## 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 a medium possessing a transversely
isotropic or orthorhombic symmetry locally. We have expressed the material
properties and the ray-tracing quantities (e.g., the ray-velocity and slowness
vectors) in a local anisotropy coordinate system with axes changing directions
continuously within the model. In this manner, a transversely isotropic medium
is described locally by five elastic moduli and two angles; an orthorhombic
medium is described locally by nine elastic moduli and three angles. The new
formulation is advantageous with respect to user-friendliness, efficiency, and
memory usage. Concerning improved efficiency, a prominent role is played by the
cross product of the ray-velocity and slowness vectors specified in the local
coordinate system. 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

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In: Seismic Waves in Complex 3-D Structures, Report 16,
pp. 47-76, Dep. Geophys., Charles Univ., Prague, 2006.

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