A specific type of modeling error evaluation for viscoelastic full waveform inversion

Tianze Zhang, Kristopher A. Innanen, Daniel O. Trad

In the traditional viscoelastic Full Waveform Inversion (FWI) approach based on the generalized standard linear solid (GSLS) model, the quality factor, denoted as Q, is not directly inverted. Instead, it is first converted into a set of relaxation variables. Thus, the viscoelastic FWI process involves inverting for both elastic and relaxation models and sub-sequently translating the relaxation variables back into the Q model to achieve the final inversion results. This indirect approach is necessitated partly because the relationship be-tween the Q values and relaxation variables is complex, and the functions that map between these two domains are not straightforward inverses of each other. In this report, we propose a novel method where a Multi-Layer Perceptron (MLP) is pre-trained to learn the mapping from Q values to relaxation variables. This trained MLP is then integrated into a Recur-rent Neural Network (RNN)-based GSLS viscoelastic FWI framework, creating a com-plete computational graph. This graph connects Q models, relaxation variables, and elastic models directly to the synthetic data, thereby enabling the direct inversion of the Q model.Additionally, we employ the Monte Carlo dropout technique within the neural network to quantify the uncertainty associated with the MLP’s learning process for mapping Q values to relaxation variables. In our assumption of a constant Q model, this uncertainty quantification reflects the relaxation variables’ limited capacity to accurately represent a constant Q model. The potential of extending this approach to variable Q models is straightforward.The impact of this limited capacity, which is effectively a modeling error, on the forwardm odeling data is also examined. For our experiments, we employ simple layered models as well as subsets of the Marmousi models. Our observations suggest that this modeling error primarily affects the attenuation models. The findings of this study have significant implications for improving the accuracy and reliability of viscoelastic FWI, particularly in how we understand and handle the inherent uncertainties in modeling complex subsurface properties