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We give a set of definitions that will help us to better understand
the properties of the noise and signal filters in equation (5).

**Definition 1:** an operator **P** is a projector if

| |
(22) |

**Definition 2:** two operators **P** and **Q** are
complementary operators if
| |
(23) |

**Definition 3:** two operators **P** and **Q** are
mutually orthogonal if
| |
(24) |

**Definition 4:** the norm of a vector **v**
is
| |
(25) |

or
| |
(26) |

where (') is the adjoint.

**Definition 5:** two vectors **u** and **v** are
orthogonal if
| |
(27) |

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Stanford Exploration Project

4/29/2001