## 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**
### Summary

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.

### Whole paper

The paper is available in
PDF (2414 kB).

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