ML-misfit: A neural network formulation of the misfit function for full-waveform inversion

A robust misfit function is essential for mitigating cycle-skipping in full-waveform inversion (FWI), leading to stable updates of the velocity model in this highly nonlinear optimization process. State-of-the-art misfit functions, including matching filter or optimal transport misfits, are all hand-crafted and developed from first principles. With the growth of artificial intelligence in geoscience, we propose learning a robust misfit function for FWI, entitled ML-misfit, based on machine learning. Inspired by the recently introduced optimal transport of the matching filter objective function, we design a specific neural network architecture for the misfit function in a form that allows for global comparison of the predicted and measured data. The proposed neural network architecture also guarantees that the resulting misfit is a pseudo-metric for efficient training. In the framework of meta-learning, we train the network by running FWI to invert for randomly generated velocity models and update the parameters of the neural network by minimizing the meta-loss, which is defined as the accumulated difference between the true and inverted velocity models. The learning and improvement of such an ML-misfit are automatic, and the resulting ML-misfit is data-adaptive. We first illustrate the basic principles behind the ML-misfit for learning a convex misfit function using a travel-time shifted signal example. Furthermore, we train the neural network on 2D horizontally layered models and apply the trained neural network to the Marmousi model; the resulting ML-misfit provides robust updating of the model and mitigates the cycle-skipping issue successfully.