Two-point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media: tests of accuracy

Umair bin Waheed, Ivan Psencik, Vlastislav Cerveny, Einar Iversen & Tariq Alkhalifah


On several simple models of isotropic and anisotropic media we study the accuracy of the two-point paraxial traveltime formula designed for the approximate calculation of the traveltime between points S' and R' located in the vicinity of points S and R on a reference ray. The reference ray may be situated in a 3D inhomogeneous isotropic or anisotropic medium with or without smooth curved interfaces. The two-point paraxial traveltime formula has the form of the Taylor expansion of the two-point traveltime with respect to spatial Cartesian coordinates up to quadratic terms at points S and R on the reference ray. The constant term and the coefficients of the linear and quadratic terms are determined from quantities obtained from ray tracing and linear dynamic ray tracing along the reference ray. The use of linear dynamic ray tracing allows the evaluation of the quadratic terms in arbitrarily inhomogeneous media and, as shown by examples, it extends the region of accurate results around the reference ray between S and R (and even outside this interval) obtained with the linear terms only. Although the formula may be used for very general 3D models, in this paper we concentrate on simple 2D models of smoothly inhomogeneous isotropic and anisotropic (~ 8% and ~ 20% anisotropy) media only. On tests, in which we estimate two-point traveltimes between a shifted source and a system of shifted receivers, we show that the formula may yield more accurate results than numerical solution of an eikonal-based differential equation. The tests also indicate that the accuracy of the formula depends primarily on the length and the curvature of the reference ray and only weakly depends on anisotropy. The greater is curvature of the reference ray, the narrower its vicinity, in which the formula yields accurate results.

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In: Seismic Waves in Complex 3-D Structures, Report 23, pp. 149-184, Dep. Geophys., Charles Univ., Prague, 2013.