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

(33)

and

(34)

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 (33) and
(34), we have for the data vector d the following equalities:

(35)

and

(36)

In the following equations, we prove that :

(37)

Similarly, we have . If we use equations (35) and (36),
the last two equalities can be written as follows:

(38)

Hence, , and d form a
right triangle with hypotenuse d and legs and , as depicted in Figure 13;
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 14). Similarly, n is in the
null space of and .

geom11
Figure 13 A geometric interpretation of the
noise filter when n and s are not orthogonal.

geom21
Figure 14 A geometric interpretation of the
noise filter when n and s are orthogonal.