单程波算子积分解的象征表示
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摘要
单程波波场延拓算子在地震偏移成像中有重要应用.单程波波场延拓算子按其实现方式可分为Kirchhoff积分、空间隐式有限差分和Fourier变换方法,他们代表了算子的不同表示方法,当截断使用这些方法时会得到不同的精度.象征表示对这些方法的导出和精度分析有重要作用.算子作用于正弦波函数所得函数称为算子的象征.算子的象征是褶积算子Fourier变换的推广.Fourier变换方法则直接用象征函数的可分表示求出.空间隐式有限差分则可以用象征函数的Padè近似或部分分式导出.单程波算子在深度域的积分称为单程波算子积分解.本文推导了单程波算子积分解的象征表达式,给出了算子象征的代数运算的头几阶表达式,这些表达式还未在前人文献中发现.Kirchhoff积分所需格林函数可以通过象征函数和鞍点法导出.基于积分解的象征表达式给出了非对称走时公式,对改善Kirchhoff积分的聚焦性能有重要意义.
Extrapolation operator of one-way wave field has great application in seismic migration imaging.According to their implement,these operators can be classified into three kinds: Kirchhoff integral method,implicit definite difference in space domain and Fourier transform method.They represent different operator expression methods.Different precisions will be got while using the operators in these methods in different truncate order.Represent in symbol in the method of derivation and accuracy analysis.Symbol of operator is the function of operator acting on sine function.Symbol of operator is the extension of convolution operator's Fourier transform.Fourier transform method can be directly derived from symbol function's separable approximation.Implicit space difference can be solved by padè approximation or part fraction.Integral of one way wave operator in depth domain.The symbol solution of one-way wave integral is deduced,the first several order operator expressions of symbol's algebra operation are given.These expressions haven't been found in any previous reference.Green function that adapts to Kirchhoff integral method could be derived from symbol function and saddle point method.The symbolic expression,which based on integral solution give the equation of asymmetric traveltime,have great significance on focusing capability of Kirchhoff integral method.
Extrapolation operator of one-way wave field has great application in seismic migration imaging.According to their implement,these operators can be classified into three kinds: Kirchhoff integral method,implicit definite difference in space domain and Fourier transform method.They represent different operator expression methods.Different precisions will be got while using the operators in these methods in different truncate order.Represent in symbol in the method of derivation and accuracy analysis.Symbol of operator is the function of operator acting on sine function.Symbol of operator is the extension of convolution operator's Fourier transform.Fourier transform method can be directly derived from symbol function's separable approximation.Implicit space difference can be solved by padè approximation or part fraction.Integral of one way wave operator in depth domain.The symbol solution of one-way wave integral is deduced,the first several order operator expressions of symbol's algebra operation are given.These expressions haven't been found in any previous reference.Green function that adapts to Kirchhoff integral method could be derived from symbol function and saddle point method.The symbolic expression,which based on integral solution give the equation of asymmetric traveltime,have great significance on focusing capability of Kirchhoff integral method.
引文
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[20]Wu R S.Wide-angle elastic wave one-way propagation in het-erogeneous media and an elastic wave complex-screen method[J],Journal of Geophysical Research,1994,99(B1):751~766.
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[22]刘礼农,高红伟,刘洪,张剑锋.三维VTI介质中波动方程深度偏移的最优分裂Fourier方法[J].地球物理学报,2005,48(2):406~414.
[23]张致付,刘洪,陈景波,李幼铭.地震偏移的最优可分近似算法实现[J].地球物理学报,2005,48(6):1422~1427
[24]宋健,高维函数和流形在低维可视空间中的最优表达[J].科学通报,2001,46(12):977~984.
[25]刘洪,刘礼农,杨辉.波场延拓算子的最佳乘积基函数展开方法研究[A],见:中国大陆地球深部结构与动力学研究[C],北京:科学出版社,2004:483~490.
[26]吕小林,刘洪.波动方程深度偏移波场延拓算子的快速重建[J].地球物理学进展,2005,20(1):24~28.
[27]齐民有,徐超江,王维克.现代偏微分方程引论[M].武汉:武汉大学出版社,2005.
[28]杨文采,范尚武.基于声波方程的油储反射地震[M].见:刘光鼎等主编.陆相薄互层油储地球物理学导论(第四章).北京:科学出版社,1998.
[29]张关泉.波场分裂、平方根算子与偏移,反射地震学论文集[M].上海:同济大学出版社,2000.
