Research On Modeling And Inversion Methods For Acoustic And Viscous Acoustic Wave Equations

The wave equation inversion methods take full advantages of the kinematic and dynamic characteristics of the wavefields and thus become an effective way of acquiring the reservoir parameters in the subsurface.At present,the acoustic wave equation waveform inversion method shows significant potential in estimating the reservoir parameters like velocity and has attracted more and more attention.One key issue of the acoustic waveform inversion lies in solving the wave equation,and increasing the modeling accuracy and efficiency will help improve the inversion accuracy and efficiency.Since the media in the subsurface is not elastic,the study on modeling and inversion methods for viscous acoustic equation will further increase the accuracies of wavefield simulation and inversion.Focusing on the acoustic and viscous acoustic wave equations,this dissertation mainly studies the highly accurate and efficient simulation methods,reverse time migration(RTM)method,least-squares RTM(LSRTM)method and full waveform inversion(FWI)method.The main achievements are as follows:Firstly,for second-order acoustic wave equation,we develop a new highly accurate time-space domain finite difference(FD)method through combining the cross and rhombus FD stencils.The conventional method adopts the cross FD stencil and can only achieve second order accuracy along arbitrary directions when using space-domain FD with operator length M.The improved time-space domain scheme can achieve(2M)th-order accuracy along eight directions.However,along other directions,the accuracy is still second-order.The recently developed rhombus FD stencil can achieve(2M)th-order accuracy along all directions.However,it is computationally expensive and inefficient.In this dissertation,we combine the cross and rhombus stencils together.The temporal derivative is solved by the rhombus stencil with operator length N and the spatial derivatives are solved by the cross stencil with operator length M.By combining these two stencils,the new method can achieve(2N)th-order temporal accuracy and(2M)th-order spatial accuracy,and obtain(2N)th-order overall accuracy along all directions.Starting from the dispersion relation of the new scheme,we derive the Taylor expansion(TE)based and least square(LS)based FD coefficients.Numerical examples suggest that the new method is more accurate and stable than conventional space-domain and time-space domain methods.The LS-based coefficients suppress the dispersion better than the TE-based coefficients.The modeling accuracy and efficiency can be significantly improved by using the new scheme with LS-based FD coefficients and variable operator length schemes.Secondly,for second-order acoustic wave equation,we further develop new FD schemes with high-order temporal and implicit spatial accuracies.The conventional implicit FD schemes solve the temporal derivative using explicit second-order FD and solve the spatial derivatives using the implicit high-order FD.This results in low accuracy in time and will be prone to dispersion.In this dissertation,we introduce to use the rhombus FD stencil to approximate the temporal derivative in order to increase the temporal accuracy.When solving the spatial derivatives,apart from the Taylor-expansion based implicit spatial FD coefficients,we further derive the least-square optimal based FD coefficients.Numerical examples suggest that both the temporal and spatial dispersions are well suppressed by adopting the new method.Compared with conventional implicit spatial scheme,the new method can adopt much larger time step,thus can increase the efficiency.Thirdly,for second-order acoustic and nearly constant-Q viscous acoustic equations,we study the one-step and two-step simulation methods that are based on the lowrank decomposition.We further develop new absorbing boundary conditions(ABCs)for one-step simulation methods.Numerical examples suggest that the spectrum methods are much more stable and accurate than FD methods.The one-step methods are much more stable and can adopt much larger time step than two-step methods.The proposed hybrid ABCs can effectively absorb the boundary reflections with 15 absorbing layers.Fourthly,we study the acoustic RTM and LSRTM methods.The one-step extrapolated complex wavefields are directly used to implement the up/down and left/right-going wavefield decomposition and the decomposed RTM images are obtained using the correlation image condition.Numerical examples suggest that the methods achieve images that are nearly free from low-frequency noises,and can help recognize the image artifacts in the presence of inaccurate velocity model.The LSRTM method using zero-lag normalized cross-correlation objective function can balance the amplitude difference between seismic traces and generate accurate images even when the first-order velocity-pressure acoustic data is used.The source-independent LSRTM methods can still achieve reliable results when the migration wavelet is inaccurate.The plane-wave LSRTM method can significantly increase the computational efficiency and reduce the total CPU time.Fifthly,the amplitude attenuation and phase change characteristics for nearly constant-Q equations and standard linear solid body(SLS)equations are studied.The linear and non-linear optimization algorithms used to approximate the constant-Q theory based on the SLS theory are compared.The Q-compensate RTM and LSRTM methods based on these two kinds of equations are studied as well.Numerical examples suggest that,the viscous acoustic equations that separate the amplitude attenuation and phase change functions can well describe the attenuation characteristics.The nonlinear algorithm is more accurate than the linear algorithm when used to approximate the constant-Q theory.The Q-compensate RTM methods can compensate the attenuated energy effectively and increase the imaging resolution significantly.The Q-compensate LSRTM can further improve the quality of Q-compensate RTM images especially in illustration and resolution.Sixthly,the FWI methods for acoustic and viscous acoustic equations are studied.We study the multi-scale inversion method,plane-wave multi-scale inversion method and the source-independent inversion method.Numerical examples suggest that the multi-scale strategy can help ease the dependence of the initial model and speed up the convergence rate of the objective function.The plane-wave inversion method can shorten the computational time significantly.The source-independent inversion method can still achieve acceptable results even when inaccurate wavelet is adopted.The viscous acoustic waveform inversion method can achieve more accurate inversion results than the acoustic waveform inversion method when the attenuated data is used because it considers the attenuation effect of the media in the subsurface.