Sa is the class of all the subsets of ÿ S having a certain fixed number p of elements. YS[Sa represents the final sample observation vector. The amount of information on X in YS[Sa is given by IYS[Sa ; X HX ÿ HX jYS[Sa : Then, an optimum design minimizes HX jYS[Sa HX ; YS[Sa ÿ HYS[Sa : In the joint multivariate Gaussian case, the mean conditional entropy to be minimized takes the form HX jYS[Sa

N 1 1 log2 log j CX jYS[Sa j; 2 2


33

where N is the cardinality of X , and CX jYS[Sa is the conditional covariance of X given YS[Sa . As N is fixed, the problem is reduced to finding min log j CX jYS[Sa j : Sa

Now, the YS part in the conditioning subvector YS[Sa is common to all the choices Sa 2 Sa . So, for computational purposes, it is advantageous to express CX jYS[Sa in terms of CX jYS . This is given by the following relation: CX jYS[Sa CX jYS ÿ CX ;YSa jYS CYÿ1Sa jYS CYSa ;X jYS ;

7

where ÿ1 CYS ;YSa ; CX ;YSa jYS CX ;YSa ÿ CX ;YS CY S

CYSa jYS CYSa ÿ CYSa ;YS CYÿ1S CYS ;YSa ;

CYSa ;X jYS CX ;YSa jYS 0

(here CA;B denotes the cross-covariance matrix of A and B, and CA;BjC the conditional cross-covariance matrix of A and B given C). Thus, assuming that CX , CYÿ1S , and CX ;YS are available from previous computations, we only need the additional computation of the matrix CYSa ;X ;YS[Sa ; for each Sa 2 Sa , where X ; YS[Sa is the joint vector of X , YS , and YSa . (As mentioned in section 2.1, the parameter estimates are here assumed to be the true parameter values, and the problem of sensitivity of the latter matrices as estimated from the data is not considered.) In section 2.1 we propose that the inversion of CYSa jYS can be made easier by orthogonalizing YS[Sa , which can be done using a fixed orthogonalization for YS for all Sa . 2.2.2 Reducing a pre-existing network Let us now assume that we are interested in finding an optimal reduction of a preexisting sampling network S , by deleting some of the locations in S. Let Sd be the set of sites to be removed from S, and let Sd be the class of all the possible admissible choices for Sd . For example, Sd might consist of the subsets of S with a certain fixed number of elements. YSÿSd represents the final sample observation vector. The structure of the problem is basically similar to that presented in section 2.2.1. Except for particular specifications of Sd , it would seem that the advantage of having fixed common information might not be applicable in this case. However, as we show below, this is indeed possible by substituting computations involving inverted matrices of dimension m ÿ d by the computation of a single inverted matrix of dimension m and inverted matrices of dimension d (note that in practice usually d m). The amount of information on X in YSÿSd } is given by IYSÿSd ; X HX ÿ HX jYSÿSd : Then, an optimum design for the reduced network minimizes HX jYSÿSd HX ; YSÿSd ÿ HYSÿSd :


34

In the joint multivariate Gaussian case, the problem is reduced to finding min log j CX jYSÿSd j : Sd

Again, CX jYSÿSd can be related to CX jYS , which is given by the following expression: CX jYSÿSd CX jYS CX ;YSd jYSÿSd CYÿ1S jYSÿS CYSd ;X jYSÿSd ; d

8

d

where CX ;YSd jYSÿSd CX ;YSd ÿ CX ;YSÿSd CYÿ1SÿS CYSÿSd ;YSd ; d

CYSd jYSÿSd CYSd ÿ CYSd ;YSÿSd CYÿ1SÿS CYSÿSd ;YSd ; d

0

CYSd ;X jYSÿSd CX ;YSd jYSÿSd : In this case, matrices CX ;YSd and CX ;YSÿSd are (complementary) blocks in CX ;YS . The same is true for CYSd , CYSÿSd , CYSd ;YSÿSd , and CYSÿSd ;YSd with respect to CYS . With regard

