In seismology, the basic problem is that of deducing some knowledge of the geological structure of portions of the Earth from observed seismic signals. This leads to the concepts of seismic interpretation, or more mathematically, the formulation of inverse problems.
Some aspects of seismic wave propagation can be interpreted in terms of asymptotic ray theory. In Chapter 1 of Part I, we describe the numerical ray tracing algorithm we developed for layered media with interfaces that can vary in three dimensions. We describe in Chapter 2, how this ray tracing method is implemented in an inversion procedure. This method is based on the theory of non-linear least-squares inversion.
In Part II of the thesis, we discuss two formulations of seismic inverse problems, which are more analytical in nature. Chapter 1 deals with the use of inverse scattering theory for the Schroedinger operator in the seismological problem. In chapter 2 of Part II, we develop the theory of the tomographical inversion of travel time anomalies to determine velocity anomalies within the Earth. Here, we have extended, in an approximate sense, the Inverse Radon Transform to situations where the "background" velocity field varies with depth.
