利用伪谱法模拟横向各向同性介质中的波(英文)
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摘要
介质的弹性常数为三维四阶张量的分量，共有８１个，由于应力张量和应变张量的对称性及能量密度是应变的二次函数，一般各向异常性介质的独立弹性常数可减为２１个，如果介质具有较高的对称性，独立弹性常数的数目会更少。 对于地壳和上地幔，具有５个独立弹性常数的横向各向同性介质是一个非常好的近似，本研究中横向各向同性介质的对称轴方向可以是任意的（即对称轴可以不平等于铅直方向），在此情况下，需要进行坐标变换，如果已知介质在某一坐标系（其坐标轴平行或垂直于介质的对称轴）中的弹性常数，我们能够容易地利用变换公式得到变换后新坐标系中的弹性常数。 本文提出了一种方案，利用伪谱法既能模拟横向各向同性介质中的平面波，也能模拟点源激发的波场。在勘探地球物理和地震学中，模拟横向各向同性介拮中传播的平面波及区域源产生的波是最重要的研究课题之一。然而在一般各向异性介质中，很难或不可能确定弹性波的相速度和偏振方向，但在横向各向同性介质中，则可以通过坐标变换来实现，这里我们所提出的方法可以用于横向各向同性介质中弹性波的模拟。
The elastic constant is a component of a three-dimensional fourth-rank tensor, hav- ing 81 components, in all. According to the symmetry of both the stress and strain ten- sors and the existence of a density energy function, which is quadratic in strain, the number of independent constants is 21 for general anisotropic media. The number of in- dependent elastic constants can be reduced still mord if media have higher symmetry. Transversely isotropic medium, which has only five indepindent constants, is a good ap- proximation of rocks in the crust and the upper mantle of the earth.In this paper, we are concerned about transversely isotropic media with an arbitiary direction of symmetrical axis(i. e., the symmetrical axis may not be parallel to the verticalaxis) . In this case we need to change coordinates from one system to another. If we know the elastic constants in one particular coordinate system, for example, whose axes are parallel or perpendicular to the symmetrical axis of the midia, we can easily obtain these elastic constants in new coordinate system by using the transformation formula. In this paper we present an approach for modeling wave-fields excited by not only a source but also a plane-wave incidence in transversely isotiopic media mentioned above by the pseudo-spectral method. Modeling of plane waves propagating in transversely isotropic media is one of the most important subjects as well as that of waves emitted from a localized source in exploration geophsics and seismology. While it is deffcult or even impossible to determine the phase velocity and the polarization direction of plane waves in general an anisotropic media, in the case of transversely isotropic media we can achieve this purpose through coordinate transformation. We here develop a scheme that can be used for plane-wave modeling in travsversely isotropic media,
The elastic constant is a component of a three-dimensional fourth-rank tensor, hav- ing 81 components, in all. According to the symmetry of both the stress and strain ten- sors and the existence of a density energy function, which is quadratic in strain, the number of independent constants is 21 for general anisotropic media. The number of in- dependent elastic constants can be reduced still mord if media have higher symmetry. Transversely isotropic medium, which has only five indepindent constants, is a good ap- proximation of rocks in the crust and the upper mantle of the earth.In this paper, we are concerned about transversely isotropic media with an arbitiary direction of symmetrical axis(i. e., the symmetrical axis may not be parallel to the verticalaxis) . In this case we need to change coordinates from one system to another. If we know the elastic constants in one particular coordinate system, for example, whose axes are parallel or perpendicular to the symmetrical axis of the midia, we can easily obtain these elastic constants in new coordinate system by using the transformation formula. In this paper we present an approach for modeling wave-fields excited by not only a source but also a plane-wave incidence in transversely isotiopic media mentioned above by the pseudo-spectral method. Modeling of plane waves propagating in transversely isotropic media is one of the most important subjects as well as that of waves emitted from a localized source in exploration geophsics and seismology. While it is deffcult or even impossible to determine the phase velocity and the polarization direction of plane waves in general an anisotropic media, in the case of transversely isotropic media we can achieve this purpose through coordinate transformation. We here develop a scheme that can be used for plane-wave modeling in travsversely isotropic media,
引文
[1] Bamford D Pn velocity anisotropy in a continental upper mantle[J] . Geophs. J R. astr.Soc, 1977, 49 29-48.
[2]Makeeva L I, Plesinger A, Horalek J. Azimuthal anisotropy beneath the Bohemian massif from broad-band seismograms of SKS waves [J] Phys. Earth Planet. Int. , 1990, 62:298~ 306 .
