Fourier Approximations for Estimation And Smoothing of Irregularly Observed Spatial Processes

Abstract

We consider estimation and smoothing for irregularly observed two dimensional spatial processes, assumed to be generated by certain stochastic partial differential equations. Such processes have been considered by Jones (1989) who gives an example arising in hydrology. Jones’ work, in turn, is descended from the seminal paper by Whittle (1954) where he studied a two-dimensional Laplace equation and data observed over a complete grid. Our objective here is to adapt the approaches considered in the above papers to processes satisfying more general stochastic partial differential equations where such processes are observed over an irregular grid. We consider various Fourier approximations for the covariance matrix, expressed in terms of the discrete Fourier transform of the spectral density, and apply them to problems of approximating the likelihood function and to minimum mean square error smoothing when the spatial processes are observed with error. The approximations are applied to groundwater data consisting of observations from 93 wells in the Saratoga Aquifer.