Comparison of two formulations of ray tracing for inhomogeneous anisotropic media, study of their behaviour in singular regions of S waves[MSc. project] Supervisor : RNDr. Ivan Pšenčík, CSc.
There are two possible formulations of the ray-tracing equations for inhomogeneous anisotropic media. One, proposed by Červený (1972), uses subdeterminants of the matrix of the Christoffel equation, see Gajewski & Pšenčík (1987). The other uses eigenvectors of the Christoffel matrix, see Červený (2001). Program ANRAY (Gajewski & Pšenčík, 1990), which will be used in this study, uses the first formulation. The first goal of this project is to develop an alternative of the ANRAY program, whose ray-tracing equations use the eigenvectors of the Christoffel matrix. The former and new versions will then be tested on accuracy and effectivity of S-wave ray computations. Tests will be made first in strongly anisotropic media, outside singular directions (i.e. directions, in which the phase speeds of both S waves coincide). Afterwards, accuracy and effectivity of both programs will be tested in a vicinity of singularities, for example, in a vicinity of axis of symmetry of a TI medium or in a vicinity of an intersection singularity, see Vavryčuk (2001). This study can be extended first to the development of the dynamic-ray-tracing equations using the eigenvectors of the Christoffel matrix, see Gajewski & Pšenčík (1990), and then to similar tests of accuracy and effectivity as in the case of kinematic ray tracing.
References:
Červený,V., 1972. Seismic rays and ray intensities in inhomogeneous anisotropic media. Geophys. J.R. astr. Soc. 28, 1-13.
Červený,V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge.
Gajewski,D. & Pšenčík,I., 1987. Computation of high-frequency seismic wavefileds in 3-D laterally inhomogeneous anisotropic media. Geophys.J.R.astr.Soc., 91, 383-411.
Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315.
Vavryčuk, V., 2001. Ray tracing in anisotropic media with singularities. Geophys.J.Int. 145, 265-276.
Construction of the complete propagator matrix of the dynamic ray tracing in the ANRAY package. Applications.[MSc. project] Supervisor : RNDr. Ivan Pšenčík, CSc.
Present version of the program package ANRAY (Gajewski & Pšenčík, 1990) provides only a part of the propagator matrix of the dynamic ray tracing (DRT) system corresponding to the point-source initial conditions. The propagator matrix of such a system is not symplectic. This considerably reduces applicability of the present version of the ANRAY package. The first goal of the project is to complete the used DRT propagator matrix to its 6x6 size, see Červený (2001). The complete propagator matrix can then be used for construction and testing of surface-to-surface propagator matrices (Červený & Moser, 2006) for inhomogeneous isotropic/anisotropic layered media. The surface-to-surface propagator matrices can be effectively used in paraxial ray methods (in which the ray quantities are calculated in a vicinity of a reference ray) in solutions of various boundary-value problems, to the computation of a two-point eikonal, of Fresnel zones, etc. in inhomogeneous isotropic/anisotropic layered structures.
References:
Červený,V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge.
Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315.
Moser,T.J. & Červený, V.: Paraxial ray methods for anisotropic inhomogeneous media. Geophysical Prospecting, 2006, in press.
Construction of the code for the R/T coefficients in anisotropic media, using ANRAY package. Study of behaviour of R/T coefficients.[MSc. project] Supervisor : RNDr. Ivan Pšenčík, CSc.
An important part of the program package ANRAY (Gajewski & Pšenčík, 1987) is the procedure for the evaluation of the plane-wave displacement R/T coefficients at plane interfaces separating homogeneous isotropic and/or anisotropic media, see, e.g., Červený (2001), Chapman (1994). The main goal of this project is a separate use of this procedure to investigate behaviour of R/T coefficients in various situations, which may occur in practical applications. Besides study of behaviour of the R/T coefficients in regular situations, study of effects of substitution of isotropic properties by anisotropic ones, etc., attention should also be devoted to behaviour of R/T coefficients in a vicinity of singular directions of S waves, i.e., directions, in which the two S waves propagate with nearly the same phase velocity. Relation of displacement and energy and/or normalized displacement R/T coefficients can be also studied.
References:
Červený,V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge.
Chapman,C.H., 1994. Reflection/transmission coefficient reciprocities in anisotropic media. Geophys.J.Int., 116, 498-501.
Gajewski,D. & Pšenčík,I., 1987. Computation of high-frequency seismic wavefileds in 3-D laterally inhomogeneous anisotropic media. Geophys.J.R.astr.Soc., 91, 383-411.
Introduction of a weak attenuation into the ANRAY package, tests of its accuracy and study of its effects[MSc. project] Supervisor : RNDr. Ivan Pšenčík, CSc.
