地震波的场方程矩阵和能量的正定二次型及其意义
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摘要
场方程是能够表述半空间地震波场的整体特征.波动方程、速度方程和能量方程.通过分析可知每个场方程都具有各自的“场方程矩阵”.能量方程能够对所有场方程矩阵进行综合和贯通,给出了能量方程以“弹性矩阵”为核心的普适性表达形式.最后,运用矩阵的正定二次型理论阐述了“能量矩阵与弹性矩阵”之间一致的对称性和正定性.能量矩阵蕴含的动态力的平衡关系、速度的时间_空间分布和能量的传播及变化的物理意义,能够从能量矩阵的正定二次型特性表述出来.本文研究分析问题的方法完全适用于复杂介质模型,相关的认识和结论可以拓展到均匀黏弹性各向同性介质、均匀弹性各向异性介质、均匀黏弹性各向异性介质以及比奥饱和流体介质.
"Field equations"-wave equations,velocity equations and energy equations-can represent the overall characteristic of the half space seismic wave field.This paper firstly pointed out that there is a field equation matrix for each of the equations.Secondly we analyzed and revealed that energy equations can be integrated into all field equation matrixes,and presented the generally adaptive expression of the energy equations based on the elastic matrix.At last the consistently symmetrical and positive definite properties between the energy matrices and the elastic matrices are clarified by using the theory of positive definite quadratic form of matrices.The physical meaning of the dynamical equilibrium relationships,velocity space-time distribution and energy propagation involved in the energy matrices can be described by the positive definite quadratic form of the energy matrices.The analytical methods and conclusions used and presented in this paper can be applied to such complex models as porous,viscoelastic and anisotropic medium.
"Field equations"-wave equations,velocity equations and energy equations-can represent the overall characteristic of the half space seismic wave field.This paper firstly pointed out that there is a field equation matrix for each of the equations.Secondly we analyzed and revealed that energy equations can be integrated into all field equation matrixes,and presented the generally adaptive expression of the energy equations based on the elastic matrix.At last the consistently symmetrical and positive definite properties between the energy matrices and the elastic matrices are clarified by using the theory of positive definite quadratic form of matrices.The physical meaning of the dynamical equilibrium relationships,velocity space-time distribution and energy propagation involved in the energy matrices can be described by the positive definite quadratic form of the energy matrices.The analytical methods and conclusions used and presented in this paper can be applied to such complex models as porous,viscoelastic and anisotropic medium.
引文
[1]杨桂通.弹性力学(第4版)[M].北京:高等教育出版社,2003.Yang G T.Elastic Mechanics(4th Edition)[M].Beijing:Higher Education Press,2003.
[2]Auld B A.Acoustic field and waves in solid.Vol.1[M].HohnWiley and Sons Inc.1973.
[3]牛滨华,孙春岩.半空间均匀各向同性单相固体弹性介质与地震波传播[M].北京:地质出版社,2005.Niu B H,Sun C Y.Half-space Homegeneous Isotropic Sin-gle-phase Solid Elastic Medium and Seismic Wave Propagation.Beijing:Geological Publishing House,2005.
[4]牛滨华,孙春岩,李明.各向同性固体连续介质与地震波传播.北京?:石油工业出版社,2002.Niu B H,Sun C Y,Li M.Isotropic Solid Continuum andSeismic Wave Propagation.Beijing:Petroleum IndustryPress,2002.
[5]牛滨华,孙春岩.半空间介质与地震波传播[M].北京:石油工业出版社,2002.Niu B H,Sun C Y.Half-space Medium and Seismic WavePropagation[M].Beijing:Petroleum Industry Press,2002.
[6][美]J.L.罗斯著.固体中的超声波,何存富,吴斌,王秀彦等译[M].北京:科学出版社,2004.J.L.Rose.Ultrasonic in Solid,He C F,Wu B,Wang X Y,et.al.translated[M].Beijing:Science Press,2004.
[7]黄筑平.连续介质力学基础[M].北京:高等教育出版社,2003.Huang Z P.Foundations of Continuum Mechenics[M].Bei-jing:Higher Education Press,2003.
