Inverting absorptive reflections: an inverse series tutorial
Kristopher A. H. Innanen
Non-linear inverse scattering, when it is applied to the big interesting seismic imaging and inversion problems, can be very complicated, and in a tutorial setting one risks losing all the basic concepts and insights in a morass of arithmetic. We can usefully proceed, however, by considering direct inversion within a highly simplified model: reflections from a single interface at a known depth, with known medium parameters above the interface and unknown medium parameters below. The seismic data reduce in this case to a single reflection coefficient, and the medium to be solved for reduces to a few scalar values. We consider the case of an absorptive target medium. A simple absorptive reflection coefficient may be expanded about small parameter contrasts and incidence angles, and used, angle by angle (AVA), or frequency by frequency (AVF), to directly determine simultaneous wavespeed and Q contrasts. Linear and non-linear inversion may occur algebraically or using an inverse series. The latter is evidently the better approach of the two, with the former becoming less tractable for cases involving large angles and large contrasts.