Full Waveform Inversion Using a Physics-based Recurrent Neural Network

Full-waveform inversion (FWI) is an advanced technique used to estimate the elastic properties of subsurface environment based on the seismic data. This method incorporates all available information from the amplitude, phase, and frequency of seismic waves, also referred to as the full-waveform. By considering the full waveform, FWI reconstructs the high-resolution models, which play a significant role in seismic imaging and are crucial for accurate subsurface characterization. However, achieving such high-resolution models comes with its own set of challenges, including the need for extensive computational power, long processing times, high dependency to the initial model, and stability and non-uniqueness problems which can sometimes limit the accuracy of the inversion process.
Considering the area of machine learning (ML) and deep learning (DL) which have been revolutionized in the recent years, researchers have increasingly concentrated on these methods to improve the efficiency and accuracy of FWI. Wavefield simulation in time domain is intrinsically recursive, so that wavefield in a certain point on time axis depends on the past history of the propagation. Considering this specification, we propose a deep recurrent neural network (RNN) block, corresponding to wave equation which can be used for forward modeling. In this approach, the velocity model of the medium is proportional to a learnable weight matrix in one of the deep network layers. The proposed method operates in an iterative scheme in which the deep learning block is used to predict seismic data. The difference between the predicted data and the observed seismic data is then computed at each iteration, and the gradient of the loss function with respect to the learnable parameters is calculated. This gradient is then used to update the model, effectively refining the velocity model to better match the observed seismic data. This process continues until the model converges to a solution that best fits the observations. One of the most important advantages of the proposed method is to increase the calculation speed. By leveraging the parallel computing capabilities of Graphics Processing Units (GPUs) and by mapping different seismic sources onto the mini-batch property of the deep recurrent neural network, the computation time is decreased by a factor hundreds of times.
The proposed algorithm was applied to the Marmousi model, both for synthetic data simulation and for full-waveform inversion. The results showed that the method was capable of accurately reconstructing the subsurface velocity model. The algorithm was assessed using quantitative metrics including L1 norm, L2 norm, Peak signal to noise Ratio (PSNR), and Structural Similarity Index Measure (SSIM), demonstrating a high degree of accuracy in model reconstruction. Conventional optimization methods which are commonly used in training of deep learning methods including Stochastic Gradient Descent Method (SGDM), RMSProp, and Adaptive Momentum (ADAM), were also applied to this problem. Among them, ADAM showed the best performance in terms of fitting the model to the observed data and its power in searching the model space. This optimization technique helped ensure that the inversion process converged more quickly and accurately. So that, SSIM between reconstructed and true models increased from 0.73 for Gradient Descent method (which is conventional optimization algorithm in FWI) to 0.77 for ADAM.