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Appendix A

Conjugate-gradient algorithm

A complete algorithm for conjugate-gradient iterative inversion with shaping regularization is given below. The algorithm follows directly from combining equation 13 with the classic conjugate-gradient algorithm of Hestenes and Steifel (1952).

\begin{algorithm}{Conjugate gradients with shaping}{\mathbf{L},\mathbf{H},\mathb...
...ight] \\
\hat{\rho} \= \rho
\end{FOR} \\
\RETURN \mathbf{m}

The iteration terminates after $N$ iterations or upon reaching convergence to the specified tolerance $tol$. It uses auxiliary vectors $\mathbf{p}$, $\mathbf{r}$, $\mathbf{s}_p$, $\mathbf{s}_m$, $\mathbf{s}_r$, $\mathbf{g}_p$, $\mathbf{g}_m$, $\mathbf{g}_r$ and applies operators $\mathbf{L}$, $\mathbf {H}$ and their adjoints only once per each iteration.

Appendix B

Combining shaping operators

General rules can be developed to combine two or more shaping operators for the cases when there are several features in the model that need to be characterized simultaneously. A general rule for combining two different shaping operators $\mathbf{S}_1$ and $\mathbf{S}_2$ can have the form

\mathbf{S}_{12} =
\mathbf{S}_{1} + \mathbf{S}_{2} -
\end{displaymath} (21)

where one adds the responses of the two shapers and then subtracts their overlap. An example is shown in Figure B-1, where an impulse response for oriented smoothing in two different directions is constructing from smoothing in each of the two directions separately.

Figure B-1.
Impulse response for a combination of two shaping operators smoothing in two different directions.
[pdf] [png] [scons]

Combining two operators that work in orthogonal directions can be accomplished with a simple tensor product, as follows:

\mathbf{S}_{xy} = \mathbf{S}_{x}\,\mathbf{S}_{y}\;,
\end{displaymath} (22)

where $\mathbf{S}_{x}$ and $\mathbf{S}_{y}$ are shaping operators that apply in orthogonal $x$- and $y$-directions, and $\mathbf{S}_{xy}$ is a combined operator that works in both directions. An example is shown in Figure B-2, where two two-dimensional shapers working in orthogonal directions are combined to produce an impulse response of 3-D shaping operator that applies smoothing along a three-dimensional plane.

Figure B-2.
3-D impulse response for a combination of two 2-D shaping operators smoothing in in-line and cross-line directions.
[pdf] [png] [scons]

Constructing multidimensional recursive filters for helical preconditioning (Fomel and Claerbout, 2003) is significantly more difficult. It involves helical spectral factorization, which may create long inefficient filters (Fomel et al., 2003).

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