Upward continuation

Upward continuing the synthetic data removes the distortions. Figures b through d are the result of upward continuing the synthetic of Figure a with the Kirchhoff, finite-difference, and phase-shift operators. The output datum for all these upward continuations is just above the highest topography at a datum elevation of 210 m. The results exhibit different types of artifacts, but they are all kinematically correct. The Kirchhoff result is the most pleasing, with the fewest artifacts.

The Kirchhoff result is generated using the one-step time domain formulation of equation (). In this instance, the near-field term has been retained and the $D_{\frac{1}{2}}$ operator has been replaced by a convolutional operator which depends on the delay $\tau$.The corresponding matrix formulation is the time domain equivalent of equation () with H operator submatrices replaced by $\tau$-dependant convolutional operators.

The finite-difference result is generated using the $45^{\circ}$equation and corresponds to the matrix formulation of equation (). This result exhibits boundary artifacts and finite-difference dispersion. The phase-shift result is generated using an implementation corresponding to the matrix formulation of equation (). It is kinematicly equivalent to the other two upward continuations, but exhibits Fourier wrap-around artifacts. The result is more aesthetically pleasing than the finite-difference result, but has more artifacts than the output of the Kirchhoff method.

The results of upward continuation can be compared with Figure , which is the result of calculating the synthetic result directly at the datum elevation of 210 m. This Figure represents the desired output of the datuming algorithms, and it is best matched by the Kirchhoff datuming result. There are minor differences between Figure  and Figure b. There is an imperceptible phase shift between the two figures, and there are a few added artifacts in the datuming result. For the most part, the Kirchhoff datuming result matches Figure  very well.

 

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Figure 7 (a) Synthetic data, and upward continuation with (b) Kirchhoff, (c) finite-difference, and (d) phase-shift methods.

 

syn21 Figure 8 Synthetic data generated at the flat output datum, at an elevation of 210 m. Compare to Figures b through d.

In Figure a, the synthetic data are migrated using the adjoint of the program which generated the synthetic. This migration requires knowledge of the subsurface velocity. In Figures b through c the upward continued synthetics of Figures b through c are migrated from the flat datum. These three migrations were performed with the same Kirchhoff imaging algorithm. In these migrations it is evident how the different dip components of the wavefield are transformed by the different datuming operators. The migration after finite-difference datuming results in the poorest image because of the dip limitations of the $45^{\circ}$ approximation to the one-way wave equation. The flanks of the anticline/syncline are not imaged and the point diffractors are poorly imaged. The migration after Kirchhoff datuming (Figure b) images all the features of the subsurface model. This is because in the Kirchhoff datuming step, much of the high dip energy has been captured during the upward continuation by padding traces on either side of the section. The migration after phase-shift datuming (Figure d) results in a satisfactory image that compares favorably with the result of imaging from the topography (Figure a). The phase-shift method preserves the steep dips, and performs much better than the finite-difference method. In the phase-shift method, some of the steep dip energy is wrapped rather than propagated off the grid. So, if a phase-shift migration had been used, the high dip resolution would be similar to the Kirchhoff method.

In Chapter  I will demonstrate the utility of upward continuing data acquired along a rugged topography to a flat processing datum in order to facilitate wave-based processing and to better determine subsurface velocity.

 

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Figure 9 (a) Migration of synthetic data and migration of synthetic data upward continued with (b) Kirchhoff, (c) finite-difference, and (d) phase-shift methods.