Smooth profiles from τ(p) and X(p) data

Stark, PB, Parker RL.  1987.  Smooth profiles from τ(p) and X(p) data. Geophysical Journal of the Royal Astronomical Society. 89:997-1010.: Blackwell Publishing Ltd


Inverse theory, regularization, seismic refraction


Summary. We reduce the problem of constructing a smooth, 1-D, monotoni-cally increasing velocity profile consistent with discrete, inexact τ (p) and X(p) data to a quadratic programming problem with linear inequality constraints. For a finite-dimensional realization of the problem it is possible to find a smooth velocity profile consistent with the data whenever such a profile exists. We introduce an unusual functional measure of roughness equivalent to the second central moment or ‘Variance’ of the derivative of depth with respect to velocity for smooth profiles, and we prove that its minimal value is unique. In our experience, solutions minimizing this functional are very smooth in the sense of the two-norm of the second derivative and can be constructed inexpensively by solving one quadratic programming problem. Still smoother models (in more traditional measures) may be generated iteratively with additional quadratic programs. All the resulting models satisfy the τ (p) and X(p) data and reproduce travel-time data remarkably well, although sometimes τ (p) data alone are insufficient to ensure arrivals at large X; then an X(p) datum must be included.