Integrated Networks for Viscoelastic FWI: Mapping From Q to Relaxation Variables and Quantifying Modeling Error

In the conventional approach to viscoelastic full waveform inversion (FWI) utilizing the generalized standard linear solid (GSLS) model, the quality factor (Q) is usually not directly inverted. Instead, this method converts Q into a suite of relaxation variables before inversion. Consequently, viscoelastic FWI entails the inversion of both elastic and relaxation models, with a subsequent conversion of relaxation variables back to the Q model to obtain final results. This method is partially due to the accurate intricate relationship between Q values and relaxation variables, where the mapping functions between these two sets are not simple inverses of each other. We introduce an approach that incorporates a multilayer perceptron (MLP) that is pretrained to learn complex mapping from Q values to relaxation variables. This MLP is then seamlessly integrated into a recurrent neural network (RNN)-based GSLS viscoelastic FWI framework, establishing a computational graph, linking Q models and elastic models to data discrepancies, facilitating the direct inversion of both Q and elastic models. To address the inherent uncertainty in the MLP’s mapping process, we apply Monte Carlo (MC) dropout within the neural network, quantifying the uncertainty of converting Q values into relaxation variables. Assuming a constant Q model, this uncertainty quantification method highlights the limited ability of relaxation variables to represent a constant Q model precisely. The feasibility of adapting our method for frequency-variant Q values appears straightforward. We also explore how this limitation, essentially a modeling error, influences the accuracy of forward modeling data. Our numerical results reveal that this modeling error has apparent impacts on the inversion outcomes for both elastic and attenuation models, underscoring the critical nature of this error in the viscoelastic FWI processes.