Seismic surveys generally have irregular areas where data cannot be acquired. These data should often be interpolated. A projection onto convex sets (POCS) algorithm using Fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high-quality results. The original 2D image restoration method, the Gerchberg-Saxton algorithm, is extended easily to higher dimensions, and the 3D version of the process used here produces much better interpolations than typical 2D methods. The only parameter that makes a substantial difference in the results is the number of iterations used, and this number can be overestimated without degrading the quality of the results. This simplicity is a significant advantage because it relieves the user of extensive parameter testing. Although the cost of the algorithm is several times the cost of typical 2D methods, the method is easily parallelized and still completely practical.

Attenuating random noise with a prediction filter in the time‐space domain generally produces results similar to those of predictions done in the frequency‐space domain. However, in the presence of moderate‐ to high‐amplitude noise, time‐space or t-x prediction passes less random noise than does frequency‐space, or f-x prediction. The f-x prediction may also produce false events in the presence of parallel events where t-x prediction does not. These advantages of t-x prediction are the result of its ability to control the length of the prediction filter in time. An f-x prediction produces an effective t-x domain filter that is as long in time as the input data. Gulunay’s f-x domain prediction tends to bias the predictions toward the traces nearest the output trace, allowing somewhat more noise to be passed, but this bias may be overcome by modifying the system of equations used to calculate the filter. The 3-D extension to the 2-D t-x and f-x prediction techniques allows improved noise attenuation because more samples are used in the predictions, and the requirement that events be strictly linear is relaxed.

Prestack depth migration uses two imaging conditions, zero time and zero offset, during downward continuation to form a migrated depth section. When the migration velocities are exact, the two imaging conditions act in a complementary fashion to yield a focused image. When the migration velocities are in error, reflected energy collapses to zero offset at depths that are inconsistent with the zero‐time imaging condition. The result is a deteriorated seismic image. However, by interpreting the nonzero times at which focusing actually occurs, the migration velocities can be updated iteratively in a process called depth‐focusing analysis. To produce a well‐focused seismic image, the goal of depth‐focusing analysis must be the elimination of focusing errors; however, practical considerations can prevent this goal from being achieved. Therefore, to relax the sensitivity of the migrated image to focusing errors, we introduce a nonzero‐time imaging condition by extracting the data along the interpreted surface of focusing from the depth‐focusing analysis volume. This method, called focal‐surface imaging, estimates the results of prestack depth migration using the updated velocities. Depth‐focusing analysis is shown to be a robust approach to velocity estimation and imaging. Limitations arising from constant‐velocity and low‐dip approximations are reduced in the presence of increasing velocities with depth. Lateral velocity errors, sources of exaggerated focusing errors and diverging velocity solutions, can also be addressed by applying a damping factor to the interpreted depth errors. Velocity estimation and focal‐surface imgaging, using iterative prestack depth migration, were applied to a southern North Sea data set. Starting with a regional velocity function, the first iteration provided an updated velocity field that more accurately conformed to the known lithologies. The focal‐surface image, formed from the same iteration, contained significantly more focused energy than the conventional section formed by prestack depth migration. However, structural differences between the two sections indicated the need for another iteration of migration using the updated velocities. The second iteration indicated smaller velocity errors and enough similarity between the migrated section and the new focal‐surface image to indicate that further iterations were unnecessary.

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