[8]蒋尔雄,高坤敏,吴景琨.线性代数[M].北京:人民教育出版社,1979.Jiang E X,Gao K M,Wu J K.Linear Algebra[M].Beijing:People’s Education Press,1979.
[9]陈维新.线性代数简明教程(第2版)[M].北京:科学出版社,2005.Chen W X.A Brief Tutorial of Linear Algebra(2nd Edition)[M].Beijing:Science Press,2005.
[10]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.Zhang X D.The Analysis and Application of Matrix[M].Beijing:Tsinghua University Press,2004.
[11]欧维义.场的数学描写方法[M].长春:吉林人民出版社,1982.Ou W Y.Descriptive Mathematic Methods of Fields[M].Changchun:Ji Lin People’s Publishing House,1982.
[12]余天庆.张量分析及演算[M].武汉:华中理工大学出版社,1996.Yu T Q.Tensor Analysis and Calculation[M].Wuhan:HuaZhong University of Technology Press,1996.
[13]George E.Backus.Long-Wave elastic Anisotropy Producedby Horizontal Layering[J].Journal of Geophysical Rerserce.67,(11),4427~4440.
[14]Fdorov F I.Theory of elastic waves in crystal[M].NewYork:Plenum Press.1968.
[15]Payton R G.Elastic wave propagation in transversely iso-tropic media[M].Mertinus Nijhoff publishers P4~5,1983.
[16]马在田.地震成像技术[M].北京:石油工业出版社,1989.Ma Z T.Seismic Imaging Techniques[M].Beijing:Petrole-um Industry Press,1989.
[17]何樵登.地震波理论[M].北京:地质出版社,1988.He Q D.Theory of Seismic Wave[M].Beijing:GeologicalPublishing House,1988.
[18]杨宝俊,牛滨华,阎贫.勘探地震学导论(下卷)[M].长春:吉林科技出版社,1992.Yang B J,Niu B H,Yan P.Introduction of Exploration Seis-mology,Vol.1[M].Changchun:Ji Lin Science and Tech-nology Press,1992.
[19]黄新武,孙春岩,牛滨华,王焕弟,曾闽山.基于数据一致性预测与压制自由表面多次波——理论研究与试处理[J].地球物理学报,2005,48(1):173~180.Huang X W,Sun C Y,Niu B H,Wang H D,Zeng M S.Surface-related multiple prediction and suppression based ondata-consistence:A theoretical study and test[J].ChineseJ.Geophys.(in Chinese),2005,48(1):173~180.
[20]王德利,何樵登,韩立国.裂隙型单斜介质中多方位地面三分量记录模拟[J].地球物理学报,2005,48(2):386~393.Wang D L,He Q D,Han L G.Multi-azimuth three-compo-nent surface seismic modeling for cracked monoclinic media[J].Chinese J.Geophys.(in Chinese),2005,48(2):386~393.
[21]秦义龙,张中杰,Shin Changsoo,Seungwon Ko,陈赟,陈竞一,Uk Han.利用单频双程波动方程计算初至走时及其振幅[J].地球物理学报,2005,48(2):423~428.Qin Y Long,Zhang Z J,Shin Changsoo,Seungwon Ko,Chen Y,Chen J Y,Uk Han.First-arrival traveltime and am-plitude calculation from monochromatic two-way wave equa-tion in frequency domain[J].Chinese J.Geophys.(in Chi-nese),2005,48(2):423~428.
[22]郑海山,张中杰.横向各向同性(VTI)介质中非线性地震波场模拟[J].地球物理学报,2005,48(3):660~671.Zheng H S,ZHANG Zhong-Jie.Synthetic seismograms ofnonlinear seismic waves in anisotropic(VTI)media[J].Chi-nese J.Geophys.(in Chinese),2005,48(3):660~671.
