INTERNATIONAL JOURNAL OF CLIMATOLOGY
Int. J. Climatol. 30: 2289–2298 (2010)
Published online 24 November 2009 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/joc.2045
Periodicities in Indian monsoon rainfall over spectrally
homogeneous regions
Sarita Azad,a T. S. Vigneshb and R. Narasimhaa *
a
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur PO, Bangalore 560 064, India
b GE Global Research, John F Welch Technology Centre, Bangalore 560066, India
ABSTRACT: This work presents results of a sharper search for significant periodicities in Indian monsoon rainfall,
based on the recognition of the area’s meteorological heterogeneity. Towards this end, a quantitative definition of spectral
homogeneity is proposed, and the concept is used to classify India into distinct spectrally homogeneous regions (SHR) by
two independent methods. The analysis is then carried out for each of the 10 SHRs, which may cut across or be subsets
of homogeneous-rainfall zones defined earlier by various workers based on different criteria. A particularly interesting
region is SHR7, the largest spectrally homogeneous cluster identified by both methods, which includes sub-divisions from
west central and peninsular India. The spectrum here shows a significant dip in the frequency band 0.2–0.31 per year,
flanked on either side by a rich structure characterised by nearly coincident spectral peaks in all the seven sub-divisions
constituting the region. The significant peaks (confidence level ≥99%) in SHR7 are 3.0, 5.7, 10.9, 13.3, 24.0, 30.3 and
60.6 years. The spectral dip is conjectured to be associated with the ENSO (EI Niño-Southern Oscillation) phenomenon,
which occurs on the period scales of 3–5 years and is known to be anti-correlated with monsoon rainfall. Copyright 
2009 Royal Meteorological Society
KEY WORDS
Indian monsoon rainfall; spectrally homogeneous zones; periodicities; reference spectrum
Received 23 August 2008; Revised 24 August 2009; Accepted 1 October 2009
1.
Introduction
The problem of predicting seasonal monsoon rainfall, and
indeed of assessing the degree of predictability in the
monsoon, continues to be of great fundamental and practical importance (Webster and Yasunari, 1998; Rajeevan
et al., 2006, 2007). Detection of significant periodicities
in the available rainfall data can be of great value in
prediction and has attracted much attention for nearly
a century. Jagannathan and Bhalme (1973) as well as
Jagannathan and Parthasarathy (1973) used mainly classical correlation and power spectral analysis techniques
to identify significant periodicities in Indian rainfall. The
homogeneous regions identified by Parthasarathy et al.
(1993) have been analysed by various workers (Munot
and Kothawale, 2000; Narasimha and Kailas, 2001; Bhattacharyya and Narasimha, 2005, 2007). A detailed analysis by Kumar (1997) showed a significant periodicity
of 2.8 years at 95% confidence level in the homogeneous Indian monsoon (HIM) region of Parthasarathy
et al. (1993). More recently, periodicities in the HIM
annual rainfall time series have been studied by Azad
et al. (2007, 2008) using multi-resolution analysis.
One persistent question that arises in the quest for identifying periodicities is the heterogeneity of the Indian
* Correspondence to: R. Narasimha, Jawaharlal Nehru Centre for
Advanced Scientific Research, Jakkur PO, Bangalore 560 064, India.
E-mail: [email protected]
Copyright  2009 Royal Meteorological Society
monsoon. In spite of many studies of the problem
(Bhalme et al., 1987; Annamalai, 1995; Kulkarni, 2000),
and the considerable evidence we have for the presence
of such heterogeneity, all-India indices still continue to be
analysed. Such indices represent a mix of diverse rainfall regimes (from the wettest in the world to some of
the driest) and different dynamical factors (Bay of Bengal, Arabian Sea, Indian Ocean, the Himalayas etc.), and
hence also presumably of different potentially present
periodicities. Thus, even those periodicities present in any
one regime can be missed because of poor signal-to-noise
ratios. Furthermore, even the homogeneous regions identified by different workers may contain heterogeneities
not considered in the criteria laid down for determining the degree of homogeneity. For example, Azad et al.
(2008) found that only seven out of the 14 sub-divisions
constituting the HIM region exhibit a characteristic spectral dip around the frequency 0.25 per year. Using heterogeneous data will mask or mix periodicities characteristic of any specific mechanism. At the other extreme,
analysing data for an individual station or sub-division
has to face the disadvantages of the influence of purely
local factors (e.g. topography) and consequently not benefitting from the smoother data and effectively larger sample sizes associated with a larger homogeneous region.
We seek to balance these factors by identifying what we
shall call spectrally homogeneous regions (SHRs).
