Identification and optimal estimation of random fields from scattered point-wise data

A~traet--A self-contained presentation of the interpolation We consider a 2-D RF Z(x, y) over a domain problem in two-dimensional spatial random fields is given. We ~: (x, y) e f2 ___ R z, and we assume that a realization investigate the case where the random field is not necessarily stationary, where the data are so scarce and so scattered in space of this RF is available in the form of a that sample covariance function estimates are not meaningful finite-dimensional vector of measurements and where, therefore, an analytical parametric 'variogram' model Z = (zl . . . . . zN), where zi = z(xi, Yi) is a realization is used in lieu of the covariance. We discuss several variogram parameter estimation techniques (LS, GLS, ML, Interpolation of Z(xi, Yi ). The N locations are randomly scattered Error Method) and we show how to derive estimates of various in the domain f~ (Fig. 1). We want to construct an functionals of the random field from the variogram. A numerical optimal linear est imatorf(z) for various functionals simulation and two typical engineering applications illustrate the variogram estimation techniques and provide a good measure of f (z) of the RF their respective performances. N