Theory

One of the most common processing steps taken with data from complex areas is migration. Unfortunately, it is also well-known that migration does not provide the best possible model Duquet and Marfurt (1999); Ronen and Liner (2000). It is possible to get a better model by inversion. However, inversion alone is not enough to fill in areas of poor illumination. Fortunately, we often have some idea of how we think events should behave in areas of poor illumination, so we can impose our idea through a regularization operator. This can be represented by these familiar equations:

$$\begin{eqnarray} {\bf d \approx Lm} \\ 0 \approx \epsilon {\bf A m} \nonumber\end{eqnarray}$$ (1)

where d and m are the data and model, respectively, L is the angle domain modeling operator described by Prucha et al. (1999), and A is the regularization operator. A simple substitution of variables can help reduce the number of iterations necessary Fomel et al. (1997). This substitution is ${\bf m = A^{-1}p}$ where m is the model and p is our new variable. This gives us a new set of equations:

$$\begin{eqnarray} {\bf d \approx LA^{-1}p} \\ 0 \approx \epsilon {\bf p}. \nonumber\end{eqnarray}$$ (2)

The inverse operator ${\bf A^{-1}}$ is applied using helical polynomial division Claerbout (1998). We chose to make the operator ${\bf A^{-1}}$ a smoothing operator that would act along a specified dip. This operator was constructed from dip penalty, or steering, filters Clapp et al. (1997). We implemented these filters in 2-D by cascading them Clapp (2000). This cascaded method means that our preconditioning operator ${\bf A^{-1}}$ is a combination of two preconditioning operators in different dimensions:

$${\bf A^{-1}\ = \ A^{-1}_{\alpha z}A^{-1}_{xz}}$$ (3)

where ${\bf A^{-1}_{xz}}$ are filters that are constructed from dips along reflectors in the CMP-depth plane. These filters smooth along chosen reflectors (Figures 4 and 12). The second term, ${\bf A^{-1}_{\alpha z}}$, are filters in the angle-depth plane and, because we assume we are using the correct velocity, we simply smooth horizontally.

One potential drawback to this cascading method is that it can introduce anisotropy to the events, depending on the strength and direction of the dip penalty filters in each dimension. Fomel 2000 describes a method using spectral factorization that may eliminate this anisotropy, but at this early stage we have not determined if such a solution is necessary.

By imposing these smoothing conditions on the model and doing iterative inversion, we hope to fill in areas that do contain real information while smoothing and removing the noise.