Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 134

First summing over j in expression (2.181), and using the definitions (2.185),
the objective function I becomes

g j g ∗ j
σ

L N N N N

1
M 2

1
M 2

1
M 2

G ∗ l d l − G k d ∗ k + G ∗ l I =

j −

G k C l − k .

j =− L l =− N k =− N l =− N k =− N

(2.188)

The objective function will take on its minimum value, I min , when the conditional
equations (2.186) hold, and thus,

g j g ∗ j
σ

L N N N

1
M 2

1
M 2

1
M 2

G ∗ l d l − G k d ∗ k + G ∗ l d l

I min =

j −

2

j =− L l =− N k =− N l =− N

g j g ∗ j
σ

L N

1
M 2

G k d ∗ k =

j −

j =− L k =− N

g j g ∗ j
σ

g ∗ j
σ

L L N

1
M

G k e i 2π ( k / M ) t j =

j −

. (2.189)

j

j =− L j =− L k =− N

Finally, with substitution from the representation (2.178), we have that

g j g ∗ j
σ

g j g ∗ j
σ

L L

I min =

j − j .

(2.190)

j =− L j =− L

If the representation (2.178) is an exact fit to the time sequence, then this minimum
value of the objective function will be zero.
In the case when each observation is taken to have equal error, or when no
error estimate is available, the objective function (2.180) becomes simply the error
energy

j =− L j ∗ j .

L

I = (2.191)

Expressions (2.185) then reduce to

j =− L g j e − i 2π( m / M ) t j

L L

e − i 2π( m / M ) t j C m = , d m = M . (2.192)

j =− L

In the equally spaced case, t j can be replaced by j Δ t , and by the orthogonality

relation (2.139),

m

C m = (2 L + 1)δ (2.193)