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As usual we
precondition by changing variables so
that the regularization operator becomes an identity matrix.
The gradient in equation (21) has no inverse, but its
spectrum ,can be factored () into triangular parts
and where is the helix derivative.
This is invertible by deconvolution.
The quadratic form
suggests the new preconditioning variable .The fitting goals in equation (21) thus become
| |
(22) |

with the residual for the new variable .Experience shows that an iterative solution for converges much
more rapidly than an iterative solution for ,thus showing that is a good choice for preconditioning.
We could view the estimated final map ,however in practice, because the depth function is so smooth,
we usually prefer to view the roughened depth .
There is no simple way of knowing beforehand the best value of .Practitioners like to see solutions for various values of .Practical exploratory data analysis is pragmatic.
Without a simple, clear theoretical basis, analysts
generally begin from and then abandon the fitting goal
.Effectively, they take .Then they examine the solution as a function
of iteration, imagining that the solution at larger iterations
corresponds to smaller and that the solution at smaller iterations
corresponds to larger .In all our explorations, we follow this approach
and omit the regularization in the estimation of the depth maps.
Having achieved the general results we want,
we should include the parameter and adjust it until
we see a pleasing result at an ``infinite'' number of iterations.
We should but usually we do not.

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Stanford Exploration Project

4/27/2004