Dimitri Bevc, 3DGeo Development Inc. Summary Seismic imaging algorithms are generally applied to data which is redatumed to a planar surface. In regions of mild topography where the near-surface velocity is much slower than the subsurface velocity, a static shift is adequate for the transformation. However, when the necessary shift increases in magnitude and when the near surface velocity is comparable to the subsurface velocity, the static approximation becomes inadequate. Under these circumstances, a static shift distorts the wavefield and degrades the velocity analysis and imaging. In this case it is necessary to propagate the wavefield numerically to some level datum. This wave-equation datuming process may be used to ``flood'' the topography by filling it with a replacement velocity and upward continuing the data through it.

Introduction
Unlike datuming with static shifts, wave-equation datuming (Berryhill, 1979) removes the distortions caused by topography in a manner consistent with wavefield propagation. This insures that subsequent processing steps which assume hyperbolic form, or even more complicated trajectories consistent with wave propagation, can be accurately applied (Bevc, 1995).

Wiggins (1984) uses a Kirchhoff formulation to directly incorporate topography in prestack migration from a rugged surface. Reshef (1991) performs phase-shift downward extrapolation from a flat datum above the topography and adds data to the extrapolated wavefield each time the topographic surface is intersected. Beasley and Lynn (1992) introduced an elegant and simple algorithm to correct for the error caused by the static time shift based on the ``zero-velocity layer'' concept. In their finite-difference migration, Beasley and Lynn migrate data after static shift by setting the velocity in the diffraction term to zero above the topography.

Schneider et al. (1995) use refraction analysis to estimate velocity in the weathering layer, followed by wave-equation datuming to downward continue the data to the base of the layer with the estimated velocity, and then upward continue with a replacement velocity. Their results show improvement over refraction statics, but there is some error associated with the refraction velocity and layer determination. Zhu et. al, (1995) use turning ray tomography to estimate the near surface velocity used for datuming. In their examples, they downward continue the data to a flat processing surface, and obtain improved depth images after migration from the flat surface.

In order for all of the above methods to work, the near-surface velocity must be known or reliably estimated. The more general and problematic situation is one for which the velocity structure is unknown or difficult to estimate. In this situation, I upward continue the data to some planar datum above the topography with a replacement velocity. This unravels the distortions caused by the rugged acquisition topography and allows standard velocity estimation and imaging techniques to be more accurately applied to the data.

I demonstrate how Kirchhoff wave-equation datuming can be used to improve and simplify the early stages of processing and imaging by applying the process to an overthrust data set from the Canadian Rockies. In this example, wave-equation datuming provides a clear improvement over conventional statics processing. Because the data are regridded onto a flat evenly sampled datum, further processing is streamlined and structural interpretations are easier to incorporate into the analysis.

Overthrust Example
The Husky--Talisman Canadian overthrust data used for this study has excellent signal quality and a very interesting complex structure. High near-surface velocity and topography variation in excess of 200 m makes this data set an excellent candidate for wave-equation datuming. After wave-equation datuming the distorting effects of the topography are removed and the reflection events have laterally continuous trajectories consistent with wave propagation. The datuming operation also serves to regrid the data onto a uniformly sampled output mesh, fill in the shot gap, and attenuate steep-dip noise.

Figures 1 and 2 are the result of stacking the data after elevation statics and after wave-equation datuming using an approximate stacking velocity. The reflection event at about 2.5 s is the most prominent feature in the data. The stack displays better lateral continuity after wave-equation datuming (Figure 2) than after elevation statics (Figure 1). This is particularly evident between CMP 1000 and 1200 along the 2.5 s reflector. The dipping reflectors above 1 s and to the right of CMP 1300 are much better defined after wave-equation datuming than after static shift. Two other prominent features that look better after wave-equation datuming are the flat reflection event running across the section at about 2 s, and the dipping event at about 1.5 s, running from CMP 200 to 500. The diffraction events in the middle of the stacked sections are harder to evaluate, but it is evident that there are more diffractions at early time in the wave-equation datumed stack.

The result of migration after static shift is displayed in Figure 3 and the result of migration after wave-equation datuming is displayed in Figure 4. As before, better event continuity is preserved in Figure 4 after wave-equation datuming. The 2 s and 2.5 s reflectors running along the length of the section are more continuous, and the dipping events in the upper right corner of the image are better imaged. The dipping event extending from CMP 200 to 500 is more continuous after datuming. In Figure 4, this event can be followed to CMP 1000 at 1.8 s where it pinches out. This interpretation could not be confidently made using the image in Figure 3.

The migration after static shift does not image the complex structures in the middle of the section as well as the migration after wave-equation datuming. The diffraction events in Figure 2 have collapsed to image the near-surface structure. One of the most prominent structural features that is imaged in Figure 4 is the dipping spoon-like event extending from coordinates (750, 1.2) to (950, 1.4). The whole structure in this region and above is fairly complex and is much better imaged in Figure 4 than in Figure 3.

Figure 4 is a good starting image that is much easier to interpret than Figure 3. Geological boundaries could be readily defined and used to build a preliminary interval velocity model which could then be further refined by prestack depth migration. Better imaging of this data could be achieved by prestack migration or even by DMO before stack. However, the point here is to compare the effects of static shift and wave-equation datuming. The important point is that other than the datum correction step, both images in Figure 3 and 4 went through the same processing flow.

Conclusions
Wave-equation datuming cannot replace statics in all applications, and it is not a panacea to be applied under all circumstances, but under certain conditions, when topographic relief is substantial and when the assumption of large static shifts is invalid, it is the most accurate and appropriate method of datuming. This can be very helpful in the early stages of data processing when not much is known about the geological structure and the subsurface velocity distribution.

In the example I presented here, I show that upward continuing the data to a flat regularly sampled processing datum with Kirchhoff datuming is superior to elevation statics. It offers a relatively painless way of obtaining a preliminary structural image that does not require a detailed knowledge of the subsurface velocity distribution. This is because stacking and migration velocities are more accurately and easily determined without the unnecessary complication of nonhyperbolic distortion, which would arise if the data were processed from the topography or after elevation statics.

Acknowledgments This work was performed while I was at the Stanford Exploration Project. The Husky--Talisman Canadian overthrust data were kindly provided by Christof Stork of Landmark Graphics Corporation.

References
Beasley, C., and Lynn, W., 1992, The zero-velocity layer: Migration from irregular surfaces: Geophysics, 57 1435--1443.

Berryhill, J. R., 1979, Wave equation datuming: Geophysics, 44 1329--1344.

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

Reshef, M., 1991, Depth migration from irregular surfaces with depth extrapolation methods (short note): Geophysics, 56 119--122.

Schneider, W. A. Jr., Phillip, L. D., Paal, E. F., 1995, Wave-equation velocity replacement of the low-velocity layer for overthrust-belt data: Geophysics, 60 573--580.

Wiggins, J. W., 1984, Kirchhoff integral extrapolation and migration of nonplanar data: Geophysics, 49 1239--1248.

Zhu, X., Angstman, B.G., and Sixta, D.P., 1995, Overthrust imaging with tomo-datuming: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts 1397--1400.

Figure 1. NMO and stack after residual statics and elevation statics.

Figure 2. NMO and stack after residual statics and wave-equation datuming.

Figure 3. Time migration after residual statics, elevation statics, NMO, and stack. (place holder picture below)

Figure 4. Time migration after residual statics, wave-equation datuming, NMO, and stack.