Study On Numerical Simulation Of Wave Equations And Viscoacoustic Full Waveform Inversion

The traditional finite-difference method(FDM)discretizes temporal and spatial derivatives separately,and to increase spatial approximation accuracy it needs to use a long stencil length.However,the traditional FDM just adopts the second-order FD operator to approximate the temporal derivative,which leads to high-order accuracy in space,but only second-order accuracy in time.When a high frequency source or a large time step is used in the simulation,the traditional FDM suffers from obvious temporal dispersion,which further leads to phase distortion of simulated waveform.To improve temporal accuracy of FDM,this paper studies a time-space domain FDM,which determines the FD coefficients from the dispersion relationship and gets a better accuracy balance between time and space.In detail,this paper develops FDMs for both the second-and first-order wave equation systems with spatial arbitrary even-order accuracy,and temporal fourth-and sixth-order accuracies.The temporal high-order FDMs are developed based on general rectangular and cubic grids.The elastic FD modeling is developed from a new velocity-stress wave equation.Numerical simulation of the new elastic equation leads to separated P and S waves.Dispersion and numerical examples verify the higher accuracy and improved efficiency of the new FDMs.Since velocity and Q are two frequently used parameters in seismic data processing,this paper investigates feasibility of adjoint inversion of velocity and Q by using a viscoacoustic full waveform inversion(FWI)scheme.The forward kernel in FWI is a newly developed fractional constant-Q wave equation.This wave equation matches the widely used constant-Q model very well.However,it meets a problem in numerical approximation of the spatial variable-order fractional Laplacians.This paper presents two effective approaches to handle the problem,and improves viscoacoustic wave propagation accuracy and efficiency.Based on the new forward kernels,this paper further develops a viscoacoustic FWI workflow,including derivation of the gradients for velocity and Q,and suggestion of a new nonattenuating adjoint operator to avoid numerical instability during adjoint modeling.This paper also combines the source-independent inversion scheme and the plane-wave data inversion scheme together to avoid source effects on inversion results and reduce FWI computational cost.Numerical examples verify the effectiveness of the viscoacoustic FWI to invert velocity and Q separately or simultaneously.