Smoothing operators for waveform tomographic imaging

Abstract

The analytic kernel in the space‐time domain for the Frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the Born approximation solution to the integral equation of waveform scattering. Preconditioning operators in the solution of this forward problem, which may incorporate a priori information and approximate solutions, are smoothing operators in the imaging problem, the first iteration of a nonlinear inversion for the slowness model. Some preconditioning operators are determined for solutions to the parabolic wave equation, and then used to create new sensitivity functions that retain appropriate characteristics of the true Frechet kernel in forward calculations. The new sensitivity functions define near‐source, near‐receiver and far‐field kernels, as well as kernels that exhibit an amplitude decay off the ray yielding ray‐perpendicular sensitivity that scales with the Fresnel zone size. A sample calculation from a synthetic cross‐well imaging experiment shows the usefulness of introducing physically appropriate model smoothing directly into the sensitivity function of the forward problem, helping to obtain a geologically reasonable image of the velocity model when ray coverage is insufficient.