层状介质中瑞利波频散方程及其线性研究
|
摘要
瑞利波法是一种新兴的地球物理勘探方法。它主要用到了层状介质中瑞利波的频散特性,涉及到瑞利波数据的采集、频散曲线的正演理论及反演解释三个问题。由于目前瑞利波法的数学理论还不完善,还没有得到频散曲线的单独计算公式,也没有弄清横波速度与瑞利波相速度的关系,这就使得以正演为基础的反演工作受到阻碍,进而极大地限制了瑞利波法的工程应用。本课题通过理论公式的推导以及大量数值计算,对层状介质中瑞利波频散方程进行了研究,主要内容及成果如下:
成功地将广义反射-透射系数算法无量纲化,得到了更为简洁的计算瑞利波频散曲线的公式体系,提高了算法的计算精度。同时,在本文的公式体系下推导出瑞利波基阶模式的近似解。
通过高频近似方法,推导出瑞利波R1模式的单独计算公式。应用Matlab编制程序计算了单独的R1模式频散曲线,并通过与无量纲实数传递矩阵算法的计算结果比较,验证了高频近似公式的有效性。
在层状介质模型中各层泊松比一定和相邻两层的密度比不变的前提下,分析了频散方程隐含着横波速度函数和相速度函数之间的映射关系,引进算子T将此关系显式化。严格证明了算子T的数乘性、R模式的高频近似可加性;通过对四组模型的数值计算及结果分析,表明算子T对R模式满足近似线性;在层厚不变时,算子T对于R1模和S2模式都是近似线性的。
数值计算及其结果分析表明:R1模式和S2模式不可叠加;第1层的厚度不变是算子T对于R1模式满足近似线性的前提;在一定条件下,R1模和R1~2模可叠加。
本课题中提出的改进算法及R1模式的单独近似计算公式完善了瑞利波的正演理论,为瑞利波反演提供了理论基础;线性研究为进一步研究频散方程提供了新思路,其数值计算结果则对瑞利波法的工程应用具有一定的参考价值。故本研究无论是对瑞利波正演理论的充实,还是对瑞利波法的工程应用都将具有实际的意义。
Rayleigh waves method is a new geophysical exploration method. It mainly uses the dispersion characteristics of Rayleigh waves in the multilayered medium, including three problems: data collection, dispersion theory and inversion application. As the mathematics theory is not perfect yet, there is not a separate formula for each dispersion curve and no clear understanding of the relationship between the S-wave velocity and the phase velocity, which, to some extent, makes an obstruction against the inversion based on it, thereby greatly limits the application in enginnering.
In this thesis, by deriving the theoretical formula and using a large number of numerical caculation, we do some research on the dispersion equation of Rayleigh waves in multilayered medium. The main contents and innovations include the following:
Firstly, successfully making the generalized reflection/transmission coeffic- ients method dimensionless, which improves the caculation pricision. The new formula system is more simple. Furthermore, we derive a high frequency asymptomic solution of the fundamental Rayleigh mode.
Secondly, by a high-frequency approximation, we derive a separate formula for the R1-mode of the Rayleigh waves dispersion curves. Its feasibility is verif- ied by comparing the dispersion curves computed by the approximate formula to the ones by the dimentionless and real transfor matrix algorithm.
Thirdly, the dispersion equation of Rayleigh waves in multilayered medium implys the relationship between the S-wave velocity and the phase velocity, which is made implicit by introducing an operator T. The scalar multiplication of the operator T is proved and verifid by numerical caculation. We prove that R-mode is approximately additive in the high-frequency, which is verified by numerical caculation, too. By numerical caculation, we verify that the operator T is approximately linear for R1-mode and S2-mode with the same thickness in the same layer.
Finally, numerical caculation and its analysis results show that R1-mode and S2-mode do not satisfy superposition. While for the case of the two R1-modes, the equality of the first layer’s thickness is the prerequisite for the establishment of the approximately linear charcteristic of the operator T. Finally, we show an example of the superposition for R1-mode and R1~2-mode, for which the approximately linear characteristic of the operator T holds.
The improved generalized feflection/transmission coeffiients algorithm and the approximate formula of R1-mode can provide a theoretical basis for the inversion application. The numerical results can provide a reference and an new idea for the further study on the dispersion equation of Rayleigh waves. So it is expected to be useful and powerful not only in the theoretically enrichment, but also in engineering application.
