Probabilistic optimal transport-driven inversion of the 2012 Palisades rockfall seismic source

During rockfall events, the seismic waves are generated in response to the time-varying normal and tangential forces between the Earth and colliding and sliding mass. These forces carry information about the nature of the generative seismic source; hence, the source dynamics can be estimated. Several studies have used forward modeling to determine the amplitude and duration of these forces and, implicitly, the source process that could generate the observed seismic waves. Through running multiple forward models, the force history inversion involves adjusting the force amplitude and duration to minimize the misfit between the proposed source model, convoluted with the force-impulse Green’s functions, and the observations. In the Bayesian framework, the normal likelihood function is traditionally used to measure the misfit between the observed and predicted waveforms with respect to amplitude. However, the normal likelihood function is insensitive to the potential misalignment of the waveforms in time. Moreover, the relevant parameter space often exhibits multiple local minima, which may lead to a convergence to a minimum that does not present the global optimum. Optimal transport distances-driven exponential likelihoods were recently proposed as alternatives thanks to their ability to capture the time structure of the signals. We employed a Metropolis-Hastings sampling strategy in the probabilistic framework to reconstruct the 2012 Palisades rockfall seismic source using two implementations of the Wasserstein distance-based exponential likelihood function. The first implementation transforms between density functions, which are always positive and integrate to one. Therefore, it requires the transformation of the signals into probability density functions, which is done here via a modified graph-space transform scheme. The second method is applied directly to the signals. We evaluated the robustness of the two implementations of the Wasserstein distance-based exponential likelihood function in simulating the source characteristics with respect to the normal likelihood. Preliminary results show that contrary to the expectations, using optimal transport distances-driven exponential likelihoods leads to negligible improvement in the fit to the observed waveform.