Joint Minimization of the Mean and Information Entropy of the Matching Filter Distribution for a Robust Misfit Function in Full-Waveform Inversion

A full-waveform inversion (FWI) is a highly nonlinear inversion methodology. FWI tends to converge to a local minimum rather than a global one. We refer to this phenomenon as ``cycle skipping'' in FWI. A cost-effective solution for resolving this issue is to design a more convex misfit function for the optimization problem. A global comparison based on using a matching filter (MF) admits more robust misfit functions. In this case, we would compute an MF first by deconvolving the predicted data from the measured ones. When the velocity model is accurate, the predicted data resemble the measured ones, and the resulting MF would be an approximated Dirac delta function. If the velocity produces data that are different from the observed ones, a misfit function can be formulated by penalizing the energy away from the zero-lag time (the center). Here, we develop a general mechanism for an evolution of the MF to our objective in FWI. Specifically from the statistics point of view, rather than using a penalty, we propose a novel misfit by minimization of the mean and information entropy of the MF distribution. We show that the resulting misfit function can mitigate the ``cycle skipping'' as well as reduce the mean and variance of the resulting MF distribution. We use a modified Marmousi example to demonstrate the features of the proposed misfit. We also evaluate the robustness of the proposed method using inaccurate (rotation in phase) source wavelets and measured data with different levels of Gaussian random noise.