共反射面叠加属性研究
摘要
共反射面叠加(CRS)提供一种从多次覆盖反射数据进行零偏移距模拟的方法,它不依赖于先验速度信息。
对二维资料应用共反射面叠加,可产生CRS叠加面,它依赖于三个参数,它们是零偏移距射线的出射角α,以及与零偏射线有关的两个波前曲率半径R N和R NIP,即法向波和法向入射点波的波前曲率半径。对自激自收剖面上的每一个样点可确定一组最佳参数,这组参数能使CRS走时面最好地拟合反射同相轴。
确定三参数的过程是反复搜索和寻优的过程,所以这里将涉及到从一维到三维的寻优算法。
此文以两种途径,分别应用多种最优化方法进行三参数的寻优,并对结果、实现效率进行分析,得出其一种途径因应用了全局最优化算法而更可取,并将其应用到实际数据,得出的剖面浅层效果比较好。
The seismic survey is the most effective prospecting geophysical method during exploration and development of oil/gas. The structure and the lithology of the geological body become increasingly complex now. So it must assure that the seismic section own upper resolution if we need accurately describe the targets. High signal/noise ratio is the precondition of high-resolution.
Common-Reflection-Surface (CRS) stack is the best way at present to simulate zero-offset section. Common Reflection Surface is a circle segment around an underground reflector. Its travel time response in the time domain, which is CRS stack surface, can be regraded as a combination of all Common Reflection Point (CRP) trajectores in the segment. CRS stack can focus more energy in the vicinity of the reflector, therefore can obtain a better zero offset section by stacking events with same phase.
The notable advantage of CRS stack Fresnel zone, and select method and it is the valid seismic data is that it can calcalate the stack area using Projected to stack. This process of selection is like usual DMO method.
At first, based on wave equation, high-frequency ray solution and its character are given to clarify theoretical foundation of the method. The hyperbolic and parabolic travel time of the reflection in layer media are presented in expression of matrix with paraxial ray theory. With geometrical optics, the relationship between object point in model and image point in image space is built for the complex subsurface. The travel time formula of reflective point in the monunifonn media is deduced. Also the formula of reflective segment of zero-offset and nonzero offset section is provided.
For 2-D acquisition, the CRS stack leads to a stacking surface depending on three search parameters. The optimum stacking surface needs to be determined for each point of the simulated zero-offset section. For a given primary reflection, these are the emergence angleαof the zero-offset ray, as well as two radii of wavefront curvatures R N and R NIP.They all are associated with two hypothetical waves:the so-called normal wave and the normal-incidence-point wave. We also addressthe problem of determining an optimal parameter triplet(α, RNIP, RN) inorder to construct the sample value(i.e.,the CRS stack value)for each point in the desired simulated zero-offset section.This optimal triplet is expected to determine for each point the best stacking surface that can be fitted to the multicoverage primary reflectionevents.
To make the CRS stack attractive in terms of computational costs, a suitable strategy is described to determine the optimal parameter triplets for all points of the simulated zero-offset section. For the implementation of the CRS stack, we make use of the hyperbolic second-order Taylor expansion of the stacking surface.This representation is not only suitable to handle irregular multicoverage acquisition geometries but also enables us to introduce simple and efficient search strategies for the parameter triple.
In specific subsets of the multicoverage data(e.g.,in the common-midpoint gathers or the zero-offset section),the chosen representation only depends on one or two independent parameters,respectively.
So we present two optimization strategy for estimating these three parameters and simulating a zero-offset section through CRS stacking formalism.by different optimized methods such as variational simulated annealing algorithm,patternsearch algorithm and so on.Finally, the processing of practical data is successful though existing the dip discrimination phenomenon.
对二维资料应用共反射面叠加,可产生CRS叠加面,它依赖于三个参数,它们是零偏移距射线的出射角α,以及与零偏射线有关的两个波前曲率半径R N和R NIP,即法向波和法向入射点波的波前曲率半径。对自激自收剖面上的每一个样点可确定一组最佳参数,这组参数能使CRS走时面最好地拟合反射同相轴。
确定三参数的过程是反复搜索和寻优的过程,所以这里将涉及到从一维到三维的寻优算法。
此文以两种途径,分别应用多种最优化方法进行三参数的寻优,并对结果、实现效率进行分析,得出其一种途径因应用了全局最优化算法而更可取,并将其应用到实际数据,得出的剖面浅层效果比较好。
The seismic survey is the most effective prospecting geophysical method during exploration and development of oil/gas. The structure and the lithology of the geological body become increasingly complex now. So it must assure that the seismic section own upper resolution if we need accurately describe the targets. High signal/noise ratio is the precondition of high-resolution.