ÿ1 to the inverse matrix CY , and matrix CYÿ1SÿS CYSÿSd ;YSd , they can be obtained by S jYSÿS d

d

d

easy manipulations of CYÿ1S . In fact, CYÿ1S jYSÿS is the matrix obtained by deleting the rows d

d

and columns corresponding to S ÿ Sd in CYÿ1S . In addition, CYÿ1SÿS CYSÿSd ;YSd can be d

obtained as ÿ1 CYÿ1SÿS CYSÿSd ;YSd ÿCY CYÿ1S ÿ1 Sd ; S SÿSd ;Sd

9

d

where both factors in the right member are matrices corresponding to the indicated blocks in CYÿ1S , obtained by deleting the respective complementary indexes in S. Usually, Sd have a small cardinality in comparison to that of S, so that the inversion required in the last term of (9) for obtaining CYÿ1SÿS CYSÿSd ;YSd is relatively easier than d directly computing CYÿ1SÿS , for each Sd . For example, if m 200 and d 10, instead of d inverting as many matrices of dimension 190 as different sets of locations to be checked, we only need to invert one matrix of dimension 200 and the same number of matrices of dimension 10, which obviously means a drastic reduction in computational burden. (If observations YS have been previously orthogonalized, it is necessary in this case to recover the original CYS by using the inverse basis change matrix, for the orthogonalized variables are not individually associated to single locations.)

2.3 Exact optimum design algorithm In the Gaussian case, Ko et al. (1995) have provided an upper bound for the entropy. Based on this bound they have developed an exact algorithm for the maximum entropy sampling design. This algorithm is not directly applicable to a spatial state-space framework since the covariance matrix involved is conditional. In this section we adapt this algorithm to the minimization problem considered. A lower bound for the minimum is given by the following expression: min jCX jYS j dCYF [E ; F ; E; s : jCX jYF j

S:jSjs F SF [E

sÿf Y eÿsif CYE jX ;YF i1

i CYE jYF

;


35

where i C is the ith eigenvalue of C (eigenvalues are taken in decreasing order). In the algorithm proposed here we start with an initial location set Sc (for example, the solution obtained by the greedy algorithm) and the upper bound UB : jCX jYSc j. Consider the initial set of active subproblems L fCYF [E ; F ; E; sg and the lower bound LB : dCYF [E ; F ; E; s. The algorithm consists of the following steps: 1.

If LB < UB, remove an active subproblem CYF [E ; F 0 ; E 0 ; s from L and select i 2 E 0 as a branching index. (a) Consider subproblem CYF [E ; F 0 ; E 0 ni; s. i. If jF 0 j jE 0 j ÿ 1 > s, append CYF [E ; F 0 ; E 0 ni; s to L and calculate dCYF [E ; F 0 ; E 0 ni; s. ii. If jF 0 j jE 0 j ÿ 1 s, define S : F 0 [ E 0 ni. If jCX jYS j < UB, replace Sc with S and set UB : jCX jYS j. (b) Consider subproblem CYF [E ; F 0 [ i; E 0 ni; s. i. If jF 0 j 1 < s, append CYF [E ; F 0 [ i; E 0 ni; s to L and calculate dCYF [E ; F 0 [ i; E 0 ni; s. ii. If jF 0 j 1 s, define S : F 0 [ i. If jCX jYS j < UB, replace Sc with S and set UB : jCX jYS j. (c) Update LB : min dL.

2.

L2L

Otherwise, Sc is an optimal solution.

3. An application to piezometric data We illustrate the method described above by an application using piezometric data from the VeÂlez aquifer (MaÂlaga, Spain), consisting of observations from 66 wells. The data have been collected by the Instituto del Agua (Water Institute) at the University of Granada (Spain). The observations represent water heights in metres above sea level and are shown in Table 1. The s1 and s2 coordinates are in UTM (Universal Transverse Mercator). Figure 1 shows a contour-level plot of piezometric heads obtained by ordinary isotropic kriging with a linear variogram. We assume that the random component of the piezometric random field X satisfies a stochastic partial differential equation given by the following expression: @ 2 xs @ 2 xs ÿ xs s; @s21 @s22

10

where s s1 ; s2 0 is the continuous coordinate vector, is a positive parameter and s is white noise with variance 2 . Equation (10) is the stochastic Laplace equation considered by Whittle (1954) for data observed on a complete grid, and by Jones (1989) and Angulo et al. (1994) for irregularly observed data (extensions of this equation have been proposed by Vecchia, 1988, and by Jones and Vecchia, 1993). A justification of the meaning and use of model (10) for the representation of piezometric head data is provided, for example, in Jones (1989). Process xs in equation (10) has an isotropic correlation structure in space given by (Whittle, 1954)


