Fractional Constant Q Viscous Wave Equation Numerical Method Study

Due to the viscous effect of the ground,seismic waves propagate underground with energy loss and phase distortion.Accurate simulation of frequency-dependent absorption attenuation of seismic waves is essential for understanding the seismic wave propagation law and developing high-precision seismic imaging algorithms based on it.Numerically solving the fractional viscoelastic wave equation to simulate the absorption attenuation of seismic waves has been a topic of great interest in recent years.Solving the timefractional viscoelastic wave equation requires storing the wave fields of all historical moments,which consumes a lot of memory space and computation time.Therefore,this paper studies the fractional Laplacian operator viscoelastic wave equation and its numerical solution.Based on the detailed introduction of the derivation process of the fractional Laplacian operator viscoacoustic wave and viscoelastic wave equations,and in order to avoid the periodic boundary problem of traditional pseudospectral method,this paper studies the matrix transformation method of approximating the fractional Laplacian operator in the spatial domain,and proposes a split matrix transformation method that can avoid the decomposition of large matrix eigenvalues,which splits the high-dimensional fractional Laplacian operator into multiple directions of one-dimensional fractional partial derivatives.This method can significantly improve the computational efficiency.On this basis,this paper further investigates the method of solving the difference coefficients using fast Fourier transform and optimizes the difference operator for approximating the fractional partial derivatives using the least-squares method to achieve the local finite difference operator approximation of the fractional partial derivatives and improve the computational efficiency.The numerical simulation results confirm that the simulation results of the proposed split matrix transformation method and the optimized coefficient finite difference method agree well with those of the traditional pseudospectral method,but the computational efficiency is improved.In addition,based on the existing highly accurate temporal extrapolation scheme for the fractional viscoacoustic wave equation,this paper establishes a highly accurate temporal extrapolation scheme for the fractional Laplacian operator viscoelastic wave equation.Compared with the time second-order accuracy of the traditional pseudospectral method,the new recurrence format is developed from the general solution of the wave equation with smaller time dispersion and better stability,which allows the use of larger time steps for wavefield extrapolation and the computational efficiency is improved.In this paper,the numerical solution of the fractional Laplacian operator for the viscoacoustic and viscoelastic wave equations can provide a basis for further research on high-precision imaging and inversion methods for viscoelastic media.