2000a): The multiresolution character of collocation

Abstract. An interesting theoretical connection between the statistical (non-stochastic) collocation principle and the multiresolution/wavelet framework of signal approximation is presented. The rapid developments in multiresolution analysis theory over the past few years have provided very useful (theoretical and practical) tools for approximation and spectral studies of irregu-larly varying signals, thus opening new possibilities for ‘non-stationary ’ gravity field modeling. It is demon-strated that the classic multiresolution formalism ac-cording to Mallat’s pioneering work lies at the very core of some of the general approximation principles traditionally used in physical geodesy problems. In particular, it is shown that the use of a spatio-statistical (non-probabilistic) minimum mean-square-error criteri-on for optimal linear estimation of deterministic signals, in conjunction with regularly gridded data, always gives rise to a generalized multiresolution analysis in the Hilbert space L2\u85R, under some mild constraints on the spatial covariance function and the power spectrum of the unknown field under consideration. Using the theory and the actual approximation algorithms associated with statistical collocation, a new constructive frame-work for building generalized multiresolution analyses in L2\u85R is presented, without the need for the usual dyadic restriction that exists in classic wavelet theory. The multiresolution and ‘non-stationary ’ aspects of the statistical collocation approximation procedure are also discussed, and finally some conclusions and recommen-dations for future work are given