A Hamiltonian approach to asymptotic seismic reflection and diffraction modelling

Abstract

A model‐independent Hamiltonian formulation of paraxial and diffracted ray‐tracing equations is presented. It is applied to asymptotic Green's function computations. The medium can have an arbitrary number of interfaces, possibly intersecting at diffracting edges and vertices. Continuously varying model parameters and anisotropy are allowed. The algorithm for elastic waves, involving accuracy control and amplitude computation, is implemented in a platform‐independent object‐orientated C++ package. Numerical tests and modelling examples are presented.

Decoupled elastic prestack depth migration

“…tracing (Gajewski and Pšencík, 1990;Hanyga et al, 2001). In this notation, the absolute geometrical spreading factor is given by J ðmÞ ðSÞ ¼ ð½B 1 x ðmÞ ðSÞ Â B 2 x ðmÞ ðSÞ Á l ðmÞ ðSÞÞ ;…”

“…We compute both FD and ray-trace travel times for the whole model. The Hamiltonian system of kinematic ray equations (Č ervený, 2001;Hanyga et al, 2001) is solved numerically using the Runge -Kutta method of order k z 2 (Cheney and Kincaid, 1999). Existing FD eikonal solvers (Podvin and Lecompte, 1991) are used to produce first-arrival travel time maps on the migration grid.…”

“…These travel times are interpolated from ray end points onto the migration grid by B-splines. The eikonal equation is invoked to control errors of ray tracing (Gajewski and Pšencík, 1990;Hanyga et al, 2001). We examine reciprocity T (m) (S,Q) = T (m) ( Q,S) to check accuracy of FD travel time computations.…”

“…See Appendix D for a detail. Also, the dynamic ray tracing system (DRT) is solved by tracing rays through the model over spec- ified range of ray take-off angles according to the ray history code (Gajewski and Pšencík, 1990;Hanyga et al, 2001). We examine the reciprocity relation G (v) ( Q,S) = G (v) (S,Q) to estimate numerical errors of amplitude computations.…”

Paraxial ray methods in inhomogeneous anisotropic media: Initial conditions

“…For example, the initial conditions for dynamic ray equations at a point source in an anisotropic medium were given by Pšenčík and Teles (1996), and by others. Point-source solutions were also treated by Hanyga et al (2001), who considered a very general anisotropic medium with an arbitrary number of interfaces, possibly intersecting at edges and vertices.…”