While these finite-difference methods are enjoying popularity in more and more areas of seismic data processing, their limitations are also being revealed. Although these methods give accurate traveltimes, they may produce large errors in calculating traveltime gradients when velocity varies rapidly. Because the geometrical amplitudes honor the zeroth order transport equation that involves traveltime gradients, these errors are reflected in the amplitude calculation. The large errors in traveltime gradients are caused by the discontinuous representation of a velocity field that a finite-difference method assumes, as well as by the first-order finite-difference approximations that the method makes.
The traveltime field in a varying velocity medium is generally a multi-valued function of subsurface positions. But all existing finite-difference methods compute only the traveltime of the first arrival at each subsurface point. One reason is that the wavefront of a first arrival is always continuous; hence finite-difference approximations to the eikonal equation are valid. Another reason is that among different arrivals at a subsurface point, the first arrival is easy to identify because it has the minimum traveltime. When imaging reflection seismic data, the events corresponding to energetic arrivals should be used. However, for a complex velocity model, the energy carried by first arrivals may become considerably weaker than that carried by other later arrivals. In such a situation, the traveltimes of more energetic, later arrivals need to be computed.
This paper introduces the local ray-tracing method for wavefront propagation. Because the method assumes a continuous velocity representation and propagates local circular wavefronts by following local rays, it gives more accurate traveltimes and amplitudes than a finite-difference method. The method computes the traveltimes of multiple arrivals by disassembling a multi-valued traveltime field into several single-valued fields, and then computing each of them separately. These single-valued fields may include discontinuities that a finite difference method cannot handle. The local ray-tracing method can propagate wavefronts of discontinuities.
This paper begins with the description of a continuous representation of a velocity field. It then presents the details of the local ray-tracing method, including a local wavefront propagation scheme, a global updating scheme, and initialization. Several issues on multiple arrivals are also addressed. Finally, the accuracy of the method is examined, and several synthetic examples are used to demonstrate the method.