Basic theory

My inversion scheme is based on the downward continuation 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 reflection point (CRP), and offset ray parameter (p_h) space. Offset ray parameter is related to the reflection angle ($\theta$) and the dip angle of the reflector ($\phi$)in 2-D as:

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

In complex areas, the image produced by downward continuation migration will suffer from poor illumination. To compensate for this, I use the migration as an operator in a least-squares inversion. The inversion procedure used in this paper can be expressed as fitting goals as follows:

$$\begin{eqnarray} {\bf 0} & \approx & {\bf Lm}\ -\ {\bf d} \\ {\bf 0} & \approx & \epsilon {\bf A m}. \nonumber\end{eqnarray}$$ (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 a regularization operator and $\epsilon$ controls the strength of the regularization.

Unfortunately, the inversion process described by fitting goals (2) can take many iterations to produce a satisfactory result. I can reduce the necessary number of iterations by making the problem a preconditioned one. I use the preconditioning transformation ${\bf m = A^{-1}p}$ Fomel et al. (1997); Fomel and Claerbout (2003) to give us these fitting goals:

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

${\bf A^{-1}}$ is obtained by mapping the multi-dimensional regularization operator ${\bf A}$ to helical space and applying polynomial division Claerbout (1998). I call this minimization scheme Regularized Inversion with model Preconditioning (RIP).

The question now is what the regularization operator ${\bf A}$ is. In this paper, I will use two different regularization schemes. The first, which I call geophysical regularization, acts horizontally along the offset ray parameter axis. Rather than using the derivative operator used by Kuehl and Sacchi (2001) or the steering filter used by Prucha et al. (2000), I have created a symmetrical filter by cascading two steering filters that are mirror images of each other. The other regularization scheme that will be demonstrated in this paper is called geological regularization. This operator acts along user-specified dips in the CRP-depth plane. It is a steering filter constructed from the dips of picked reflectors.