Kirchhoff migration using eikonal-based computation of traveltime source-derivatives

Discussion

The proposed approach could be implemented either along with a finite-difference eikonal solver or separately. Our current implementation outputs both traveltime and source-derivative at the same time, with a roughly 30% extra cost per eikonal solve compared to a FMM solver without the source-derivative functionality. An interpolation with these source-derivatives is superior to the ones without and thus enables an accurate traveltime-table compression. For 3D datasets, as both inline and crossline directions may benefit from the source-derivative and interpolation, the overall data compression could be significant. For instance, interpolating $10$ shots within each sparse source sampling interval in both inline and crossline directions leads to an approximately $100$-fold savings in traveltime storage. The method could be further combined with an interpolation within each source, for example from a coarse grid to a fine grid, for a greater data compression.

While our implementation is for first-arrivals only, the governing equations are valid also for other characteristic branches, for example the most energetic arrivals. However, an underlying assumption of the proposed method is a continuous change in the wave-front of selected arrivals within individual sources. For first-arrivals, this condition always holds valid. However, the most energetic wave-front can be more complicated than that of first-arrival, for example only piece-wise continuous, which may lead to a potential degradation in accuracy. For example, Nichols (1994) showed the most energetic wave-fronts in the Marmousi model. Another assumption is that the traveltime source-derivatives are continuous between nearby sources. This condition breaks down when multi-pathing takes place. Vanelle and Gajewski (2002) suggested to smooth traveltimes around the discontinuities in order to overcome this limitation. In theory, one can try to identify the discontinuities and only perform interpolation within individual continuous pieces by using the eikonal-based source-derivatives. By doing so, one should be able to recover branch jumping in interpolated traveltimes, but only for those locations within the identified continuous pieces. For the discontinuities themselves as well as the gaps between them, additional eikonal solving may be required. An efficient implementation of this strategy remains open for future research.

Kirchhoff migration using eikonal-based computation of traveltime source-derivatives