Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 208

Taking complex conjugates of both sides and using the Hermitian property of the
autocorrelations gives
⎝

⎠ ⎝ ⎠ ⎝ ⎠

Δ ∗ 2
0
P 2

φ ff (0) φ ff ( − 1) φ ff ( − 2) 0

γ ∗ 1,1
1

φ ff (1) φ ff (0) φ ff ( − 1) = . (2.487)

φ ff (2) φ ff (1) φ ff (0)

Now, we multiply the system (2.487) by the parameter γ 2,2 and add it to the system
(2.485) to get

⎝ ⎠ ⎝ ⎝ ⎠ ⎝ ⎠ ⎠

φ ff (0) φ ff ( − 1) φ ff ( − 2) 1

γ 1,1
0

0
γ ∗ 1,1
1

φ ff (1) φ ff (0) φ ff ( − 1) + γ 2,2

φ ff (2) φ ff (1) φ ff (0)

⎝ ⎝ ⎠ ⎝ ⎠ ⎠

Δ ∗ 2
0
P 2

P 2
0
Δ

= + γ 2,2 . (2.488)

2

If this system is to match the prediction error equations (2.80) for N = 2,

Δ ∗ 2 = P 2 + γ 2,2 P 3 , (2.489)

+ γ 2,2 γ ∗ 1,1 = γ 1,2 , γ 1,1 (2.490)

Δ + γ 2,2 P 2 = 0. (2.491) 2

γ 2,2 is then the only free parameter left to be determined. From equation (2.491),
Δ 2 =− γ 2,2 P 2 . Then, from equation (2.489),

1

− γ 2,2 γ ∗ 2,2 P 2 . P 3 = (2.492)

The third equation of the system (2.488) yields

φ ff (2) =− γ 1,2 φ ff (1) − γ 2,2 φ ff (0). (2.493)

Once again we take P 3 to be given by the average of the forward and backward
prediction errors,

F
j + 2,2

= f j + 2 + γ 1,2 f j + 1 + γ 2,2 f j , j = 1,..., M − 2

γ 1,1

+ γ 2,2 γ ∗ 1,1 f j + 1 = f j + 2 + + γ 2,2 f j

F
j + 2,1

B

= + γ 2,2

j ,1 ,

(2.494)

j ,2

f j + γ ∗ 1,2 f j + 1 + γ ∗ 2,2 f j + 2 , = j = 1,..., M − 2

γ ∗ 1,1 + γ ∗ 2,2 γ 1,1 f j + 1

+ γ ∗ 2,2 f j + 2 = f j +

B
j ,1

+ γ ∗ 2,2

F

=

j + 2,1 .

(2.495)