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

Abstract

E. Iversen et al. SUMMARYWith a Hamilton-Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order phase-space perturbation derivatives along a reference ray. Such derivatives can be exploited for calculation of geometrical spreading on the reference ray, and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of the first-order phase-space perturbation derivatives has historically been referred to as dynamic ray tracing.The reason for this is its importance in the process of calculating amplitudes along the reference ray. We extend the standard dynamic ray tracing scheme to include higher orders in the phase-space perturbation derivatives. The main motivation is to extrapolate and interpolate important amplitude and phase properties of highfrequency Green's functions with better accuracy. Principal amplitude coefficients, geometrical spreading factors, traveltimes, slowness vectors, and curvature matrices are examples of quantities for which we enhance the computation potential. This, in turn, has immediate applications in modelling, mapping, and imaging. Numerical tests for 3D isotropic and anisotropic heterogeneous models yield clearly improved extrapolation results for traveltime and geometrical spreading. One important conclusion is that the extrapolation function for geometrical spreading must be at least third order to be appropriate at large distances away from the reference ray. E. Iversen et al.The developed methodology has the following main applications:• Fast computation of high-frequency elastic-wave Green's functions through (Hermite or spline) interpolation and extrapolation of amplitude and phase with derivatives. Our procedure holds in generally anisotropic media, leading to systems of equations describing the propagation of elastic waves, of principal type.

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

“…where H(x, p) is the Hamiltonian (see, e.g. Červený 2001;Iversen et al 2019). As indicated by the form of eq.…”

“…We comment on a special case, for which the mapping of the derivatives of traveltime becomes particularly simple, namely, the traveltime function arising as a result of an initial plane wave. Iversen et al (2019) describes in detail how one can define the initial condition for the dynamic ray tracing quantities in Cartesian coordinates, referred to as derivatives of phase-space perturbations, and also how to derive from these quantities the derivatives of traveltime up to order four: p i , M ij , M ijk and M ijkl . Essential in this setup is the use of constraint relations of the standard type ( Červený 2001) and of higher order (Iversen et al 2019).…”

“…Iversen et al (2019) describes in detail how one can define the initial condition for the dynamic ray tracing quantities in Cartesian coordinates, referred to as derivatives of phase-space perturbations, and also how to derive from these quantities the derivatives of traveltime up to order four: p i , M ij , M ijk and M ijkl . Essential in this setup is the use of constraint relations of the standard type ( Červený 2001) and of higher order (Iversen et al 2019). In particular, when using the plane-wave initial condition, one can from these constraint relations alone obtain the complete set of dynamic ray tracing quantities.…”

“…To establish the fifth-order derivatives M ijklm , we therefore have available the required model input data, but there are two minor extra issues to be handled, relative to the description in Iversen et al (2019): (1) the set of constraint relations must be extended with one additional order; (2) the set of equations for the derivatives of traveltime must be extended with one additional equation, thus involving M ijklm . We do not write the specific equations here, as they would need a formal introduction of the derivatives of perturbations in phase space, which is outside the scope of the paper.…”

Taming the divergent terms in the scattering series of Born by renormalization

“…Most traveltime approximations employ perturbation theory to calculate the traveltimes in anisotropic media (Ursin, 1982;Gjøystdal et al, 1984;Červenỳ et al, 1984;Alkhalifah, 2011a,b;Červenỳ et al, 2012;Iversen et al, 2018). Alkhalifah (2011a,b) derived the traveltime approximations and used them for scanning anisotropic parameters in transversely isotropic media with a vertical-symmetry axis (VTI) and transversely isotropic media with a tilted symmetry axis (TTI) media.…”