| Abstrakt: |
Stochastic process theory involves integrals of measurable functions over probability measure spaces. One of these is the ensemble space, Ω, whose members are sample functions on Euclidean spaceRk and the other isRk itself. What geostatisticians call the “theory of regionalized variables” is said to based on stochastic theory. A recent paper inMathematical Geology proclaims a distinction between “probabilistic” and “deterministic” geostatistics. The former is said to rely on “ensemble integrals” over Ω and the latter on “spatial integrals” overRk. This study shows that the proposed distinction rests on an arbitrary choice between two estimators for the covariance of a stochastic process; neither is an ensemble integral, both are spatial integrals, and both are Kolmogorov inconsistent. The “deterministic” estimator is identical with that of classical bivariate least-squares regression in which “spatial structure” is of no consequence. This study shows that both stochastic models are suboptimal approximations to the unique nonstationary classical statistical multivariate regression model generated by each sample pattern. The stochastic process model and its “spatial continuity measures,” thus, appear as questionable mathematical embellishments on suboptimal estimates, correspondence with geomorphic reality is tenuous, and estimates are biased and distorted. Various related misconceptions in the paper are also discussed. |
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