Step-wise solutions to random field prediction problems

The approximation of functionals of second order random fields from observations is a method widely used in gravity field modelling. This procedure is known as collocation or Wiener – Kolmogorov technique. A drawback of this theory is the need to invert matrices (or solve systems) as large as the number of observations. In order to overcome this difficulty, it is common practice to decimate the data or to average them or (as in the case of this study) to produce more manageable gridded values. Gridding the data has sometimes the great advantage of stabilizing the solution. Once such a procedure is envisaged, several questions arise: how much information is lost in this operation? Is rigorous covariance propagation necessary in order to obtain consistent estimates? How do different approximation methods compare? Such questions find a clear formulation in the paper: some of them (the simplest ones) are answered from the theoretical point of view, while others are investigated numerically. From this study it results that no information is lost if the intermediate grid has the same dimension of the original data, or if the functionals to be predicted can be expressed as linear combinations of the gridded data. In the case of a band-limited signal, these linear relations can be exploited to obtain the final estimates from the gridded values without a second step of collocation. A similar result can be obtained even in the case of non-band limited signal, due to the low pass filtering of the signal, along with the noise, performed by the intermediate gridding.