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

TL;DR

This paper introduces a smooth, differentiable version of the dynamic time warping (DTW) distance as a misfit function for wave-equation inversion, improving convergence by providing a continuous gradient.

Contribution

The paper presents the first implementation of a differentiable DTW distance in geophysics, enabling better optimization in seismic inversion tasks.

Findings

Penalized differentiable DTW outperforms traditional DTW in numerical tests.

Smooth DTW provides continuous derivatives, aiding convergence.

The method effectively mitigates cycle-skipping issues in seismic inversion.

Abstract

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