Summation integrals for a Green function in a 3-D inhomogeneous anisotropic medium

Vlastislav Cerveny & Ivan Psencik


Summation integrals for time-harmonic Green function at an arbitrary point of a 3D inhomogeneous anisotropic medium containing smooth curved interfaces are studied. Formulae for the summation of paraxial Gaussian beams from contributions stored along rays in a vicinity of a receiver point or from contributions stored along a target surface are presented. The summation is done over the ray parameters of rays of elementary waves shot from a point source. The ray parameters may be defined, for example, as the take-off angles at the source or as two components of the slowness vector at the source. In the Gaussian beam formulae, the complex-valued travel time is used. In the limiting case of real-valued travel time, the integrals represent the summation over paraxial ray approximations, including Maslov-Chapman integrals. The computation of summation integrals requires the initial value ray tracing (no two-point ray tracing is required) and dynamic ray tracing with resulting values available at points along rays in a vicinity of the receiver point or along a target surface, with the receiver point on it or in its close vicinity. Dynamic ray tracing is performed in Cartesian coordinates. In dynamic ray tracing, computation of only the 3 x 2 parts of the 3 x 3 paraxial matrices is sufficient. The applicability of summation integrals is extended to inhomogeneous, weakly anisotropic media.

Whole paper

The paper is available in PDF (169 kB).

Seismic Waves in Complex 3-D Structures, 24 (2014), 131-158 (ISSN 2336-3827, online at