## Ray tracing for continuously rotated local
coordinates belonging to a specified anisotropy

**Einar Iversen** ** & **
**Ivan Psencik**
### Summary

Conventional ray tracing for arbitrarily
anisotropic and heterogeneous media is
expressed in terms of 21 elastic moduli
belonging to a fixed, global, Cartesian coordinate
system. Our principle objective is to obtain
a new ray-tracing formulation, which takes
advantage of the fact that the number of
independent elastic moduli is often less than 21,
and that the anisotropy thus has a simpler
nature locally, as is the case for transversely
isotropic and orthorhombic media.
We have expressed material properties and ray-tracing
quantities (e.g., ray-velocity and slowness
vectors) in a local anisotropy
coordinate system with axes changing directions
continuously within the model. In this
manner, ray tracing is formulated in terms
of the minimum number of required elastic
parameters, e.g., four and nine parameters
for P-wave propagation in transversely
isotropic and orthorhombic media, plus
a number of parameters specifying the rotation
matrix connecting local and global coordinates.
In particular, we parameterize this
rotation matrix by one, two, or three Euler angles.
In the ray-tracing equations, the
slowness vector differentiated with respect
to traveltime is related explicitly to the
corresponding differentiated slowness vector
for non-varying rotation and the cross
product of the ray-velocity and slowness vectors.
Our formulation is advantageous with
respect to user-friendliness, efficiency,
and memory usage. Another important aspect is
that the anisotropic symmetry properties
are conserved when material properties are
determined in arbitrary points by linear
interpolation, spline function evaluation, or by
other means.

### Keywords

Ray tracing, rotated anisotropy, transverse isotropy, orthorhombic
symmetry, TTI medium.

### Whole paper

The reprint is available in
PDF (283 kB).

*Stud. geophys. geod.*, **51 (2007)**, 37-58.

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