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In seismic processing we often transform data into
equivalent data using linear operators. Among these operators, we have the
Fourier transform, the Radon transform, the migration operator etc.
Some of these operators are **unitary** (Fourier transform), meaning that
the input data are perfectly recoverable using the adjoint.
Mathematically speaking, unitarity implies
| |
(3) |

where **H** is the operator, the adjoint, and **I**
the identity matrix.
Unfortunately, most of the operators are not unitary, meaning that
one can't go back and forth between the model **m** and
the data **d** without losing information or resolution.
Mathematically speaking, non-unitarity implies
| |
(4) |

This loss of information can be overcome using inverse theory.
The goal of inverse theory is to find a model **m**
that optimally represents the input data **d** given an operator **H** and
given a definition of optimality (minimum energy residual-*l*^{2} for example):
| |
(5) |

** Next:** The classical approach: least-squares
** Up:** Guitton: Coherent noise attenuation
** Previous:** Why two methods?
Stanford Exploration Project

9/5/2000