频率域波动方程正演基于多重网格预条件的迭代算法研究
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摘要
频率域全波形反演是重要的地震成像方法,而频率域波动方程数值模拟是频率域全波形反演的基础.对于大规模的问题,由于受存储和计算量的限制,基于LU分解的直接方法一般不再适用,而是采用迭代方法.基于多重网格预条件的双共轭梯度稳定化方法是一种重要的迭代方法.本文重点讨论了多重网格预条件求解过程中的松弛因子选择方法,研究结果表明,(1)对于一般选取的松弛因子,随模型复杂性的增加,所能计算的重数逐渐下降,方法的实用性也随之下降;(2)对于复杂模型,采用局部模式分析方法选取松弛因子,提高了所能计算的重数,保证了多重网格方法的收敛性和实用性.这些研究成果对基于多重网格预条件的迭代算法的实际应用具有重要意义.
Frequency-domain full waveform inversion(FWI)is an important method for seismic imaging,and frequency-domain modeling is the basis of frequency-domain FWI.For large-scale problems,the LU-decomposition-based direct method is no longer applicable due to limitations of storage and computational time.Instead,iterative methods are employed.The bi-conjugategradient-stabilized method,which is based on multi-grid precondition,is an important iterative method.The preconditioner based on a heavily damped wave equation is approximately solved with one multi-grid cycle.The multi-grid method is implemented with a weighted Jacobi smoothing,a standard full-weighting coarsening,a linear interpolation,and a matrix-free implementation.To make it convergent for complex model,local model analysis is used to obtain the relaxation factor in the implementation of weighted Jacobi smoothing.Numerical experiments reveal:(1)for a generally-chosen relaxation factor,the levels of themulti-grid decrease as the complexity of the model increases,and,accordingly,the method becomes less practical;(2)for complex models,the relaxation factor obtained by local mode analysis increases the levels and reduces the number of iterations for each single frequency.The bi-conjugate-gradient-stabilized method based on multi-grid precondition obtains its efficiency and precision by using one multi-grid cycle for the approximate inversion of the preconditioner.To obtain reasonably fast convergence of the multi-grid method for complex model,local model analysis is applied in the relaxation factor selection.Compared with a generally-chosen relaxation factor,the relaxation factor obtained by local mode analysis increases the levels and guarantees the convergence and practicality of the multi-grid method.These conclusions are of great significance for application of the multi-grid-precondition-based iterative algorithm.
Frequency-domain full waveform inversion(FWI)is an important method for seismic imaging,and frequency-domain modeling is the basis of frequency-domain FWI.For large-scale problems,the LU-decomposition-based direct method is no longer applicable due to limitations of storage and computational time.Instead,iterative methods are employed.The bi-conjugategradient-stabilized method,which is based on multi-grid precondition,is an important iterative method.The preconditioner based on a heavily damped wave equation is approximately solved with one multi-grid cycle.The multi-grid method is implemented with a weighted Jacobi smoothing,a standard full-weighting coarsening,a linear interpolation,and a matrix-free implementation.To make it convergent for complex model,local model analysis is used to obtain the relaxation factor in the implementation of weighted Jacobi smoothing.Numerical experiments reveal:(1)for a generally-chosen relaxation factor,the levels of themulti-grid decrease as the complexity of the model increases,and,accordingly,the method becomes less practical;(2)for complex models,the relaxation factor obtained by local mode analysis increases the levels and reduces the number of iterations for each single frequency.The bi-conjugate-gradient-stabilized method based on multi-grid precondition obtains its efficiency and precision by using one multi-grid cycle for the approximate inversion of the preconditioner.To obtain reasonably fast convergence of the multi-grid method for complex model,local model analysis is applied in the relaxation factor selection.Compared with a generally-chosen relaxation factor,the relaxation factor obtained by local mode analysis increases the levels and guarantees the convergence and practicality of the multi-grid method.These conclusions are of great significance for application of the multi-grid-precondition-based iterative algorithm.
引文
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Pratt R G.1990.Frequency-domain elastic wave modeling by finitedifferences:A tool for cross-hole seismic imaging.Geophysics,55(5):626-632.