[23]陈祖安,白武明,徐文跃.多组分天然气水合物在海底沉积层中稳定区及存在区的预测[J].地球物理学报,2005,48(4):870~875.Chen Z A,Bai W M,Xu W Y.Predication of stability zonesand occurrence zones gas hydrate of multiple-composition nat-ural in seafloor sediments[J].Chinese J.Geophys.(in Chi-nese),2005,48(4):870~875.
[24]刘恩儒,岳建华,刘彦.具有离散裂缝空间分布的二维固体中地震波传播的有限差分模拟[J].地球物理学报,2006,49(1):180~188.Liu E R,Yue J H,Liu Y.Finite difference simulation ofseismic wave propagation in 2-D solids with spatial distribu-tion of discrete fractures[J].Chinese J.Geophys.(in Chi-nese),2006,49(1):180~188.
[25]王德利,何樵登,韩立国.单斜介质中方位NMO速度Thomsen参数反演方法研究[J].地球物理学报,2006,49(1):249~255.Wang D L,He Q D,Han L G.Thomsen parameter inversionwith azimuthal NMO velocity in monoclinic media[J].Chi-nese J.Geophys.(in Chinese),2006,49(1):249~255.
[26]杨文采,杨午阳,程振炎.中国大陆科学钻探孔区的地震波速模型[J].地球物理学报,2006,49(2):477~489.Yang W C,Yang W Y,Cheng Z Y.Seismic velocity model ofthe Chinese continental scientific drilling site[J].Chinese J.Geophys.(in Chinese),2006,49(2):477~489.
[27]王秀娟,吴时国,刘学伟.天然气水合物和游离气饱和度估算的影响因素[J].地球物理学报,2006,49(2):504~511.Wang X J,Wu S G,Liu X W.Factors affecting the estima-tion of gas hydrate and free gas saturation[J].Chinese J.Geophys.(in Chinese),2006,49(2):504~511.
[28]王东,张海澜,王秀明.部分饱和孔隙岩石中声波传播数值研究[J].地球物理学报,2006,49(2):524~532.Wang D,Zhang H L,Wang X M.A numerical study of a-coustic wave propagation in partially saturated poroelasticrock[J].Chinese J.Geophys.(in Chinese),2006,49(2):524~532.
[29]郑海山,张中杰,田小波.VTI介质中非线性对地震波频率和频宽变化的约束[J].地球物理学报,2006,49(3):885~894.Zheng H S,Zhang Z J,Tian X B.Constraints of the nonlin-earity in VTI media on seismic wave frequency and bandwidthchanges[J].Chinese J.Geophys.(in Chinese),2006,49(3):885~894.
[30]王大兴,辛可锋,李幼铭,高静怀,伍向阳.地层条件下砂岩含水饱和度对波速及衰减影响的实验研究[J].地球物理学报,2006,49(3):908~914.Wang D X,Xin K F,Li Y M,Gao J H,Wu X Y.An experi-mental study of influence of water saturation on velocity andattenuation in sandstone under stratum conditions[J].Chi-nese J.Geophys.(in Chinese),2006,49(3):908~914.
[31]刘恩儒,岳建华,潘冬明.地震各向异性——多组裂隙对横波偏振的影响[J].地球物理学报,2006,49(5):1401~1409.Liu E R,Yue J H,Pan D M.Frequency-dependent anisotro-py:Effects of multiple fracture sets on shear-wave polariza-tions[J].Chinese J.Geophys.(in Chinese),2006,49(5):1401~1409.
[2]Auld B A.Acoustic field and waves in solid.Vol.1[M].HohnWiley and Sons Inc.1973.
[3]牛滨华,孙春岩.半空间均匀各向同性单相固体弹性介质与地震波传播[M].北京:地质出版社,2005.Niu B H,Sun C Y.Half-space Homegeneous Isotropic Sin-gle-phase Solid Elastic Medium and Seismic Wave Propagation.Beijing:Geological Publishing House,2005.
[4]牛滨华,孙春岩,李明.各向同性固体连续介质与地震波传播.北京?:石油工业出版社,2002.Niu B H,Sun C Y,Li M.Isotropic Solid Continuum andSeismic Wave Propagation.Beijing:Petroleum IndustryPress,2002.
