Optimal transport for seismic inverse problems
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Seismic data contains interpretable information about subsurface properties, which are important for exploration geophysics. Full waveform inversion (FWI) is a nonlinear inverse technique that inverts the model parameters by minimizing the difference between the synthetic data from numerical simulations and the observed data at the surface of the earth. The least-squares norm of this difference is the traditional objective function for FWI, but it is sensitive to the initial model, the data spectrum, the noise in the measurement, and other issues related to optimization. The least-squares norm is a point-by-point comparison. Other misfit functions with a global feature have been proposed in the literature to achieve better convexity, but none of them is technically a metric. Here we apply the quadratic Wasserstein metric of the optimal transport theory to FWI. Both the amplitude differences and the phase mismatches are considered in this new misfit function. Mathematically, we prove the convexity of the quadratic Wasserstein metric concerning shift, dilation, and partial amplitude changes of data as well as its insensitivity to noise. Despite these good properties of the quadratic Wasserstein metric, solving optimal transport problem in higher dimension is challenging. We first compute the misfit globally by regarding it as a 2D optimal transport problem. Since the Monge-Ampère equation is rigorously related to the quadratic Wasserstein metric, we solve the 2D optimal transport problem by solving a fully nonlinear Monge-Ampère equation based on a monotone finite difference solver which has been proved to converge to the viscosity solution. To increase the resolution of the inversion, we further develop another method to compute the quadratic Wasserstein metric: trace-by-trace comparison based on the 1D optimal transport. The 1D technique can be solved accurately and efficiently and thus is more robust to handle more complicated problems with less computational cost. We also explore the connections between optimal transport and other misfit functions and explain the intrinsic features of the transport-based idea. Since optimal transport problem concerns nonnegative measures, we will also investigate the critical data normalization step which transforms the sign-changing wavefields into probability densities. This is the most important topic to address in applying optimal transport to seismic inversion. With the least-squares norm being the misfit function, FWI using the reflection data often results in migration-like features in the model updates. We argue that it is the inherent nonconvexity that prevents it from updating the kinematics with high-frequency data. Through numerical examples and discussions, we demonstrate that the better convexity of the quadratic Wasserstein metric can tackle the local minima generated by the high-wavenumber update which appears in addition to the known cycle-skipping issues caused by phase mismatches.