Matching-filter based extended full waveform inversion

Extended waveform inversion provides an effective way to mitigate cycle skipping that usually occurs in conventional full waveform inversion (FWI), resulting in an inaccurate local minimum model. A matching filter between the predicted and observed data can provide an additional degree of freedom to avoid the cycle skipping. We extend the search space to treat the matching filter as an independent variable that we use to bring the compared data within a half cycle to obtain accurate direction of velocity updates. In this case, the objective function with a reasonable penalty parameter has a larger region of convexity compared to conventional FWI. The normalization of the data can bring us an equivalent normalization of the filter, and a more effective convergence. A Marmousi example demonstrates these features.