Minimum norm quadratic estimation of spatial variograms

Abstract The estimation of spatial variograms, a measure of spatial correlation, is a critical problem in the implementation of kriging, a method for interpolating random fields. We consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters. We investigate the asymptotic behavior of such estimators for Gaussian processes as the number of observations within some bounded region increases. The basic conclusion is that we can estimate consistently those functions of the parameters, and only those functions, that have a nonnegligible impact asymptotically on the kriging procedure. In general, the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important. As an example, consider a Gaussian process z(·) on the real line with unknown constant mean and , where γ(·) is known as the semivariogram of the process and θ is a finite vector of unknown parameters. Suppose, for 0...

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