Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 182

axis, with a spectral estimate based on a window w n + m ( t ) of the data centred at
b n + m on the time axis,
E S gg, n ( t ) S gg, n + m ( f )

I 2 E H n ( f ) H ∗ n ( f ) H n + m ( f ) H ∗ n + m ( f ) ,

1

= (2.366)

where H n ( f )and H n + m ( f ) are Fourier transforms of the respective windowed data
segments. Then,
E S gg, n ( f ) S gg, n + m ( f )

I 2 ∞

∞ ∞ ∞

1

e − i 2π f ( s − t + u − v) =

−∞ −∞ −∞ −∞

· w n ( s )w ∗ n ( t )w n + m ( u )w ∗ n + m (v)
· E g( s )g ∗ ( t )g( u )g ∗ (v) ds dt dud v. (2.367)

The evaluation of this expected value requires the expected value of a fourth moment
of the process generatingg( t ). The realisations of the process are complex numbers.
If they are taken to have real and imaginary parts that jointly are normally distrib-
uted with zero mean, at a number of points N along the time axis, the probability
density has been given by Wooding (1956) as

exp

jk g k ,

1

− g ∗ j c − 1 p (g 1 ,g 2 ,...,g N ) = (2.368)

N

π | C |

where summation is implied over the repeated subscripts j , k , while C is the Her-
mitian variance-covariance matrix,

is its determinant and c − 1

jk are the elements
of its inverse. Wooding (1956) also gives the characteristic function for this distri-
bution as

| C |

exp

4 ω ∗ j c jk ω k

1

Φ= − (2.369)

exp i ω ∗ j g j

+ ω j g ∗ j /2 p (g 1 ,g 2 ,...,g N ) d = V , (2.370)

with the integral over all volume elements d of the complex realisation space.

Again, summation is implied over repeated subscripts, and c jk are the elements of
the variance-covariance matrix C . Labelling the realisations at locations s , t , u , von
the time axis g 1 , g 2 , g 3 , g 4 , respectively, power series expansion of the exponential
in the integral (2.370) for the characteristic function, and integration term by term,
yield

V

4 pd

1
4!

1
16

···+ ω ∗ 1 g 1 + ω 2 g ∗ 2 + ω ∗ 3 g 3 + ω 4 g ∗ 4 +··· Φ= 1 +···+ V+···

1
16

ω ∗ 1 g 1 ω 2 g ∗ 2 ω ∗ 3 g 3 ω 4 g ∗ 4 pd = 1 +···+ V+···

16 ω ∗ 1 ω 2 ω ∗ 3 ω 4 E g 1 g ∗ 2 g 3 g ∗ 4

1

= 1 +···+ +··· , (2.371)