Rotation Parameters for Model Building and Stable Parameter Inversion in Orthorhombic Media

Symmetry groups commonly used to describe seismic anisotropy include transverse isotropy, orthorhombic, monoclinic, and triclinic. For all but the last, the choice of particular orientations of the coordinate system can substantially simplify parameterization of the anisotropy. Choosing a coordinate system requires defining a rotation in three-dimensional space relative to fixed world coordinates. We discuss here two major families of rotation parameterizations: Euler angles and axis/angle quaternion representations. Either method can work well for forward modeling. However, for traveltime tomography or full-waveform inversion, the inverse problem formulated in Euler angles can become ill-posed because very different choices of angle parameters can yield nearly identical data. In the worst case, there can be a complete loss of a degree of freedom, making it difficult to invert for important model parameters such as fracture orientation. Quaternions provide an alternative representation for 3D rotations that avoid these inversion problems. Quaternions also provide efficient and well-behaved interpolation of rotation angles, as well as differentials of data with respect to rotation parameters.