Towards autonomous event identifications in wave-equation traveltime inversion

We present a method to automatically relate events between observed and synthetic data for wave-equation traveltime (WT) inversion. The scheme starts with local similarity measurements by applying cross-correlation to localized traces. We then develop a differentiable alternative of the argmax function. By applying the differentiable argmax to each time slice of the local similarity map, we build a traveltime difference function but keep this process differentiable. WT inversion requires only the traveltime difference of related events through a phase shift. Thus, we must reject events that are not apparently related between observed and synthetic data. The local similarity map implies the possibility of doing so but shows abrupt changes with time and offset. To mitigate this problem, we introduce a dynamic programming algorithm to define a warping function. Wave packets between observed and synthetic data are assumed to be related if they are connected by this warping function, and they also exhibit high local similarity. Only such related events are considered in the subsequent calculation of misfit and adjoint source. Numerical examples demonstrate that the proposed method successfully retrieves an informative model from roughly selected data. In contrast, WT inversion based on cross-correlation or deconvolution fails to do so.