[3] Crampin S,Lovell J . A decade if shear wave splitting in the earth's crust:what does it mean? what use can we make of it ?andwhat should we do next? [J] Geophys.J.Int .,1991. 107:387-407
[4] Xiaodong Song, Paul G Richards Seismological evidence for differential rotation of the earth' s inner core[J] . Nature,1996382 : 221 -224.
[5]Bennett A M. Inner core rotation rate from small-scale heterogeneity and time-varying travel times[J] . Science,1997, 278: 1284 ~ 1288 .
[6] Crampin S.Booth D C. Shear wave polarization near the north/Analia fault,II,Interpretation in terms of crack-inducedanisotropy[J] . Geophys.J . R. astr. Soc. , 1985 , 83 75 ~ 92 .
[7] Hudson J A. Wave speeds and attenuation of elastic waves in material containing cracks[J] Geophys, : J R. astr. Soc . 1981 64 133 -150 .
[8 ]Babuska V, Gra M . Seismic anisotropy in the earth[ M] . Kluwer Academic, the Netherlands, 1991 .
[9]Takenaka B Modeling seismic wave propagation in complex media[J] J . Phys. Earth, 1995, 43: 351 ~ 368.
[10] Min Lou, Rial J A. Modelling elastic-wave propagation in inhomogeneous anisotropic media by the pseudo-spectral method[J ]. Geophys J . Int . 1995 , 120 :60 ~ 72.
[11 ] Furumura T, Takenaka H 2.5-D modelling of elastic waves using the psudospectral.method[J]. Geophys. J . Int. , 1996,124: 820 ~ 832
[12] Furumura T, Kennett B L N, Takenaka H.Parallel 3-D psudospectral simulation of seismic wave propagation[J ]Geophysics, 1988. 63 :279 ~ 288
[13]Shu-Huei Hung, Donald W Forsyth. Modellirg anisotropic wave propagation in oceanic inhomogeneous structures using theparallel multidomain pseudo-spectral method[J].Geophys. J . Int . 1998, 133:726 ~ 740
[14] Chapman C H, Drummond R. Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory[J].BullSeis.Soc Am.,1982,72:S227 ~ S317 .
[15] Alterman 2 Karal F C Jr. Propagation of elastic waves in layered media by finite difference methods[J ] , Bull.Seis SocAm. , 1968, 58 : 367 ~ 398 .
[16] Kelly K R, Ward R W, Sven Treitel, Alford R M . Synthetic seismograms: a finite difference approach[J] . Geophysics,1976,41 :2-27 .
[17] Virieux J.P-SV wave propagation in heterogeneous media:velocity stress finite -difference method[J].Geophysics,1986, 51 889 ~ 901
[18] Smith W D. The application of finite element analysis to body wave proaagation problems[J].Geophys.J.R astr.Soc.,1976. 42 747~ 768 .
[19] Kosloff D, Baysal E. Forward modeling by a Fourier method[J]. Geophysics,1980, 47:1402 ~ 1412 .
[20 ] Kosloff D, Reshef M, Loewenthal D.Elastic wave calculations by the Fourier method[J]. Bull.seis Soc. Am.,1984, 74 :875-891.
[21] Fomberg J. The pseudo-spectral method:comparisons with finite differences for the elastic wave equation[J].Geophysics,1987,52:483-501.
[22] Reshef M, Kosloft D, Edwards M HsingC. Three-dimension acoustic modelling by the Fourier method [J ] . Geophysics1988a,53 :1175 ~ 1183.
[23] Reshef M. Kosloff D, Edwards M, Hsing C. Three-dimension acoustic modelling by the Fourier method[J] . Geophysics,1988b, 53: 1184 ~ 1193.
[24] Gazdag J . Modeling of the acoustic wave equation with transform methods[J] . Geophysics. 1981 . 46:854 ~ 859 .
[25] Daudt C R, Braile L W, Nowack R L, Chiang C S. A comparison of finite-difference and Fourier method calculations of syn-thetic seismograms[J ] . Bull. seis. Soc . am., 1989, 79 : 1210 ~ 1230 .
[26] Cerjan C, Kosloff D, Kosloff R,Reshef M . A nonreflecting boundary condition for discrete acoustic and elastic equations[J]Geophysics, 1985, 50 705 ~ 708
[27] Fedorov F I. The theory of elastic waves in crystals[M] . Plenum Press, New York 1968,
[28] Adi K, Richards P G. Quatitative seismology theory and methods[ M] Freeman, San Francisco, 1980 .
[2]Makeeva L I, Plesinger A, Horalek J. Azimuthal anisotropy beneath the Bohemian massif from broad-band seismograms of SKS waves [J] Phys. Earth Planet. Int. , 1990, 62:298~ 306 .
