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

The dynamic time warping (DTW) misfit function has been employed for full-waveform inversion to mitigate the local minima issue. However, it is not a smooth measurement and can yield strong discontinuities in the adjoint source. Such a weakness hampers the convergence of waveform inversion to informative minima. We introduce a smooth DTW misfit function to remedy this shortcoming. The fundamental idea of the smooth DTW is to replace the {min} operator with its smooth relaxation. This replacement results in differentiable misfit measurements. Moreover, considering that the optimal warping plan indicates the traveltime difference between the observed and synthetic trace, we can construct a penalization term such that the penalized differentiable DTW distance highlights the traveltime difference further. Numerical examples demonstrate the advantage of the penalized differentiable DTW misfit function over the conventional non-differentiable one.