Analysis of optimal transport related misfit functions in seismic imaging

Abstract

We analyze different misfit functions for comparing synthetic and observed data in seismic imaging, for example, the Wasserstein metric and the conventional least-squares norm. We revisit the convexity and insensitivity to noise of the Wasserstein metric which demonstrate the robustness of the metric in seismic inversion. Numerical results illustrate that full waveform inversion with quadratic Wasserstein metric can often effectively overcome the risk of local minimum trapping in the optimization part of the algorithm. A mathematical study on Fréchet derivative with respect to the model parameters of the objective functions further illustrates the role of optimal transport maps in this iterative approach. In this context we refer to the objective function as misfit. A realistic numerical example is presented.