[30]刘超颖,王成祥.非稳态相移法叠前深度偏移[J].地球物理学进展,2004,19(3):543~546.
[31]De h M,Le R J,Wu R S.Generalization of the phase-screenapproximation for the scattering of acoustic waves[J].WaveMotion,2000,31:43~70.
[32]Hoop M V D.Generalization of the Bremmer coupling series[J].J.Math.Phys.1996,37:3246~3282.
[33]刘洪,袁江华,勾永峰,张向君,李幼铭.地震逆散射波场和算子的谱分解[J].地球物理学报,2007,50(1):240~247.
[34]崔兴福,张关泉,吴雅丽.三维非均匀介质中真振幅地震偏移算子研究[J].地球物理学报,2004,47(3):509~513.
[35]张关泉.一维波动方程的反演问题[J],中国科学,1988,7:707~721.
[36]Zhang Y,Zhang G,Theory of true-amplitude one-way waveequations and true-amplitude common-shot migration[J].Ge-ophysics,2005,70:1~10.
[37]张平平,常旭.保幅偏移中的权函数[J].地球物理学进展,2006,21(1):203~207.
[38]刘洪,袁江华,陈景波,首皓,李幼铭.大步长波场深度延拓的理论[J].地球物理学报,2006,49(6):1779~1793.
[39]Feng K.On difference schemes and symplectic geometry[A].In Feng ed.:Symp Diff Geometry and Diff Equations[C].Beijing:Science Press,1985.
[40]Chen J B,Munthe K H,Qin M Z,.Square-conservativeschemes for a class of evolution equations using Lie-groupmethods[J].SIAM J.Numer.Anal.,2002,39:2164~2178.
[41]Munthe K H.High order Runge-Kutta methods on manifolds[J].Appl.Numer.Math.1999,29:115~127.
[42]McLachlan RI,Quispel G R W.Splitting methods[J],ActaNumerica,2002:341~434,DOI:10.1017/S0962492902000053.
[43]Celledoni E,Iserles A.Methods for the approximation of thematrix exponential in a Lie-algebraic setting[J].IMA Journalof Numerical Analysis 2001,21:463~488.
[44]Moler C.C.Van Loan,Nineteen dubious ways to computethe exponential of a matrix[J].SIAM Review,1978,20(4):801~836.
[2]Benson A K,Residual Depth-Migration:Enhancing Dip[J].Geophysical Prospecting 1989,31:771~779.
[3]Castle R J.Wave-equation migration in the presence of lateralvelocity variations[J].Geophysics,1982,47:1001~1011.
[4]Hubral.Time migration-some ray theoretical aspects.Geo-physi.Prosp,1975,25:738~745.
[5]Dix C H.Seismic Velocities from Surface Measurements[J].Geophysics,1955,20,68~86.
[6]Larner K L.Hatton L.Gibson B S.et,Depth migration of im-aged time section[J].Geophysics,1981,46(5):734~750.
[7]Rocca F.And Salvawrl,Residual migration[A].In:52nd SEGmeeting Expanded Abstracts[C].Soc.of Expl.Geophys,1982,4~7.
[8]Rothman D H,Levins A.And Roccaf,Residual migration:applications and limitations[J].Geophysics,1985,50:110~126.
[9]王华忠,徐蔚亚,徐兆涛,等.DMO和叠前时间偏移的共同起点[J].石油地球物理勘探,2002,37(3):224~229.
[10]Sun C,Martinez R,3D Kirchhoff PS-wave prestack time mi-gration for V(z)and VTI media[A].In:73rd SEG meetingExpanded Abstracts[C].Soc.of Expl.Geophys,2003,957~960.
[11]Claerbout J F,Doherty S M.Downward continuation ofmoveout-corrected seismograms[J].Geophysics,Soc.of Ex-pl.Geophys,1972,37:741~768.
[12]马在田.高阶方程偏移的分裂算法[J].地球物理学报,1983,26(4):377~388.
[13]Zhang G.A new Algorithm for finite-difference migration ofsteep dips[J].Geophysics,1988,53:167~175.
[14]Ristow D.Ruhl T,Fourier finite-difference migration[J].Geophysics,1994,59(12):1882~1893.
[15]Claerbout J F.Multidimensional recursive filters via a helix[J].Geophysics,1998,63:1532~1541.
[16]Ristow D,Ruhl T.3-D implicit finite-difference migration bymulti-way splitting[J].Geophysics,1997,62(2):554~567
[17]杨辉,刘洪,李幼铭.单程波方程偏移算子辛格式[J].地球物理学报,2003,46(4):533~538.
