Least-squares inversion for velocity profiles from band-limited data

Seismic record sections are always bandlimited, having very little energy below 4-10 Hz. The corresponding low frequency band in the parameter distribution contains, for instance, velocity trend information which is crucial in correctly placing major lithologic changes (reflectors). The velocity analysis procedures practiced by the petroleum industry fill this gap. These procedures arewidely used and often effective, but are usually justified only within an approximate and not entirely consistent description of the seismic reflection experiment. On the other hand, numerical experiments by McAulay (1885), Kolb (1886) and others indicate that the low frequency content of a layered velocity profile is determined by a bandlimited seismogram, at least under some circumstances, through output least squares inversion. This seems to be a difficult optimization problem: convergence appears to require either a very good initial guess or some clever continuation tricks. We will combine some simple examples, asymptotic analyses, and numerical experiments to explain the success and the difficulty of this approach to the velocity inversion problem in the layered case. Our conclusions accord well with conventional wisdom regarding velocity analysis, with the observations of McAulay and Kolb et al., and with the singular value analysis of seismic reflection tomography undertaken by Bube et al. (1985). We also exhibit a successful nonlinear leastrsquares inversion algorithm based on the principles explained in the text.