Linear estimation of random fields with second-order increments and its application

The purpose of this paper is to develop a mathematical model of a class of nonstationary random fields with second-order increments and to consider the estimation problem of the random field from noisy observation data taken at scattered locations. First, a theory of random field which is nonstationary with second-order increments is developed concerning to several mathematical properties, extending the discussion to the intrinsic random field and its ergodic property. Secondly, minimum variance estimators for a state of the random field at an arbitrary location in the two-dimensional domain are considered for both cases where the information about statistical properties of the random field is a priori given and not given. The estimators obtained are compared with the kriging which is a useful interpolation method used in geostatistics and hydrosciences. Finally, a numerical example is presented for examining the estimators and for identifying the variogram of the given random field, and the proposed estimators are tested for the real meteorological data to establish a weather map.