[5]牛滨华,孙春岩.半空间介质与地震波传播[M].北京:石油工业出版社,2002.Niu B H,Sun C Y.Half-space Medium and Seismic WavePropagation[M].Beijing:Petroleum Industry Press,2002.
[6][美]J.L.罗斯著.固体中的超声波,何存富,吴斌,王秀彦等译[M].北京:科学出版社,2004.J.L.Rose.Ultrasonic in Solid,He C F,Wu B,Wang X Y,et.al.translated[M].Beijing:Science Press,2004.
[7]黄筑平.连续介质力学基础[M].北京:高等教育出版社,2003.Huang Z P.Foundations of Continuum Mechenics[M].Bei-jing:Higher Education Press,2003.
[8]蒋尔雄,高坤敏,吴景琨.线性代数[M].北京:人民教育出版社,1979.Jiang E X,Gao K M,Wu J K.Linear Algebra[M].Beijing:People’s Education Press,1979.
[9]陈维新.线性代数简明教程(第2版)[M].北京:科学出版社,2005.Chen W X.A Brief Tutorial of Linear Algebra(2nd Edition)[M].Beijing:Science Press,2005.
[10]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.Zhang X D.The Analysis and Application of Matrix[M].Beijing:Tsinghua University Press,2004.
[11]欧维义.场的数学描写方法[M].长春:吉林人民出版社,1982.Ou W Y.Descriptive Mathematic Methods of Fields[M].Changchun:Ji Lin People’s Publishing House,1982.
[12]余天庆.张量分析及演算[M].武汉:华中理工大学出版社,1996.Yu T Q.Tensor Analysis and Calculation[M].Wuhan:HuaZhong University of Technology Press,1996.
[13]George E.Backus.Long-Wave elastic Anisotropy Producedby Horizontal Layering[J].Journal of Geophysical Rerserce.67,(11),4427~4440.
[14]Fdorov F I.Theory of elastic waves in crystal[M].NewYork:Plenum Press.1968.
[15]Payton R G.Elastic wave propagation in transversely iso-tropic media[M].Mertinus Nijhoff publishers P4~5,1983.
[16]马在田.地震成像技术[M].北京:石油工业出版社,1989.Ma Z T.Seismic Imaging Techniques[M].Beijing:Petrole-um Industry Press,1989.
[17]何樵登.地震波理论[M].北京:地质出版社,1988.He Q D.Theory of Seismic Wave[M].Beijing:GeologicalPublishing House,1988.
[18]杨宝俊,牛滨华,阎贫.勘探地震学导论(下卷)[M].长春:吉林科技出版社,1992.Yang B J,Niu B H,Yan P.Introduction of Exploration Seis-mology,Vol.1[M].Changchun:Ji Lin Science and Tech-nology Press,1992.
[19]黄新武,孙春岩,牛滨华,王焕弟,曾闽山.基于数据一致性预测与压制自由表面多次波——理论研究与试处理[J].地球物理学报,2005,48(1):173~180.Huang X W,Sun C Y,Niu B H,Wang H D,Zeng M S.Surface-related multiple prediction and suppression based ondata-consistence:A theoretical study and test[J].ChineseJ.Geophys.(in Chinese),2005,48(1):173~180.
[20]王德利,何樵登,韩立国.裂隙型单斜介质中多方位地面三分量记录模拟[J].地球物理学报,2005,48(2):386~393.Wang D L,He Q D,Han L G.Multi-azimuth three-compo-nent surface seismic modeling for cracked monoclinic media[J].Chinese J.Geophys.(in Chinese),2005,48(2):386~393.
[21]秦义龙,张中杰,Shin Changsoo,Seungwon Ko,陈赟,陈竞一,Uk Han.利用单频双程波动方程计算初至走时及其振幅[J].地球物理学报,2005,48(2):423~428.Qin Y Long,Zhang Z J,Shin Changsoo,Seungwon Ko,Chen Y,Chen J Y,Uk Han.First-arrival traveltime and am-plitude calculation from monochromatic two-way wave equa-tion in frequency domain[J].Chinese J.Geophys.(in Chi-nese),2005,48(2):423~428.
