Research On High-Accuracy Finitedifference Scheme And Parallel Modeling Method For Frequencydomain Wave Equation

As the easily accessible mineral deposits are exhausted gradually and the technology keeps advancing,the prospecting techniques ask for higher efficiency and accuracy.Most of the existing seismic imaging techniques require an efficient and accurate forward modeling method,one of which is the frequency-domain wave modeling.The discretization of the wave equation could be conducted by the finite difference(FD)method,the finite element method,the discontinuous Galerkin method or the spectral element method.Among these methods,the FD scheme has been gaining popularity in seismic prospecting due to its simplicity and computational efficiency.Developing a highly accurate FD scheme can yield a reduction for spatial sampling or increase the frequency used in the wave modeling.This may be very beneficial for increasing the image accuracy.In addition,because the frequency-domain wave modeling after discretization is equivalent to solving a large sparse linear system,a fast and stable linear solver is a cornerstone of the frequency-domain implementation of many seismic imaging techniques,such as the full waveform inversion and reverse time migration.Incorporating the highly heterogeneity,anisotropy and other complex properties of the subsurface media will generate from the discretization of wave equation a linear system with an indefinite and ill-conditioned coefficient matrix.Solving such a linear system is a great challenge.The main content of this thesis is thus inspired based on two mentioned issues:Firstly,the second-order FD scheme,the mixed-grid FD scheme and the fourth-order FD scheme are investigated for the frequency-domain acoustic wave forward modeling.The convergence order of these schemes are analysed theoretically and verified through numerical experiments.By the comparison between the numerical and analytical solutions,the absolute errors of different schemes are checked.A comparison of the running time gives a test on the efficiency of each scheme.The results show that the mixed-grid method is a compromise between the other two schemes.It could keep the computational cost at the same level as the second-order scheme while guaranteeing a satisfactory accuracy.For the case of 2D frequency-domain elastic wave modeling,a trial experiment of second-order staggered-grid scheme is carried out preliminarily and the accuracy is testified by a comparison with the time-domain seismograms.The anti-lumped mass strategy is introduced to the fourth-order staggered-grid scheme.Within the framework of dispersion analysis,the optimal fourth-order staggered-grid scheme is obtained through minimizing the misfit between the normalized phase velocity and the unity.Compared with the conventional fourth-order scheme,the dispersion of the optimal scheme is drastically reduced.Therefore,the numerical accuracy could be retained while sampling fewer gridpoints per minimum wavelength.This will lead to a great reduction of computational cost for wave modeling.The complex Marmousi2 model is used to compare the numerical results with time-domain seismograms.The good agreement between these results shows a high accuracy of the optimal fourth-order staggered-grid scheme.Furthermore,in order to derive the optimal fourth-order staggered-grid scheme for the 3D case,the dispersion analysis for 3D elastic wave is developed.The LevenbergMarquardt method and the simulated annealing algorithm on the continuous coefficient range combined with the downhill simplex method are utilized to solve the minimization of the misfit between the normalized phase velocities and the unity.The optimal mass weighting coefficients and optimal FD coefficients are obtained consequently.The dispersion curves shows that the optimal scheme is relatively insensitive to the Poisson's ratio.With the upper limit of group velocities being 1 %,the number of grid-points required per minimum wavelength is just 3.7.The memory cost complexity is analyzed for the conventional fourth-order scheme and the optimal scheme using both the direct and iterative methods.The memory cost for wave modeling using the optimal scheme is reduced significantly.By comparing the numerical results with the analytical solutions of both the acoustic and elastic wave equation,the superior accuracy of the optimal scheme is confirmed.The optimal scheme results conform to the analytical solutions well even with only 3.3 grid-points per minimum wavelength.Finally,the highly parallel CARP-CG method is applied to the 2D and 3D elastic wave modeling to deal with the challenge from the large scale applications.Numerical experiments are conducted to investigate the influence of the Poisson's ratio,free surface boundary condition and the seismic attenuation.Several experiments with different frequencies and different scales are carried out to investigate the scalability on multiple computing cores.The feasibility of CARP-CG on large scale applications is also discussed.By analyzing the computational complexity,using the CARP-CG method for frequency-domain modeling is shown to have a smaller complexity than the time-domain modeling.In addition,the CARP-CG method is compared with other iterative methods,such as GMRES,CGNR and BiCGSTAB.The results indicate that CARP-CG could always converge for different kind of media and different experimental settings,which emphasizes the good stability of the CARP-CG method.Thus the CARP-CG method could be an efficient candidate for large scale forward modeling algorithms in the framework of seismic imaging techniques.