The operators and are the noise and signal
resolution operators. They describe how well the predictions match the
noise and signal Menke (1989). In the following equations, we consider that
(114)
and
(115)
meaning that each component of the data has been predicted. These
equalities will help us to build a comprehensive geometric interpretation
for the different operators. Based on equations () and
(), we have for the data vector d the following equalities:
(116)
and
geom11
Figure 1 A geometric interpretation of the
noise filter when n and s are not orthogonal.
geom21
Figure 2 A geometric interpretation of the
noise filter when n and s are orthogonal.
(117)
In the following equations, we prove that :
(118)
Similarly, we have . If we use equations () and (),
the last two equalities can be written as follows:
(119)
Hence, , and d form a
right triangle with hypotenuse d and legs and , as depicted in Figure ;
similarly, , and d form a
right triangle with hypotenuse d and legs and .If n and s are orthogonal, s is in the
null space of and (Figure ). Similarly, n is in the
null space of and . Next:Estimation of nonstationary PEFs Up:Geometric interpretation of the Previous:General properties of the
Stanford Exploration Project
5/5/2005