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

Marmousi Model

The Marmousi model (Versteeg, 1994) has large velocity variations and is challenging for Kirchhoff migration with first-arrivals (Geoltrain and Brac, 1993). We apply a single-fold 2D triangular smoothing of radius $20$ m to the original model (see Figure 9) to remove only sharp velocity discontinuities but retain the complex velocity structures. Because wave-fronts change shapes rapidly, the traveltime interpolation may be subject to inaccurate source-derivatives and provide less satisfying accuracy compared to that in a simple model. Although the derivative computation in the proposed eikonal-based method is source-sampling independent, in practice we should limit the interpolation interval to be sufficiently small, so that the traveltime curve could be well represented by a cubic spline. For the smoothed Marmousi model, we use a sparse source sampling of $0.2$ km based on observations of the horizontal width of major velocity structures. Figures 9 and 10 compare the traveltime interpolation errors of three methods as in Figure 3 for a source located at $(0,3.1)$ km from nearby source samples at $(0,3)$ km and $(0,3.2)$ km. Figure 11 plots a reference traveltime curve for the fixed subsurface location $(2,3.3)$ km computed by a dense eikonal solving of $4$ m source spacing against curves produced by the interpolations. While these comparisons vary between different source intervals and subsurface locations, the cubic Hermite interpolation out-performs the linear and the shift interpolations except for the source singularity region. However in Figure 9 the errors are relatively large in the upper-left region. These errors occur due to the collapse of overlapping branches of the traveltime field (Xu et al., 2001) that causes wave-front discontinuities and undermines the assumptions of the proposed method.

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Figure 9. (Top) the smoothed Marmousi model. The model has a $4$ m fine grid. (Bottom) the traveltime error by the cubic Hermite interpolation.

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Figure 10. The traveltime error by (top) the linear interpolation and (bottom) the shift interpolation.

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Figure 11. Traveltime interpolation for a fixed subsurface location. Compare between the result from a dense source sampling (solid blue), cubic Hermite interpolation (dotted magenta), linear interpolation (dashed cyan) and shift interpolation (dashed black). The $l2$ norm of the error (against the dense source sampling results) of $49$ evenly interpolated sources between interval $(0,3)$ km and $(0,3.2)$ km for all locations but the top $100$ m source singularity region are $3.9$ s, $9.2$ s and $11.6$ s respectively.

One strategy for imaging multi-arrival wavefields with first-arrival traveltimes is the semi-recursive Kirchhoff migration (Bevc, 1997). It breaks the image into several depth intervals, applies Kirchhoff redatuming to the next interval, performs Kirchhoff migration from there, and so on. The small redatuming depth effectively limits the maximum traveltime and the evolving of complex waveforms before the most energetic arrivals separate from first-arrivals. Since Kirchhoff redatuming also relies on traveltimes between datum levels, our method can be fully incorporated into the whole process. Again, for simplicity, we do not consider amplitude factors during migration. We use the Marmousi dataset with a source/receiver sampling of $25$ m. Due to the source and receiver reciprocity, the receiver side interpolations are equivalent to those on the source side. Figure 12 is the result of a Kirchhoff migration with eikonal solvings at each source/receiver location, i.e. no interpolation performed. Only the upper portion is well imaged. Figure 13 shows the image after employing the cubic Hermite interpolation with a $0.2$ km sparse source/receiver sampling, which means $7$ source interpolations within each interval. Even though a $7$ times speed-up is not attainable in practice due to the extra computations in source-derivative and interpolation, we are still able to gain an approximately $5$-fold cost reduction in traveltime computations, while keeping the image quality comparable between Figures 12 and 13. Next, following Bevc (1997), we downward continue the data to a depth of $1.5$ km in three datuming steps. The downward continued data are then Kirchhoff migrated and combined with the upper portion of Figure 13. We keep the same $0.2$ km sparse source/receiver sampling whenever eikonal solvings are required in this process. Figure 14 shows the image obtained by the semi-recursive Kirchhoff migration. The target zone at approximately $(2.5,6.5)$ km appears better imaged.

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Figure 12. Image of Kirchhoff migration with first-arrivals (no interpolation).

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Figure 13. Image of Kirchhoff migration with first-arrivals and a sparse source/receiver sampling.

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Figure 14. Image of semi-recursive Kirchhoff migration with a three-step redatuming from top surface to $1.5$ km depth and a $0.5$ km interval each time. The sparse source/receiver sampling is the same as in Figure 13.

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