Eigenvalues and eigenvectors of linearized elastic inversion

Abstract

Small contrasts in the parameters allow the linearization of elastic modeling and inversion. Although this assumption simplifies parameter estimation, it also impoverishes information: a linearized model, besides being an approximation, requires shrinking the data space to precritical angles. This work investigates the two basic questions regarding the type of information that can be retrieved and how to retrieve it. The method is the singular value decomposition of the transfer function between data and parameters. This tool allows a simple interpretation of the information contained in reflections, and it outlines a general estimation procedure. A medium characterized by a uniform background is considered; reflections are linearized according to the Born approximation. P-P reflections are analyzed. The simplicity of the model allows a theoretical study of the effects of velocity errors in the overburden. The analysis is performed for two different sets of elastic parameters. In both cases the conclusion is that only the estimate of one parameter can be accurate while the quality of the estimate of a second parameter critically depends upon the velocity uncertainty and the maximum incident angle; estimation of a third parameter is very difficult.