成功地将广义反射-透射系数算法无量纲化,得到了更为简洁的计算瑞利波频散曲线的公式体系,提高了算法的计算精度。同时,在本文的公式体系下推导出瑞利波基阶模式的近似解。
通过高频近似方法,推导出瑞利波R1模式的单独计算公式。应用Matlab编制程序计算了单独的R1模式频散曲线,并通过与无量纲实数传递矩阵算法的计算结果比较,验证了高频近似公式的有效性。
在层状介质模型中各层泊松比一定和相邻两层的密度比不变的前提下,分析了频散方程隐含着横波速度函数和相速度函数之间的映射关系,引进算子T将此关系显式化。严格证明了算子T的数乘性、R模式的高频近似可加性;通过对四组模型的数值计算及结果分析,表明算子T对R模式满足近似线性;在层厚不变时,算子T对于R1模和S2模式都是近似线性的。
数值计算及其结果分析表明:R1模式和S2模式不可叠加;第1层的厚度不变是算子T对于R1模式满足近似线性的前提;在一定条件下,R1模和R1~2模可叠加。
本课题中提出的改进算法及R1模式的单独近似计算公式完善了瑞利波的正演理论,为瑞利波反演提供了理论基础;线性研究为进一步研究频散方程提供了新思路,其数值计算结果则对瑞利波法的工程应用具有一定的参考价值。故本研究无论是对瑞利波正演理论的充实,还是对瑞利波法的工程应用都将具有实际的意义。
Rayleigh waves method is a new geophysical exploration method. It mainly uses the dispersion characteristics of Rayleigh waves in the multilayered medium, including three problems: data collection, dispersion theory and inversion application. As the mathematics theory is not perfect yet, there is not a separate formula for each dispersion curve and no clear understanding of the relationship between the S-wave velocity and the phase velocity, which, to some extent, makes an obstruction against the inversion based on it, thereby greatly limits the application in enginnering.
In this thesis, by deriving the theoretical formula and using a large number of numerical caculation, we do some research on the dispersion equation of Rayleigh waves in multilayered medium. The main contents and innovations include the following:
Firstly, successfully making the generalized reflection/transmission coeffic- ients method dimensionless, which improves the caculation pricision. The new formula system is more simple. Furthermore, we derive a high frequency asymptomic solution of the fundamental Rayleigh mode.
Secondly, by a high-frequency approximation, we derive a separate formula for the R1-mode of the Rayleigh waves dispersion curves. Its feasibility is verif- ied by comparing the dispersion curves computed by the approximate formula to the ones by the dimentionless and real transfor matrix algorithm.
Thirdly, the dispersion equation of Rayleigh waves in multilayered medium implys the relationship between the S-wave velocity and the phase velocity, which is made implicit by introducing an operator T. The scalar multiplication of the operator T is proved and verifid by numerical caculation. We prove that R-mode is approximately additive in the high-frequency, which is verified by numerical caculation, too. By numerical caculation, we verify that the operator T is approximately linear for R1-mode and S2-mode with the same thickness in the same layer.
Finally, numerical caculation and its analysis results show that R1-mode and S2-mode do not satisfy superposition. While for the case of the two R1-modes, the equality of the first layer’s thickness is the prerequisite for the establishment of the approximately linear charcteristic of the operator T. Finally, we show an example of the superposition for R1-mode and R1~2-mode, for which the approximately linear characteristic of the operator T holds.
The improved generalized feflection/transmission coeffiients algorithm and the approximate formula of R1-mode can provide a theoretical basis for the inversion application. The numerical results can provide a reference and an new idea for the further study on the dispersion equation of Rayleigh waves. So it is expected to be useful and powerful not only in the theoretically enrichment, but also in engineering application.
