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Regularized linear least squares estimation
problems can be written as minimizing the quadratic
function
| |
(1) |

where is our data, is our modeling operator,
is our regularization operator and we are inverting
for a model .Alternately, we can write them
in terms of fitting goals,
| |
(2) |

| |

For the purpose of this paper I will refer to the first
goal as the *data fitting goal* and the second as
the *model styling goal*.
Normally we think of *data fitting goal * as describing
the physics of the problem. The *model styling goal* is suppose
to provide information about the model character. Ideally
should be the inverse model covariance. In practice
we don't have the model covariance so we attempt to approximate it through
another operator. At SEP the regularization operator is typically one
of the following:
**Laplacian or gradient**
- a simple operator that assumes nothing about the model
**Prediction Error Filter (PEF)**
- a stationary operator estimated from known portions of the model or some field with the same properties as the model Claerbout (1999)
**steering filter**
- a non-stationary operator built from minimal information about the model Clapp et al. (1997)
**non-stationary PEF**
- a non-stationary operator built from a field with the same properties as the model Crawley (2000).

A problem with the first three operators is
that while they approximate the model covariance, they
have little concept of model variance. As a result our model estimates tend
to have the wrong statistical properties.

** Next:** Missing Data
** Up:** Clapp: Multiple realizations
** Previous:** Introduction
Stanford Exploration Project

9/5/2000