# 3-D residual statics corrections

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The surface-consistent residual statics model that we discussed in residual statics corrections does not restrict shot and receiver stations to a survey line; they can be situated anywhere as in a 3-D survey. Hence, equation (3-25), which is re-written below, can be used to model the static shifts at all shot and receiver stations over the entire 3-D survey area as

 ${\displaystyle t_{ij}^{\prime }=s_{j}+r_{i}+G_{k}+4M_{k}h_{ij}^{2}.}$ (10)

Here, ${\displaystyle t_{ij}^{\prime }}$ are the modeled traveltime deviations associated with a moveout-corrected reflection event within a specified time gate, sj is the shot residual statics, ri is the receiver residual statics, Gk is the structure term at the kth midpoint location, where k = (i + j)/2, and ${\displaystyle 4M_{k}h_{ij}^{2}}$ is the residual parabolic moveout term, with 2hij being the shot-receiver separation. The idea is to decompose the reflection time deviations picked from moveout-corrected cell-gathers into surface-consistent shot and receiver residual statics shifts, as well as structure and residual moveout terms. The procedure is based on a formulation (residual statics corrections), where the residual statics terms in equation (10) are estimated such that the difference between the modeled times t′ and the actual times t picked from moveout-corrected cell-gathers is minimum in the least-squares sense.

As it is for the 2-D case, the critical step is in picking the traveltime deviations from the common-cell gathers. For 3-D seismic data, an initial pilot trace is built from the common-cell gather of a selected line where the signal-to-noise ratio is good. Pilot traces for other common-cell gathers are computed from both local input traces and neighboring pilot traces. Note that land recording geometry must accommodate some overlap between adjacent swaths to ensure statics coupling between the swaths.

A fundamental question arises regarding 3-D DMO correction in relation to residual statics corrections. Specifically, which one should follow the other? We note from Figure 7.2-8 and Levin’s equation (2) that reflection times need to be corrected for the source-receiver azimuth effect by way of the 3-D DMO process. If 3-D DMO correction were to follow residual statics estimation, traveltime variations caused by the source-receiver azimuth effect would influence the statics estimates. In contrast, if 3-D DMO were to precede residual statics estimation, the moveout-corrected times used in DMO correction would be contaminated by statics errors. Nevertheless, this problem is of a lesser degree than the problem caused by the effect of traveltimes that were not corrected for source-receiver azimuth on statics estimates. Therefore, it is advisable to apply 3-D DMO correction prior to residual statics estimation.

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {4h^{2}(1-\sin ^{2}\phi \cos ^{2}\theta )}{v^{2}}},}$ (2)

Figures 7.2-6b and 7.2-7b show the shot and receiver residual statics over the survey area. Figure 7.2-13 shows a crossline from the 3-D survey of Figure 7.2-1 with and without residual statics corrections. Note the improvement in the reflection continuity at times below 2 s.

Following 3-D DMO correction and residual statics corrections, a final velocity analysis is carried out using a grid of the size that may be as small as 500 × 500 m. Figure 7.2-14 shows velocity panels from selected analysis locations over the survey area. Velocity analysis for land data often requires more than just a velocity spectrum (Figure 7.2-14d). A set of guide velocity functions that differ by a certain percentage from one another is specified. These functions are used to apply moveout correction to a set of CMP gathers (along, say, the inline direction) on both sides of the analysis location. The center CMP gather (Figure 7.2-14a) after moveout correction using the guide functions constitute one part of the velocity panel (Figure 7.2-14b). The CMP stack of the selected gathers at the analysis location with the guide functions constitute another part of the panel (Figure 7.2-14c). The velocity spectrum associated with the center CMP gather also is included in the panel (Figure 7.2-14d). Combined analysis of the gather, stack, and spectrum enables reliable picking of a velocity function at the analysis location.

By combining the velocity functions picked at analysis locations over the survey area, a 3-D velocity field is created. Figure 7.2-15 shows selected time slices from the 3-D velocity field. A thorough check on the velocity field is essential for stacking and time migration. For the latter, DMO-stacking velocity field (Figure 7.2-15) usually is smoothed spatially so as to eliminate lateral velocity variations that are judged to be unacceptable for time migration.

Figure 7.2-16 shows selected crosslines from the volume of stacked data without and with 3-D DMO correction. For the location of these crosslines, refer to the base map in Figure 7.2-1. Note that the differences between the sections with and without 3-D DMO correction are seen to the right of inline location 300. Specifically, note the enhanced flanks of diffractions associated with the tight imbricate structure that conflict with the near-flat reflections on the DMO-stacked data.

## References

1. Levin (1971), Levin, F. K., 1971, Apparent velocities from dipping interface reflections: Geophysics, 36, 510–516.