Mathematical Statistics for Spatial Data; The Use of Geostatistics for Geodetic Purposes

Abstract

In geodesy long experience exists in using procedures to predict the earth’s gravitational field at unvisited locations. A well known quantitative approach, least-squares collocation, has been proposed (Krarup, 1969) and has been used extensively in the past (see Moritz 1980a, which contains a detailed list of references). Least-squares collocation provides a linear unbiased predictor to predict e.g. the geoid on the basis of a number of observations on gravity anomalies. Fundamental aspects of Hilbert spaces and of approximation theory have been formulated (Meissl, 1976; Tscherning, 1978; Dermanis, 1977). Recent research has focused on application of the theory in many different areas (Haagmans and Van Gelderen, 1991) as well as on problems of taking the spherical shape of the earth into account (Schaffrin, 1992). Quite remarkably, a parallel rise may be observed in developing and using geostatistical theory, in particular inspired by the French work in the nineteen seventies. (Matheron, 1973; Delfiner, 1976; Journel and Huijbregts, 1978) and continuing in the nineties (Cressie, 1992). Geostatistics also provides a linear unbiased predictor of any spatial phenomenon, taking the spatial dependencies into account. A typical example is given on the land quality moisture deficit (Stein et al., 1991a), which is correlated with the mean highest groundwater level, leading to the use of universal cokriging as an appropriate prediction technique (Stein and Corsten, 1991). Modern approaches apply geostatistical procedures as well for geodetic purposes, using either generalized covariance functions (Blais, 1984) or anisotropy (Hansen, 1993).