Born Waveform Inversion via Variable Projection and Shot Record Model Extension

Abstract

Summary Born waveform inversion is a partially linearized version of full waveform inversion based on Born (linearized) modeling, in which the earth model is separated into a smooth background model and a short scale reflectivity, and both are updated to fit observed trace data. Because kinematic variables (velocity) are updated, the possibility of cycle-skipping and consequent trapping at local minimizers exists for Born waveform inversion, just as it does for full waveform inversion. Extended Born waveform inversion allows reflectivity to depend on additional parameters, potentially minimizing the likelihood of cycle skipping by permitting data fit throughout the inversion process. Extended or not, the Born waveform inversion objective function is quadratic in the reflectivity, so that a nested optimization approach is available: minimize over reflectivity in an inner stage, then minimize the background-dependent result in a second, outer stage. This paper uses a 2D acoustic modeling, reflectivity permitted to depend on shot coordinates (shot record extension), a differential semblance penalty to control this dependence, and the variable projection variant of nested optimization. Our examples suggest that neither extended modeling nor variable projection alone are sufficient to enable convergence to a global best-fitting model, but the two together are quite effective.