引文
1 L. Rayleigh. On Waves Propagation along the Plane Surface of an Elastic Solid. Proc. London Math. Soc.1887, 17:4~11
2 N. A. Haskell. The Dispersion of Surface Waves on Multilayered Media. Bull. Seism. Soc. Am.1953, 43:17~34
3 L. Knopoff. A Matrix Method for Elastic Wave Problems. Bull. Siesm. Soc. Am. 1964, 54:431~438
4 M. J. Randall. Fast Programs for Layered Half-space Problems. Bull. Seism Soc. Am.1967, 57:1299~1316
5 F. Schwab and L. Knopoff .Surface-wave Dispersion Computations. Bull. Seism. Soc. Am. 1970, 60:321~344
6 F. Schwab and L. Knopoff. Fast Surface Wave and Free Mode Computations. Bull. Seism. Soc. Am. 1972, 2:87~180
7 F. Schwab. Surface Wave Dispertion Computation: Knopoff’s method. Bull. Seism. Soc. Am. 1970, 60:1491~1520
8 E. Pestel and F. Leckie. Matrix Methods in Elasto-mechanics. Me Graw-Hill, New York
9 J. W. Dunkin. Computation of Modal Solutions in Layered Elastic Media at High Frequencies. Bull. Seism. Soc. Am. 1965, 55:335~358
10 T. H. Watson. A Note on Fast Computation of Rayleigh Wave Dispersion in the Multilayered Clastic Half-space. Bull. Seism. Soc.Am.1970, 60:161~166
11 P. W. Buchen and R. Ben-Hador. Free-mode Surface-wave Computations. Geophysics. J. Int. 1996, 124:869~887
12 S. Ivansson. Comment on Free-mode Surface-wave Computations by P. W. Buchen, and R. Ben-Hador. Geophys. J. Int. 1998, 132:725~727
13 A. Abo-Zena. Dispersion Function Computations for Unlimited Frequency Values. Geophys. J. R. Astr. Soc. 1979, 58:91~105
14 W. Menke. Comment on“Dispersion Function Computations for Unlimited Frequency Values’’by Anas Abo-Zena. Geophys. J. R. Astr. Soc. 1979, 95:315~323
15 A. Ben-Menahem and S. J. Singh. Multipolar Elastic Eield in a Layered Half-space. Bull Seism. Soc. Am. 1968, 58:1519~1572
16 B. L. N. Kennett. Reflectin Rays and Reverberations. Bull. Seism. Soc. Am. 1974, 64:1685~1696
17 B. L. N. Kennett and N. J. Kerry. Seismic Waves in a Stratified Halfspace. Geophys. J.R. Astr. Soc. 1979, 57:557~583
18李幼铭,束沛镒.层状介质中地震面波频散函数和体波广义反射系数的计算.地球物理学报. 1982, 25(2):130~139
19 T. Kundu, A. K. Mat and R. D. Wehleih. Calculation of the Acoustic Material Signature of a Layered Solid. J. Acoust. Soc. Am. 1985, 77:353~361
20 T. Kundu and A. K. Mat. Elastic Wabes in Multi-layered Solid Due to a Dislocation Source. Wave Motion. 1985, 7:459~497
21 X. F. Chen. A Systematic and Efficient Method of Computing Normal Modes for Multilayered Half-space. Geophys. J. Int. 1993, 115:391~409
22 B. X. Zhang, M. Yu, C. Q. Lan and W Xiong. Elastic Wave Excitation Mec- hanism of Surface Waves in Multilayered Media, J. Acoust. Soc Am. 1996, 100(6):3527~3538
23张碧星,喻明.层状介质中的声波场及面波研究.声学学报. 1997, 22(3):230~241
24张碧星,兰从庆,喻明等.分层介质中面波的能量分布.声学学报. 1998, 23(2): 97~106
25林维正,苏勇. Rayleigh表面波频散曲线的特征方程的计算方法.同济大学学报. 1997, 25(5):570~575
26 J. F. Coste. Approximate Dispersion Foemulae for Rayleigh-like Waves in a Layered Medium. Ultrasonics. 1997, 35:431~440
27凡友华,刘家琦.层状介质中瑞雷面波的频散研究.哈尔滨工业大学学报. 2001, 33(5):577~581.
