黏弹性介质中瑞雷波有限差分数值模拟与波场分析
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摘要
瑞雷面波经常被用来反演地表浅层横波速度,受到越来越广泛的关注。对瑞雷波的研究一般都基于完全弹性介质,而实际地层更接近黏弹性介质,对黏弹性介质中的瑞雷面波进行模拟更具实际意义。本文采用广义标准线性体模型来描述黏弹性介质,并采用交错网格有限差分法对考虑水平自由表面的黏弹介质进行正演模拟,再与弹性介质中的结果进行对比分析。首先采用非线性最优化算法根据期望常数品质因子直接求取松弛时间来拟合常Q模型,并给出广义标准线性固体的具体算例,实施自由表面条件时采用声学-弹性边界近似法,通过剪切模量不变来考虑自由表面上、下横向应力保持连续的条件。对于非自由表面,采用非分裂的多轴卷积完全匹配层来吸收波场。然后对几种典型的数值模型进行正演模拟计算,数值解与解析解的对比验证了本文方法的准确性与有效性,正演结果的对比表明波场尤其是面波频散会受黏弹性影响,因此有必要在面波勘探中考虑黏弹性因素。
Rayleigh surface wave has been used to research the character of shallow subsurface widely, and the real geological medium is close to viscoelastic media. This paper models Rayleigh wave in viscoelastic media with planar free surface by using the staggered-grid finite-difference method based on the generalized standard linear body. And we compared the results of the viscoelastic media with the elastic one. In this paper the Levenberg-Marquarat algorithm is adopted to compute the relaxation time to fitting the constant Q model firstly.Acoustic-elastic boundary approximation method is used to implement free surface conditions. The condition that transverse stress on and under the free surface remain continuous is considered by keeping the shear modulus unchanged. And for other boundaries the unsplit multiaxial convolutional perfectly matched layer is chosen to absorb waves. Then the wave fields are calculated in several typical models. The comparisons of numerical and analytical solutions confirm the veracity and validity of the method in this paper. And the computed results indicate that viscoelasticity can influence the surface wave dispersion. So the viscoelastic factors should be considered in the surface wave exploration.
Rayleigh surface wave has been used to research the character of shallow subsurface widely, and the real geological medium is close to viscoelastic media. This paper models Rayleigh wave in viscoelastic media with planar free surface by using the staggered-grid finite-difference method based on the generalized standard linear body. And we compared the results of the viscoelastic media with the elastic one. In this paper the Levenberg-Marquarat algorithm is adopted to compute the relaxation time to fitting the constant Q model firstly.Acoustic-elastic boundary approximation method is used to implement free surface conditions. The condition that transverse stress on and under the free surface remain continuous is considered by keeping the shear modulus unchanged. And for other boundaries the unsplit multiaxial convolutional perfectly matched layer is chosen to absorb waves. Then the wave fields are calculated in several typical models. The comparisons of numerical and analytical solutions confirm the veracity and validity of the method in this paper. And the computed results indicate that viscoelasticity can influence the surface wave dispersion. So the viscoelastic factors should be considered in the surface wave exploration.
引文
[1]PARK C B, MILLER R D, XIA J. Imaging dispersion curves of surface waves on multi-channel record[C]//68th Annual International Meeting, Expanded Abstracts, SEG, 1998:1377-1380.
[2]LUOY,XIAJ,MILLERRD,etal.Rayleigh-wavedispersiveenergyimagingusinga high-resolution linear radon transform[J]. Pure and Applied Geophysics, 2008, 165:903-922.
[3]PARK C B, MILLER R D, XIA J. Multi-channel analysis of surface waves(MASW)[J]. Geophysics,1999, 64:800-808.
[4]MITTET R. Free-surface boundary conditions for elastic staggered-grid modeling schemes[J].Geophysics, 2002, 67(5):1616-1623.
[5]KRISTEK J, MOCZO P, ARCHULETA R J. Efficient methods to simulate planar free surface in the3D 4th-order staggered-grid finite-difference schemes[J]. Study Geophysics, 2002, 46:355-381.
[6]ROBERTSSON J O A. A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography[J]. Geophysics, 1996, 61(6):1921-1934.
