Wavelets and Collocation: An Interesting Similarity

The rapid developments in the fields of multiresolution appro-ximation theory and wavelets over the past few years have cre-ated an enormous amount of important theoretical knowledge and useful practical tools to be used for various signal proc-essing applications. One of the most attractive properties of multiresolution/wavelet theory is the ability to study the details of a signal locally in various scale levels, according to a zoom-in/zoom-out approach. In gravity field modelling, on the other hand, we are used to employing (both theoretically and practi-cally) the concept of collocation in order to approximate and study the behavior of unknown signals based on discrete data. One of the standard formulations of collocation, as a spatio-statistical linear approximation problem in purely deterministic fields, requires the use of a covariance function which is de-fined via a certain spatial averaging operator over the signal’s domain. The “stationary ” form of this CV function is often thought to provide strong limitations in the approximation framework, because the actual behavior of the gravity field is “non-stationary”. The aim of this paper is to show that there does not really exist any “signal stationarity restriction ” prob-lem in statistical collocation, since it can be proven to be equivalent to a wavelet-type expansion. The connection be-tween the two concepts is discussed, and also some recom-mendations for further work are given.