Common-Reflection-Surface (CRS) stack is the best way at present to simulate zero-offset section. Common Reflection Surface is a circle segment around an underground reflector. Its travel time response in the time domain, which is CRS stack surface, can be regraded as a combination of all Common Reflection Point (CRP) trajectores in the segment. CRS stack can focus more energy in the vicinity of the reflector, therefore can obtain a better zero offset section by stacking events with same phase.
The notable advantage of CRS stack Fresnel zone, and select method and it is the valid seismic data is that it can calcalate the stack area using Projected to stack. This process of selection is like usual DMO method.
At first, based on wave equation, high-frequency ray solution and its character are given to clarify theoretical foundation of the method. The hyperbolic and parabolic travel time of the reflection in layer media are presented in expression of matrix with paraxial ray theory. With geometrical optics, the relationship between object point in model and image point in image space is built for the complex subsurface. The travel time formula of reflective point in the monunifonn media is deduced. Also the formula of reflective segment of zero-offset and nonzero offset section is provided.
For 2-D acquisition, the CRS stack leads to a stacking surface depending on three search parameters. The optimum stacking surface needs to be determined for each point of the simulated zero-offset section. For a given primary reflection, these are the emergence angleαof the zero-offset ray, as well as two radii of wavefront curvatures R N and R NIP.They all are associated with two hypothetical waves:the so-called normal wave and the normal-incidence-point wave. We also addressthe problem of determining an optimal parameter triplet(α, RNIP, RN) inorder to construct the sample value(i.e.,the CRS stack value)for each point in the desired simulated zero-offset section.This optimal triplet is expected to determine for each point the best stacking surface that can be fitted to the multicoverage primary reflectionevents.
To make the CRS stack attractive in terms of computational costs, a suitable strategy is described to determine the optimal parameter triplets for all points of the simulated zero-offset section. For the implementation of the CRS stack, we make use of the hyperbolic second-order Taylor expansion of the stacking surface.This representation is not only suitable to handle irregular multicoverage acquisition geometries but also enables us to introduce simple and efficient search strategies for the parameter triple.
In specific subsets of the multicoverage data(e.g.,in the common-midpoint gathers or the zero-offset section),the chosen representation only depends on one or two independent parameters,respectively.
So we present two optimization strategy for estimating these three parameters and simulating a zero-offset section through CRS stacking formalism.by different optimized methods such as variational simulated annealing algorithm,patternsearch algorithm and so on.Finally, the processing of practical data is successful though existing the dip discrimination phenomenon.
引文
[1] Hubral. Macro model independent seismic reflection imaging. J. Appl. Geoph.1999,(42):3-4
[2] Wave Inversion Technology, annual report 2001,page:4
[3] de Bazelaire,E. ,Normal moveout revisited-inhomogeneous media and curved interfaces:Geophysics,1988,(53):143-157
[4] de Bazelaire,E.,and Viallix,J.R. Normal moveout in focus. Geophys, Prosp.,1994,(42):477-499.
[5] Berkovitch,A.,Gelchinsky,B.,and Keydar,S.Basic formulae for multifocusing stack:56th Mtg.Eur.Assoc.Expl.Geophys.,Extended Abstracts.1994
[6] Berkovitch, A., Gelchinsky, B. Inversion of common reflection element (cre) data (migration combined with interval velocity datermination).59th Annual Internat,Mtg., Soc. Expl. Geophys., Expanded Abstracts,1989,140
[7] Thore, P. D., de Bazelaire, E., and Ray, M. P.Three-parameter equation:An efficient tool to enhance the stack.Geophysics,1994,(59):297-308
[8] Berkovitch,A.,Keydar,S.,landa, E., and trachtman, P.,Multifocusing in practice. 68th Annual Internat,Mtg., Soc. Expl. Geophys.,Expanded Abstracts,1998,pl748-1751
[9] Berkhout. A. J.,A real shot record technology.Journal seismic exploration,1992,(1):251-264
[10] Berkhout. A. J,Pushing the limits of seismic imaging, Part Ι: Prestack migration in terms of double dynamic focusing. Geophysics,1997, 62(3):937-953
[11] Berkhout. A. J., Pushing the limits of seismic imaging, Part П: Integration of prestack migration, velocity estimation, and AVO analysics., Geophysics, 1997,62(3):954-969
[12] Gelchinsky, B., The common-reflection-element (CRE) method:ASEG/SEG Intemat. Geophys. Conf, Expl. Geophys., Extended Abstracts,1998. 71-75.
