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One method of solving
so-called ``mixed-determined'' problems is to force the problem to be purely
overdetermined by applying regularization, in which case Equation (1)
becomes
| |
(3) |

is the regularization operator; usually convolution with a compact differential filter.
is a scaling factor. The least squares inverse is then
| |
(4) |

The regularization term, , is nonsingular with positive eigenvalues,
so it stabilizes singularities in , but it is poorly-conditioned for many
common choices of , i.e., Laplacian or gradient. The smallest eigenvalues of
correspond to smooth (low-frequency) model components,
so iterative methods of solving equation (4), including the conjugate-direction method
used in this paper, require many iterations to obtain smooth estimates of the model Shewchuk (1994).

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Stanford Exploration Project

9/5/2000