Research On Forward Modeling And Reverse Time Migration Of Wave Equation Based On High-precision Finite Difference Method

The finite difference method(FDM)is widely used in forward modeling of various wave equations,and is the basis of reverse-time migration(RTM)and full waveform inversion(FWI).However,the numerical dispersion caused by the discretization of the grid is an inevitable problem of the FDM.The numerical dispersion of FDM caused by the low-order,coarse grid or high frequency will seriously affect the accuracy of the forward modeling of wave equations.At present,many methods for suppressing numerical dispersion have been proposed and developed,but it is a bit regretful that most of them cannot take into account the calculation efficiency and accuracy at the same time.How to effectively suppress the numerical dispersion in the forward modeling of the wave equation of the FDM without sacrificing additional computational efficiency is still a problem that needs to be solved further.This article mainly focuses on the simulation accuracy and calculation efficiency of the wave equation forward modeling of the finite difference method.Starting from the basic theory of the time-domain FDM,the constant-coefficient optimization method based on the Remez algorithm is used to derive the explicit and implicit difference coefficients of the spatial derivative with equal ripple characteristics,and the time dispersion transform(TDTs)is introduced to solve the time dispersion error.In addition,the pre-stack RTM imaging based on the high-precision FDM is studied by taking the acoustic wave equation as an example.The main work and research content of the paper are as follows.(1)Based on the Taylor-series expansion method(TE),the explicit difference coefficients(EFDCs)and implicit difference coefficients(IFDCs)of the spatial derivatives are solved separately,and their accuracy is compared.Theoretical analysis shows that at the same order,the accuracy of using IFDCs to approximate the spatial derivative is significantly higher than that of EFDCs.In addition,under similar accuracy requirements,the required order of IFDCs is significantly less than EFDCs.(2)The differential discretization schemes for forward modeling of acoustic wave equation,elastic wave equation and two-phase medium wave equation of finite-difference time-domain method are derived respectively,and the finite-difference expressions under PML absorbing boundary conditions are respectively given.In addition,the numerical dispersion and stability conditions of the forward simulation of acoustic wave equations,elastic wave equations and two-phase medium wave equations in traditional finite-difference time-domain methods are studied.Numerical examples show that PML absorbing boundary conditions can solve the boundary reflection problem well.The spatial dispersion error of the wave field obtained based on IFDCs under the same order is smaller than that of EFDCs,but the time dispersion error of the former will be more serious.The stability conditions of IFDCs are more stringent than those of EFDCs.(3)The specific process of optimizing the finite difference coefficients based on the L?norm,the L₂norm and the L₁ norm are studied respectively,and the simulated annealing(SA),least squares(LS)and alternating direction multiplier method(ADMM)are used to solve the difference coefficient of spatial derivative.In this paper,the Remez algorithm is extended to solve the implicit difference coefficients of spatial derivatives.Numerical dispersion curve analysis shows that the difference coefficients solved by the new method can obtain wave number coverage close to spectral accuracy with the smallest order(2M?20).In addition,the constant coefficient optimization method and the time dispersion transform(TDTs)are combined to simultaneously solve the time dispersion and spatial dispersion of the forward modeling of the acoustic wave equation.Numerical experiments have proved that when there is a numerical dispersion error in the space direction,the use of TDTs can not completely remove the time numerical dispersion error,and even aggravate the dispersion error in the space direction.Then the higher the spatial accuracy,the better the time dispersion error removal effect of TDTs.In addition,the implicit finite difference method(OIFDM)optimized based on the Remez algorithm has the smallest spatial dispersion error and dispersion anisotropy,and the additional spatial numerical error added after combining with TDTs is also the smallest.In this paper,the implicit difference coefficients obtained by the optimization of Remez algorithm are extended to be used in the forward simulation of elastic wave equation and wave equation of two-phase medium respectively.(4)The basic principles of RTM imaging are studied.The imaging conditions at the time of excitation,amplitude ratio imaging conditions and cross-correlation imaging conditions are briefly introduced.In addition,this article discusses the low-frequency noise generation mechanism of RTM.We use the proposed implicit finite difference coefficients based on Remez optimization for pre-stack RTM imaging of acoustic wave equations of different models.Numerical experiments show that the new method can significantly improve the imaging accuracy and calculation efficiency compared with the traditional TE method.In this paper,starting from the numerical solution of the FDM of the seismic wave field derivatives,the basic framework and method theory of the FDM wave equation forward simulation are systematically studied.Then,the forward modeling and reverse time migration imaging based on high-precision finite-difference methods are realized.The research results show that these original and generalized work have significantly improved the accuracy and efficiency of forward modeling and RTM.