Imaging multivalued arrivals using semi-recursive Kirchhoff


Dimitri Bevc, 3DGeo Development Inc., 465 Fairchild Drive, Suite 226, Mountain View, CA 94043, USA, Email:



Many researchers have discovered that Kirchhoff algorithms do a poor job of imaging complex structures. Geoltrain and Brac (1993) propose to remedie this by either ray tracing to find the most energetic arrivals, or by calculating dynamically correct multivalued Green's functions. Nichols (1994) calculates band-limited Green's functions to estimate the most energetic arrivals. My approach is simpler: by breaking up the velocity structure, I am able to calculate traveltimes in subsets of the velocity model where finite-differencing the eikonal equation is valid. My approach therefore has the advantage of being able to use any simple first-arrival traveltime algorithm, thus benefiting from the computational efficiency, robustness, and simplicity of such methods.

The semi-recursive migration produces accurate images of complex structures by downward continuing the data and imaging from a lower datum. The method alternates steps of datuming and imaging. Because traveltimes are computed for each step, the adverse effects of caustics, headwaves, and multivalued arrivals do not develop.


The Marmousi velocity model generates complex propagation paths in which late energetic arrivals are not fit well by first-arrival finite-difference traveltimes. In the presentation I will demonstrate that as the wavefield evolves, complex propagation effects begin to manifest themselves and the arrivals become multivalued, so that the first-arrival traveltimes no longer match the most energetic wavefront. I will show that by starting the traveltime calculation from a depth level deeper in the velocity model, it is possible to compute first-arrival traveltime tables which match the acoustic wavefield propagation and accurately parameterize the asymptotic Green's functions required for Kirchhoff imaging (Bevc, 1995).

The top image in Figure 1 is the result of standard Kirchhoff migration of the Marmousi synthetic using first-arrival traveltimes calculated with a finite-difference eikonal equation solver. The faults and beds in the upper portion are well imaged; however, the anticlinal structure below 2300 m and the target zone at a lateral position of about 6500 m and depth of 2500 m are not well imaged. The central portion of the image, at the target zone, corresponds to regions where the acoustic wavefield and first-arrival traveltime do not match. The propagation here is complicated by the overlying faults which contain fast and slow velocity regions, and by the high velocity salt which partially overlays the target, resulting in multivalued arrivals.

The result of applying the semi-recursive algorithm to the Marmousi synthetic data set is displayed in the bottom image of Figure 1. It is generated by downward continuing the data to a depth of 1500 m in three datuming steps. The downward continued data are then migrated and combined with the previous image of the upper 2000 m. The bottom image in Figure 1 is a clear improvement over the previous migration result. The anticlinal structure below the salt and the target are now clearly imaged. Events which unconformably define the top of the anticline, the anticline events themselves, and the target events, are clearly imaged. The resolution is so good that the flat spot in the reservoir and the strongly reflective cap stand out clearly.


I obtain excellent imaging results in the presence of multivalued arrivals by combining wave-equation datuming and Kirchhoff migration into a semi-recursive migration method. In this case, first-arrival traveltimes produce satisfactory images because the velocity model is subdivided and traveltimes are calculated under conditions where finite-differencing the eikonal equation is valid and where traveltimes correspond to energetic arrivals.


Bevc, D., 1995, Imaging under rugged topography and complex velocity structure: Ph.D. Thesis, Stanford University.

Geoltrain, S. and Brac, J., 1993, Can we image complex structure with first-arrival traveltime?, Geophysics 58, 564--575

Nichols, D., 1994, Imaging complex structures using band limited Green's functions: Ph.D. Thesis, Stanford University.

Figure 1. Migrated image using traveltimes calculated from the surface, and traveltimes calculated from a depth of 1500 m. The lower part of the image was obtained by migrating data which was redatumed to a depth of 1500 m in three steps of 500 m each.

For a more detailed article on this subject, download my Geophysics preprint, or thesis.

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