Determination of a Seismic Wave Velocity from the Travel-Time Curve

A solution of the following problem is presented: given a travel-time curve of a seismic wave, to determine the corresponding velocity distribution. This is a generalization of the Herglotz—Wiechert method to a medium with low-velocity zones. The velocity depends only on depth and is a piecewise double-smooth function with a finite number of waveguides. A complete mathematical description of this solution is presented. In the presence of wave-guides the solution is ambiguous. Necessary and sufficient conditions for a velocity to be a solution are formulated and the set formed by plots of solutions is obtained. The ambiguity arising from waveguides is reduced by a joint analysis of travel-time curves from surface and deep sources. In particular the following theorem is proved: If the travel times for a source between any adjacent waveguides as well as for a surface source are known, then the velocity between these waveguides can be determined uniquely.