36

Table 1. Piezometric data from 66 wells in the VeÂlez aquifer (MaÂlaga, Spain) s1

s2

Water heights

s1

s2

Water heights

395000 397028 396920 397028 397065 397235 397823 398108 398978 399434 399452 399359 399378 399438 399465 399540 399890 399470 399448 399640 399600 399930 399875 400423 400780 400925 400050 400710 400713 400210 400453 400458 400478

4075788 4074353 4074295 4074318 4074220 4074390 4073703 4073688 4072778 4075893 4075898 4075228 4074843 4074685 4074235 4073320 4072990 4072265 4072303 4072320 4072375 4071465 4070850 4071088 4071150 4071040 4070760 4070098 4069963 4069970 4069543 4069448 4069303

71.509 44.853 44.793 43.334 43.161 41.921 36.06 33.845 25.39 47.545 46.189 37.063 34.163 32.81 30.298 27.369 26.554 23.215 23.265 23.117 23.972 16.849 15.781 16.069 16.06 17.254 15.722 12.976 12.758 12.319 11.166 10.476 9.93

400530 400530 400578 400713 400888 401164 400865 400938 400878 400715 400978 400815 400793 400560 400726 400778 400545 400620 400753 400830 401100 401002 400790 400925 400635 400710 400662 400690 400425 400220 401645 401412 401420

4069555 4069670 4069333 4069058 4068802 4068658 4068540 4068353 4068118 4068020 4067983 4067903 4067302 4067100 4066843 4066678 4066480 4066360 4066213 4066275 4066365 4065793 4065330 4065515 4065820 4065650 4065490 4065325 4065535 4065655 4066125 4065890 4065715

11.585 12.066 10.093 8.892 7.064 6.339 6.27 5.358 4.015 2.938 3.191 1.204 ÿ0.226 0.81 ÿ0.043 ÿ0.396 ÿ2.839 ÿ0.387 ÿ0.268 ÿ0.275 ÿ0.325 ÿ0.039 0.051 ÿ0.005 ÿ0.181 ÿ0.073 0.205 0.076 ÿ0.463 ÿ0.191 0.254 0.28 0.203

p p r r K1 r ; where r is the distance between points and K1 is the modified Bessel function of the second kind order 1. This equation establishes the meaning of parameter with regard to spatial dependence in model (10). We consider the sample information to be given by observations at n locations s1 ; . . . ; sn , of a process, ys, related to process xs by the observation equation ysi xsi esi ;

i 1; . . . ; n;

11

where esi i 1; . . . ; n are the measurement errors with zero mean and variance 2e . The unknown parameters are , 2 and 2e . We assume that xs and ys are Gaussian processes.


37

Figure 1. Contour-level plot of piezometric heads in the VeÂlez aquifer (MaÂlaga, Spain), and subregion of interest (coordinates are in UTM).

In order to apply the method described in section 2, we need to compute the conditional covariance matrix of xs given fysi ; i 1; . . . ; ng (for s in ), according to equation (8). We estimate the covariance matrix using the maximum-likelihood estimates obtained for the parameters from observations at the 66 wells. For the purposes of illustration, we consider reducing the pre-existing network. First, a deterministic trend is removed from the original data by fitting a quadratic surface, the residuals obtained then being considered as the values for ys in the observation equation (11). We estimate the values of the unknown parameters using the approach in Jones (1989), which results in ^ 0:00000239, ^e 0:462925, and r^ 0:1063, with r 2 =2x . Assuming these values of estimates to be true values for the parameters, a reduction of the network is performed. To that end, a FORTRAN 77 program has been developed. In all the cases studied here, a certain subregion of the aquifer domain defined by a discrete mesh (see Fig. 1) is considered as set . First, we consider a sequential (optimal in each step) reduction of the network. The initial network and the prediction error

38


Figure 2. Initial network for 66 observed locations in the VeÂlez aquifer (MaÂlaga, Spain).

Figure 3. Contour-level map of prediction error standard deviations for the initial network.


39

Figure 4. Resulting network after (sequentially) deleting 44 sites.

Figure 5. Contour-level map of prediction error standard deviations after (sequentially) deleting 44 sites.