28凡友华.层状介质中瑞利面波频散曲线的正反演研究.哈尔滨工业大学博士学位论文. 2001:11~32,35~43,95~113
29张碧星,肖柏勋.瑞利波勘探中“之”形频散曲线形成机理及反演研究.地球物理学报. 2000, 43(4):558~569
30张碧星,鲁来玉,鲍光淑.瑞利波勘探中“之”字形频散曲线研究.地球物理学报. 2002, 45(2):263~274
31杨天春.瑞利波勘探中频散曲线的正演计算.工程地球物理学报. 2004, 1(6):469~473
32杨天春.瑞利波泄露模式的模拟研究.湖南大学学报. 2005, 32(4):12~17
33肖翔,徐果明,朱良保.面波频散曲线快速追踪算法.地震地磁观测与研究. 2004, 25(3):1~7
34张碧星,鲁来玉.用频率-波数法分析瑞利波频散曲线.工程地球物理学报. 2005, 2(4): 246~257
35张金清,梁青,陈超.软弱夹层瑞利波频散曲线特征.工程地球物理学报. 2005, 2(3):209~217
36梁志强.层状介质中多模式面波频散曲线研究.长安大学硕士学位论文. 2006:20~35
37何耀锋,陈蔚天,陈晓非.利用广义反射-透射系数方法求解含低速层介质模型中面波频散曲线问题.地球物理学报. 2006, 49(4):1047~1081
38凡友华,陈晓非. Rayleigh波的频散方程高频近似分解和多模式激发数目.地球物理学报. 2007, 50(1):233~239
39周新民,夏唐代.半空间准饱和土中瑞利波的传播特性研究.岩土工程学报. 2007, 29(5):750~754
40杨天春,吴燕清,孙新华.对瑞利波频散曲线计算中高频数值溢出的处理.煤炭学报. 2007, 32(10):1041~1045
41林秀巧,黄模佳. Rayleigh波传播速度的积分确定.南昌大学学报. 2007, 29(1):36~42
42 N. Gucunski and R. D. Woods. Inversion of Rayleigh Wave Dispersion Cur- ve for SASW Test. First Iternational Conference on Soil Dynamics and Earthquake Engeering. 1991:127~138
43 J. K. Chung and Y. T. Yeh. Shallow Crustal Structure from shot-period Rayleigh-wave dispersion data in southwesten Taiwan. Bull. Seism. Soc. Am. 1977, 87(2):370~382
44 L. Malagnini, R. Herrmann and A. Mercuri. Shear Wave Velocity Structure of Sediments Form the Inversion of Explosion-induced Rayleigh Waves: Comparison with Cross-hole measurements. Bull. Seism. Soc. Am. 1997, 87(6):1413~1421
45 G. J. Rix and E. A. Leipski. Accurasy and Resolution of Surface Wave Invertion. Geotechnical Special Publication. 1991, 11:17~32
46裴江云,吴永刚,刘英杰.近地表低速带反演.长春地质学院学报. 1994, 24(3):317~320
47石耀霖,王文.面波频散反演地球内部构造的遗传算法.地球物理学报. 1995, 38(2):189~198
48 H. Yamanaka and H. Ishida. Application of Genetic Algorithms to an Inversion of Surface-wave Dispersion Data. Bull. Seism. Soc. Am. 1996, 86(2):436~444
49 R. W. Meier and G. J. Rix. Initial Study of Surface-wave Inversion Using Aritificial Neural Networks. Geotechnical Testing Journal. 1993, 16(4):425~431
50 N. Gucuncki, T. P. Willialms and V. Kristic. Surface Wave Testing Inversion by Neural networks. Computing in Civil Engineering. 1995:574~581
51肖柏勋.高模式瑞利面波及其正反演研究.中南大学博士学位论文,2000:75~99
52鲁来玉,张碧星,汪承灏.基于瑞利波高阶模式反演的实验研究.地球物理学报. 2006, 49(4):1082~1091
53丁彦礼,单娜琳.瑞利面波最大模频散曲线的反演解释方法. 2007, 27(3):322~328
54孟小红,郭良辉.利用地震瑞利波速度反演求取P-SV波横波静校正量.石油地球物理勘探. 2007, 42(4):448~453
55朱良保.区域面波群速度反演的球谐函数法.地球物理学报. 1997, 40(4):503~511
56 H. Maruering and R. Snieder. Shear-wave Velocity Structure beneath Europe, the Northeasten Atlantic and Westen Asia from Waveform Inversion including Surface Mode Coupling. Geophys. J. Int. 1996, 127(2):283~288
57罗银河.基阶与高阶瑞利波联合反演研究.地球物理学报. 2008, 5(1):243~149
58 J. H. Xia, R. Miller and C. B. Park. Estimation of Near-surface Shear-wave Velocity by Inversion of Rayleigh Waves. Geophysics. 1999, 64(3):692~700
