Seismic Tomography with Random Batch Gradient Reconstruction

Abstract: Seismic tomography solves high-dimensional optimization problems to image subsurface structures of Earth. In this paper, we propose to use random batch methods to construct the gradient used for iterations in seismic tomography. Specifically, we use the frozen Gaussian approximation to compute seismic wave propagation, and then construct stochastic gradients by random batch methods. The method inherits the spirit of stochastic gradient descent methods for solving high-dimensional optimization problems. The proposed idea is general in the sense that it does not rely on the usage of the frozen Gaussian approximation, and one can replace it with any other efficient wave propagation solvers, e.g., Gaussian beam methods and spectral element methods. We prove the convergence of the random batch method in the mean-square sense, and show the numerical performance of the proposed method by two-dimensional and three-dimensional examples of wave-equation-based travel-time inversion and full-waveform inversion, respectively. As a byproduct, we also prove the convergence of the accelerated full-waveform inversion using dynamic mini-batches and spectral element methods.