In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, that is the spatial distribution is unchanged when the origin of the index set is translated and the process is stationary under rotations about the origin. However in environmental problems, it is not very realistic to make such assumptions since local influences in the correlation structure of the spatial process may be clearly found in the data. This paper proposes a Bayesian model wherein the main aim is to address the anisotropy problem. Following Sampson and Guttorp (1992), we define the correlation function of the spatial process by reference to a latent space, denoted by D, where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by G. We adopt a Bayesian approach in which the mapping between G space and D space is represented by an unknown function d(:). A Gaussian process prior distribution is defined for d(:). ...

Spatial processes are an important modeling tool for many problems of environmental monitoring. Classical geostatistics is based on processes which are stationary and isotropic, but it is widely recognized that real environmental processes are rarely stationary and isotropic. In this paper, a new class of nonstationary processes is proposed, based on a convolution of local stationary processes. This model has the advantage that the model is simultaneously de ned everywhere, unlike \moving window" approaches, but it retains the attractive property that locally in small regions, it behaves like a stationary spatial processes. We discuss model tting through exact and approximate likelihood maximization, and propose a hierarchical Bayes approach to allow predictive inference when the parameters of the model are unknown. Applications include obtaining the total loading of sulfur dioxide concentrations over dierent geo-political boundaries.

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For modeling spatial processes, we propose rich classes of range anisotropic covariance structures that greatly increase the scope of variogram contours in R² and include geometric anisotropy and isotropy as special cases. We demonstrate how the class of all completely monotonic isotropic variograms can be extended to capture range anisotropy and illustrate with two examples, the Matérn and the general exponential. We adopt a Bayesian perspective and fit these range anisotropic covariance models using sampling-based methods. In the presence of measurement error/microscale effect, we develop the noiseless predictive distribution. We analyze a data set of scallop catches, withholding ten sites, to compare the accuracy and precision of the standard and noiseless predictive distributions.

Many environmental processes are heterogeneous in space (spatially non-stationary), due to factors such as topography, local pollutant emissions, and meteorology. Much of the commonly used spatial statistical methodology depends on simplifying assumptions such as spatial isotropy. Violations of these assumptions can cause problems, including incorrect error assessment of spatial estimates. This paper demonstrates important properties of the spatial deformation model of Sampson and Guttorp (1992) and Guttorp and Sampson (1994) for heterogeneous anisotropic spatial correlation structure. The modeling approach utilizes a deformation of the geographic coordinate space into a new coordinate system (known as the D-space, or D-plane in two dimensions) where isotropic spatial correlation structure is modeled. We provide proofs of two fundamental properties of the model: validity and invariance of the modeled correlations to translating, scaling and rotating operations on a D-space representati...

Environmental monitoring networks are recording pollutant levels, weather, and a myriad of other factors. It is often of interest to estimate these values at locations where records are not available. Many spatial estimation procedures rely on spatial covariance models. Assumptions of spatial isotropy or stationarity may be violated due to factors such as topography and local emissions structures. In this paper we discuss computational issues for a heterogeneous (spatially non-stationary) model for spatial correlations between point monitoring sites. The modeling procedure involves deforming the geographic space into a new space (D-space) where inter-site correlations depend only on distances. Correlations between unmonitored sites are then estimated as a function of distance in the D-space. The estimation of the D-space locations of the monitoring sites, and of the parameters of the isotropic D-space variogram model is a difficult multidimensional problem. The dimensionality increases...

assessment of numerical models

this paper we give a new methodology for spatial interpolation of nonstationary processes using spectral methods. The interpolation procedure presented here concentrates on the high frequency values of the process. More specifically, we represent the process locally as a stationary isotropic random field, but only the parameters of the stationary process that describe the behavior of the process at high frequencies 2

Many environmental data sets have a continuous domain, in time and/or space, and complex features that may be poorly modeled with a stationary Gaussian process (GP). We adapt stochastic volatility modeling to this context, resulting in a stochastic heteroscedastic process (SHP), which is unconditionally stationary and non-Gaussian. Conditional on a latent GP, the SHP is a heteroscedastic GP with nonstationary covariance structure. The realizations from SHP are versatile and can represent spatial inhomogeneities. The unconditional correlation functions of SHP form a rich isotropic class that can allow for a smoothed nugget effect. We apply an importance sampling strategy to implement pseudo maximum likelihoodparameter estimation fortheSHP.Topredicttheprocessat unobservedlocations, we develop a plug-in best predictor. We extend the single-realization SHP model to handle replicates across time of SHP realizations in space. Empirical results with simulated data show that SHP is nearly as efficient as GP in out-of-sample prediction when the true process is stationary GP, and outperforms GP substantially when the true process is SHP. The SHP methodology is applied to enhanced vegetation index data and U.S. NO3 deposition data for illustration.