In Section 2, we give a brief description of the data
analysed. In Section 3, the spectrum of HIM rainfall
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S. AZAD et al.
annual time series is estimated and the significance of
peaks tested against the classical reference spectrum
proposed by Gilman et al. (1963). In Section 4, two
different techniques for analysing spectral homogeneity
are proposed and used. The first is based on the mean
square deviation of the spectrum among normalised subdivisional rainfall spectra. Using this definition, certain
sub-divisions showing significantly low spectral variability are grouped into SHRs. The second method utilises
the cross-correlation coefficient between pairs of rainfall spectra to define a ‘separation metric’, which can be
used, in an automated and objective process, to identify
those time series that are spectrally close. In Sections 5
and 6, SHRs over the entire country are identified and
the significance levels of periodicities detected in these
regions are assessed. These methods help us to identify
the largest spectrally homogeneous region (called SHR7)
within India. This is proposed as a good candidate for
further analysis, as likely to provide the strongest evidence for any periodicity in Indian rainfall. Section 7
summarises our conclusions.
2.
The data analysed
Many attempts have been made to classify India into
different regions that can be considered as homogeneous with respect to the variation of rainfall (Gadgil
and Iyengar, 1980; Gregory, 1989; Gadgil et al., 1993;
Parthasarathy et al., 1993; Guhathaakurta and Rajeevan,
2007). The techniques most commonly used for grouping stations or grid points into homogeneous regions
are cluster, principal component, and correlation analyses. The India Meteorological Department (IMD) divides
the country into 35 meteorological sub-divisions based
on data from 306 well-distributed rain-gauge stations.
Omitting the island sub-divisions and hilly areas, we are
left with 29 sub-divisions in the main land, as listed
in Table I; we consider only these for analysis. Out of
these, 14 sub-divisions in the central and north-western
parts of India covering 55% of the total land area of
the country, namely, Haryana, Punjab, West Rajasthan,
East Rajasthan, East Madhya Pradesh, West Madhya
Pradesh, Gujarat, Konkan, Madhya Maharashtra, Marathwada, Telangana, Vidarbha, Saurashtra and North interior Karnataka, are grouped into the HIM region by
Parthasarathy et al. (1993). This region may be considered to be dominated by the Arabian Sea limb of the
southwest monsoon. The data have been taken from the
website of the Indian Institute of Tropical Meteorology
(http://www.tropmet.res.in). The data used in the first part
of the present analysis consist of the rainfall time series of
these 14 sub-divisions over the period 1871–1990. (More
recent data have not been used because it is not clear that
they have been processed the same way as Parthasarathy
et al. (1993) did.) The HIM rainfall is an area-weighted
average over the ensemble of these 14 rainfall time series.
Each time series is henceforth normalised by subtracting
its mean and dividing by its standard deviation, as the
Copyright  2009 Royal Meteorological Society
Table I. The 29 meteorological sub-divisions considered for
analysis.
Region
Abbreviation
Bihar plains
Bihar plateau
Coastal Andhra Pradesh
Coastal Karnataka
East Madhya Pradesh
East Rajasthan
East Uttar Pradesh
Gangetic West Bengal
Gujarat
Haryana
Sub-Himalayan West Bengal
Kerala
Konkan
Madhya Maharashtra
Marathwada
North Assam
North interior Karnataka
Orissa
Punjab
Rayalaseema
South Assam
Saurashtra
South interior Karnataka
Telangana
Tamil Nadu
Vidarbha
West Madhya Pradesh
West Rajasthan
West Uttar Pradesh
BPL
BPT
CAP
CKA
EMP
ERA
EUP
GWB
GUJ
HAR
HWB
KER
KNK
MMH
MTW
NA
NIK
ORS
PUN
RAY
SA
SAU
SIK
TEL
TN
VDA
WMP
WRA
WUP
present analysis is concerned with spectral structure and
not absolute rainfall.
Parthasarathy et al. (1995) identify five homogeneous
regions over the country on the basis of the following
criteria: (1) contiguity of area, (2) contribution of monsoon seasonal rainfall to the annual amount, (3) intercorrelations of sub-divisional and all-India monsoon rainfall and (4) relationships between sub-divisional monsoon rainfall and regional/global circulation parameters. (Note that criterion (3) above gives weightage to
correlation with an all-India index, rather than closeness among the members of the putative homogeneous
region.) The five homogeneous regions so identified
are: (1) Northwest India (NWI); (2) West Central India
(WCI); (3) Central Northeast India (CNEI); (4) Northeast
India (NEI) and (5) peninsular India (PENSI); (1) and
(2) together comprise the HIM region.
3.
Identifying and testing for periodicities
Estimation of the power spectral density (PSD) of a time
series is usually based on procedures employing the fast
Fourier transform (FFT). For a discrete-time series r(t)
with unit time interval (so the Nyquist frequency is 1/2)
Int. J. Climatol. 30: 2289–2298 (2010)
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PERIODICITIES IN INDIAN MONSOON RAINFALL
the spectral representation is a periodogram defined as
N−1
2
2 −iωk t r(t)e
(1)
r̂(ωk ) =
N
where k = 0, 1, . . . , N /2 is the frequency index and α is
the lag-1 autocorrelation coefficient
where ω = 2πk/N , N is the sample size and k =
0, 1, . . . , N /2 is the frequency index. In the meteorological literature, spectra have often been estimated by
such techniques as the Blackman-Tukey algorithm (e.g.
