Geoscience Reference
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present (Kottegoda, 1980). Therefore, if one or more persistence are present
in the time series, spectral technique is often used. A non-stationary periodic
time series x ( t ) may be expanded into a Fourier series using following
expression:
Ç
A
[
A
sin (2
S
k f t
)
B
cos (2
S
k f t
)]
x ( t ) =
(88)
0
k
k
k
1
where f = frequency and rest of the parameters are the same as expressed in
Eqn. (75). The frequency can be given as:
1
Z
f =
(89)
P
2
S
where P = period of the function (base period) and Z = angular frequency.
With reference to Eqn. (89), two spectral density functions S (Z) and S ( f ) can
be related as follows:
Sf S (90)
The density function S (Z) is related to Fourier transformation F (Z) as
follows:
S (Z) = 2()
F
()
Z
S (Z) =
(91)
Z
Similarly,
S ( f ) =
Ff
()
(92)
f
Considering an infinitesimal portion of frequency in the range of ( f ,
f + df ), the spectrum S ( f ) df represents the contribution of components with
frequencies in the range ( f , f + df ) to the total variance.
The spectral density function S ( f ) for a discrete process can be written as
(Box and Jenkins, 1976):
Ç
J J
2
cos (2
S
fk
)
S ( f ) =
- < f <
(93)
0
k
k
1
However, the spectral density function for a continuous process can be
written as:
U
Ô
()cos(2
k
S
f k dk
)
S ( f ) =
-< f <
(94)
Negative frequencies obtained with above expressions can be avoided
and integral of the normalized spectral density function can be maintained
over the entire range at one, a one-sided spectral density function G ( f ) =
2 S ( f ) can be written, where f varies only over (0, ) and zero elsewhere
(Shahin et al., 1993). The function G ( f ) may be expressed as:
 
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