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

Abstract

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.