Misfit functions based on differentiable dynamic time warping for waveform inversion

Misfit functions based on differentiable dynamic time warping (DTW) have demonstrated an excellent performance in various sequence-related tasks. We introduce this concept in the context of waveform inversion and discuss a fast algorithm to calculate the first and second derivatives of such misfits. The DTW distance is originally calculated by solving a series of min sorting problems, thus it is not differentiable. The fundamental idea of differentiable DTW is replacing the min operator with its smooth relaxation. It is then straightforward to solve the derivatives following the differentiation rules, which results in both the differentiable misfit measurements and warping path. This path weights the traveltime mismatch more than its amplitude counterpart. We can construct a penalization term based on this warping path. The penalized misfit function is adaptable to measure traveltime and amplitude separately. Numerical examples show the flexibility and the performance of the proposed misfit function in mitigating local minima issues and retrieving the long-wavelength model features