Present version of the program package ANRAY (Gajewski & Pšenčík, 1990) can be used for computation of seismic wavefields in elastic, layered, laterally inhomogeneous isotropic and/or anisotropic structures. The goal of this project is to generalize the considered model by allowing weak attenuation, see Gajewski & Pšenčík (1992). An approximate consideration of weak attenuation requires only an additional integration along a real-valued ray (attenuation is generally connected with complex-valued rays), see, e.g., Červený (2001). The resulting code can be then used to study effects of weak attenuation on seismic waves propagating in inhomogeneous, anisotropic, weakly attenuating media. Possible additional goal is to test the accuracy of the generalized package with respect to available codes based on more accurate methods. e.g., matrix methods (see Wang, 1999).
References:
Červený,V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge.
Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315.
Gajewski,D. & Pšenčík,I.,1992. Vector wave fields for weakly attenuating anisotropic media by the ray method. Geophysics, 57, 27-38.
Wang,R., 1999. A simple orthonormalization method for the stable and efficient computation of Green's functions, Bull.seismol.Soc.Am., 89, 733-741.
Comparison of results of quasi-isotropic ray approximation with results of more accurate methods [MSc. project]
Supervisor : RNDr. Ivan Pšenčík, CSc.
Present version of the program package ANRAY (Gajewski & Pšenčík, 1990) allows approximate computation of coupled shear waves in inhomogeneous, weakly anisotropic media, which is based on the quasi-isotropic (QI) approximation (Pšenčík, 1998) of the coupling ray theory, see Coates & Chapman (1990), Klimeš & Bulant (2006), Červený (2001) and references there. Principal goal of this project is to test accuracy of the QI approximation in various models of inhomogeneous, weakly anisotropic media by comparing the results obtained with ANRAY package with results of available codes based on more accurate methods like, e.g., Chebyshev spectral method, see Tessmer (1995).
References:
Červený,V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge.
Coates,R.T. & Chapman,C.H., 1990. Quasi-shear wave coupling in weakly anisotropic 3-D media, Geophys.J.Int., 103, 301-320.
Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315.
Klimeš,L. & Bulant,P., 2006. Errors due to the anisotropic-common-ray approximation of the coupling ray theory. Stud.Geophys.Geod., 50, 463-477.
Pšenčík,I., 1998. Green's functions for inhomogeneous weakly anisotropic media. Geophys.J. Int., 135, 279-288. Tessmer,E., 1995. 3-D seismic modelling of general material anisotropy in the presence of the free surface by a Chebyshev spectral method. Geophys.J.Int., 121, 557-575.
Tessmer,E., 1995. 3-D seismic modelling of general material anisotropy in the presence of the free surface by a Chebyshev spectral method. Geophys.J.Int., 121, 557-575.
Generalization of the quasi-isotropic approach for the reference rays computed in the studied anisotropic medium[PhD. project] Supervisor: RNDr. Ivan Pšenčík, CSc. Present version of the program package ANRAY (Gajewski & Pšenčík, 1990) allows approximate computation of coupled shear waves in inhomogeneous, weakly anisotropic media, which is based on the quasi-isotropic (QI) approximation of the coupling ray theory, see Coates & Chapman (1990), Klimeš & Bulant (2006), Červený (2001) and references there. The amplitude contribution of the two coupled shear waves is calculated by integrating two coupled ordinary differential equations along a reference ray in a reference isotropic medium, see Kravtsov & Orlov (1980), Pšenčík (1998), Pšenčík & Delinger (2001). As shown, e.g., by Bulant & Klimeš (2004), the choice of the reference ray in a reference isotropic medium may have a substantial effect on accuracy of the QI approximation. The goal of this project is to substitute the presently used reference ray in an isotropic medium by a reference ray in the anisotropic medium, in which the QI computations are to be performed. The corresponding coupling equations should be modified accordingly, then coded and resulting code should be tested by comparing its results with results of available codes based on more accurate methods like, e.g., Chebyshev spectral method, see Tessmer (1995). References: Červený,V., 2001. Seismic Ray Theory: Cambridge University Press, Cambridge. Bulant & Klimeš, 2004. Comparison of quasi-isotropic approximations of the coupling ray theory with the exact solution in the 1-D anisotropic "oblique twisted crystal" model. Stud.Geophys.Geod., 48, 97-116. Coates,R.T. & Chapman,C.H., 1990. Quasi-shear wave coupling in weakly anisotropic 3-D media, Geophys.J.Int., 103, 301-320. Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315. Klimeš,L. & Bulant,P., 2006. Errors due to the anisotropic-common-ray approximation of the coupling ray theory. Stud.Geophys.Geod., 50, 463-477. Kravtsov, Yu.A. & Orlov,Yu.I., 1980. Geometrical optics of inhomogeneous media, Nauka, Moscow (in Russian). Pšenčík,I., 1998. Green's functions for inhomogeneous weakly anisotropic media. Geophys.J. Int., 135, 279-288. Pšenčík,I. & Dellinger.J., 2001. Quasi-shear waves in inhomogeneous weakly anisotropic media by the quasi-isotropic approach: a model study. Geophysics, 66, 308-319. Tessmer,E., 1995. 3-D seismic modelling of general material anisotropy in the presence of the free surface by a Chebyshev spectral method. Geophys.J.Int., 121, 557-575.
