# Residual statics estimation by stack-power maximization

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Estimation of traveltime deviations from NMO-corrected CMP gathers may fail with land data which have low fold and poor signal-to-noise ratio. As a result, residual statics solution by traveltime decomposition can be erratic and unstable. A more robust alternative for surface-consistent estimates of shot and residual static shifts is based on minimizing the difference between modeled and actual traveltime deviations (equation **26**) associated with a reflection event on moveout-corrected gathers. Specifically, surface-consistent static shifts also can be determined by maximizing the power of stacked traces ^{[1]}.

**(**)

The conceptual basis of the method of stack power maximization is intuitively simple. Consider determining the residual static at a shot station. As in the case of residual statics estimation by traveltime decomposition, this method also is applied to moveout-corrected data.

- Apply a static shift to all the traces in the common-shot gather associated with the station under consideration.
- Stack over a time gate the CMP gathers that include traces from that shot gather.
- Compute the cumulative energy of the stacked traces from step (b) by summing the squared amplitudes.
- Repeat steps (a), (b), and (c) for a range of static shifts.
- Choose the static shift that yields the highest stack power and assign it to the shot location under consideration.
- Apply the shot residual static shift associated with the highest stack power to all the traces in the shot gather.
- Stack the CMP gathers that include traces from this shot gather.
- Move to the next shot station and repeat steps (a) through (g).

The process is then repeated for the receiver stations using common-receiver gathers.

This formal recipe for stack-power maximization is intensive both computationally and in terms of data movement. A practical alternative involves creating two supertraces — one from the traces of the common-shot or common-receiver gather under consideration, and a second one from the traces of the stacked traces associated with the common-shot or common-receiver gather ^{[1]}. A supertrace is created by augmenting the individual segments of traces within the specified time gate in a gather, one followed by the other with a zone of zero-amplitude samples between them. The subtlety of the method to keep in mind is that the stack supertrace does not include the contribution of the traces from the common-shot or common-receiver gather.

Define the shot and stack supertraces by the time series *F*(*t*) and *G*(*t*), respectively. The stack power defined as the power of the sum of these two traces over the time gate *t* is

**(**)

where Δ*t* is the trial static shift applied to the shot supertrace *F*(*t*). By expanding the squared term, we obtain

**(**)

The first two terms are the powers of the two supertraces that can be defined by a constant, and the third term is the crosscorelation of the two supertraces. Therefore, maximizing the stack power is equivalent to maximizing the crosscorrelation ^{[1]}.

Now, consider, again, determining the residual static at a shot station.

- Create the shot supertrace. To circumvent end effects in step (c), place zero-amplitude samples between the trace segments when creating the supertraces.
- Create the stack supertrace.
- Crosscorrelate the two supertraces.
- Determine the correlation lag associated with the peak crosscorrelation value — this is the shot residual static shift.
- Apply the shot residual static shift associated with the highest correlation value to all the traces in the shot gather.
- Stack the CMP gathers that include traces from this shot gather.
- Move to the next shot station and repeat steps (a) through (g).
- Repeat steps (a) through (h) for all receiver stations.

Steps (a) through (i) usually are applied iteratively to converge to a solution of shot and residual static shifts.

## See also

- Residual statics estimation by traveltime decomposition
- Traveltime decomposition in practice
- Maximum allowable shift
- Correlation window
- Other considerations
- Stack-power maximization in practice
- Exercises
- Topics in moveout and statics corrections

## References

- ↑
^{1.0}^{1.1}^{1.2}Ronen and Claerbout, 1985, Ronen, J. and Claerbout, J. F., 1985, Surface-consistent residual statics estimation by stack-power maximization: Geophysics, 50, 2759–2767.