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Least-squares inversion can be expressed simply as the minimization of
this objective function:
| |
(1) |

where **L** is a linear modeling operator, **d** is the data, and
**m** is the model. This minimization can be expressed more concisely
as a fitting goal:

| |
(2) |

However, in areas of poor illumination, this problem will have a large
null space. The null space is partially caused by the fact that our
survey can not have infinite extents and infinitely dense source and receiver
grids. Any noise that exists within the null space can grow with
each iteration until the problem becomes unstable. Fortunately, we can
stabilize this problem with Tikhonov regularization Tikhonov and Arsenin (1977).
The regularization adds a second fitting goal that we are minimizing:
| |
(3) |

| |

The first fitting goal is the ``data fitting goal,'' meaning that it is
responsible for making a model that is consistent with the data. The
second fitting goal is the ``model styling goal,'' meaning that it allows us
to impose some idea of what the model should look like using the
regularization operator . The strength of the regularization
is controlled by the regularization parameter .
Unfortunately, the inversion process described by fitting goals (3)
can take many iterations to produce a satisfactory result.
We can reduce the necessary number of iterations by making the problem
a preconditioned one. We use the preconditioning transformation
Fomel et al. (1997); Fomel and Claerbout (2002) to give us these
fitting goals:

| |
(4) |

| |

is obtained by mapping the multi-dimensional regularization
operator to helical space and applying polynomial division
Claerbout (1998). After obtaining from the fitting goals in
(4), it is simple to transform back to . Now that our
inversion is defined, we can take a closer
look at and at the number of iterations (*n*_{iter}).

** Next:** The question of and n_{iter}
** Up:** M. Clapp: The oddities
** Previous:** Introduction
Stanford Exploration Project

7/8/2003