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.
Highlights
For more than 40 years, dynamic ray tracing has been a powerful method to compute important amplitude and phase attributes of high-frequency Green’s functions
Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems
Such transformation can be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation
Summary
For more than 40 years, dynamic ray tracing has been a powerful method to compute important amplitude and phase attributes of high-frequency Green’s functions. The derivatives of the phase-space perturbations considered for transformation may depend explicitly on the higher-order derivatives of the model parameters These perturbation quantities typically correspond to initial conditions or end results of dynamic ray tracing. We utilize the properties of basic splines, or B-splines (de Boor 1972), to ensure a consistent higher-order transformation of phase-space perturbations between Cartesian and ray-centred coordinates. In a numerical examples section we show applications of the derived theory, for isotropic and vertical transversely isotropic (VTI) versions of the Marmousi model In these examples, we (1) visualize the validity limits of the ray-centred coordinate system and (2) show some subtleties of the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates. For overview of the mathematical symbols used in the paper, see Table 1
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