Characterizing moveout errors

Tomography requires us to provide moveout errors. It is unreasonable to hand pick every reflector at every CRP gather in 2-D and inconceivable in 3-D. As a result, people have tried to find alternate methods to pick moveout errors. Clark et al. (1996) used a neural network approach to pick CRP gathers and many people have suggested seeding-based approaches to pick the gathers. Both approaches describe complicated moveouts, but they suffer from cycle skipping and have problems in areas where the S/N ratio is not very high. An alternative approach is to characterize the moveout in CRP gathers by a single parameter Biondi (1990); Etgen (1990). A single parameter is a much more robust estimator. It requires less human involvement (less picking and/or QA is necessary) and is less sensitive to signal to noise problems.

At early iterations a single parameter is especially valuable. All that can be resolved at early iterations are gross features. A single parameter can capture these where picking the entire CRP gather is likely to cause the inversion to be overwhelmed small features that are not resolvable at early iterations. When we were close to the correct velocity allowing freedom in moveout behavior is desirable and beneficial.

For the tomography problem we will begin with a migrated image d at a depth z, angle $\theta$, at CRP location x. For estimating the residual moveout in the CRP gathers by calculating semblance s in terms of some curvature parameter $\gamma$, 

$$s(z,c,x)={\left[ \sum_{\theta} d(z+\theta \gamma^2,\theta,x)\right]^2 \over n(z,x) \sum_{j=0} d(z+\theta \gamma^2,x)^2 } ,$$ (6)

where n(z,x) is the number non-zero samples summed over for each semblance calculation.

For this dataset hand picking the semblance along each reflector would not be too tedious, but in 3-D it would quickly become so. As a result, we wanted to come up with a simple way for the computer to do most of the work. One option would be to just pick the maximum semblance at each location, but we can get an unrealistic, high spatial wavenumber behavior for $\gamma(x)$.When doing convention semblance analysis we are confronted with a similar problem, that picking the maximum semblance at each time could result in an unreasonable velocity function. Clapp et al. (1998b) proposed a method to avoid hand picking that still led to a reasonable velocity model. We can adapt that work by starting with the maximum curvature value at each CRP ${\boldsymbol \gamma_{\bf max}}$,the semblance at the maximum curvature value $\bold W$,and a derivative operator $\bold D$.We can find a smooth curvature function ${\boldsymbol \gamma}$by setting up a simple set of equations

$$\begin{eqnarray} \bf 0&\approx&\bold W( {\boldsymbol \gamma_{\bf max}} - {\bolds... ...umber \\ \bf 0&\approx&\epsilon \bold D {\bf \boldsymbol \gamma} .\end{eqnarray}$$ (7)

By increasing $\epsilon$ we get smoother ${\boldsymbol \gamma}$ values while small $\epsilon$ values honor more our maximum semblance picks.