[7]XU Y, XIA J, MILLER R D. Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach[J]. Geophysics, 2007, 72(5):147-153.
[8]张伟.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京:北京大学, 2006.ZHANG W. Finite difference seismic wave modelling in 3D heterogeneous media with surface topographyanditsimplementationinstronggroundmotionstudy[D].Beijing:Peking University, 2006.(in Chinese).
[9]凡友华,陈晓非,刘雪峰,等. Rayleigh波的频散方程高频近似分解和多模式激发数目[J].地球物理学报, 2007, 50(1):233-239.FAN Y H, CHEN X F, LIU X F, et al. Approximate decomposition of the dispersion equation at highfrequenciesandthenumberofmultimodesforRayleighwaves[J].ChineseJournal Geophysics, 2007, 50(1):233-239.(in Chinese).
[10]周竹生,刘喜亮,熊孝雨.弹性介质中瑞雷面波有限差分法正演模拟[J].地球物理学报, 2007,50(2):567-573.ZHOU Z S, LIU X L, XIONG X Y. Finite-difference modelling of Rayleigh surface wave in elastic media[J]. Chinese Journal Geophysics, 2007, 50(2):567-573.(in Chinese).
[11]张大洲,熊章强,顾汉明.高精度瑞雷波有限差分数值模拟及波场分析[J].地球物理学进展, 2009,24(4):1313-1319.ZHANG D Z, XIONG Z Q, GU H M. Numerical modeling of Rayleigh wave using high-accuracy finite-difference method and wave field analysis[J]. Progress in Geophysics, 2009, 24(4):1313-1319.(in Chinese).
[12]裴正林.任意起伏地表弹性波方程交错网格高阶有限差分法数值模拟[J].石油地球物理勘探, 2004,39(6):629-634.PEI Z L. Staggered-grid high-order finite-difference numerical simulation of elastic wave equation with arbitrary irregular surface[J]. Oil Geophysical Prospecting, 2004, 39(6):629-634.(in Chinese).
[13]秦臻,张才,郑晓东,等.高精度有限差分瑞雷面波模拟及频散特征提取[J].石油地球物理勘探,2010, 45(1):40-46.QIN Z, ZHANG C, ZHENG X D, et al. High-accuracy finite-difference modeling of Rayleigh surface wave and extracting of dispersion characteristic[J]. Oil Geophysical Prospecting, 2010,45(1):40-46.(in Chinese).
[14]潘冬明,胡明顺,崔若飞,等.基于拉东变换的瑞雷面波频散分析与应用[J].地球物理学报, 2010,53(11):2760-2766.PAN D M, HU M S, CUI R F, et al. Dispersion analysis of Rayleigh surface waves and application based on Radon transform[J]. Chinese Journal Geophysics, 2010, 53(11):2760-2766.(in Chinese).
[15]高静怀,何洋洋,马逸尘.黏弹性与弹性介质中Rayleigh面波特性对比研究[J].地球物理学报,2012, 55(1):207-218.GAO J H, HE Y Y, MA Y C. Comparision of the Rayleigh wave in elastic and viscoelastic media[J].Chinese Journal Geophysics, 2012, 55(1):207-218.(in Chinese).
[16]张维,何正勤,胡刚,等.用面波联合勘探技术探测浅部速度结构[J].地球物理学进展, 2013,28(4):2199-2206.ZHANG W, HE Z Q, HU G, et al. Detection of the shallow velocity structure with surface wave prospection method[J]. Progress in Geophysics, 2013, 28(4):2199-2206.(in Chinese).
[17]张立,刘争平.水平层状介质中基阶瑞利面波椭圆极化特征数值分析与研究[J].地球物理学报,2013, 56(5):1686-1695.ZHANG L, LIU Z P. A study of the elliptic polarization characteristics of fundamental mode Rayleigh wave based on numerical simulation[J]. Chinese Journal Geophysics, 2013, 56(5):1686-1695.(in Chinese).