Pratt R G,Worthington M H.1990.Inverse theory applied tomulti-source cross-hole tomography,Part 1:Acoustic waveequation method.Geophysical Prospecting,38(3):287-310.
Pratt R G,Shin C,Hicks G J.1998.Gauss-Newton and fullNewton methods in frequency-space seismic waveform inversion.Geophysics,133:341-362.
Riyanti C D,Erlangga Y A,Plessix R E,et al.2006.A newiterative solver for the time-harmonic wave equation.Geophysics,71(5):E57-E63.
tekl I,Pratt R G.1998.Accurate viscoelastic modeling byfrequency-domain finite differences using rotated operators.Geophysics,63(5):1779-1794.
Tarantola A.1984.Inversion of seismic reflection data in theacoustic approximation.Geophysics,49(8):1259-1266.
Van der Vorst H A.1992.Bi-CGSTAB:A fast and smoothlyconverging variant of bi-CG for the solution of nonsymmetriclinear systems.Society of Industrial and Applied Mathematics(SIAM)Journal on Scientific and Statistical Computing,13(2):631-644.
Virieux J,Operto S.2009.An overview of full-waveform inversionin exploration geophysics.Geophysics,74(6):WCC1-WCC26.
马召贵,王尚旭,宋建勇.2010.频率域波动方程正演中的多网格迭代算法.石油地球物理勘探,45(1):1-5.
Chen J B.2012.An average-derivative optimal scheme forfrequency-domain scalar wave equation.Geophysics,77(6):T201-T210.
Erlangga Y A.2008.Advances in iterative methods andpreconditioners for the Helmholtz equation.Arch.Comput.Methods Eng.,15(1):37-66.
Jo C H,Shin C,Suh J H.1996.An optimal 9-point,finitedifference,frequency-space,2-D scalar wave extrapolator.Geophysics,61(2):529-537.
Kim S,Kim S.2002.Multigrid simulation for high-frequencysolutions of the Helmholtz problem in heterogeneous media.Society of Industrial and Applied Mathematics(SIAM)Journal on Scientific Computing,24(2):684-701.
Marfurt K J.1984.Accuracy of finite-difference and finite-elementmodeling of the scalar and elastic wave equations.Geophysics,49(5):533-549.
Ma Z G,Wang S X,Song J Y.2010.Multigrid iterative algorithmin frequency domain wave equation forward modeling.OilGeophysical Prospecting(in Chinese),45(1):1-5.
Plessix R E.2007.A Helmholtz iterative solver for 3D seismicimaging problems.Geophysics,72(5):SM185-SM194.
Pratt R G.1990.Frequency-domain elastic wave modeling by finitedifferences:A tool for cross-hole seismic imaging.Geophysics,55(5):626-632.
Pratt R G,Worthington M H.1990.Inverse theory applied tomulti-source cross-hole tomography,Part 1:Acoustic waveequation method.Geophysical Prospecting,38(3):287-310.
Pratt R G,Shin C,Hicks G J.1998.Gauss-Newton and fullNewton methods in frequency-space seismic waveform inversion.Geophysics,133:341-362.
Riyanti C D,Erlangga Y A,Plessix R E,et al.2006.A newiterative solver for the time-harmonic wave equation.Geophysics,71(5):E57-E63.
tekl I,Pratt R G.1998.Accurate viscoelastic modeling byfrequency-domain finite differences using rotated operators.Geophysics,63(5):1779-1794.
Tarantola A.1984.Inversion of seismic reflection data in theacoustic approximation.Geophysics,49(8):1259-1266.
Van der Vorst H A.1992.Bi-CGSTAB:A fast and smoothlyconverging variant of bi-CG for the solution of nonsymmetriclinear systems.Society of Industrial and Applied Mathematics(SIAM)Journal on Scientific and Statistical Computing,13(2):631-644.
Virieux J,Operto S.2009.An overview of full-waveform inversionin exploration geophysics.Geophysics,74(6):WCC1-WCC26.
马召贵,王尚旭,宋建勇.2010.频率域波动方程正演中的多网格迭代算法.石油地球物理勘探,45(1):1-5.
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