Generalization of the quasi-isotropic approach for layered media [PhD. project] Supervisor: RNDr. Ivan Pšenčík, CSc. Present version of the program package ANRAY (Gajewski & Pšenčík, 1990) allows approximate computation of coupled shear waves in smooth inhomogeneous, weakly anisotropic media, which is based on the quasi-isotropic (QI) approximation (Kravtsov & Orlov 1980, Pšenčík, 1998) of the coupling ray theory, see Coates & Chapman (1990), Klimeš & Bulant (2006), Červený (2001) and references there. The amplitude contribution of the two coupled shear waves is calculated by integrating two coupled ordinary differential equations along a reference ray in a reference isotropic medium, see Kravtsov & Orlov (1980), Pšenčík (1998), Pšenčík & Delinger (2001). The principal goal of this project is to generalize the QI approximation, which works, at present, in smooth media without interfaces, to layered media. This requires derivation of the transformation relations at interfaces for the two coupled differential equations, their coding and testing the resulting code by comparing its results with results of available codes based on more accurate methods like, e.g., Chebyshev spectral method, see Tessmer (1995). References: Červený,V., 2001. Seismic Ray Theory: Cambridge University Press, Cambridge. Coates,R.T. & Chapman,C.H., 1990. Quasi-shear wave coupling in weakly anisotropic 3-D media, Geophys.J.Int., 103, 301-320. Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315. Klimeš,L. & Bulant,P., 2006. Errors due to the anisotropic-common-ray approximation of the coupling ray theory. Stud.Geophys.Geod., 50, 463-477. Kravtsov, Yu.A. & Orlov,Yu.I., 1980. Geometrical optics of inhomogeneous media, Nauka, Moscow (in Russian). Pšenčík,I., 1998. Green's functions for inhomogeneous weakly anisotropic media. Geophys.J. Int., 135, 279-288. Pšenčík,I. & Dellinger.J., 2001. Quasi-shear waves in inhomogeneous weakly anisotropic media by the quasi-isotropic approach: a model study. Geophysics, 66, 308-319. Tessmer,E., 1995. 3-D seismic modelling of general material anisotropy in the presence of the free surface by a Chebyshev spectral method. Geophys.J.Int., 121, 557-575.
Coding and testing the first-order ray tracing [PhD. project] Supervisor: RNDr. Ivan Pšenčík, CSc. Pšenčík & Farra (2005) proposed an approximate P-wave ray tracing for smooth, inhomogeneous, weakly anisotropic media. It is of the first order with respect to deviations of anisotropy from isotropy. In contrast to standard ray tracing, which depends on 21 elastic parameters, the approximate P-wave ray tracing depends on only 15 weak-anisotropy parameters (non-dimensional parameters obtained by normalization of differences of elastic parameters or their combinations and squares of velocity of a reference isotropic medium). The equations are considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the approximate ray tracing equations differ only slightly from those for isotropic media. The first goal is to reformulate the ray-tracing equations of Pšenčík & Farra (2005) specified for transversely isotropic (TI) and orthorhombic (OR) media according to Iversen & Pšenčík (2006) so that the resulting ray tracer allows consideration of TI and/or OR media with varying orientation of symmetry axes. The next step of this project is to substitute the standard ray-tracing equations in the present version of the program package ANRAY (Gajewski & Pšenčík, 1990) by the newly developed P-wave ray-tracing equations. The resulting code should be tested on accuracy and speed with the standard version of ANRAY. The final, but not necessary, step is the corresponding modification of the dynamic-ray-tracing equations (Červený, 2001) and generalization of the procedure for layered media. References: Červený,V., 2001. Seismic Ray Theory: Cambridge University Press, Cambridge. Gajewski,D. & Pšenčík,I., 1990. Vertical seismic profile synthetics by dynamic ray tracing in laterally varying layered anisotropic structures. J.geophys.Res., 95, 11301-11315. Iversen,E. & Pšenčík,I., 2006. Ray tracing for continuously rotated local coordinates belonging to a specified anisotropy. In: Seismic Waves in Complex 3-D Structures, Report 16, pp.47-76, Dept.of Geophysics, Charles University Prague, online at http://sw3d.mff.cuni.cz. Pšenčík,I., Farra,V., 2005. First-order ray tracing for qP waves in inhomogeneous weakly anisotropic media. Geophysics, 70, D65-D75. |
