3-DIMENSIONAL PRIMARY ZERO-OFFSET REFLECTIONS

Zero-offset reflections resulting from point sources are often computed on a large scale in three-dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical-spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so-called dynamic ray-tracing system down and up to the two-way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray-tracing system in a downward direction with only the initial condition of a point source at the earth's surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero-offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth's surface) the complete Fresnel volume around the two-way normal ray can be found. This should be known to ascertain the validity of the computed zero-offset event. A careful analysis of the problem as performed here shows that round-trip integrations of the dynamic ray-tracing system following the actually propagating wavefront along the two-way normal ray need never be considered. In fact some useful quantities related to the two-way normal ray (e.g., the normal-moveout velocity) require only one single integration in one specific direction only. Finally, a two-point ray tracing for normal rays can be derived from one-way dynamic ray tracing.