** Next:** Geometric interpretation of the
** Up:** Least-squares solution of the
** Previous:** Inversion of a 22

Using the results above, the least-squares estimate of
in equation () is derived.
Assuming that , the fitting goal is
| |
(97) |

with and .The normal equations are given by
| |
(98) |

where and are the unknowns.
The least-square estimate of can be
derived from the bottom row of equation ().
The least-square estimate of can be
derived from the top row of equation ().
We have, then,
| |
(99) |

| |

| (100) |

| |

which can be simplified as follows:
| |
(101) |

| (102) |

is the coherent noise resolution matrix,
whereas is the signal resolution
matrix Tarantola (1987).
Denoting and yields the following simplified expression for and
:
| |
(103) |

By property of the resolution operators, and
perform noise and signal filtering, i.e.,
| |
(104) |

if the noise and signal are well predicted by the noise
and signal modeling operators. Nemeth (1996) demonstrates
that the inverse of the Hessian in equation ()
is well conditioned if the noise and signal operators are orthogonal,
meaning that they predict distinct parts of the model space without
overlapping. If overlapping occurs, a model regularization term can
improve the signal/noise separation.

** Next:** Geometric interpretation of the
** Up:** Least-squares solution of the
** Previous:** Inversion of a 22
Stanford Exploration Project

5/5/2005