Geoscience Reference
In-Depth Information
P
Ç
(2/
Px
)
cos (2
S
kt P
/
),
B k =
k = 1, 2, ..., P /2-1
(78)
t
t
1
For the monthly hydrologic records, P = 12, and hence h = 6. As it may
not always be necessary to expand the Fourier series up to the maximum
number of harmonics, the maximum number of significant harmonics ( h *)
can be obtained by examining the cumulative periodogram, which determines
the relative significance of each harmonic. To accomplish this task, firstly the
variations caused by a periodic component ( I k ), say k th harmonic, is computed
as (Shahin et al., 1993; Haan, 2002):
2
2
I k =
(79)
(1/ 2) (
AB
)
k
k
Secondly, the periodogram is obtained by plotting I k against 2S k / P for k
= 1, 2, ......, N /2. Thirdly, the cumulative periodogram ( P j ) shows a rapidly
rising part up to h * and then gradually increases up to its maximum value of
unity and is calculated as (Jayawardena and Lai, 1989):
j
P
Ç
Ç
2
2
2
P j =
(
AB
)/2
(
x
P
) /
P
(80)
k
k
t
k
1
t
1
where P is the mean of t x It should be noted that in the numerator of Eqn.
(80), the terms under summation should be arranged in decreasing order of
their magnitudes. Finally, the significance of different harmonics is tested
using the Fisher's g -statistic, which is given as (Yevjevich, 1972; Shahin et
al., 1993):
P
k
/2
2
2
g k =
I
(
A
B
)/2
(81)
k
k
k
1
When the g -statistic (i.e., Eqn. 81) is multiplied by 100, it gives the
percent contribution of the k th harmonic. It is worth mentioning that the quantity
/2
P
Ç
2
2
(
AB
)
of Eqn. (81), when multiplied by ' P ' is used as a measure of the
k
k
k
1
variation caused by a given periodic component (Shahin et al., 1993). On the
P
Ç
2
(
x
P
) ,
other hand, the total variations in a series are expressed as
which
t
t
1
is the denominator of Eqn. (80) multiplied by ' P '. Thus, if these two terms are
equal, the total variations in the series can be explained by the periodic
components only considering its all harmonics, and under such circumstances
the cumulative periodogram is given as follows:
j
– Ç
100
g
P j =
(82)
k
k
1
 
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