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The dynamic time warping (DTW) distance has been used as a misfit functionfor wave-equation inversion to mitigate the local minima issue. However, theoriginal DTW distance is not smooth; therefore it can yield a strongdiscontinuity in the adjoint source. Such a weakness does not help nonlinearinverse problems converge to a plausible minimum by any means. We thereforeintroduce for the first time in geophysics the smooth DTW distance, which hasdemonstrated its performance in time series classification, clustering, andprediction as the loss function. The fundamental idea of such a distance is toreplace the $\min$ operator with its smooth relaxation. Then it becomespossible to define the analytic derivative of DTW distance. The new misfitfunction is entitled to the differentiable DTW distance. Moreover, consideringthat the warping path is an indicator of the traveltime difference between theobserved and synthetic trace, a penalization term is constructed based on thewarping path such that the misfit accumulation by the penalized differentiableDTW distance is weighted in favor of the traveltime difference. Numericalexamples demonstrate the advantage of the penalized differentiable DTW misfitfunction over the conventional non-differentiable one.

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