Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 133

The conditions for a minimum of I are that its partial derivatives with respect to
the real and imaginary parts of G m vanish for m =− N ,..., N .Ondi ff erentiation of

(2.181), we find that

⎩ −

g ∗ j

L

g j

∂ I
∂Re G m =

1
σ

M e − i 2π( m / M ) t j M e i 2π( m / M ) t j

−

j

j =− L

G k e i 2π(( k − m )/ M ) t j ⎭ =

N N

1
M 2

1
M 2

G ∗ l e i 2π(( m − l )/ M ) t j + + 0,

l =− N k =− N

(2.182)

and

⎩ −

g ∗ j

L

i ∂ I

1
σ

g j

M e − i 2π( m / M ) t j M e i 2π( m / M ) t j ∂Im G m =

+

2
j j =− L

G k e i 2π(( k − m )/ M ) t j ⎭ =

N N

1
M 2

1
M 2

G ∗ l e i 2π(( m − l )/ M ) t j − + 0.

l =− N k =− N

(2.183)

On addition of (2.182) and (2.183), we find that
N

L L

g j
σ

1
σ

e i 2π(( k − m )/ M ) t j e − i 2π( m / M ) t j G k = M , m =− N ,..., N .

2
j

2
j k =− N j =− L j =− L

(2.184)

With

L L

g j
σ

1
σ

e − i 2π( m / M ) t j e − i 2π( m / M ) t j C m = , d m = M , (2.185)

2
j

2
j j =− L j =− L

the conditional equations (2.184) become
N

G k C m − k = d m , m =− N ,..., N . (2.186)

k =− N

These conditional equations may be written in array form as (Rochester et al. ,
1974, Appendix B),
⎝

⎠ ⎝ ⎠ ⎝ ⎠

C 0 C − 1 ··· C − 2 N G − N

G − N + 1
. . G N

d − N
d − N + 1
. . . d N

C 1 C − 2 N + 1

. . . . . . . . . . . .
C 2 N C 2 N − 1 ··· C 0

C 0 ···

= . (2.187)