# Residual statics estimation by stack-power maximization

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

 ${\displaystyle E=\sum \limits _{ij}(t_{ij}-t'_{ij})^{2}.}$ (26)

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.

1. Apply a static shift to all the traces in the common-shot gather associated with the station under consideration.
2. Stack over a time gate the CMP gathers that include traces from that shot gather.
3. Compute the cumulative energy of the stacked traces from step (b) by summing the squared amplitudes.
4. Repeat steps (a), (b), and (c) for a range of static shifts.
5. Choose the static shift that yields the highest stack power and assign it to the shot location under consideration.
6. Apply the shot residual static shift associated with the highest stack power to all the traces in the shot gather.
7. Stack the CMP gathers that include traces from this shot gather.
8. 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

 ${\displaystyle P(\Delta t)=\sum _{t}{[F(t-\Delta t)+G(t)]}^{2},}$ (39a)

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

 ${\displaystyle P(\Delta t)=\sum _{t}F^{2}(t-\Delta t)+\sum _{t}G^{2}(t)+2\sum _{t}F(t-\Delta t)G(t).}$ (39b)

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.

1. Create the shot supertrace. To circumvent end effects in step (c), place zero-amplitude samples between the trace segments when creating the supertraces.
2. Create the stack supertrace.
3. Crosscorrelate the two supertraces.
4. Determine the correlation lag associated with the peak crosscorrelation value — this is the shot residual static shift.
5. Apply the shot residual static shift associated with the highest correlation value to all the traces in the shot gather.
6. Stack the CMP gathers that include traces from this shot gather.
7. Move to the next shot station and repeat steps (a) through (g).
8. 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.