[18]赵海波,陈百军,李奎周,等.黏弹性介质VSP记录模拟及在估算Q值研究中应用[J].地球物理学报, 2011, 54(2):329-335.ZHAO H B, CHEN B J, LI C Z, et al. VSP record numerical modeling in viscoelastic media and its application in the study of Q-value estimation method[J]. Chinese Journal Geophysics,2011, 54(2):329-335.(in Chinese).
[19]严红勇,刘洋.黏弹TTI介质中旋转交错网格高阶有限差分数值模拟[J].地球物理学报, 2012,55(4):1354-1365.YAN H Y, LIU Y. Rotated staggered grid high-order finite-difference numerical modeling for wave propagation in viscoelastic TTI media[J]. Chinese Journal Geophysics, 2012, 55(4):1354-1365.(in Chinese).
[20]杨仁虎,常旭,刘伊克.基于非均匀各向同性介质的黏弹性波正演数值模拟[J].地球物理学报,2009, 52(9):2321-2327.YANG R H, CHANG X, LIU Y K. Viscoelastic wave modeling in heterogeneous and isotropic media[J].Chinese Journal Geophysics, 2009, 52(9):2321-2327.(in Chinese).
[21]BLANCH J O, ROBERTSSON J O A, SYMES W W. Modeling of a constant Q:Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique[J]. Geophysics, 1995,60(1):176-184.
[22]TAL-EZER H, CARCIONE J M, KOSLOFF D. An accurate and efficient scheme for wave propagation in linear viscoelastic media[J]. Geophysics, 1990, 55(10):1366-1379.
[23]孙成禹,印兴耀.三参数常Q粘弹性模型构造方法研究[J].地震学报, 2007, 29(4):348-357.SUN C Y, YIN X Y. Research on construction method of constant Q viscoelastic model with three parameters[J]. Acta Seismologica Sinica, 2007, 29(4):348-357.(in Chinese).
[24]CAODP,YINXY.QreflectionsmodelingwithgeneralizedMaxwellmodelintime domain[C]//81th Annual International Meeting, Expanded Abstracts, SEG, 2011:2819-2823.
[25]DAY S M, MINSTER J B. Numerical simulation of attenuated wavefields using a Padéapproximant method[J]. Geophysical Journal of the Royal Astronomical Society, 1984, 78:105-118.
[26]EMMERICH H, KORN M. Incorporation of attenuation into time-domain computations of seismic wave fields[J]. Geophysics, 1987, 52:1252-1264.
[27]CARCIONE J M, KOSLOFF D, KOSLOFF R. Wave propagation simulation in a linear viscoelastic medium[J]. Geophysical Journal International, 1988, 95:597-611.
[28]MOCZO P, KRISTEK J. On the rheological models used for time-domain methods of seismic wave propagation[J]. Geophysical Research Letters, 2005, 32:L01306.
[29]TIAN K, HUANG J P, LI Z C, et al. Simulation of the planar free surface in the finite-difference modelingofhalf-spaceviscoelasticmedium[C]//75thEAGEConference&Exhibition Incorporating SPE EUROPEC, 2013:Tu-P13-03.
[30]LIU H D, ANDERSON L, KANAMORI H. Velocity dispersion due to anelasticity; Implications for seismology and mantle composition[J]. Geophysical Journal International, 1976, 47:41-58.
[31]张光澄.非线性最优化计算方法[M].北京:高等教育出版社, 2005.ZHANG G C. Computational methods for nonlinear optimization[M]. Beijing:Higher Education Press, 2005.(in Chinese).
[32]MARTIN R, KOMATITSCH D. An unsplit convolutional perfectly matched layer technique improved atgrazingincidencefortheviscoelasticwaveequation[J].GeophysicalJournal International, 2009, 179:333-344.
[33]MEZA-FAJARDO K C, PAPAGEORGIOU A S. A non-convolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media:Stability analysis[J].Bulletin of the Seismological Society of America. 2008, 98(4):1811–1836.
[34]李振春,田坤,黄建平,等.多轴完全匹配层的非分裂实现[J].地球物理学进展, 2013, 28(6):2984-2992.LI Z C, TIAN K, HUANG J P, et al. An unsplit implementation of the multiaxial perfectly matched layer[J]. Progress in Geophysics, 2013, 28(6):2984-2992.(in Chinese).
