Tomographic Reconstruction of Velocity Anomalies by John A. Fawcett and Robert W. Clayton

Tomography refers to the technique of reconstructing a field from line or surface integrals of it. In medical X-ray tomography, for example, the tissue density field is deduced from measurements of X-ray attenuation through the patient. In this case the data are regularly sampled line integrals, so they are discrete values of a Radon transform. In seismology, the determination of slowness (inverse of velocity) and attenuation fields can also be viewed in a tomographic framework. The travel times or the amplitude decay along rays connecting the sources and receivers are the projections of the fields. Here, the problem is complicated by the fact that the rays are curved and that the ray path depends on the slowness field itself. The problem is in general nonlinear. One can linearize the problem about a reference slowness which essentially decouples the ray paths from the unknown slowness field. However, this leaves the problem of tomographic reconstruction from line integrals along curved rays. Our methods discussed below will be thought of in terms of a reflection seismology experiment. However, the results are also applicable to transmission problems. The goal of our tomographic reconstruction is to identify from travel-time information (source positions known) areas of relatively high and slow velocity (with respect to a known background field) within a layer of the earth. We examine the theory of tomographic reconstruction when the reference slowness is taken to be a known function of depth. Also, although we specialize the problem to depth-dependent background velocities and flat reflectors, we hope that our results for this case will indicate the concepts to apply for more general situations. The generalized Inverse Radon Transform which we will derive for curved ray projections is similar to that derived independently by G. Beylkin (1982).