Steep-dip implicit methods
Figure 4.4-1 shows the impulse responses of a series of implicit frequency-space finite-difference schemes with different degrees of dip accuracy. Whether it is implemented in the time-space domain (Figure 4.3-1) or frequency-space domain (Figure 4.4-1), the 15-degree algorithm yields an elliptic impulse response. The 45-degree algorithm yields an impulse response in the shape of a heart.
The 15-degree equation is derived from the Taylor expansion of the dispersion relation (equation 14a). The 45-degre equation is based on the continued fractions expansion (equation 18), which allows wider angle approximations. Kjartansson  implemented the 45-degree equation for migration of stacked data.
The 45-degree equation (18) can be upgraded to be accurate for dips up to 65 degrees by tuning some coefficients (Section D.4). Higher-order operators can be obtained by the successive application of a number of operators like the 45-degree operator  with a different set of coefficients . As shown in Figure 4.4-1, with increasing dip accuracy, the impulse responses of the algorithms approach the shape of a semicircle. However, branches in the impulse response associated with evanescent energy remain.
Figure 4.4-2 shows migration of a constant-velocity diffraction hyperbola using the 65-, 80-, 87-, and 90-degree implicit schemes (Section D.4). While the focusing is better than that achieved by the 15-degree implicit scheme (Figure 4.3-4c), note that the dispersive noise still persists in the image obtained from the 65-degree implicit scheme (Figure 4.4-2c). It is evident that the quality of focusing from the 80-degree implicit scheme is superior (Figure 4.4-2d). The 87- and 90-degree schemes have caused overmigration of the diffraction hyperbola (Figures 4.4-2e,f).
The overmigration effect also can be observed on the results from the constant-velocity dipping events model in Figure 4.4-3. In fact, the dispersive noise that accompanies the steeply dipping events is present in all cases. The response of an implicit scheme is the product of a complicated interplay of various parameters (Section D.6) — depth step size, sampling intervals in space and time, dip angle, velocity and frequency. Dispersive noise, and under- or overmigration characteristics of implicit schemes depend on the specific implementation.
Figure 4.4-4 shows the stacked data migrated using three different approximations in frequency-space domain — 15, 45, and 65 degrees. Note that by higher-degree approximation, the collapse of the diffraction becomes complete, and the steeply dipping event is migrated more accurately. Compare these results with the desired migration in Figure 4.3-2b. Also note the similar results obtained from the 15-degree time-space (t − x) algorithm (Figure 4.3-2c) and the 15-degree frequency-space (ω − x) algorithm (Figure 4.4-4a).
Figure 4.4-5 shows a field data example of frequency-space implicit finite-difference migrations with different degrees of accuracy. Compare the results with the desired migration in Figure 4.2-15b and note that the 80-degree scheme probably produces the most preferred image of the salt dome as compared to that from the 65-degree scheme. The schemes with steeper dip accuracy (87- and 90-degree schemes), however, yield marginal improvements over the 80-degree scheme. Often the 65-degree scheme produces acceptable results, and the 80-degree scheme, which requires twice the computational effort, is used occasionally in practice.
Figure 4.3-22 (a) A CMP-stacked section, (b) reverse time migration.
- Kjartansson, 1979, Kjartansson, E., 1979, Modeling and migration by the monochromatic 45-degree equation: Stanford Exploration Project Report No. 15, Stanford University.
- Ma, 1981, Ma, Z., 1981, Finite-difference migration with higher-order approximation: Presented at the 1981 Joint Mtg. Chinese Geophys. Soc. and Soc. Explor. Geophys.
- Lee and Suh, 1985, Lee, M.W. and Suh, S.H., 1985, Optimization of one-way wave equations: Geophysics, 50, 1634–1637.