Cycle-skipping mitigation using misfit measurements based on differentiable dynamic time warping

The dynamic time warping (DTW) distance has been used as a misfit function for wave-equation inversion to mitigate the local minima issue. However, original DTW is not smooth; therefore it can yield strong discontinuity in adjoint source. Such weakness does help nonlinear inverse problems converge plausible minimum by any means. We introduce first geophysics smooth distance, which demonstrated its performance series classification, clustering, and prediction loss function. fundamental idea of such replace $\min$ operator with relaxation. Then becomes possible define analytic derivative distance. new entitled differentiable Moreover, considering that path an indicator traveltime difference between observed synthetic trace, penalization term constructed based on accumulation penalized weighted favor difference. Numerical examples demonstrate advantage over conventional non-differentiable one.