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The basic formulation of a geophysical estimation problem
consists of setting up two goals,
one for data fitting,
and the other for model smoothing.
We have data ,a map or model ,a transformation
and a roughening filter ,like the gradient or the helix derivative .The two goals may be written as:
| |
(12) |

| (13) |

which defines two residuals,
a data residual ,and a model residual , which
are usually minimized by iterative conjugate-gradient, least-squares methods.
A common case is ``minimum wiggliness''
when is taken to be the gradient operator.
The gradient is not invertable so we cannot precondition with its inverse.
On the other hand,
since ,taking to be the helix derivative ,we can invert and proceed with the preconditioning:
We change the free variable in the
fitting goals from to (by inverting (13))
with and substituting into both goals getting new goals
| |
(14) |

| (15) |

In my experience,
iterative solvers find convergence much more quickly
when the free variable is the roughened map rather than the map itself.
Figure 11 (left) shows ocean depth
measured by a Seabeam apparatus.

**seapef
**

Figure 11
Filling empty bins with a prediction-error filter.

Locations not surveyed are evident as the homogeneous gray area.
Using a process akin to ``blind deconvolution'' a
2-D prediction error filter is found.
Then missing data values are estimated and shown on the right.
Preconditioning with the helix speeded this estimation by a factor of about 30.
The figure required a few seconds of calculation for about 10^{5} unknowns.

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Stanford Exploration Project

6/2/1998