Extended full waveform inversion with matching filter

Full waveform inversion has shown its huge potentials in recovering a high-resolution subsurface model. However, conventional full waveform inversion usually suffers from cycle skipping, resulting in an inaccurate local minimum model. Extended waveform inversion provides an effective way to mitigate cycle skipping. A matching filter between the predicted and observed data can provide an additional degree of freedom to improve the data fitting and 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. We formulate the objective function using the penalty method by linearly combining a data-misfit term and a penalty term. The objective function with a reasonable penalty parameter has a larger region of convergence compared to conventional full waveform inversion. We search the optimal solution over the extended model by updating the matching filter and the velocity in a nested way. The normalization of the data can bring us an equivalent normalization to the filter, and a more effective convergence. In the synthetic Marmousi model, the proposed inversion method recovers the velocity model stably and accurately starting from a linearly increasing model in the case of lack of low frequencies below 3 Hz, in which conventional full waveform inversion suffers from cycle skipping. We also use a marine field data to demonstrate the effectiveness and practicality of the proposed method.