40


Figure 6. Conditional entropy (except for constant terms) and rate of information vs. number of (sequentially) deleted sites.

standard deviations are shown in Figs 2 and 3, respectively. The resulting network after deleting 44 sites and the corresponding prediction error standard deviations are displayed in Figs 4 and 5, respectively. In Fig. 6, the resulting conditional entropy (except for constant terms) and the rate of information IX ; YSÿSd =IX ; YS are represented with respect to the number of deleted sites. By using this ratio, we can determine the maximum number of locations to be observed to maintain a certain rate of information. For example, to retain 80% of the amount of information con-


41

Figure 7. Comparison of the conditional entropy (except for constant terms) in sequential reduction of network obtained for the entropy-based criterion and other related criteria based on alternative measurements of the covariance matrix (trace, maximum eigenvalue and maximum element of the diagonal).

tained in the initial network on the region of interest, we require at least 22 locations to be observed. Finally, in Fig 7, we compare the entropy-based criterion with other related criteria based on alternative measurements of the covariance matrix such as the trace, the maximum eigenvalue, or the maximum element of the diagonal (see, for example, Mardia and Goodall, 1993). In our context, these criteria are respectively formulated as follows:


42

Figure 8. Map containing the locations: non-included sites, forced sites, nonselected sites, and ? selected sites in the network.

min trCX jYSÿSd Sd

d minmax SÿS i

Sd

i

minmax iSÿSd ; Sd

i

d SÿS i

and iSÿSd , for i 1; . . . ; N, being the eigenvalues and the elements of the with diagonal of CX jYSÿSd , respectively. In the second example studied, we consider only 45 of the 66 available sites as potential locations to be observed. The non-included sites are located in the north of the aquifer, far away from the region of interest, and their influence is negligible. We force 25 predetermined sites to be in the network (see Fig. 8), completing it by addition of 15 more sites. The optimal design has been achieved using the exact algorithm presented in section 2.3. The sequential solution has been taken as the starting solution for the algorithm, which in the end turned out to be optimal. The results of this example are shown in Fig. 8.

4. Conclusion The objective of this work is to present a methodology to design or redesign a spatial network when the underlying process of interest and the observation is defined by means of a state-space model. The main advantage of working within this framework is given by the fact that in many practical situations the available data may not correspond to the variable of interest. In addition, the potentially observable locations may be different from the set of interest sites for the variable X.


43

The entropy-based approach to spatial sampling design has been extensively applied in the literature. In the state-space-model context, the procedure consists in minimizing the conditional entropy. In the Gaussian case, this is equivalent to minimizing the logarithm of a conditional covariance matrix. A detailed formulation for extending or reducing a pre-existing network is shown. When the set of locations is increased (decreased) by sequentially adding (deleting) locations, the procedure is quite simple by appropriately handling blocks of a certain covariance matrix. When the problem is of a large dimension, the achievement of an optimal network is costly in computing time. An exact algorithm for finding an optimal design, adapting the algorithm proposed by Ko et al. (1995) to the space-state-model framework, is presented. In the example studied here the optimal solution obtained coincides with the sequential one. The model proposed for the observed variable in the examples treated in this paper consists in adding an observation error to the variable of interest. More complex models may be considered as the procedure only requires, in the Gaussian case, the covariance structure between the involved variables.

Acknowledgements We thank the editor and the three referees for their helpful comments and suggestions, which have significantly improved this paper. This work has been supported in part by the Plan Nacional de I+D (Project AMB93± 0932) of the ComisioÂn Interministerial de Ciencia y TecnologõÂa, Ministerio de EducacioÂn y Ciencia, Spain. We are also grateful to the Instituto del Agua (Water Institute), Universidad de Granada (Spain), in particular to JoseÂ, L. GarcõÂa-AroÂstegui for support in preparing the piezometric data of VeÂlez aquifer and graphics.

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Biographical sketches The authors are members of the Departmento de EstadõÂstica e InvestigacioÂn Operativa of Universidad de Granada, Spain, and collaborate on a regular basis with the Instituto del Agua of this university on stochastic modelling and applications in hydrology, currently under project AMB93-0932, of Environment and natural Resources Planning, of the ComisioÂn Interministeral de Ciencia y TechnologõÂa, Ministerio de EducacioÂn y Ciencia, Spain. JoseÂ M. Angulo, who is Associate Professor, is the person responsible for the above mentioned project, and heads the research group on space-time stochastic modelling. Francisco J. Alonso is Assistant Professor, and did his Ph.D. on estimation and prediction of spatial processes. MariaÂ C. Bueso is Assistant Professor and her research is related to spatial sampling design problems.