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

| |
(5) |

| (6) |

[5] is the ``data matching'' goal, which states that the model must match the known data , while [6] is the ``model smoothness''
goal, where is an arbitrary roughening operator. To combat slow convergence,
Claerbout 1998a preconditions with the inverse of the convolutional
operator (multidimensional *de*convolution). Provided that
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 , we have the equivalent preconditioned problem:
| |
(7) |

| (8) |

The operator 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 , so
our choice of wholly determines the behavior of the missing model points, i.e.,
their *texture* Fomel et al. (1997).
The PEF is a perfect choice for , 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
, 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 regularization, data in-filled using preconditioned PEF regularization.

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Stanford Exploration Project

4/20/1999