[35]田坤,李振春,黄建平,等.多轴卷积完全匹配层吸收边界条件[J].石油地球物理勘探, 2014,49(1):143-152.TIAN K, LI Z C, HUANG J P, et al. Multi-axial convolution perfectly matched layer absorbing boundary condition[J]. Oil Geophysical Prospecting, 2014, 49(1):143-152.(in Chinese).
[36]袁士川,宋先海,蔡伟,等.瑞雷波有限差分数值模拟中不同自由表面边界条件的对比[J].石油地球物理勘探, 2017, 52(6):1156-1169.YUAN S C, SONG X H, CAI W, et al. Comparison of different free-surface boundary conditions for Rayleigh waves finite-difference modeling[J]. Oil Geophysical Prospecting, 2017, 52(6):1156-1169.(in Chinese).
[37]GUTOWSKI P R, HRON F, WAGNER D E, et al. S*[J]. Bulletin of the Seismological Society of America, 1984, 74:61-78.
[38]张学涛,田坤,黄建平.黏弹性介质中瑞雷面波正演模拟及频散特征分析[J].地球物理学进展,2016, 31(2):861-868.ZHANG X T, TIAN K, HUANG J P. Forward modeling and dispersion properties analysis of Rayleigh surface wave in viscoelastic media[J]. Progress in Geophysics, 2016, 31(2):861-868.(in Chinese).
[39]BERG P, IF F, NILSEN P, et al. Analytical reference solutions:Advanced seismic Modeling[M]//HELBIG K. Modeling the earth for oil exploration. Pergamon Press, 1994:421-427.
[2]LUOY,XIAJ,MILLERRD,etal.Rayleigh-wavedispersiveenergyimagingusinga high-resolution linear radon transform[J]. Pure and Applied Geophysics, 2008, 165:903-922.
[3]PARK C B, MILLER R D, XIA J. Multi-channel analysis of surface waves(MASW)[J]. Geophysics,1999, 64:800-808.
[4]MITTET R. Free-surface boundary conditions for elastic staggered-grid modeling schemes[J].Geophysics, 2002, 67(5):1616-1623.
[5]KRISTEK J, MOCZO P, ARCHULETA R J. Efficient methods to simulate planar free surface in the3D 4th-order staggered-grid finite-difference schemes[J]. Study Geophysics, 2002, 46:355-381.
[6]ROBERTSSON J O A. A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography[J]. Geophysics, 1996, 61(6):1921-1934.
[7]XU Y, XIA J, MILLER R D. Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach[J]. Geophysics, 2007, 72(5):147-153.
[8]张伟.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京:北京大学, 2006.ZHANG W. Finite difference seismic wave modelling in 3D heterogeneous media with surface topographyanditsimplementationinstronggroundmotionstudy[D].Beijing:Peking University, 2006.(in Chinese).
[9]凡友华,陈晓非,刘雪峰,等. Rayleigh波的频散方程高频近似分解和多模式激发数目[J].地球物理学报, 2007, 50(1):233-239.FAN Y H, CHEN X F, LIU X F, et al. Approximate decomposition of the dispersion equation at highfrequenciesandthenumberofmultimodesforRayleighwaves[J].ChineseJournal Geophysics, 2007, 50(1):233-239.(in Chinese).
[10]周竹生,刘喜亮,熊孝雨.弹性介质中瑞雷面波有限差分法正演模拟[J].地球物理学报, 2007,50(2):567-573.ZHOU Z S, LIU X L, XIONG X Y. Finite-difference modelling of Rayleigh surface wave in elastic media[J]. Chinese Journal Geophysics, 2007, 50(2):567-573.(in Chinese).
[11]张大洲,熊章强,顾汉明.高精度瑞雷波有限差分数值模拟及波场分析[J].地球物理学进展, 2009,24(4):1313-1319.ZHANG D Z, XIONG Z Q, GU H M. Numerical modeling of Rayleigh wave using high-accuracy finite-difference method and wave field analysis[J]. Progress in Geophysics, 2009, 24(4):1313-1319.(in Chinese).
