Seismic borehole tomography—Theory and computational methods

Tomographic inversion can be applied to seismic traveltime data to obtain a map of seismic velocities in a rock volume, although the field geometries will not, in general, allow a complete ray sampling. This paper deals with theoretical and computational aspects of tomographic inversion under such circumstances. It is shown that some usual measurement geometries could, in theory, be sufficient for determining the seismic velocity distribution uniquely (under the straight-ray approximation). From the computational point of view, however, the inverse problem can be poorly conditioned. The usual transform methods of reconstruction, developed for medical tomography, are not applicable. A brief discussion is, therefore, given of a few relevant geophysical inverse techniques and the CG-algorithm applied to a certain quadratic functional is shown to give reasonable images. A convergence result that contains "ART with smoothing" as a special case is also obtained. Techniques to correct for ray bending are discussed.