by Einar Iversen, Bjørn Ursin, Teemu Saksala, Joonas Ilmavirta, Maarten V. de Hoop
Abstract:
With 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 high-frequency 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.
Reference:
Higher-order Hamilton-Jacobi perturbation theory for anisotropic heterogeneous media: Dynamic ray tracing in Cartesian coordinates (Einar Iversen, Bjørn Ursin, Teemu Saksala, Joonas Ilmavirta, Maarten V. de Hoop), Geophysical Journal International, volume 216, number 3, pp. 2044–2070, 2019.
[show abstract]
[hide abstract]
With 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 high-frequency 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.
Bibtex Entry:
@article{hj-high-1,
author = {Einar Iversen and Bj{\o}rn Ursin and Teemu Saksala and Joonas Ilmavirta and Maarten V. de Hoop},
title = {{Higher-order Hamilton-Jacobi perturbation theory for anisotropic heterogeneous media: Dynamic ray tracing in Cartesian coordinates}},
month = mar,
year = {2019},
url={http://users.jyu.fi/~jojapeil/pub/hj-high-1.pdf},
journal = {Geophysical Journal International},
volume = {216},
number = {3},
pages = {2044--2070},
doi = {10.1093/gji/ggy533},
gsid = {5106977690071546183},
abstract = {With 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 high-frequency 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.}
}