[12]裴正林.任意起伏地表弹性波方程交错网格高阶有限差分法数值模拟[J].石油地球物理勘探, 2004,39(6):629-634.PEI Z L. Staggered-grid high-order finite-difference numerical simulation of elastic wave equation with arbitrary irregular surface[J]. Oil Geophysical Prospecting, 2004, 39(6):629-634.(in Chinese).
[13]秦臻,张才,郑晓东,等.高精度有限差分瑞雷面波模拟及频散特征提取[J].石油地球物理勘探,2010, 45(1):40-46.QIN Z, ZHANG C, ZHENG X D, et al. High-accuracy finite-difference modeling of Rayleigh surface wave and extracting of dispersion characteristic[J]. Oil Geophysical Prospecting, 2010,45(1):40-46.(in Chinese).
[14]潘冬明,胡明顺,崔若飞,等.基于拉东变换的瑞雷面波频散分析与应用[J].地球物理学报, 2010,53(11):2760-2766.PAN D M, HU M S, CUI R F, et al. Dispersion analysis of Rayleigh surface waves and application based on Radon transform[J]. Chinese Journal Geophysics, 2010, 53(11):2760-2766.(in Chinese).
[15]高静怀,何洋洋,马逸尘.黏弹性与弹性介质中Rayleigh面波特性对比研究[J].地球物理学报,2012, 55(1):207-218.GAO J H, HE Y Y, MA Y C. Comparision of the Rayleigh wave in elastic and viscoelastic media[J].Chinese Journal Geophysics, 2012, 55(1):207-218.(in Chinese).
[16]张维,何正勤,胡刚,等.用面波联合勘探技术探测浅部速度结构[J].地球物理学进展, 2013,28(4):2199-2206.ZHANG W, HE Z Q, HU G, et al. Detection of the shallow velocity structure with surface wave prospection method[J]. Progress in Geophysics, 2013, 28(4):2199-2206.(in Chinese).
[17]张立,刘争平.水平层状介质中基阶瑞利面波椭圆极化特征数值分析与研究[J].地球物理学报,2013, 56(5):1686-1695.ZHANG L, LIU Z P. A study of the elliptic polarization characteristics of fundamental mode Rayleigh wave based on numerical simulation[J]. Chinese Journal Geophysics, 2013, 56(5):1686-1695.(in Chinese).
[18]赵海波,陈百军,李奎周,等.黏弹性介质VSP记录模拟及在估算Q值研究中应用[J].地球物理学报, 2011, 54(2):329-335.ZHAO H B, CHEN B J, LI C Z, et al. VSP record numerical modeling in viscoelastic media and its application in the study of Q-value estimation method[J]. Chinese Journal Geophysics,2011, 54(2):329-335.(in Chinese).
[19]严红勇,刘洋.黏弹TTI介质中旋转交错网格高阶有限差分数值模拟[J].地球物理学报, 2012,55(4):1354-1365.YAN H Y, LIU Y. Rotated staggered grid high-order finite-difference numerical modeling for wave propagation in viscoelastic TTI media[J]. Chinese Journal Geophysics, 2012, 55(4):1354-1365.(in Chinese).
[20]杨仁虎,常旭,刘伊克.基于非均匀各向同性介质的黏弹性波正演数值模拟[J].地球物理学报,2009, 52(9):2321-2327.YANG R H, CHANG X, LIU Y K. Viscoelastic wave modeling in heterogeneous and isotropic media[J].Chinese Journal Geophysics, 2009, 52(9):2321-2327.(in Chinese).
[21]BLANCH J O, ROBERTSSON J O A, SYMES W W. Modeling of a constant Q:Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique[J]. Geophysics, 1995,60(1):176-184.
[22]TAL-EZER H, CARCIONE J M, KOSLOFF D. An accurate and efficient scheme for wave propagation in linear viscoelastic media[J]. Geophysics, 1990, 55(10):1366-1379.
[23]孙成禹,印兴耀.三参数常Q粘弹性模型构造方法研究[J].地震学报, 2007, 29(4):348-357.SUN C Y, YIN X Y. Research on construction method of constant Q viscoelastic model with three parameters[J]. Acta Seismologica Sinica, 2007, 29(4):348-357.(in Chinese).
