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Article Excerpt
1. INTRODUCTION
Many naturally occurring spatial processes exhibit highly irregular (i.e., nonsmooth) behavior in their covariance structure. For instance, in geostatistical applications where the response variable of interest is highly dependent on rock type, we can expect to see sharp transitions in the covariance between points across strata (i.e., rock boundaries). In such cases conventional spatial models built on stationarity assumptions are inappropriate. Moreover, most nonstationary models, in which the covariance between points depends on location, also implicitly assume that the covariance dies off smoothly as a function of distance.
In this article we propose a model that can deal with sharp transitions in covariance structure. The method that we adopt is a random (Bayesian) partition model in which the spatial domain is partitioned using a Voronoi tiling such that within regions (tiles), the process is assumed to be stationary, whereas across regions, we assume independence. By treating the model structure within a Bayesian framework, we are able to assess uncertainty in the position of the boundaries as well as the form of the model within regions. This leads us to model averaging when making predictive statements, which tends to smooth the nonsmooth individual models in areas of greatest uncertainty.
Our motivating example is taken from the Schneider Buda oil field in the Wood County, Texas. The response variable of interest is soil permeability, and the task is to recover a two-dimensional surface over the field based on 92 measurement of permeability taken at distinct locations. The ability to accurately model the permeability of the oil field is a vital task for petroleum engineers, because the profitability of secondary stage oil extraction from the field is a function of the permeability.
Although we concentrate on this example from petroleum engineering, our approach is generic and applicable to any situation in which sharp changes in covariance are suspected. For instance, whenever a natural or man-made boundary exists in the spatial field, there is likely to be a possibility of highly irregular changes in covariance between points.
In this article we assume that the reader is familiar with conventional approaches to modeling Gaussian spatial fields using stationary kriging methods. Cressie (1993), Stein (1999), Handcock and Wallis (1994) have provided comprehensive overviews of this approach. The seminal articles by Le and Zidek (1992). Handcock and Stein (1993), and Brown, Le, and Zidek (1994) detail Bayesian alternatives to kriging. The rest of this article is organized as follows. Section 2 overviews current approaches to nonstationary spatial processes and highlights why they would be inappropriate for data where the covariance structure changes sharply across boundaries. Section 3 describes our procedure using a piecewise Gaussian process within a Bayesian hierarchical model and the computation strategies. Section 4 applies the method to several simulated datasets and shows that it results in good consequences. Section 5 analyzes permeability data from the Schneider Buda field in Wood County. Texas. Section 6 ends the article with a discussion.
2. EXISTING APPROACHES FOR NONSTATIONARY SPATIAL PROCESSES
In this section we briefly review some current approaches to model nonstationary spatial fields.
2.1 Moving-Window Approaches
Haas (1990, 1995) proposed the idea of a moving-window approach where to predict at a spatial location s, one first constructs a local spatial window around it and then assumes a stationary random field model within the window and performs kriging. This operation is reperformed at every point in which one is interested. The size of the window is chosen by cross-validation, although there is always a question as to whether using a particular cross-validation criterion is the best or most reasonable approach. The window is by design symmetric around the point s and hence is unable to capture very sharp changes in covariance structure. In addition, by specifically defining a local model in this way, joint predictions at, say, two locations do not take correlation into account.
2.2 The Method of Empirical Orthogonal Functions
This method is based on Karhunen-Loeve expansions of the covariance function introduced by Cohen and Jones (1969) and have been used by geophysicists (Creutin and Obled 1982). Nychka, Wikle, and Royle (1999) have modified the model using wavelet basis functions. Although this is clearly a very general and flexible approach, it has not proved popular, due to a lack of connection with traditional approaches based on kriging and variograms. Our piecewise approach is more interpretable, because we use conventional kriging models within each region, and is more connected with traditional approaches.
2.3 Deformation Approaches
The very popular idea of the spatial deformation approach is to perform a change of coordinates that results in isotropic spatial correlation. The principal advocates of this methodology are Sampson and Guttorp (1992) and Guttorp and Sampson (1994). Usually the mapping is done from the original geographical space to an alternative dispersion space where stationarity is assumed, and usually the transformation functions are modeled by thin plate splines (Sampson and Guttorp 1992). This method works only with multiple observations. Moreover, the single smooth mapping will not be well suited to handling sharply changing covariance structures, because the thin-plate spline implicitly penalizes this. Sampson, Damian, and Guttorp (2001) have described more recent developments. Smith (1996) pointed out that spatial correlations of temperature data had different structures in the western and eastern United States and that the deformation approach does not handle this situation very well. Recently, Schmidt and O'Hagan (2003) proposed a Bayesian version of the deformation approach by developing a fully Bayesian model and using Markov chain Monte Carlo (MCMC) based computation. They modeled the transformation function using Gaussian processes rather than thin-plate splines.
2.4 Models Using Kernels
A stationary Gaussian process Z(s) can be expressed as the convolution of a Gaussian white noise process X(s) and convolution kernel K (s) as Z (s) = [[integral].sub.[R.sup.2]] K (s - u)X(u) du. Higdon. Swall, and Kern (1999) extended this method with the novel idea of a spatially evolving kernel, where they considered the process Z(s) = [[integral].sub.[R.sup.2]] [K.sub.s](u)X(u) du, where the kernel [K.sub.s] depends on position s. They treated [K.sub.s] as an unknown smooth function and used a Bayesian hierarchical model to estimate it with other model parameters. This method is ideally suited for smoothly varying spatial covariance structure and may not be that successful in capturing sharp changes.
Other recent developments for modeling nonstationary processes include those of Nott and Dunsmuir (2002) and Fuentes and Smith (2001). Nott and Dunsmuir (2002) extended the deformation approach of Sampson and Guttorp (1992), which is more useful for smoothly varying structures. The method suggested by Fuentes and Smith (2001) proposes a superposition of locally stationary processes, which again are assumed to evolve smoothly over the spatial domain.
2.5 Multiple Models
The work most closely related to our approach is that of Fuentes (2001), who used a mixture of locally stationary process defined within disjoint regions to model the globally nonstationary process. Fuentes (2001) assumed that the shape of the regions are known a priori and the number of regions is estimated using the Akaike information criterion or Bayes information criterion for model choice. Kernels are used to weight the influence of the local processes outside of their regions.
A key feature of our method is the treatment of model uncertainty through a prior distribution on the shape and the number of regions. Model uncertainty relates to the fact that many different spatial models may offer nearly equally plausible representations of the data. The uncertainty in the partition locations also allows us to achieve data-adaptive smooth transitions in the covariance structure, as well as sharp changes. Although our model is motivated mainly for data with sharp changes, the marginal predictive densities can capture smoothly changing processes through model averaging (Draper...
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