[13] Gelchinsky, B., Homemorphic imaging in processing of seismic data (fundamentals and schemes): 59th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts,1989, 983-988.
[14] Gelchinsky, B., Berkovitch, A., and Keydar, S.. Multifocusing homeomorphic imaging. Part 1:Basic concepts and formulas. Journal of Applied Geophysics, 1999a,(42):229-242.
[15] Gelchinsky, B., Berkovitch, A., and Keydar, S., Multifocusing homeomorphic imaging. Part 2: Multifold data set and multifocusing. Journal of Applied Geophysics, 1999b,(42):243-260.
[16] Keydar, S,Gelchinsky, B.and Berkovitch, A., The combined homeomorphic stacking. 63th Annual. Internat. Mtg. Soc. Expl. Geophys.,Expanded Abstracts, 1993, p1141-1144
[17] Keydar, S., Gelchinsky, B., and Berkovitch, A.Common shot point stacking and imaging. Journal of Seismic Exploration, 1996,(5):261-274
[18] Keydar, S., Gelchinsky, B., Shtivelman, V., and Berkovitch, A.,Common evolute element (cee) stack and imaging (zaeo offset stack).60th Annual. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts,1990,p1719-1722
[19] Hubral,R.and Krey, T., Interval velocities from seismic reflection traveltime measurements: SEG.,1980
[20] Hubral,R.and Krey, T., Interval velocities from seismic reflection traveltime measurements: SEG..1980
[21] Hubral, P., Schleicher, J., and Tygel, M., Three-dimensional paraxial ray properties, PART I: Basic relations: Journal of Seismic Exploration, 1992a, 265-279.
[22] Hubral,P.,Schleicher,J.,and Tygel, M., Three-dimensional paraxial ray properties,PART II: Applications: Journal of Seismic Exploration, 1992b,(1):347-362.
[23] Hubral,R,Schleicher, J.,and Tygel, M., A unified approach to 3D seismic reflection imaging, PART I: Basic Concepts:Geophysics, 1996,(61):742-758
[24] Hubral, P., Tygel M., and Schleicher, J., Seismic image waves:Geophys. Joum.Intern., 1996, (125):431-442.
[25] Hubral,P.,et al, The Common Reflection Surface (CRS) stack----structural resolution in Time Domain beyond the conventional NMO/DMO stack.2001
[26] Hubral,P,Hoecht,G., Jaeger, R., Seismic illumination, The Leading Edge, November,1999,1268-1271
[27] Hubral,P. and Mann,J, The overview of Common Reflection Surface (CRS) stack, The proceedings CNPC technology communication conference.2002
[28] Hoecht, G., Perroud,and Hubral, P., Migration around on hyperbolas and parabolas:The Leading Edge, MAY,1997, 437-476
[29] Hoecht, G., de Bazelaire,E., Majer, P., Hubral, P., Seismics and Optics:hyperbolae and curvatures: Journal of Applicied Geophysics, 1999,(42):261-281.
[30] Jager, R.,. The Common Reflection Surface Stack-Theory and Application. thesis, University of Karlsruhe, 1999
[31] Jager, R,et al,Common-Reflection-Surface stack: image and attributes. Geophysics, 2001,(66):97-109
[32] Birgin, E., et al, Restricted optimization: a clue to a fast and accurate implementation of the Common Reflection Surface Stack method: Jouranl of Applied Geophysices,1999, (42):143-155.
[33] Bergler, S., Duveneck, E., Hoecht, G., Zhang, Y., and Hubral, P., Common-Reflection-Surface stack for converted waves: Wave Inversion Technology, Report No.(5):24-31.
[34] Biloti, R., Santos, L., and Tygel, M., Macro-Velocity Model Inversion from CRS Attributes:The Hubral and Krey algorithm revisited: Wave InversionTechnology, Report No.(4):49-62.
