Cell ray tracing for smooth, isotropic media: a new concept based on a generalized analytic solution

For an isotropic medium characterized by constant second-order partial derivatives of squared slowness, the analytic solution of the associated initial-value ray tracing problem is derived in two and three dimensions. Formulae for both kinematic and dynamic quantities are provided as a function of a suitably chosen curve parameter. These formulae represent a generalization of the well-known constant-gradient case. Model design for cell ray tracing applications consists of two steps. First, the smooth medium must be suitably subdivided into triangular (or tetrahedral) cells. As a rule of thumb, the dimensions of the cells should be taken roughly one order of magnitude smaller than the smallest spatial wavelength represented in the structure to be modelled. Examples of suitable subdivisions for 2-D applications are given. Second, to approximate a given geological structure, cell constants are adjusted such that both the velocity and the velocity gradient are continuous at internal cell boundaries. Consequently, no phase matching is required at these boundaries in dynamic applications. For the problem of finding the intersection between the ray and the boundaries of a cell, an iterative procedure is proposed. Numerical tests prove that this procedure converges quickly, even for fairly complex structures.