Layered Velocity Inversion: A Model Problem from Reflection Seismology

A simple model problem in exploration seismology requires that a depth-varying sound velocity distribution be estimated from reflected sound waves. For various physical reasons, these reflected signals or echoes have very small Fourier coefficients at both very high and very low frequencies. Nonetheless, both geophysical practice, based on heuristic considerations, and recent numerical evidence indicate that a spectrally complete estimate of the velocity distribution is often achievable. We prove a theorem to this effect, showing that “sufficiently rough” velocity distributions may be recovered from reflected waves under some restrictions, independently of the very low- or high-frequency content of the data. The main restriction is that the velocity depend only on a single (depth) variable; only in this case are sufficiently refined propagation-of-singularity results available. The proof is based on a novel variational principle, from which numerical algorithms have been derived. These algorithms have been implemented and used to estimate velocity distributions from both synthetic and field reflection seismograms.