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 ≤ d ≤ 1, that we model

We observe z(x) at N equally spaced locations on [0, 1] and consider what happens as N increases. Then the minimum norm quadratic estimator of θ is not consistent for θ₁ or θ₂ but is consistent for θ₁ + θ₂. We note that γ′(0⁺) = θ₁ + θ₂; that is, we can consistently estimate the behavior at the origin of the semivariogram. Stein (in press) showed that θ₁ + θ₂ is the only function of the parameters that needs to be estimated well. That is, suppose that we are trying to predict z(x) (0 < x < 1) and we assume that θ₁ = 2 and θ₂ = 0 when in fact θ₁ = θ₂ = 1. Then as N → ∞, we will obtain an asymptotically efficient predictor of z{x) (relative to using the correct value of θ). We also obtain a consistent (in a relative sense) estimator of the variance of the kriging predictor for z(x).

Key Words:

• Kriging
• Equivalence of Gaussian measures
• Gaussian process
• Spatial statistics
• Geostatistics