Seismic wavefield and traveltime modeling using machine learned functions

Abstract:

Forward modeling lies at the heart of all inversion algorithms. The majority of the computational cost associated with seismic inversion is in solving the forward problem. These inversion algorithms are often computationally bottlenecked due to repeated modeling needed for perturbations in the velocity model and/or source location, particularly in large 3D models with complex physics needed to model the observed data. While many algorithms have been proposed over the years to efficiently compute seismic traveltimes and wavefields, these algorithms typically require the same amount of computational effort even for small changes in the velocity model or the source location. We employ the emerging paradigm of physics-informed neural networks (PINNs) to solve the eikonal and Helmholtz equations for computing seismic traveltime and wavefield solutions, respectively. By minimizing a loss function formed by the underlying partial differential equation (PDE), we train a neural network to produce solutions consistent with the given physical equation. For a spatial location in the model space as input, the network learns to predict the traveltime or wavefield value at that location and its partial derivatives using automatic differentiation that satisfies the corresponding PDE. Feeding in a reasonable number of randomly chosen points from the model space ultimately trains a fully connected deep neural network to predict the solution for the entire model space. Moreover, we observe that once a network is trained for a given velocity model and source location, the computational effort needed to predict the solution for perturbations in the velocity model and/or source location is significantly reduced, thanks to transfer learning. Another major advantage of such a formulation is the ease with which complex physics can be incorporated into the modeling framework just by updating the loss function for the neural network. Through tests on benchmark synthetic isotropic and anisotropic models, we demonstrate the potential of the proposed approach in massively speeding up seismic modeling and in solving computationally intractable inverse problems.