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Spatial continuity measures for probabilistic and deterministic geostatistics. (English) Zbl 0970.86516

Summary: Geostatistics has traditionally used a probabilistic framework, one in which expected values or ensemble averages are of primary importance. The less familiar deterministic framework views geostatistical problems in terms of spatial integrals. This paper outlines the two frameworks and examines the issue of which spatial continuity measure, the covariance \(C(h)\) or the variogram \(\gamma(h)\), is appropriate for each framework. Although \(C(h)\) and \(\gamma(h)\) were defined originally in terms of spatial integrals, the convenience of probabilistic notation made the expected value definitions more common. These now classical expected value definitions entail a linear relationship between \(C(h)\) and \(\gamma(h)\); the spatial integral definitions do not. In a probabilistic framework, where available sample information is extrapolated to domains other than the one which was sampled, the expected value definitions are appropriate; furthermore, within a probabilistic framework, reasons exist for preferring the variogram to the covariance function. In a deterministic framework, where available sample information is interpolated within the same domain, the spatial integral definitions are appropriate and no reasons are known for preferring the variogram. A case study on a Wiener-Levy process demonstrates differences between the two frameworks and shows that, for most estimation problems, the deterministic viewpoint is more appropriate. Several case studies on real data sets reveal that the sample covariance function reflects the character of spatial continuity better than the sample variogram. From both theoretical and practical considerations, clearly for most geostatistical problems, direct estimation of the covariance is better than the traditional variogram approach.

MSC:

86A32 Geostatistics
86-08 Computational methods for problems pertaining to geophysics
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