Abstract
The dynamic time warping (DTW) misfit function has been used 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 have introduced 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 further highlights the traveltime difference. Numerical examples demonstrate the advantage of the penalized differentiable DTW misfit function over the conventional nondifferentiable one.