Theory

Our inversion scheme is based on the angle-domain wave-equation migration explained by Prucha et al. (1999a). To summarize, this migration is carried out by downward continuing the wavefield in frequency space, slant stacking at each depth, and extracting the image at zero time. The result is an image in depth (z), common midpoint (CMP), and offset ray parameter (p_h) space. The offset ray parameter can be easily related to the reflection angle by:

$${\partial t \over \partial h}=p_{h}=\frac{2 \sin \theta \cos\phi}{V\left(z,cmp\right)},$$ (1)

where $\theta$ is the reflection angle, $\phi$ is the geologic dip, and $V\left(z,cmp\right)$ is the velocity function in depth and CMP location.

The inversion procedure used in this paper can be expressed as fitting goals as follows:

$$\begin{eqnarray} {\bf d \approx Lm} \\ 0 \approx \epsilon {\bf A m}. \nonumber\end{eqnarray}$$ (2)

The first equation is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second equation is the ``model styling goal,'' meaning that it allows us to impose some idea of what the model should look like using the regularization operator ${\bf A}$. The model styling goal also helps to prevent a divergent result.

In the data fitting goal, ${\bf d}$ is the input data and ${\bf m}$ is the image obtained through inversion. ${\bf L}$ is a linear operator, in this case it is the adjoint of the angle-domain wave-equation migration scheme summarized above and explained thoroughly by Prucha et al. (1999b). In the model styling goal, ${\bf A}$ is, as has already been mentioned, a regularization operator. $\epsilon$ controls the strength of the model styling.

Unfortunately, the inversion process described by Equation 2 can take many iterations to produce a satisfactory result. We can reduce the necessary number of iterations by making the problem a preconditioned one. We use the preconditioning transformation ${\bf m = A^{-1}p}$ Fomel et al. (1997); Fomel and Claerbout (2002) to give us these fitting goals:

$$\begin{eqnarray} {\bf d \approx LA^{-1}p} \\ 0 \approx \epsilon {\bf p}. \nonumber\end{eqnarray}$$ (3)

${\bf A^{-1}}$ is obtained by mapping the multi-dimensional regularization operator ${\bf A}$ to helical space and applying polynomial division Claerbout (1998).

The question now is what the preconditioning operator ${\bf A^{-1}}$ is. We have chosen to make this operator from steering filters Clapp et al. (1997); Clapp (2001) which tend to create dips along chosen reflectors. This paper includes results from two different preconditioning schemes. One is called the 1-D preconditioning scheme and simply acts horizontally along the offset ray parameter axis. The 2-D scheme acts along chosen dips on the CMP axis and horizontally along the offset ray parameter axis. To construct the preconditioning operator along the CMP axis, we pick ``reflectors'' that represent the dip we believe should be in a certain location, then interpolate the dips between the picked reflectors to cover the whole plane.