Large holes

This set of fitting goals can runs into problems when we deal with real marine geometry. To demonstrate the problem we will look at where data was recorded for a real 3-D marine survey. We can calculate where we have traces in the $h_x, {\rm cmp}_x,{\rm cmp}_y$ plane. If our acquisition lines are perfectly straight, we are able to acquire data throughout our survey. If our grid is perfectly oriented with acquisition geometry, we should have consistent fold in this cube. Figure  shows that this is far from the case. The figure shows the result of stacking over all offsets. Note that we have some areas where we don't have any data (white). If we use fitting goals (2) to estimate our model we run into a problem. The inversion result will show a dimming of amplitudes as we move away from our known data.

 

fold Figure 3 Fold of a real marine dataset. Note how we have some regions with zero fold (white).

Figure  shows the result of applying fitting goals (2) to our synthetic. Note how the amplitude declines markedly as we move away from locations where we have data. Even more problematic than dimming is when we see significant unrealistic, brightening of amplitudes for certain ${\rm cmp}_y$. The brightening is caused by the fold pattern seen in Figure . The three panels represent the fold in the (${\rm cmp}_x,{\rm cmp}_y$) plane as we increase in offset from left to right. Note how we have fairly regular coverage at the near offsets and much more variable coverage as we move to larger offsets. This inconsistency is mainly caused by cable feathering. For some ${\rm cmp}_y$we only have near offset traces. The near offset traces tend to be of higher amplitude and are more consistent as function of h (the tops of hyperbolas are insensitive to velocity errors). Our model covariance operator puts these unrealistically large amplitudes at all offsets, resulting in a striping of the amplitudes as a function of ${\rm cmp}_y$.

 

fold-off
Figure 4 The three panels represent the fold in the (${\rm cmp}_x,{\rm cmp}_y$) plane as we increase in offset from left to right. Note how we have fairly regular coverage at the near offsets and much more variable coverage as we move to larger offsets. For some ${\rm cmp}_y$we only have near offset traces.

 

bad-syn
Figure 5 Two views of the result of applying fitting goals (2). The left panel is a three dimensional view at a fixed h_x. The right panel is a three dimensional view at a fixed ${\rm cmp}_x$.Note the inconsistent, unrealistic amplitude behavior as a function of ${\rm cmp}_y$.

Both of these problems are due to the lack of `mixing' of information along the y direction. By mixing I mean that a column of the matrix implied by fitting goals ($\ref{bob1/eq:prob}$) has very few non-zero elements at ${\rm cmp}_y$'s different from the ${\rm cmp}_y$ associated with its corresponding model point. Our regularization is just DMO, which produces no mixing in the y direction. Our zeroing operator produces a limited amount of mixing, but the range is limited due to the small offset in the h_y direction inherent in marine surveys. As a result our inversion can have realistic kinematic but unrealistic amplitude behavior as a function of ${\rm cmp}_y$. A simple solution to this problem is to introduce another operator to our model covariance description that tends to produce consistency as a function of ${\rm cmp}_y$. We must be careful to avoid introducing unrealistic smoothness in the ${\rm cmp}_y$ direction by our choice of preconditioners. I chose leaky integration along the ${\rm cmp}_y$ plane $\bf B_y$. The leaky integration will encourage the inversion to keep consistent amplitudes unless the data says otherwise. Using a relatively small leaky parameter and a very small $\epsilon$ should force it to have only an amplitude balancing effect rather than an effect on the kinematics of the solution.

Combining our two model preconditioners we get a new operator $\bf S$,

$$\bf S= \bf C_h\bf B_y,$$ (3)

and a new set of fitting goals

$$\begin{eqnarray} \bf d&\approx&\bf L\bf Z\bf S\bf p \\ \nonumber \bf 0&\approx&\epsilon \bf p.\end{eqnarray}$$ (4)

Figure  shows the result of applying fitting goals (4) to the small synthetic. Note how the amplitude behavior is much more consistent than the result shown in Figure .

 

inv-syn
Figure 6 Two views of the result of applying fitting goals (4). The left panel is a three dimensional view at a fixed h_x. The right panel is a three dimensional view at a fixed ${\rm cmp}_x$.Both views are identical the ones shown in Figure . Note how the unrealistic amplitude behavior seen in Figure  has been corrected.

Fitting goals (4) should be avoided when possible. They introduce a smoothing along the ${\rm cmp}_y$ axis that is often unrealistic. Unfortunately when encountering large acquisition holes, some additional regularization is needed.