[35] Hoecht, G., Traveltime approximations for 2D and 3D media and kinematic wavefield attributes., thesis, University of Karlsruhe.2002
[36] Bergler. S., The common-reflection-surface stack for common offset theory and application., thesis, University of Karlsruhe. 2001
[37] Koglin. ?.,Picking and Smoothing of Seismic Events and CRS Attributes , Application for Inversion, thesis, University of Karlsruhe.2001
[38] Vieth. K. U., Kinematic wavefield attributes in seismic imaging, thesis , University of Karlsruhe.2001
[39] 裴江云,地震波场延拓十菲涅尔体叠加方法及成像效果 博士学位论文,2002
[40] 裴江云,刘洪,李幼铭,朱振宇.反射元弧叠加方法在火山岩成像中的应用.地球物理学报,2004,47(l):106~111.
[41] 王华忠,杨楷,马在田,共反射面元叠加的应用理论——从共反射点到共反射面元.地球物理学报,2004,(470):137~42.
[42] 杨楷,王华忠,马在田,共反射面元叠加的应用实践.地球物理学报,2004,47(2):327~331.
[43] 杨楷,徐蔚亚,王华忠.基于模型数据的共反射面元叠加初步实践.同济大学学报,2003,31(3) :309~313
[44] 杨楷,王华忠,马在田.实现最佳零炮检距地震照明成像——CRS 叠加之几何阐述.勘探地球物理进展,2002,25(3):27~ 31.
[45] 罗银河,多聚焦(MF)成像技术综述.地球物理学进展,2003,18(4):635~ 642.
[46] Zhang Y, Bergler S, Hubral. Common-reflection-surface (CRS) stack for common offset[J]. Geophysical Prospecting, 2001,49(6):709-718.
[47] Zhang, Y., Bergler, S., Tygel, M., and Hubral, P., 2002, Model-independent Traveltime Attributes for 2D Finite-Offset Multicoverage Reflections: Pure and Applied Geophysics,159:1-16.
[48] Schleicher,J.,Tygel,M.,and Hubral,P. Parabolic and hyperbolic paraxialtwo-point traveltimes in 3D media. Geophys.Prosp.,1993,41(4):495~514.
[49] Mann,Jjager,R.,Muller,T,Hoecht,G.and Hubral,P. Common reflection surface stack- a real data example. Journal of Applied Geophysics 1999 , 42(3.4):301~318.
[50] Duveneck, E., and Hubual, P., 2002, Tomographic velocity model inversion using kinematic wavefield attributes: 727h Ann. Internat. Mtg., Soc. Expl.Geophys,Expanded Abstracts
[51] Hrbral P.Computing true amplitude reflections in a laterally inhomogeneous earth.Geophysics,1983,(48):1051~1062
[52] Bortfeld, R., Geometrical ray theory: rays and traveltimes in seismic systems (second-order approximations of the traveltime): Geophysics, 1989,(48):342-349.
[53] Cerveny, V., 2001,Seismic Ray Theory: Cambridge University Press.
[54] Beydoun, W. B., and Keho, T. H., The paraxial ray method : Geophysics , 1987,(52):1639-1653.
[55] Cerveny, V., Seismic Ray Theory: Cambridge University Press. 2001
[56] Perroud H,Hubral P,Hoecht G.Common-reflection-point stacking in laterally inhomogeneous media.Geophys.Prosp.,1999,(47):1~24
[57] Tygel M,Muller T,Hubral,P,et al.Eigenwave based multiparameter traveltime expansions:67th Annual Intermat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1997.1770~1773
[59] German Garabito,Joao C.Cruz, Common Reflection Surface Stack:A new parameter search strategy by global optimization.SEG Int’I Exposition and Annual Meeting,2001.
[60] 康立山,非数值并行算法(第一册)模拟退火算法,科学出版社,1994年 04 月第 1 版
[61] 孙振绮,丁效华,最优化方法,机械工业出版社,2004
[62] 谭未一.共反射面元叠加. 中国学位论文全文库 CDDB.2004
[2] Wave Inversion Technology, annual report 2001,page:4
[3] de Bazelaire,E. ,Normal moveout revisited-inhomogeneous media and curved interfaces:Geophysics,1988,(53):143-157
[4] de Bazelaire,E.,and Viallix,J.R. Normal moveout in focus. Geophys, Prosp.,1994,(42):477-499.
