Various topics in pseudo-differential operator theory applied to scalar qP wave propagation in a transversely isotropic medium
P. F. Daley
Scalar wave equation approximations for quasi-compressional (qP) propagation in a transversely isotropic medium are developed and solution options presented. The initial eikonals are obtained from linearization of the exact eikonal, or more correctly the linearized phase velocity, as well as from other approximations of the exact eikonal. The linearized approximate phase velocities or eikonals are used to construct partial differential equation of order four in spatial derivatives and order two in time using pseudo-differential operator theory. The assumption that some or all of the 2D model space is rotated at some angle with respect to a Cartesian model coordinate system is examined in a cursory manner. That the medium is weakly anelliptic is understood. Also, as the anisotropic parameters are usually spatially dependent this fact is taken explicitly into consideration when constructing the partial differential equation. The degenerate, or elliptical case, is also investigated as it is much simpler with results that should at least grossly approach the full scalar qP wave equation.