[24]CAODP,YINXY.QreflectionsmodelingwithgeneralizedMaxwellmodelintime domain[C]//81th Annual International Meeting, Expanded Abstracts, SEG, 2011:2819-2823.
[25]DAY S M, MINSTER J B. Numerical simulation of attenuated wavefields using a Padéapproximant method[J]. Geophysical Journal of the Royal Astronomical Society, 1984, 78:105-118.
[26]EMMERICH H, KORN M. Incorporation of attenuation into time-domain computations of seismic wave fields[J]. Geophysics, 1987, 52:1252-1264.
[27]CARCIONE J M, KOSLOFF D, KOSLOFF R. Wave propagation simulation in a linear viscoelastic medium[J]. Geophysical Journal International, 1988, 95:597-611.
[28]MOCZO P, KRISTEK J. On the rheological models used for time-domain methods of seismic wave propagation[J]. Geophysical Research Letters, 2005, 32:L01306.
[29]TIAN K, HUANG J P, LI Z C, et al. Simulation of the planar free surface in the finite-difference modelingofhalf-spaceviscoelasticmedium[C]//75thEAGEConference&Exhibition Incorporating SPE EUROPEC, 2013:Tu-P13-03.
[30]LIU H D, ANDERSON L, KANAMORI H. Velocity dispersion due to anelasticity; Implications for seismology and mantle composition[J]. Geophysical Journal International, 1976, 47:41-58.
[31]张光澄.非线性最优化计算方法[M].北京:高等教育出版社, 2005.ZHANG G C. Computational methods for nonlinear optimization[M]. Beijing:Higher Education Press, 2005.(in Chinese).
[32]MARTIN R, KOMATITSCH D. An unsplit convolutional perfectly matched layer technique improved atgrazingincidencefortheviscoelasticwaveequation[J].GeophysicalJournal International, 2009, 179:333-344.
[33]MEZA-FAJARDO K C, PAPAGEORGIOU A S. A non-convolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media:Stability analysis[J].Bulletin of the Seismological Society of America. 2008, 98(4):1811–1836.
[34]李振春,田坤,黄建平,等.多轴完全匹配层的非分裂实现[J].地球物理学进展, 2013, 28(6):2984-2992.LI Z C, TIAN K, HUANG J P, et al. An unsplit implementation of the multiaxial perfectly matched layer[J]. Progress in Geophysics, 2013, 28(6):2984-2992.(in Chinese).
[35]田坤,李振春,黄建平,等.多轴卷积完全匹配层吸收边界条件[J].石油地球物理勘探, 2014,49(1):143-152.TIAN K, LI Z C, HUANG J P, et al. Multi-axial convolution perfectly matched layer absorbing boundary condition[J]. Oil Geophysical Prospecting, 2014, 49(1):143-152.(in Chinese).
[36]袁士川,宋先海,蔡伟,等.瑞雷波有限差分数值模拟中不同自由表面边界条件的对比[J].石油地球物理勘探, 2017, 52(6):1156-1169.YUAN S C, SONG X H, CAI W, et al. Comparison of different free-surface boundary conditions for Rayleigh waves finite-difference modeling[J]. Oil Geophysical Prospecting, 2017, 52(6):1156-1169.(in Chinese).
[37]GUTOWSKI P R, HRON F, WAGNER D E, et al. S*[J]. Bulletin of the Seismological Society of America, 1984, 74:61-78.
[38]张学涛,田坤,黄建平.黏弹性介质中瑞雷面波正演模拟及频散特征分析[J].地球物理学进展,2016, 31(2):861-868.ZHANG X T, TIAN K, HUANG J P. Forward modeling and dispersion properties analysis of Rayleigh surface wave in viscoelastic media[J]. Progress in Geophysics, 2016, 31(2):861-868.(in Chinese).
[39]BERG P, IF F, NILSEN P, et al. Analytical reference solutions:Advanced seismic Modeling[M]//HELBIG K. Modeling the earth for oil exploration. Pergamon Press, 1994:421-427.
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