[3] Crampin S,Lovell J . A decade if shear wave splitting in the earth's crust:what does it mean? what use can we make of it ?andwhat should we do next? [J] Geophys.J.Int .,1991. 107:387-407
[4] Xiaodong Song, Paul G Richards Seismological evidence for differential rotation of the earth' s inner core[J] . Nature,1996382 : 221 -224.
[5]Bennett A M. Inner core rotation rate from small-scale heterogeneity and time-varying travel times[J] . Science,1997, 278: 1284 ~ 1288 .
[6] Crampin S.Booth D C. Shear wave polarization near the north/Analia fault,II,Interpretation in terms of crack-inducedanisotropy[J] . Geophys.J . R. astr. Soc. , 1985 , 83 75 ~ 92 .
[7] Hudson J A. Wave speeds and attenuation of elastic waves in material containing cracks[J] Geophys, : J R. astr. Soc . 1981 64 133 -150 .
[8 ]Babuska V, Gra M . Seismic anisotropy in the earth[ M] . Kluwer Academic, the Netherlands, 1991 .
[9]Takenaka B Modeling seismic wave propagation in complex media[J] J . Phys. Earth, 1995, 43: 351 ~ 368.
[10] Min Lou, Rial J A. Modelling elastic-wave propagation in inhomogeneous anisotropic media by the pseudo-spectral method[J ]. Geophys J . Int . 1995 , 120 :60 ~ 72.
[11 ] Furumura T, Takenaka H 2.5-D modelling of elastic waves using the psudospectral.method[J]. Geophys. J . Int. , 1996,124: 820 ~ 832
[12] Furumura T, Kennett B L N, Takenaka H.Parallel 3-D psudospectral simulation of seismic wave propagation[J ]Geophysics, 1988. 63 :279 ~ 288
[13]Shu-Huei Hung, Donald W Forsyth. Modellirg anisotropic wave propagation in oceanic inhomogeneous structures using theparallel multidomain pseudo-spectral method[J].Geophys. J . Int . 1998, 133:726 ~ 740
[14] Chapman C H, Drummond R. Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory[J].BullSeis.Soc Am.,1982,72:S227 ~ S317 .
[15] Alterman 2 Karal F C Jr. Propagation of elastic waves in layered media by finite difference methods[J ] , Bull.Seis SocAm. , 1968, 58 : 367 ~ 398 .
[16] Kelly K R, Ward R W, Sven Treitel, Alford R M . Synthetic seismograms: a finite difference approach[J] . Geophysics,1976,41 :2-27 .
[17] Virieux J.P-SV wave propagation in heterogeneous media:velocity stress finite -difference method[J].Geophysics,1986, 51 889 ~ 901
[18] Smith W D. The application of finite element analysis to body wave proaagation problems[J].Geophys.J.R astr.Soc.,1976. 42 747~ 768 .
[19] Kosloff D, Baysal E. Forward modeling by a Fourier method[J]. Geophysics,1980, 47:1402 ~ 1412 .
[20 ] Kosloff D, Reshef M, Loewenthal D.Elastic wave calculations by the Fourier method[J]. Bull.seis Soc. Am.,1984, 74 :875-891.
[21] Fomberg J. The pseudo-spectral method:comparisons with finite differences for the elastic wave equation[J].Geophysics,1987,52:483-501.
[22] Reshef M, Kosloft D, Edwards M HsingC. Three-dimension acoustic modelling by the Fourier method [J ] . Geophysics1988a,53 :1175 ~ 1183.
[23] Reshef M. Kosloff D, Edwards M, Hsing C. Three-dimension acoustic modelling by the Fourier method[J] . Geophysics,1988b, 53: 1184 ~ 1193.
[24] Gazdag J . Modeling of the acoustic wave equation with transform methods[J] . Geophysics. 1981 . 46:854 ~ 859 .
[25] Daudt C R, Braile L W, Nowack R L, Chiang C S. A comparison of finite-difference and Fourier method calculations of syn-thetic seismograms[J ] . Bull. seis. Soc . am., 1989, 79 : 1210 ~ 1230 .
[26] Cerjan C, Kosloff D, Kosloff R,Reshef M . A nonreflecting boundary condition for discrete acoustic and elastic equations[J]Geophysics, 1985, 50 705 ~ 708
[27] Fedorov F I. The theory of elastic waves in crystals[M] . Plenum Press, New York 1968,
[28] Adi K, Richards P G. Quatitative seismology theory and methods[ M] Freeman, San Francisco, 1980 .
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