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 midpoint (CMP), and offset ray parameter (p_h) space.

This migration operator is used in a Tikhonov regularized Tikhonov and Arsenin (1977) conjugate-gradient least-squares minimization:  

$$min\{Q(\bold m)\ =\ \vert\vert\bold W \left( \bold L \bold m... ...2} + \epsilon^{2}\ \vert\vert \bold A \bold m \vert\vert^{2}\}.$$ (1)

This inversion procedure can be expressed as fitting goals as follows:

$$\begin{eqnarray} {\bf 0} & \approx & \bold W \left( {\bf Lm} - {\bf d}\right) \\ {\bf 0} & \approx & \epsilon {\bf A m}. \nonumber\end{eqnarray}$$ (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, as has already been mentioned, a regularization operator. ${\bf W}$ is a weighting operator. $\epsilon$ controls the strength of the model styling.

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 (2002) to give us these fitting goals:

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

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

The question now is what the regularization operator ${\bf A}$ is. I built my regularization operator based on the same assumptions as Prucha et al. (2000). First, I assume that the correct velocity is being used in the inversion, therefore there should be no moveout along the offset ray parameter (p_h) axis. Second, I assume that the amplitudes of individual events should vary smoothly and any drastic changes in amplitude are caused by illumination problems, which are what we wish to overcome. These assumptions allow me to say that ${\bf A}$ needs to act to minimize amplitude differences horizontally along the p_h 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.