Differential equation for geometrical spreading on a ray and second derivatives of eikonal matrix structure

Dynamic ray tracing implies calculation of geometrical spreading for the purpose of the ray method amplitude obtaining. And for 3D inhomogeneous isotropic elastic media it is usually executed by solving M. Popov´s system of ordinary matrix differential equations with further determinant taking, which is exactly the geometrical spreading, or as consequence by solving the matrix Riccati differential equation followed by a special integration. The problem whether it is possible to deduce the scalar differential equation directly for the geometrical spreading is considered. It is also shown how second derivatives of eikonal matrix can be represented in terms of geometrical spreading.