Finite-difference solutions to the eikonal equation are attractive for generating traveltime tables for Kirchhoff migration because their implementations are generally fast and efficient. The traveltimes are smooth and they fill the computational grid. Throughout this study, I use an upwind finite-difference solution to the eikonal equation to generate traveltimes (van Trier and Symes, 1991; Popovici, 1991).

Green's functions based on first-arrival traveltime calculation methods result in poor images in structurally complex areas. Several reasons have been given for this failure:

- The high-frequency approximation of ray and eikonal methods breaks down in complex velocity models. In rapidly varying velocity models, different frequency components of the wavefield propagate at different velocities; therefore, summation trajectories based on only the high-frequency components may not capture the desired events.
- When high velocity zones are present, the first-arrivals may be non-energetic headwaves.
- As energy propagates in complex models, raypaths tend to cross eventually. This causes phase shifts and triplications. First-arrival traveltimes follow the fastest branch of the triplication bow-tie, which is also the low-energy branch.

Complexity of velocity models and validity of high-frequency approximations can be defined in various ways. In a large-scale depth model, there can be considerable relative changes in velocity. Changes in velocity distort the shape of the wave propagation front and create more opportunities for frequency components to separate, for headwaves to develop, and for triplications to occur. By dividing the depth model into small areas, I obtain a traveltime propagation front that better matches the wave equation propagation front. Therefore, imaging with first-arrival traveltimes is more accurate.

- Comparison of acoustic wavefields and first-arrival traveltimes
- Imaging with first-arrival traveltimes

2/12/2001