Fast viscoacoustic forward modeling method based on U-net Fourier neural operator

The utilization of a fractional visco-acoustic equation facilitates a significantly enhanced simulation of seismic wave propagation, rendering it indispensable for seismic data interpretation and processing. However, this equation poses a challenge due to the presence of two complex fractional Laplace operators, which impose difficulties in numerical solution. Recently, Fourier neural operators (FNO) have demonstrated notable efficacy in the resolution of partial differential equations (PDEs). To enhance the predictive accuracy further, we present a novel approach that leverages the U-net Fourier neural operator (U-FNO) for solving the intricate complex fractional visco-acoustic wave equation. Diverging from FNO, U-FNO incorporates an additional U-Fourier layer after the conventional Fourier layer. This augmentation empowers U-FNO to efficiently capture the high-frequency components, thus enabling the trained U-FNO to generate highly accurate solutions for the remaining time steps. Numerical examples provide evidence that U-FNO can successfully solve the fractional visco-acoustic wave equation without explicitly considering the fractional Laplace operators. Furthermore, U-FNO achieves superior prediction accuracy compared to the conventional FNO-based method.