Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

Abstract

Summary Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton-Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we 1) visualize the validity limits of the ray-centred coordinate system, 2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates, and 3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in ray-centred coordinates

“…We give a brief review of the concept ray-centred coordinates. For greater technical detail, see Iversen et al (2021).…”

“…see the initial discussion on the momentum vector in ray-centred coordinates in Iversen et al (2021). Moreover, contra-variant basis vectors denoted e 1 and e 2 are assigned, respectively, to the q 1 and q 2 coordinates; the components of these vectors form a 3 × 2 matrix E = {E i M }.…”

“…Klimeš (2006b) provides a fundamental treatment of the continuation of the bases of ray-centred coordinates along a certain reference ray, . Iversen et al (2021) obtain two equivalent sets of ODEs for the contra-variant basis E-the phase-velocity formulation…”

“…The standard option to dynamic ray tracing in ray-centred coordinates ( Červený 2001;Klimeš 2006b, section 5.4;Iversen et al 2021, Appendix A) corresponds to requiring A I J = 0 along the whole ray . This specification yields B I J constant along .…”

“…For overview of the mathematical symbols used in the paper, see Tables 1-5. Explicit results from Iversen et al (2021) are summarized in Tables 6-7.…”