2 N. A. Haskell. The Dispersion of Surface Waves on Multilayered Media. Bull. Seism. Soc. Am.1953, 43:17~34
3 L. Knopoff. A Matrix Method for Elastic Wave Problems. Bull. Siesm. Soc. Am. 1964, 54:431~438
4 M. J. Randall. Fast Programs for Layered Half-space Problems. Bull. Seism Soc. Am.1967, 57:1299~1316
5 F. Schwab and L. Knopoff .Surface-wave Dispersion Computations. Bull. Seism. Soc. Am. 1970, 60:321~344
6 F. Schwab and L. Knopoff. Fast Surface Wave and Free Mode Computations. Bull. Seism. Soc. Am. 1972, 2:87~180
7 F. Schwab. Surface Wave Dispertion Computation: Knopoff’s method. Bull. Seism. Soc. Am. 1970, 60:1491~1520
8 E. Pestel and F. Leckie. Matrix Methods in Elasto-mechanics. Me Graw-Hill, New York
9 J. W. Dunkin. Computation of Modal Solutions in Layered Elastic Media at High Frequencies. Bull. Seism. Soc. Am. 1965, 55:335~358
10 T. H. Watson. A Note on Fast Computation of Rayleigh Wave Dispersion in the Multilayered Clastic Half-space. Bull. Seism. Soc.Am.1970, 60:161~166
11 P. W. Buchen and R. Ben-Hador. Free-mode Surface-wave Computations. Geophysics. J. Int. 1996, 124:869~887
12 S. Ivansson. Comment on Free-mode Surface-wave Computations by P. W. Buchen, and R. Ben-Hador. Geophys. J. Int. 1998, 132:725~727
13 A. Abo-Zena. Dispersion Function Computations for Unlimited Frequency Values. Geophys. J. R. Astr. Soc. 1979, 58:91~105
14 W. Menke. Comment on“Dispersion Function Computations for Unlimited Frequency Values’’by Anas Abo-Zena. Geophys. J. R. Astr. Soc. 1979, 95:315~323
15 A. Ben-Menahem and S. J. Singh. Multipolar Elastic Eield in a Layered Half-space. Bull Seism. Soc. Am. 1968, 58:1519~1572
16 B. L. N. Kennett. Reflectin Rays and Reverberations. Bull. Seism. Soc. Am. 1974, 64:1685~1696
17 B. L. N. Kennett and N. J. Kerry. Seismic Waves in a Stratified Halfspace. Geophys. J.R. Astr. Soc. 1979, 57:557~583
18李幼铭,束沛镒.层状介质中地震面波频散函数和体波广义反射系数的计算.地球物理学报. 1982, 25(2):130~139
19 T. Kundu, A. K. Mat and R. D. Wehleih. Calculation of the Acoustic Material Signature of a Layered Solid. J. Acoust. Soc. Am. 1985, 77:353~361
20 T. Kundu and A. K. Mat. Elastic Wabes in Multi-layered Solid Due to a Dislocation Source. Wave Motion. 1985, 7:459~497
21 X. F. Chen. A Systematic and Efficient Method of Computing Normal Modes for Multilayered Half-space. Geophys. J. Int. 1993, 115:391~409
22 B. X. Zhang, M. Yu, C. Q. Lan and W Xiong. Elastic Wave Excitation Mec- hanism of Surface Waves in Multilayered Media, J. Acoust. Soc Am. 1996, 100(6):3527~3538
23张碧星,喻明.层状介质中的声波场及面波研究.声学学报. 1997, 22(3):230~241
24张碧星,兰从庆,喻明等.分层介质中面波的能量分布.声学学报. 1998, 23(2): 97~106
25林维正,苏勇. Rayleigh表面波频散曲线的特征方程的计算方法.同济大学学报. 1997, 25(5):570~575
26 J. F. Coste. Approximate Dispersion Foemulae for Rayleigh-like Waves in a Layered Medium. Ultrasonics. 1997, 35:431~440
27凡友华,刘家琦.层状介质中瑞雷面波的频散研究.哈尔滨工业大学学报. 2001, 33(5):577~581.
