三维VTI介质qP波正演方法研究
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摘要
地球介质的各向异性是普遍存在的,地震各向异性研究已经成为地震学研究领域中的前沿课题之一。地震波正演模拟是模拟地震波在地球介质中的传播过程,并研究地震波的传播特性与地球介质参数的关系,通过正演模拟以达到对实际观测地震记录的最优逼近。开展地震波的正演模拟研究,对人们正确认识地震波的传播规律,验证所求地球模型的正确性;进行实际地震资料的地质解释与储层预测以及地球资源开发等,均具有重要的理论和实际意义。
本文对具有垂直对称轴的三维横向各向同性(VTI)介质qP波在频率域和时间域进行了正演模拟。在频率域正演中,从三维VTI介质频率空间域qP波波动方程出发,在常规差分算子的基础上构造了适合三维VTI介质的频率空间域加权平均有限差分算子,利用Gauss-Newton优化方法算出VTI介质加权平均差分算子系数,并对常规差分算子和加权平均差分算子进行了速度频散精度对比分析。本文把螺旋边界和吸收边界应用在频率域正演上,在求取大型带状稀疏差分矩阵方面,作者提出了改进算法;并给出了三维频率域正演的数值实例。在三维VTI介质qP波时间域正演模拟方面,首先对三维VTI介质qP波时间域波动方程进行降阶处理,接着利用提出的加权平均差分算子对波动方程进行差分,并对加权平均差分算子进行了速度频散精度对比分析,还提出了一种新的分裂的完全匹配层来处理时间域边界,并进行了边界处理效果对比分析,最后进行了三维时间域正演模拟。频率域和时间域数值模拟表明:本文提出的算法有效地压制了数值频散,成功地减少了人为边界反射,具有较高的模拟精度,为三维VTI介质qP波的偏移和反演奠定了理论基础。
Anisotropy is ubiquitous in earth media. Study on seismic anisotropy has become one of the leading edge directions in the field of seismology. Forward modeling simulates wave propagate in the media of earth. We can learn the relationship of characteristic of seismic propagate and the parameter of earth media from the simulating, and get the optimal approximation of real seismic recording. Studying on seismic forward modeling can help people accurately know the rule of seismic propagation. It also can verify the validity of earth model, interpret the real seismic data, predict reservoir and develop the resource. In a word, it has very important meaning in both theory and practice.
This paper simulates the qP wave in 3-D transversely isotropy with a vertical axis of symmetry (VTI) media in both domains of frequency and time. In frequency domain modeling, stating with frequency-space domain qP wave equation in 3-D VTI media, the weighted mean finite difference operators which is based on conventional finite difference operators is presented. Comparing the precision of velocity dispersion of conventional finite difference operators and weighted mean finite difference is following. Then we get the weighted mean coefficient by the Gauss-Newton method in optimization theory to make the finite-difference equation phase velocity being equal to that of wave equation. The helix boundary condition and absorb boundary condition in frequency domain is adopted. In aspect of solving large sparse matrix this paper advances the improved arithmetic. The numerical experiments are given in the end of frequency domain modeling. In time domain modeling, we firstly deflate the time domain qP wave equation in 3-D VTI media. Secondly, apply the weighted mean finite difference operators which are presented former to the wave function in time domain. A new split perfect matched layer boundary condition is presented and is validated the effect in this paper. The complex model simulating in time domain is given at last. The model simulating of frequency and time domains show that the arithmetic that is given in this paper effectively suppress the numerical dispersion and successfully decrease the artificial boundary reflecton. It is supply the foundation for migriation and invertion of qP wave in 3-D VTI media.
