A General Approach to Seismic Inversion with Automatic Differentiation and Neural Networks

Abstract:

Imaging Earth structure or locating seismic sources is commonly solved by minimizing the misfit between recorded and predicted seismic data through gradient-based optimization. The adjoint-state method has been developed to compute the gradient efficiently; however, its implementation can be time-consuming and difficult. We develop a general seismic inversion framework to calculate gradients using reverse-mode automatic differentiation. The central idea is that adjoint-state methods and reverse-mode automatic differentiation are mathematically equivalent. The mapping between numerical PDE simulation and deep learning allows us to build a seismic inverse modeling library, ADSeismic, based on deep learning frameworks, which supports high performance reverse-mode automatic differentiation on CPUs and GPUs. We demonstrate the performance of ADSeismic on inverse problems related to velocity model estimation, rupture imaging, earthquake location, and source time function retrieval.

Automatic differentiation bridges the gap between PDE-constrained optimization and neural networks. We have developed a novel method, NNFWI, to combine deep neural networks with full-waveform inversion (FWI) by representing the velocity model with a generative neural network. In contrast to FWI, NNFWI calculates the gradients of both the neural networks and PDEs using automatic differentiation allowing the gradients to directly flow from the loss function through the PDEs to the weights and biases of the generative neural network. A key advantage of NNFWI is that the inductive bias of convolutional neural networks imposes a regularization effect, which filters out noise in the gradients and guides the optimization toward more realistic velocity models. Thus, NNFWI is more robust to local minima and significantly improves the inversion resulting from noisy seismic data. Furthermore, we analyzed the uncertainty of the inverted model by adding dropout layers during training, which acts to approximate Bayesian inference and thus provides an efficient approach for uncertainty quantification in FWI.

In summary, ADSeismic and NNFWI open a new pathway to combining deep learning framework and deep neural networks with seismic inversion while keeping both the characteristics of deep neural networks and the high accuracy of PDE solvers. This approach can be applied to a wide range of inversion problems.