28凡友华.层状介质中瑞利面波频散曲线的正反演研究.哈尔滨工业大学博士学位论文. 2001:11~32,35~43,95~113
29张碧星,肖柏勋.瑞利波勘探中“之”形频散曲线形成机理及反演研究.地球物理学报. 2000, 43(4):558~569
30张碧星,鲁来玉,鲍光淑.瑞利波勘探中“之”字形频散曲线研究.地球物理学报. 2002, 45(2):263~274
31杨天春.瑞利波勘探中频散曲线的正演计算.工程地球物理学报. 2004, 1(6):469~473
32杨天春.瑞利波泄露模式的模拟研究.湖南大学学报. 2005, 32(4):12~17
33肖翔,徐果明,朱良保.面波频散曲线快速追踪算法.地震地磁观测与研究. 2004, 25(3):1~7
34张碧星,鲁来玉.用频率-波数法分析瑞利波频散曲线.工程地球物理学报. 2005, 2(4): 246~257
35张金清,梁青,陈超.软弱夹层瑞利波频散曲线特征.工程地球物理学报. 2005, 2(3):209~217
36梁志强.层状介质中多模式面波频散曲线研究.长安大学硕士学位论文. 2006:20~35
37何耀锋,陈蔚天,陈晓非.利用广义反射-透射系数方法求解含低速层介质模型中面波频散曲线问题.地球物理学报. 2006, 49(4):1047~1081
38凡友华,陈晓非. Rayleigh波的频散方程高频近似分解和多模式激发数目.地球物理学报. 2007, 50(1):233~239
39周新民,夏唐代.半空间准饱和土中瑞利波的传播特性研究.岩土工程学报. 2007, 29(5):750~754
40杨天春,吴燕清,孙新华.对瑞利波频散曲线计算中高频数值溢出的处理.煤炭学报. 2007, 32(10):1041~1045
41林秀巧,黄模佳. Rayleigh波传播速度的积分确定.南昌大学学报. 2007, 29(1):36~42
42 N. Gucunski and R. D. Woods. Inversion of Rayleigh Wave Dispersion Cur- ve for SASW Test. First Iternational Conference on Soil Dynamics and Earthquake Engeering. 1991:127~138
43 J. K. Chung and Y. T. Yeh. Shallow Crustal Structure from shot-period Rayleigh-wave dispersion data in southwesten Taiwan. Bull. Seism. Soc. Am. 1977, 87(2):370~382
44 L. Malagnini, R. Herrmann and A. Mercuri. Shear Wave Velocity Structure of Sediments Form the Inversion of Explosion-induced Rayleigh Waves: Comparison with Cross-hole measurements. Bull. Seism. Soc. Am. 1997, 87(6):1413~1421
45 G. J. Rix and E. A. Leipski. Accurasy and Resolution of Surface Wave Invertion. Geotechnical Special Publication. 1991, 11:17~32
46裴江云,吴永刚,刘英杰.近地表低速带反演.长春地质学院学报. 1994, 24(3):317~320
47石耀霖,王文.面波频散反演地球内部构造的遗传算法.地球物理学报. 1995, 38(2):189~198
48 H. Yamanaka and H. Ishida. Application of Genetic Algorithms to an Inversion of Surface-wave Dispersion Data. Bull. Seism. Soc. Am. 1996, 86(2):436~444
49 R. W. Meier and G. J. Rix. Initial Study of Surface-wave Inversion Using Aritificial Neural Networks. Geotechnical Testing Journal. 1993, 16(4):425~431
50 N. Gucuncki, T. P. Willialms and V. Kristic. Surface Wave Testing Inversion by Neural networks. Computing in Civil Engineering. 1995:574~581
51肖柏勋.高模式瑞利面波及其正反演研究.中南大学博士学位论文,2000:75~99
52鲁来玉,张碧星,汪承灏.基于瑞利波高阶模式反演的实验研究.地球物理学报. 2006, 49(4):1082~1091
53丁彦礼,单娜琳.瑞利面波最大模频散曲线的反演解释方法. 2007, 27(3):322~328
54孟小红,郭良辉.利用地震瑞利波速度反演求取P-SV波横波静校正量.石油地球物理勘探. 2007, 42(4):448~453
55朱良保.区域面波群速度反演的球谐函数法.地球物理学报. 1997, 40(4):503~511
56 H. Maruering and R. Snieder. Shear-wave Velocity Structure beneath Europe, the Northeasten Atlantic and Westen Asia from Waveform Inversion including Surface Mode Coupling. Geophys. J. Int. 1996, 127(2):283~288
57罗银河.基阶与高阶瑞利波联合反演研究.地球物理学报. 2008, 5(1):243~149
58 J. H. Xia, R. Miller and C. B. Park. Estimation of Near-surface Shear-wave Velocity by Inversion of Rayleigh Waves. Geophysics. 1999, 64(3):692~700