## Transformation
of spatial and perturbation derivatives of travel time
at a general interface between two general media

**Ludek Klimes**
### Summary

We consider the partial derivatives of travel time with respect
to both spatial coordinates and perturbation parameters.
We derive the explicit equations for transforming
these travel-time derivatives of arbitrary orders
at a general smooth interface between two general media.
The equations are applicable
to both real-valued and complex-valued travel time.
The equations are expressed in terms of
a general Hamiltonian function and are applicable
to the transformation of travel-time derivatives
in both isotropic and anisotropic media.
The interface is specified by an implicit equation.
No local coordinates are needed for the transformation.

### Keywords

Ray theory, Hamilton-Jacobi equation, eikonal equation,
travel time (action), spatial derivatives of travel time,
perturbation derivatives of travel time,
reflection or refraction at curved interfaces,
anisotropy, heterogeneous media,
paraxial approximation, Gaussian beams, wave propagation.

### Whole paper

The paper is available in
PDF (120 kB).

In: Seismic Waves in Complex 3-D Structures, Report 20,
pp. 103-114, Dep. Geophys., Charles Univ., Prague, 2010.