本文对具有垂直对称轴的三维横向各向同性(VTI)介质qP波在频率域和时间域进行了正演模拟。在频率域正演中,从三维VTI介质频率空间域qP波波动方程出发,在常规差分算子的基础上构造了适合三维VTI介质的频率空间域加权平均有限差分算子,利用Gauss-Newton优化方法算出VTI介质加权平均差分算子系数,并对常规差分算子和加权平均差分算子进行了速度频散精度对比分析。本文把螺旋边界和吸收边界应用在频率域正演上,在求取大型带状稀疏差分矩阵方面,作者提出了改进算法;并给出了三维频率域正演的数值实例。在三维VTI介质qP波时间域正演模拟方面,首先对三维VTI介质qP波时间域波动方程进行降阶处理,接着利用提出的加权平均差分算子对波动方程进行差分,并对加权平均差分算子进行了速度频散精度对比分析,还提出了一种新的分裂的完全匹配层来处理时间域边界,并进行了边界处理效果对比分析,最后进行了三维时间域正演模拟。频率域和时间域数值模拟表明:本文提出的算法有效地压制了数值频散,成功地减少了人为边界反射,具有较高的模拟精度,为三维VTI介质qP波的偏移和反演奠定了理论基础。
Anisotropy is ubiquitous in earth media. Study on seismic anisotropy has become one of the leading edge directions in the field of seismology. Forward modeling simulates wave propagate in the media of earth. We can learn the relationship of characteristic of seismic propagate and the parameter of earth media from the simulating, and get the optimal approximation of real seismic recording. Studying on seismic forward modeling can help people accurately know the rule of seismic propagation. It also can verify the validity of earth model, interpret the real seismic data, predict reservoir and develop the resource. In a word, it has very important meaning in both theory and practice.
This paper simulates the qP wave in 3-D transversely isotropy with a vertical axis of symmetry (VTI) media in both domains of frequency and time. In frequency domain modeling, stating with frequency-space domain qP wave equation in 3-D VTI media, the weighted mean finite difference operators which is based on conventional finite difference operators is presented. Comparing the precision of velocity dispersion of conventional finite difference operators and weighted mean finite difference is following. Then we get the weighted mean coefficient by the Gauss-Newton method in optimization theory to make the finite-difference equation phase velocity being equal to that of wave equation. The helix boundary condition and absorb boundary condition in frequency domain is adopted. In aspect of solving large sparse matrix this paper advances the improved arithmetic. The numerical experiments are given in the end of frequency domain modeling. In time domain modeling, we firstly deflate the time domain qP wave equation in 3-D VTI media. Secondly, apply the weighted mean finite difference operators which are presented former to the wave function in time domain. A new split perfect matched layer boundary condition is presented and is validated the effect in this paper. The complex model simulating in time domain is given at last. The model simulating of frequency and time domains show that the arithmetic that is given in this paper effectively suppress the numerical dispersion and successfully decrease the artificial boundary reflecton. It is supply the foundation for migriation and invertion of qP wave in 3-D VTI media.
引文
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[4]Virieux J.P. SV wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics, 1986, 51 (4):889-901
[5]Levander A R. Fourth-order finite-difference P-SV seismograms[J]. Geophysics, 1988, 53 (11):1425-1436
[6]何兵寿,张会星.VTI介质中准P波方程的数值解法[J].煤炭学报,2006,4:446-450
[7]Marfurt K.J. Accuracy of finite-difference and finite-element modeling of scalar and elastic wave equations[J]. Geophysics, 1984, 49, 533-549
[8]Pratt, R.G.. Frequency-domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging[J]. Geophysics, 1990, 55, 626-632
[9]杜世通.地震波动力学[M].山东东营:石油大学出版社,1996:1-45,257-304
[10]孙成禹.地震波动力学理论[M].山东东营:石油大学出版社,2004:7-29,133-171
[11]Crampin, S. An introduction to wave propagation in anisotropic media[J]. Geophysical Journal of the Royal Astronomical Society, 1984, 76:17-28
[12]Thomsen, L. Weak elastic anisotropy[J]. Geophysics, 1986, 51:1954-1966.
