In the preceding section I demonstrated that wave-equation extrapolation is superior to static shift when raypaths are not vertical. This conclusion in itself is not new and has been demonstrated by other authors Reshef (1991); Shtivelman and Canning (1988); Wiggins (1984). None of these authors discuss the determination of the velocity to be used in the datuming algorithm at any great length. In regions of large topographic relief the subsurface velocity structure is difficult to determine, therefore it may not always be possible to continue the data to some lower datum directly. I have shown that it is feasible to continue data upwards and perform wave-equation based processing at the new flat datum. Although I show this for time migration, the same logic indicates that velocity analysis, DMO, or pre-stack migration could be applied at the higher datum.
Some resolution of steep dips is lost due to loss of aperture as the wavefield is extrapolated upward. This suggests that it may be adequate to perform the datuming by finite-differencing with the or equations. This may be particularly applicable if it is desired to downward continue the data after the subsurface velocity structure has been determined because the finite-difference formulation allows lateral velocity variation.
Because the Kirchhoff datuming algorithm can be formulated as conjugate pairs for upward and downward continuation, it is possible to formulate re-datuming as a least-squares minimization problem using the conjugate-gradient descent method. This is done by Ji and Claerbout 1992 for a frequency domain datuming scheme using the same concept as Reshef 1991. Popovici 1992 formulates v(x,z) datuming in the frequency domain as upward and downward conjugate pairs which could also be used in a conjugate-gradient scheme. The use of such least-squares datuming may improve the method by eliminating artifacts such as those visible in Figure 11b. Least-squares datuming may also improve results where aperture is limited, particularly in pre-stack datuming.
No matter what the datuming algorithm is based on (Kirchhoff, Fourier, or finite-difference), my proposed application can be outlined in three major steps:
The next phase in this investigation is to apply wave-equation datuming to an actual land data set with irregular acquisition topography.