Multiresolution parameterization for geophysical inverse problems

Abstract

A model parameterization based on a multiresolution wavelet representation is proposed for geophysical inverse problems in this study. The matrix equation for the wavelet representation of the model is constructed by dual wavelet transforms of a readily built matrix based on the pixel parameterization. We demonstrate the merits of the proposed parameterization using the simple examples of geophysical data gridding and 3D seismic tomography, and comparing the results with those obtained by the conventional pixel parameterization. It is shown that for the conventional damped least‐squares solutions of pixel‐parameterized models, regularization by norm damping deteriorates the underlying model correlation due to sparse and heterogeneous sampling, whereas roughness damping schemes impose a priori arbitrary correlation length irrespective of the heterogeneous sampling. The scale hierarchy embedded in the proposed multiresolution parameterization facilitates the compromise between the dual spatial and scale resolution. Depending on the local richness of the data constraints across different scales at a site, model variation of various scale spectra can be robustly resolved through the scale hierarchy. There is thus no need to invoke additional smoothness regularization.