Spatial sampling design for prediction taking account of uncertain covariance structure

Jürgen Pilz 1 and Gunter Spöck 2 
1 Universität Klagenfurt, Institut für Mathematik
Universitätsstraße 65-67, 9020 Klagenfurt, Austria
Tel.: + 0043 463 2700-3113; Fax: + 0043 463 2700-3199
juergen.pilz@uni-klu.ac.at
2 Universität Klagenfurt, Institut für Mathematik
Universitätsstraße 65-67, 9020 Klagenfurt, Austria
Tel.: + 0043 463 2700-3125; Fax: + 0043 463 2700-3199
gunter.spoeck@uni-klu.ac.at
 

Abstract
This paper presents a model-based approach to the problem of the optimal choice of a spatial design in the presence of uncertainty about the distribution of the observations, a topic which has received only little attention in the geostatistics literature so far. In spatial sampling one usually starts with an initial design to estimate the covariance function, such a design is non- model-based and chosen e.g. according to principles of deterministic sampling, cluster sampling, simple random and stratified sampling etc. The basic difficulty for rigorous model- based approaches to spatial sampling is the fact that the spatial correlatedness of the observations leads to analytically intractable  design criteria, whereas for linear regression models with uncorrelated observations one obtains well-tractable objective functions which can be optimized using a rich and fully developed methodology from  convex analysis. After briefly reviewing the “classical” experimental design theory approach to regression models with uncorrelated observations, we will show how this approach can be extended to spatial sampling with correlated errors, using an approximation of the random field by linear regression models with random coefficients. In this context, we also present a useful algorithm for the iterative generation of an optimal design for spatial prediction. The results are illustrated by means of a real data set of Cs137 measurements.

Keywords: spatial model-based  design, experimental design  theory, uncertain covariance function, Bayes kriging

In: Caetano, M. and Painho, M. (eds). Proceedings of the 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences, 5 – 7 July 2006, Lisboa, Instituto Geográfico Português

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