| Assuming the convolutional model and a known wavelet, reflectivity inversion is an ill-posed problem. Existing methods mostly regularize the problem using mathematical criterion, such as minimization of the reflectivity vector norm.;This thesis presents a new reflectivity inversion method using matching pursuit, an atomic decomposition in which the atoms can be basic geological structures or derived from real geological data. This is equivalent to replacing mathematical criterion by geological ones. This method systematically incorporates all a priori knowledge in the inversion process. Numerical results show that atomic decomposition is robust to noise and is capable of delivering high resolution reflectivities.;When applying to multi-dimensional data in a trace by trace basis, atomic decomposition shows lateral instability. The solutions are non-unique. The inversion only yields one instance of all possible solutions. I propose several schemes to laterally regularize the inversion. These schemes reduce instability but do not completely solve the problem. Re-projection, which projects the data to a slowly varying interpretative basis, laterally regularizes the inversion. |
