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I remind you of the ``standard trick''
used by Bill Symes, Bill Harlan, and probably many others.
In geophysical fitting we generally have two goals,
the first being data ``fitting''
and the second being the ``damping'' or ``regularization'' goal.
| |
(1) |

| (2) |

where is a roughening operator.
Since the roughening operator is somewhat arbitrarily chosen,
presumably we can know an inverse to it,
a smoothing operator so that .In the simple one-dimensional case, the roughener
is a first or second derivative operator
and the smoother is a single or double integral.
(I'm not so sure that the multidimensional case
is as easy as we often like to pretend).
We define a new variable
by
and have a set of regression goals
for which iterative methods generally converge faster:
| |
(3) |

| (4) |

After solving for we easily find .
A problem arises when we do not start from .Then before we can get started we need to find by solving .

** Next:** HOW MUCH DAMPING?
** Up:** Claerbout: Preconditioning and scalingPreconditioning
** Previous:** ROW NORMALIZATION, NON-POSITIVE OPERATORS
Stanford Exploration Project

11/11/1997