[18]Liu Y,Wu R S.A Comparison between Phase Screen,FiniteDifference and Eigenfunction Expasion Calculation for ScalarWaves in inhomogeneous media[J].Bulletin of the seismo-logical society of America,1994,84(4):1154~1168.
[19]Wu R S.Wave Propagation,Scattering and Imaging UsingDual-domain One-way and One-return Propagators[J].Pureand Applied Geophysics,2003,160(3-4):509~539.
[20]Wu R S.Wide-angle elastic wave one-way propagation in het-erogeneous media and an elastic wave complex-screen method[J],Journal of Geophysical Research,1994,99(B1):751~766.
[21]Chen J B,Liu H,Optimization approximation with separablevariables for the one-way wave operator[J].Geophysical Re-search letter,2004,31(6),L06613,doi:10.1029/2004GL019429.
[22]刘礼农,高红伟,刘洪,张剑锋.三维VTI介质中波动方程深度偏移的最优分裂Fourier方法[J].地球物理学报,2005,48(2):406~414.
[23]张致付,刘洪,陈景波,李幼铭.地震偏移的最优可分近似算法实现[J].地球物理学报,2005,48(6):1422~1427
[24]宋健,高维函数和流形在低维可视空间中的最优表达[J].科学通报,2001,46(12):977~984.
[25]刘洪,刘礼农,杨辉.波场延拓算子的最佳乘积基函数展开方法研究[A],见:中国大陆地球深部结构与动力学研究[C],北京:科学出版社,2004:483~490.
[26]吕小林,刘洪.波动方程深度偏移波场延拓算子的快速重建[J].地球物理学进展,2005,20(1):24~28.
[27]齐民有,徐超江,王维克.现代偏微分方程引论[M].武汉:武汉大学出版社,2005.
[28]杨文采,范尚武.基于声波方程的油储反射地震[M].见:刘光鼎等主编.陆相薄互层油储地球物理学导论(第四章).北京:科学出版社,1998.
[29]张关泉.波场分裂、平方根算子与偏移,反射地震学论文集[M].上海:同济大学出版社,2000.
[30]刘超颖,王成祥.非稳态相移法叠前深度偏移[J].地球物理学进展,2004,19(3):543~546.
[31]De h M,Le R J,Wu R S.Generalization of the phase-screenapproximation for the scattering of acoustic waves[J].WaveMotion,2000,31:43~70.
[32]Hoop M V D.Generalization of the Bremmer coupling series[J].J.Math.Phys.1996,37:3246~3282.
[33]刘洪,袁江华,勾永峰,张向君,李幼铭.地震逆散射波场和算子的谱分解[J].地球物理学报,2007,50(1):240~247.
[34]崔兴福,张关泉,吴雅丽.三维非均匀介质中真振幅地震偏移算子研究[J].地球物理学报,2004,47(3):509~513.
[35]张关泉.一维波动方程的反演问题[J],中国科学,1988,7:707~721.
[36]Zhang Y,Zhang G,Theory of true-amplitude one-way waveequations and true-amplitude common-shot migration[J].Ge-ophysics,2005,70:1~10.
[37]张平平,常旭.保幅偏移中的权函数[J].地球物理学进展,2006,21(1):203~207.
[38]刘洪,袁江华,陈景波,首皓,李幼铭.大步长波场深度延拓的理论[J].地球物理学报,2006,49(6):1779~1793.
[39]Feng K.On difference schemes and symplectic geometry[A].In Feng ed.:Symp Diff Geometry and Diff Equations[C].Beijing:Science Press,1985.
[40]Chen J B,Munthe K H,Qin M Z,.Square-conservativeschemes for a class of evolution equations using Lie-groupmethods[J].SIAM J.Numer.Anal.,2002,39:2164~2178.
[41]Munthe K H.High order Runge-Kutta methods on manifolds[J].Appl.Numer.Math.1999,29:115~127.
[42]McLachlan RI,Quispel G R W.Splitting methods[J],ActaNumerica,2002:341~434,DOI:10.1017/S0962492902000053.
[43]Celledoni E,Iserles A.Methods for the approximation of thematrix exponential in a Lie-algebraic setting[J].IMA Journalof Numerical Analysis 2001,21:463~488.
[44]Moler C.C.Van Loan,Nineteen dubious ways to computethe exponential of a matrix[J].SIAM Review,1978,20(4):801~836.
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