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Following the preceding definitions, we can define the noise and
signal filters more precisely. But first, recall that
with
 

 (109) 
and are signal and
noise filtering operators respectively. If we define
 

 (110) 
with and
the signal and
noise resolution operators, we deduce that and , and are
complementary operators (definition 2).
It can be shown that , , and are projectors. Indeed, for and , we have
 

 
 (111) 
and
 

 
 (112) 
Thus, and are projectors as defined
in definition 1. The same proofs work for and .
We can prove also that and , and are mutually orthogonal. For and , we have
 

 
 (113) 
Hence, and , and are complementary, mutually orthogonal projectors.
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Stanford Exploration Project
5/5/2005