The Kirchhoff-Helmholtz integral for anisotropic elastic media. (English) Zbl 1163.74437

Summary: The Kirchhoff-Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth interface in acoustic or isotropic elastic media due to a given incident wavefield and observation points sufficiently far away from the interface. This integral makes use of the Kirchhoff approximation of the unknown scattered wavefield and its normal derivative at the interface in terms of the corresponding quantities of the known incident field. An attractive property of the Kirchhoff-Helmholtz integral is that its asymptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. Here, we extend the Kirchhoff-Helmholtz modeling integral to general anisotropic elastic media. It uses the natural extension of the Kirchhoff approximation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff-Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff-Helmholtz integral, we also derive an extension to anisotropic media of a useful decomposition formula of the geometrical spreading of a primary reflection ray.


74-XX Mechanics of deformable solids
76-XX Fluid mechanics
Full Text: DOI


[1] J.D. Achenbach, A.K. Gautesen, H. McMaken, Ray Methods for Waves in Elastic Solids, Pitman Advanced Publishing Program, Pitman, Boston, MA, 1982. · Zbl 0498.73020
[2] K. Aki, P.G. Richards, Quantitative Seismology: Theory and Methods, Vol. 1, Freeman, New York, 1980.
[3] B.B. Baker, E.T. Copson, The Mathematical Theory of Huygens’ Principle, 3rd Edition, Chelsea, New York, 1987. · Zbl 0653.01016
[4] N. Bleistein, Mathematical Methods for Wave Phenomena, Academic Press, Orlando, FL, 1984. · Zbl 0554.35002
[5] V.D. Kupradze, Potential Methods in the Theory of Elasticity (translation), Israel Program for Scientific Translations, Jerusalem, 1965.
[6] Frazer, L.N.; Sen, M.K., Kirchhoff – helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium. I. theory, Geophys. J. R. astr. soc., 89, 121-147, (1985)
[7] Tygel, M.; Schleicher, J.; Hubral, P., Kirchhoff – helmholtz theory in modelling and migration, J. seismic exploration, 3, 203-214, (1994)
[8] de Hoop, M.V.; Bleistein, N., Generalized Radon transform inversions for reflectivity in anisotropic elastic media, Inverse probl., 13, 669-690, (1997) · Zbl 0884.44003
[9] Hubral, P.; Schleicher, J.; Tygel, M., Three-dimensional paraxial ray properties. I. basic relations, J. seismic exploration, 1, 265-279, (1992)
[10] Hubral, P.; Schleicher, J.; Tygel, M., Three-dimensional paraxial ray properties. II. applications, J. seismic exploration, 1, 347-362, (1992)
[11] V. Červený, Seismic Wavefields in Three-dimensional Isotropic and Anisotropic Structures, Cambridge University Press (2001).
[12] Chapman, C.H.; Coates, R.T., Generalized Born scattering in anisotropic media, Wave motion, 19, 309-341, (1994) · Zbl 0926.74055
[13] Ursin, B.; Tygel, M., Reciprocal volume and surface scattering integrals for anisotropic elastic media, Wave motion, 26, 31-42, (1997) · Zbl 0954.74532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.