An Augmented Lagrangian Method-Based Deep Iterative Unrolling Network for Seismic Full-Waveform Inversion

Seismic full-waveform inversion (FWI) is a powerful technique for high-resolution imaging of subsurface physical properties. However, it suffers from the possibility of falling into local minimum due to the inherent nonlinearity and ill-posedness. Recently, the data-driven deep-learning approach has been used to solve ill-posed inverse problems, with the limitations of high data collection costs and poor model generalization capabilities. To alleviate these difficulties, we unroll the iterative optimization algorithm based on the augmented Lagrangian (AL) method into a layerwise network architecture to solve seismic FWI, called ALFWI-Net. The ALFWI-Net decomposes the constrained optimization problem into three unconstrained subproblems. Correspondingly, the velocity model is updated by three alternating iterative formulations implemented by four modules in the network. One of the modules uses the Lagrange multiplier method to solve for the gradient of FWI, while the others correspond to the solution of the three subproblems. A soft-threshold-based convolutional neural network is used to learn the proximal operator in the implicit regularized optimization subproblem. Therefore, ALFWI-Net alleviates limitations in applicability, since both the regularization function and parameters, step size, and thresholds are automatically learned in the training process. Experiments were conducted on two geologic models, with the Society of Exploration Geophysics (SEG) salt model posing a challenge due to its limited data samples. Nevertheless, we successfully derived improved velocity models from seismic full-waveform recordings. Numerical experiments on two kinds of geological models demonstrated that ALFWI-Net outperforms classical FWI and data-driven methods in reconstruction accuracy, convergence speed, and out-of-distribution generalization.