Geoscience Reference
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where the standardized Mann-Kendall test-statistics for
s
seasons and
t
stations
has been defined as
Z
st
. It is to note that the
Z
st
is computed by using either
Eqn. (65) for sample size (
n
) < 40 or Eqn. (66) otherwise. Here
Z
s
is the
average of
Z
st
for season
s
over the stations;
Z
t
is the average of
Z
st
for station
t
over the seasons; and
Z
is the average of
Z
st
over the season-stations. The
computed test-statistic values are compared with the critical values of the chi-
square distribution at 5% significance level and specific degree of freedom
(
df
). The
df
for all the test-statistics are denoted as subscript in left side
expression [Eqns 69 to 74], for example, the
df
for F
2
Total
is
pq
and for
F
2
Homogeneity
is
pq
-1. Finally, if the computed absolute value of a test-statistic
is greater than the critical value of the chi-square distribution, the hypothesis
of a trend cannot be rejected at the 5% significance level.
4.4 Methods for Checking Periodicity
Periodicity in the hydrologic time series can be detected if the time series are
defined at time intervals less than a year; in most cases, six and 12 months
periodicity is very common. The Fourier series has been mainly used for the
detection of periodic components in the hydrologic time series (e.g., Maidment
and Parzen, 1984; Kite, 1989; Jayawardena and Lai, 1989; Fernando and
Jayawardena, 1994; Pugacheva et al., 2003). However, some researchers have
suggested 'coherence plot' and 'periodic autocorrelation function' methods
for testing the periodically correlated time series (e.g., Hurd and Gerr, 1991;
Vecchia and Ballerini, 1991).
Periodicity is detected through harmonic analysis using the well-known
Fourier series. If a periodicity exists in a trend-free time series, it can be
represented by a Fourier series, which is expressed as follows (Stein and
Weiss, 1971; Shahin et al., 1993; Howell, 2001):
Ç
h
k
A
[ in 2/
A
S
kt P
)
B
2/
S
kt P
]
x
(
t
) =
(75)
0
k
k
1
where
x
(
t
) = harmonically fitted means at period
t
(
t
= 1, 2, ......,
P
),
A
0
= population mean,
h
= total number of harmonics [
h
=
P
/2 for even
P
and
(
P
+1)/2 for odd
P
].
P
= base period or period of the function and
A
k
and
B
k
are sine and cosine Fourier coefficients, respectively.
Here,
A
o
,
A
k
and
B
k
are computed as (Shahin et al., 1993):
P
Ç
(1/
Px
)
A
0
=
(76)
t
t
1
P
Ç
(2/
Px
)
sin (2
S
kt P
/
),
A
k
=
k
= 1, 2, ...,
P
/2-1
(77)
t
t
1
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