## 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 have studied 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 twopoint
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, we concentrated on simple 2D models of smoothly
inhomogeneous isotropic and anisotropic (~8% and ~20%
anisotropy) media only. On tests, in which we estimated twopoint
traveltimes between a shifted source and a system of shifted
receivers, we found that the formula may yield more accurate results
than the numerical solution of an eikonal-based differential
equation. The tests also indicated 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 the
curvature of the reference ray, the narrower its vicinity, in which
the formula yields accurate results.

### Whole paper

The reprint is available in
PDF (3035 kB !).

*Geophysics*, **78** (2013), C41-C56.