On the modeling and inversion of seismic data

Abstract

In this thesis we investigate some mathematical questions related to the inversion of seismic data. In Chapter 2 we review results in the literature and give some new results on wave equations with coefficients that are just bounded and measurable. We show that these equations have unique solutions and we investigate the dependence on the coefficients. We also discuss solutions in case the coefficients have a discontinuity along a smooth surface. In Chapters 3 and 4 we discuss seismic imaging using high-frequency techniques (microlocal analysis). In particular we discuss the case that multipathing (the formation of caustics) occurs. In Chapter 3 we construct an operator that maps data to an angle-dependent reflectivity function, for elastic media. This makes it possible to reconstruct the position of a reflector and the reflection coefficients, given a smooth approximation to the medium coefficients. It also gives a criterion to determine the smooth part of the medium (velocity analysis), even in the case of caustics. In Chapter 4 we investigate the linearized inverse scattering problem for acoustic media. We consider the case where the so called traveltime injectivity condition is violated. We show that generically the inversion is still possible, but that in certain special cases this is no longer the case. We also give some examples.