The variable projection method for waveform inversion with an unknown source function

This paper compares three alternative algorithms for simultaneously estimating a source wavelet at the same time as an earth model in full-waveform inversion: (i) simultaneous descent, (ii) alternating descent and (iii) descent with the variable projection method. The latter is a technique for solving separable least-squares problems that is well-known in the applied mathematics literature. When applied to full-waveform inversion, it involves making the source wavelet an implicit function of the earth model via a least-squares filter-estimation process. Since the source wavelet becomes purely a function of medium parameters, it no longer needs to be treated as a separate unknown in the inversion. Essentially, the predicted data are projected onto the measured data in a least-squares sense at every function evaluation, making use of the fact that the filter estimation problem is trivial when compared to the full-waveform inversion problem. Numerical tests on a simple 1D model indicate that the variable projection method gives the best result; actually producing results in quality that are very similar to control experiments with a known, correct wavelet.