[5] Berkovitch,A.,Gelchinsky,B.,and Keydar,S.Basic formulae for multifocusing stack:56th Mtg.Eur.Assoc.Expl.Geophys.,Extended Abstracts.1994
[6] Berkovitch, A., Gelchinsky, B. Inversion of common reflection element (cre) data (migration combined with interval velocity datermination).59th Annual Internat,Mtg., Soc. Expl. Geophys., Expanded Abstracts,1989,140
[7] Thore, P. D., de Bazelaire, E., and Ray, M. P.Three-parameter equation:An efficient tool to enhance the stack.Geophysics,1994,(59):297-308
[8] Berkovitch,A.,Keydar,S.,landa, E., and trachtman, P.,Multifocusing in practice. 68th Annual Internat,Mtg., Soc. Expl. Geophys.,Expanded Abstracts,1998,pl748-1751
[9] Berkhout. A. J.,A real shot record technology.Journal seismic exploration,1992,(1):251-264
[10] Berkhout. A. J,Pushing the limits of seismic imaging, Part Ι: Prestack migration in terms of double dynamic focusing. Geophysics,1997, 62(3):937-953
[11] Berkhout. A. J., Pushing the limits of seismic imaging, Part П: Integration of prestack migration, velocity estimation, and AVO analysics., Geophysics, 1997,62(3):954-969
[12] Gelchinsky, B., The common-reflection-element (CRE) method:ASEG/SEG Intemat. Geophys. Conf, Expl. Geophys., Extended Abstracts,1998. 71-75.
[13] Gelchinsky, B., Homemorphic imaging in processing of seismic data (fundamentals and schemes): 59th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts,1989, 983-988.
[14] Gelchinsky, B., Berkovitch, A., and Keydar, S.. Multifocusing homeomorphic imaging. Part 1:Basic concepts and formulas. Journal of Applied Geophysics, 1999a,(42):229-242.
[15] Gelchinsky, B., Berkovitch, A., and Keydar, S., Multifocusing homeomorphic imaging. Part 2: Multifold data set and multifocusing. Journal of Applied Geophysics, 1999b,(42):243-260.
[16] Keydar, S,Gelchinsky, B.and Berkovitch, A., The combined homeomorphic stacking. 63th Annual. Internat. Mtg. Soc. Expl. Geophys.,Expanded Abstracts, 1993, p1141-1144
[17] Keydar, S., Gelchinsky, B., and Berkovitch, A.Common shot point stacking and imaging. Journal of Seismic Exploration, 1996,(5):261-274
[18] Keydar, S., Gelchinsky, B., Shtivelman, V., and Berkovitch, A.,Common evolute element (cee) stack and imaging (zaeo offset stack).60th Annual. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts,1990,p1719-1722
[19] Hubral,R.and Krey, T., Interval velocities from seismic reflection traveltime measurements: SEG.,1980
[20] Hubral,R.and Krey, T., Interval velocities from seismic reflection traveltime measurements: SEG..1980
[21] Hubral, P., Schleicher, J., and Tygel, M., Three-dimensional paraxial ray properties, PART I: Basic relations: Journal of Seismic Exploration, 1992a, 265-279.
[22] Hubral,P.,Schleicher,J.,and Tygel, M., Three-dimensional paraxial ray properties,PART II: Applications: Journal of Seismic Exploration, 1992b,(1):347-362.
[23] Hubral,R,Schleicher, J.,and Tygel, M., A unified approach to 3D seismic reflection imaging, PART I: Basic Concepts:Geophysics, 1996,(61):742-758
[24] Hubral, P., Tygel M., and Schleicher, J., Seismic image waves:Geophys. Joum.Intern., 1996, (125):431-442.
[25] Hubral,P.,et al, The Common Reflection Surface (CRS) stack----structural resolution in Time Domain beyond the conventional NMO/DMO stack.2001
[26] Hubral,P,Hoecht,G., Jaeger, R., Seismic illumination, The Leading Edge, November,1999,1268-1271
[27] Hubral,P. and Mann,J, The overview of Common Reflection Surface (CRS) stack, The proceedings CNPC technology communication conference.2002
[28] Hoecht, G., Perroud,and Hubral, P., Migration around on hyperbolas and parabolas:The Leading Edge, MAY,1997, 437-476
[29] Hoecht, G., de Bazelaire,E., Majer, P., Hubral, P., Seismics and Optics:hyperbolae and curvatures: Journal of Applicied Geophysics, 1999,(42):261-281.
[30] Jager, R.,. The Common Reflection Surface Stack-Theory and Application. thesis, University of Karlsruhe, 1999
[31] Jager, R,et al,Common-Reflection-Surface stack: image and attributes. Geophysics, 2001,(66):97-109
[32] Birgin, E., et al, Restricted optimization: a clue to a fast and accurate implementation of the Common Reflection Surface Stack method: Jouranl of Applied Geophysices,1999, (42):143-155.
