Computationally Efficient Spatial Interpolators Based on Spartan Spatial Random Fields

This paper addresses the spatial interpolation of scattered data in
d
dimensions. The problem is approached using the theory of Spartan spatial random fields (SSRFs), focusing on a specific Gaussian SSRF, i.e., the fluctuation-gradient-curvature (FGC) model. A family of spatial interpolators (predictors) is formulated by maximizing the FGC-SSRF probability density function at each prediction point, conditioned by the data. An analytical expression for the general uniform bandwidth Spartan (GUBS) predictor is derived. The linear weights of this predictor involve weighted summations of kernel functions over the sample and prediction points. Approximations for the sums are obtained at the asymptotic limit of a dense sampling network, leading to simplified explicit expressions of the weights. An asymptotic locally adaptive Spartan (ALAS) predictor is defined by means of a kernel family that involves a tunable local parameter. The relevant equations are fully developed in
d
=2. Using simulated data in two dimensions, we show that the ALAS prediction accuracy is comparable to that of ordinary kriging (OK), which is an optimal spatial linear predictor (SOLP). The numerical complexity of the ALAS predictor increases linearly with the sample size, in contrast with the cubic dependence of OK. For large data sets, the ALAS predictor is shown to be orders of magnitude faster than OK at the cost of a slightly higher prediction dispersion. The performance of the ALAS predictor and OK are compared on a data set of rainfall measurements using cross validation measures.