PVM kriging with R Dr . Albrecht Gebhardt

Kriging is one of the most often used prediction methods in spatial data analysis. This paper examines which steps of the underlying algorithms can be performed in parallel on a PVM cluster. It will be shown, that some properties of the so called kriging equations can be used to improve the parallelized version of the algorithm. The implementation is based on R and PVM. An example will show the impact of different parameter settings and cluster configurations on the computing performance. 1 The classical form of kriging One of the aims of geostatistical analysis is the prediction of a variable of interest at unmeasured locations. The prediction method which is used most often is kriging. It is based on the concept of so called regionalized variables Z(x) with x ∈ D ⊂ R, d = 2, 3. Analysis usually starts with a spatial dataset consisting of measurements Z(xi), i ∈ I at a grid of observation points xi. These values are now treated as a realization of the underlying stochastical process Z(x) and modeled with a usual regression setup Z(x) = m(x) + ε(x),E(ε(x)) = 0. Of course it is not possible to make inference from only one realization. The idea of regionalized variables is now to partition the region D into nearly independent parts and to use them as different realizations. This is only valid if a stationarity assumption holds: m(x) = const, x ∈ D (1) DSC 2003 Working Papers 2 Cov(Z(xi), Z(xj)) = C(h), h = Z(xi)− Z(xj), xi, xj ∈ D (2) That means that mean and covariance function m(h) and C(h) are assumed to be translation invariant. A more general assumption, called intrinsic stationarity, only demands that the variance of the increments of Z(.) has to be invariant with respect to translation: Var(Z(xi)− Z(xj)) = 2γ(h), h = xi − xj , xi, xj ∈ D (3) Equation (3) introduces also the semivariogram γ(h). It is connected with the covariance function via γ(h) = C(0)−C(h) if C(h) exists. In simple cases γ(.) depends only on | h |= h. This leads to isotropic variograms and covariance functions. Both will later be used to determine the system matrix in the prediction step. Therefore it is necessary to estimate the semivariogram. The usual estimator in the isotropic case γ(h) = γ(h) with h = |h| is