Geoscience Reference
In-Depth Information
Ô
0
2( ) c s (2 )
U
k
S
f k dk
G
(
f
) =
0 <
f
<
(95)
The above function may be discretized as follows:
È
Ø
k
G
(
f
) =
2
J J
2
cos (2
S
fk
)
0 <
f
<
(96)
É
Ù
0
k
Ê
Ú
1
The above analysis considers entire population of the variable, and for a
finite time series of size
n
, the J
k
's are replaced with their sample estimates
C
k
's and J
0
is replaced with
C
0
. Equation (96) for a finite time series may be
reduced to the following:
n
1
È
Ø
Ç
ˆ
()
Gf
=
2
C
2
C
cos (2
S
f k
)
(97)
É
Ù
0
k
Ê
Ú
k
1
In Equation (97), an increase in value of
k
results in reduced precision in
estimating the value of
C
k
. Hence, there is a need to give more weight to
C
k
values for small
k
's and less weight to
C
k
values for larger
k
's. This is normally
done by introducing a set of weights O
k
, known as
lag window
, and by truncating
the upper limit of the summation in Equation (97) to
k
values less than (
n
-1).
Generally,
k
is chosen equal to or less than
n
/4. Equation (97) can be more
conveniently expressed in terms of
autocorrelation function
(U
k
) rather in
terms of
autocovariances
(
C
k
). The
serial correlation coefficient
(
r
k
) based on
sample data is used as an estimate of the
autocorrelation coefficient
for the
same lag
k
. Considering all these statements, Equation (97) can be rewritten
as:
m
È
Ø
Ç
2
r
O
2
r
cos (2
S
f k
)
Gf
=
(98)
()
É
Ù
0
k
k
Ê
Ú
k
1
For autocorrelation,
r
0
= 1, Equation (98) can be written as:
m
È
Ø
Ç
Gf
=
()
2 1
O
2
r
cos (2
S
f k
)
(99)
É
Ù
kk
Ê
Ú
k
1
where
¯
(
f
) = smoothed spectral ordinate.
Of the various available
lag window expressions
, the mostly used are
'Parzen window' (Parzen, 1963), 'Tukey window' (Blackman and Tukey,
1959), and 'hamming and hanning procedures' (Shahin et al., 1993).
4.6 Merits and Demerits of Time Series Methods
Based on the experiences of world-wide researchers and scientists in the area
of time series analysis, the following merits and demerits of time series analysis
methods can be identified. Firstly, the merits and demerits of time series
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