One of the most important problems in exploration seismology is to relate the surface seismic measurements with the subsurface geologic parameters. The concept of wavefront curvature has been in extensive use for this purpose. Byun (1982) developed relationships between several measurable seismic parameters (e.g., geometrical spreading and normal moveout velocity) and parameters of the media with elliptical velocity dependencies.This paper extends the wavefront curvature concept to more general, transversely isotropic media. After a brief discussion on ray tracing, a procedure is developed to describe the local properties of the ray based on an elliptical surface fit to the actual wave surface. The apparent velocities of the elliptical fit are then used to generalize the seismic parameters developed in Byun (1982).Simple numerical experiments are given to demonstrate the explorational significance of the theory. It is shown that the measurements of the normal moveout velocity are not sufficient to estimate the velocity structure of the transversely isotropic medium. The 'side-slip' effect can lead to significant errors in depth-mapping dipping reflectors.