[33] Bergler, S., Duveneck, E., Hoecht, G., Zhang, Y., and Hubral, P., Common-Reflection-Surface stack for converted waves: Wave Inversion Technology, Report No.(5):24-31.
[34] Biloti, R., Santos, L., and Tygel, M., Macro-Velocity Model Inversion from CRS Attributes:The Hubral and Krey algorithm revisited: Wave InversionTechnology, Report No.(4):49-62.
[35] Hoecht, G., Traveltime approximations for 2D and 3D media and kinematic wavefield attributes., thesis, University of Karlsruhe.2002
[36] Bergler. S., The common-reflection-surface stack for common offset theory and application., thesis, University of Karlsruhe. 2001
[37] Koglin. ?.,Picking and Smoothing of Seismic Events and CRS Attributes , Application for Inversion, thesis, University of Karlsruhe.2001
[38] Vieth. K. U., Kinematic wavefield attributes in seismic imaging, thesis , University of Karlsruhe.2001
[39] 裴江云,地震波场延拓十菲涅尔体叠加方法及成像效果 博士学位论文,2002
[40] 裴江云,刘洪,李幼铭,朱振宇.反射元弧叠加方法在火山岩成像中的应用.地球物理学报,2004,47(l):106~111.
[41] 王华忠,杨楷,马在田,共反射面元叠加的应用理论——从共反射点到共反射面元.地球物理学报,2004,(470):137~42.
[42] 杨楷,王华忠,马在田,共反射面元叠加的应用实践.地球物理学报,2004,47(2):327~331.
[43] 杨楷,徐蔚亚,王华忠.基于模型数据的共反射面元叠加初步实践.同济大学学报,2003,31(3) :309~313
[44] 杨楷,王华忠,马在田.实现最佳零炮检距地震照明成像——CRS 叠加之几何阐述.勘探地球物理进展,2002,25(3):27~ 31.
[45] 罗银河,多聚焦(MF)成像技术综述.地球物理学进展,2003,18(4):635~ 642.
[46] Zhang Y, Bergler S, Hubral. Common-reflection-surface (CRS) stack for common offset[J]. Geophysical Prospecting, 2001,49(6):709-718.
[47] Zhang, Y., Bergler, S., Tygel, M., and Hubral, P., 2002, Model-independent Traveltime Attributes for 2D Finite-Offset Multicoverage Reflections: Pure and Applied Geophysics,159:1-16.
[48] Schleicher,J.,Tygel,M.,and Hubral,P. Parabolic and hyperbolic paraxialtwo-point traveltimes in 3D media. Geophys.Prosp.,1993,41(4):495~514.
[49] Mann,Jjager,R.,Muller,T,Hoecht,G.and Hubral,P. Common reflection surface stack- a real data example. Journal of Applied Geophysics 1999 , 42(3.4):301~318.
[50] Duveneck, E., and Hubual, P., 2002, Tomographic velocity model inversion using kinematic wavefield attributes: 727h Ann. Internat. Mtg., Soc. Expl.Geophys,Expanded Abstracts
[51] Hrbral P.Computing true amplitude reflections in a laterally inhomogeneous earth.Geophysics,1983,(48):1051~1062
[52] Bortfeld, R., Geometrical ray theory: rays and traveltimes in seismic systems (second-order approximations of the traveltime): Geophysics, 1989,(48):342-349.
[53] Cerveny, V., 2001,Seismic Ray Theory: Cambridge University Press.
[54] Beydoun, W. B., and Keho, T. H., The paraxial ray method : Geophysics , 1987,(52):1639-1653.
[55] Cerveny, V., Seismic Ray Theory: Cambridge University Press. 2001
[56] Perroud H,Hubral P,Hoecht G.Common-reflection-point stacking in laterally inhomogeneous media.Geophys.Prosp.,1999,(47):1~24
[57] Tygel M,Muller T,Hubral,P,et al.Eigenwave based multiparameter traveltime expansions:67th Annual Intermat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1997.1770~1773
[59] German Garabito,Joao C.Cruz, Common Reflection Surface Stack:A new parameter search strategy by global optimization.SEG Int’I Exposition and Annual Meeting,2001.
[60] 康立山,非数值并行算法(第一册)模拟退火算法,科学出版社,1994年 04 月第 1 版
[61] 孙振绮,丁效华,最优化方法,机械工业出版社,2004
[62] 谭未一.共反射面元叠加. 中国学位论文全文库 CDDB.2004