[13]Sena, A.G., Toksoz, M.N. Kirchhoff migration and velocity analysis for converted and nonconverted waves in anisotropic media[J]. Geophysics, 1993,58:265-276
[14]Tariq Alkhalifah. Acoustic approximation for processing in transversely isotropic media[J]. Geophysics, 1998, Vol.63(2): 623-631
[15]Tariq Alkhalifah. An acoustic wave equation for anisotropic media[J]. Geophysics, 2000, Vol.65(4): 1239-1250
[16]Tariq Alkhalifah. An acoustic wave equation for orthorhombic media[J]. Geophysics, 2003, Vol.68(4): 1169-1172
[17]Alkhalifah, T. An acoustic wave equation for anisotropic media[J]. Geophysics, 2000, 65:1239-1250
[18]R.Gerhard Pratt.Frequency-domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging[J]. Geophysics, 1990; Vol. 55(5): 626-632
[19]吴国忱.各向异性介质地震波传播与成像[M].山东:中国石油大学出版社,2005:73-83
[20]Alkhalifah T. Acoustic approximation for processing in transversely isotropic media[J]. Geophysics, 1998, 63(2):623-631
[21]Alkhalifah T. An acoustic wave equation for anisotropic media[J]. Geophysics, 2000, 65(4):1239-1250
[22]Min D J, Shin C, Kwon B D, et al. Improved frequency-domain elastic wave modeling using weighted- averaging difference operators[J]. Geophysics, 2000, 65(3): 884-895
[23]Clayton, R., Engquist, B. Absorbing boundary conditions for acoustic and elastic wave equations[J]. Bull. Seis. Soc. Am., 1977, 67:1529-1540
[24]Reynolds, A.C. Boundary conditions for the numerical solutions of wave propagation problems[J]. Geophysics, 1978, 43:1099-1110
[25]Berenger J P. A perfectly matched layer for the absorption of electromagnetic wave[J]. Comput Phys, 1994, 114(2):185-200
[26]Chew, W. C., Liu, Q.H. Perfectly matched layers for elastodynamics: A new absorbing boundary condition[J]. Comp. Acoust., 1996, 4:341-359
[27]张贤达.现代数字信号处理[M].北京:清华大学出版社,1995
[28]Claerbout J. Multidimensional recursive filters via a helix[J]. Geophysics,1998, 63(5):1532-1541
[29]Claerbout J. Multidimensional recursive filters via a helix with application to velocity estimation and 3D migration[J]. 68th Ann Internat Mtg, Soc Expl Geophys, Expanded Abstracts, 1998, 1995-1998
[30]罗明秋,刘洪,李幼铭.基于螺旋线上谱因式分解的地震波场隐式辛算法[J].地球物理学报,2001,43(3):379-388
[31]罗明秋,刘洪,李幼铭.地震波传播的哈密顿表述及辛几何算法[J].地球物理学报,2001,44(1):120-128
[32]Buneman O. Advantages of Hamilton Formulation in Computer Simulations[A]. In: Tabor[C]. Mathematical Methods in Hydrodynamics and Integrability of Dynamical Systems.Amer.Inst.Phys.USA,1981.
[33]Jean Laurent, Mallet. Discrete smooth interpolation in geometric modeling[J]. Computer-aided Design, 1992, 24(4):125-128
[34]Joho F. Claerbout. geophysical estimation by Example, Environmental soundings image construction[M]. Multidimensional autoregreesion, April30, 2000.
[35]牟永光,裴正林.三维复杂介质地震数值模拟[M].石油工业出版社,2005:185-186
[36]G .Meijerink. Computer Solution of Large Linear Systems[J]. Elsvier Science B.V., North Holland, 1999.
[37]Luo Mingqiu,Gao Hongwei,Liu Hong,et al. A comparison on the LU decomposition of Laplacian matrix with spectral and graphic methods[C]. SEG 70th International Meeting.Expanded Abstracts[A]. 2000.
[38]Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operator[J]. Res.Bur.Stand., 1950,45-47
[39]Lanczos C.Solution of systems of linear equations by minimized iterations [J] .J.Res.Nat.Bur.Stand.,1952,45:33-53
[40]Basermann A. Conjugate gradient and Lanczos methods for sparse matrices on distributed memory mutiprocessors[J]. J.Para.Distri.Comput.,1997,45:46-52
[41]Arnoldi W. The principle of minimized iterations in the solution of the matrix eigenvalue problem[J]. Quart.Appl.Math., 1951,9:17-29
[42]Fletcher R. Conjugrate gradient methods for indefinite systems[J] Watsoned Numerical Analysis Dundee 1975. Berlin, New York: Springer Veriag, 1976:73-89.
[43]Saad Y, Schultz M H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linearsystems[J]. SIAM J.Sci. Stat. Comput. 1986, 7(3):854-869
[44]Vander Vorst H A, Vuik C. GMRESR:A family of nested GMRES methods[J]. Number.Lin Alg.Applic., 1994, 1:369-386
[45]Li G. A block variant of the GMRES method on massively parallel processors [J]. Parallel Comput, 1997, 23:1005-1019
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