Factors affecting spatial resolution

The theory of spatial resolution has been well-established in various papers dealing with inversion and prestack migration. Nevertheless, there is a continuing how of papers being published on spatial resolution, in particular in relation to spatial sampling. This paper continues the discussion, and deals with various factors affecting spatial resolution. The theoretical best-possible resolution can be predicted using Beylkin's formula. This formula gives answers on the effect on resolution of frequency, aperture, offset, and acquisition geometry. In this paper, these factors are investigated using a single diffractor in a constant-velocity medium. Some simple resolution formulas are derived for 2-D zero-offset data. These formulas are similar to formulas derived elsewhere in a more heuristic way, and which are in common use in the industry. The formulas are extended to 2-D common-offset data. The width of the spatial wavelet resulting from migration of the diffraction event is compared with the resolution predicted from Beylkin's formula for various 3-D single-fold data sets. The measured widths confirm the theoretical prediction that zero-offset data produce the best possible resolution and 3-D shots the worst, with common-offset gathers and cross-spreads scoring intermediate. The effects of sampling and fold cannot be derived directly from Beylkin's formula; these effects are analyzed by looking at the migration noise rather than at the width of the spatial wavelet. Random coarse sampling may produce somewhat less migration noise than regular coarse sampling, though it cannot be as good as regular dense sampling, The bin-fractionation technique (which achieves finer midpoint sampling without changing the station spacings) does not achieve higher resolution than conventional sampling. Generally speaking, increasing fold does not improve the theoretically best possible resolution. However, as noise always has a detrimental effect on the resolvability of events, fold-by reducing noise-will improve resolution in practice. This also applies to migration noise as a product of coarse sampling.