Preconditioned Missing Data Infill

To fill ``holes'' in collected data, we have the familiar SEP formulation Claerbout (1998a):

$$\begin{eqnarray} \bf Km - d &\approx& 0 \\ [0.05in] \bf \epsilon Am &\approx& 0 \end{eqnarray}$$ (5) (6)

[5] is the ``data matching'' goal, which states that the model $\bf m$ must match the known data $\bf d$, while [6] is the ``model smoothness'' goal, where $\bf A$ is an arbitrary roughening operator. To combat slow convergence, Claerbout 1998a preconditions with the inverse of the convolutional operator $\bf A$ (multidimensional deconvolution). Provided that $\bf A$ is minimum phase or factorizable into the product of minimum phase filters Sava et al. (1998), the helix transform now permits stable multidimensional deconvolution. Making the change of variables $\bf m = A^{-1}x$, we have the equivalent preconditioned problem:

$$\begin{eqnarray} \bf KA^{-1}x - d &\approx& 0 \\ [0.05in] \bf \epsilon x &\approx& 0 \end{eqnarray}$$ (7) (8)

The operator $\bf K$ effectively maps vectors in model space into a smaller-dimension ``known data space'', so it has a nonempty nullspace. Missing points in model space are completely unconstrained by $\bf K$, so our choice of $\bf A$ wholly determines the behavior of the missing model points, i.e., their texture Fomel et al. (1997). The PEF is a perfect choice for $\bf A$, as shown in Figure 10. The preconditioned, PEF-regularized result fills the hole quite believably after only 20 iterations, as opposed to the case where $\bf A = \nabla^2$, which imposes an unrealistically smooth texture on the missing model points.

 

tree-hole-filled
Figure 10 Clockwise from top left: Data with hole, impulse response of ``inverse PEF'' (deconvolution of the PEF estimated from the data and a spike), data in-filled using $\nabla^2$ regularization, data in-filled using preconditioned PEF regularization.