[22]郑海山,张中杰.横向各向同性(VTI)介质中非线性地震波场模拟[J].地球物理学报,2005,48(3):660~671.Zheng H S,ZHANG Zhong-Jie.Synthetic seismograms ofnonlinear seismic waves in anisotropic(VTI)media[J].Chi-nese J.Geophys.(in Chinese),2005,48(3):660~671.
[23]陈祖安,白武明,徐文跃.多组分天然气水合物在海底沉积层中稳定区及存在区的预测[J].地球物理学报,2005,48(4):870~875.Chen Z A,Bai W M,Xu W Y.Predication of stability zonesand occurrence zones gas hydrate of multiple-composition nat-ural in seafloor sediments[J].Chinese J.Geophys.(in Chi-nese),2005,48(4):870~875.
[24]刘恩儒,岳建华,刘彦.具有离散裂缝空间分布的二维固体中地震波传播的有限差分模拟[J].地球物理学报,2006,49(1):180~188.Liu E R,Yue J H,Liu Y.Finite difference simulation ofseismic wave propagation in 2-D solids with spatial distribu-tion of discrete fractures[J].Chinese J.Geophys.(in Chi-nese),2006,49(1):180~188.
[25]王德利,何樵登,韩立国.单斜介质中方位NMO速度Thomsen参数反演方法研究[J].地球物理学报,2006,49(1):249~255.Wang D L,He Q D,Han L G.Thomsen parameter inversionwith azimuthal NMO velocity in monoclinic media[J].Chi-nese J.Geophys.(in Chinese),2006,49(1):249~255.
[26]杨文采,杨午阳,程振炎.中国大陆科学钻探孔区的地震波速模型[J].地球物理学报,2006,49(2):477~489.Yang W C,Yang W Y,Cheng Z Y.Seismic velocity model ofthe Chinese continental scientific drilling site[J].Chinese J.Geophys.(in Chinese),2006,49(2):477~489.
[27]王秀娟,吴时国,刘学伟.天然气水合物和游离气饱和度估算的影响因素[J].地球物理学报,2006,49(2):504~511.Wang X J,Wu S G,Liu X W.Factors affecting the estima-tion of gas hydrate and free gas saturation[J].Chinese J.Geophys.(in Chinese),2006,49(2):504~511.
[28]王东,张海澜,王秀明.部分饱和孔隙岩石中声波传播数值研究[J].地球物理学报,2006,49(2):524~532.Wang D,Zhang H L,Wang X M.A numerical study of a-coustic wave propagation in partially saturated poroelasticrock[J].Chinese J.Geophys.(in Chinese),2006,49(2):524~532.
[29]郑海山,张中杰,田小波.VTI介质中非线性对地震波频率和频宽变化的约束[J].地球物理学报,2006,49(3):885~894.Zheng H S,Zhang Z J,Tian X B.Constraints of the nonlin-earity in VTI media on seismic wave frequency and bandwidthchanges[J].Chinese J.Geophys.(in Chinese),2006,49(3):885~894.
[30]王大兴,辛可锋,李幼铭,高静怀,伍向阳.地层条件下砂岩含水饱和度对波速及衰减影响的实验研究[J].地球物理学报,2006,49(3):908~914.Wang D X,Xin K F,Li Y M,Gao J H,Wu X Y.An experi-mental study of influence of water saturation on velocity andattenuation in sandstone under stratum conditions[J].Chi-nese J.Geophys.(in Chinese),2006,49(3):908~914.
[31]刘恩儒,岳建华,潘冬明.地震各向异性——多组裂隙对横波偏振的影响[J].地球物理学报,2006,49(5):1401~1409.Liu E R,Yue J H,Pan D M.Frequency-dependent anisotro-py:Effects of multiple fracture sets on shear-wave polariza-tions[J].Chinese J.Geophys.(in Chinese),2006,49(5):1401~1409.
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