Kumar, 1997). A more modern method of estimating PSD
is the Welch technique with Hanning window (Stoica and
Moses, 1997). This technique (which to the best of our
knowledge has not been previously used for analysing
monsoon rainfall characteristics) exploits the powerful
idea of the averaged periodogram of overlapped, windowed segments of a time series, and reduces the variance associated with the standard periodogram by cutting
the data into blocks and then averaging over their periodograms. Though various modern methods of spectral
analysis (Broomhead and King, 1986; Ghil and Vautard,
1991; Allen and Smith, 1996; Vaughan, 2004) have been
developed in recent years to estimate the spectrum of a
time series, Allen and Smith (1996) explain that progress
has been hindered by a lack of effective statistical tests to
discriminate between potential oscillations and anything
but the simplest form of noise, namely ‘white’ (independent, identically distributed) noise in which power
is independent of frequency. The authors have recently
shown (Azad et al., 2007) that a particularly appropriate
method for analysing PSD of rainfall data, especially for
separating closely spaced frequencies, is a combination of
wavelet-based multi-resolution analysis (MRA) and PSD
of partially reconstructed time series. In their ‘hybrid’
technique, the advantages of wavelet methods in handling nonlinear non-stationary time series are combined
with those of spectral analysis, and the method has been
found to be useful in removing interference between different scales. As the point we wish to make here concerns
chiefly the notion of spectral homogeneity, we adopt the
Welch technique that has been widely used in other applications.
The statistical significance of any peaks found in
the PSD is usually assessed by devising a reference
background spectrum and testing the hypothesis that
the estimated PSD is a statistical fluctuation from the
underlying reference spectrum for the process.
The presence of noise in a time series is an inherent
cause of unpredictability. By ‘noise’ we mean random
fluctuations, which make the spectrum continuous without sharp peaks. It has been found that most climatic
and geophysical time series tend to have larger power at
lower frequencies; hence the background spectrum often
tends to be an appropriate red noise (Gilman et al., 1963;
Thomson, 1990). This spectrum is obtained from a firstorder autoregressive (AR1) process (Gilman et al., 1963),
and is given by
To illustrate the method we consider HIM rainfall. The
PSD function is estimated using the Welch technique.
It is found that for HIM rainfall α = −0.007 at lag1; so the reference spectrum defined by Equation (2) is
very close to white. The details of significance testing
on HIM rainfall are given in Azad et al. (2008). It is
found that a 2.3-year period is statistically significant
above the 99% confidence level and a 2.8-year period
above the 95% confidence level using the reference
spectrum of Equation (2). However, Kumar (1997), using
the algorithm of Blackman and Tukey (1958), reported
only one spectral peak in HIM rainfall at 2.8 years at 95%
confidence level against a reference white noise spectrum.
(0 ≤ α ≤ 1).
t=0
Pk =
1 − α2
1 + α 2 − 2α cos(2πk/N )
Copyright  2009 Royal Meteorological Society
(2)
4. Notation and methodology for analysing spectral
homogeneity
We now introduce, in two independent ways, the notion
of spectral homogeneity among a set of M sub-divisions
constituting a region. In the first, the PSD function,
defined as r̂ (m) (ωk ), m = 1, . . . , M, k = 0, . . . , N /2, for
the sub-divisional rainfall time series ri(m) , i = 1, . . . , N
for sub-division m, is first estimated using the Welch
technique. For this analysis the time series ri(m) is normalised to have zero mean and unit standard deviation,
i.e. it is the standardised anomaly. The mean square deviation of the sub-divisional PSD from the PSD of regional
rainfall is defined as
sr̂2(m) =
1
N
−1
2
k
(r̂ (m) (ωk ) − r̂(ωk ))2
(3)
where r̂(ωk ) is the PSD function of the ensemble average
rainfall over the region,
(m)
ri (t) =
ri (t)/M
(4)
m
The principle we shall use here is to compare Equation (3) with a similar quantity for independent realisations of synthetic white noise n. For this purpose, the PSD
of M realisations of white noise ni , 1 ≤ i ≤ M each with
the same number of samples (N = 120) as rainfall, is estimated and the mean square deviation of the PSD of each
white noise realisation from the PSD of the ensemble
average is then taken as
sn̂2(m) =
1
N
−1
2
k
(n̂(m) (ωk ) − n̂(ωk )2
(5)
We now explore whether the variation among subdivisional rainfall time series differs significantly from
what may be expected in the same number of different
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S. AZAD et al.
realisations of a white noise process. The confidence level
with which deviations of sub-regional rainfall from white
noise can be identified provides a parameter relevant to
the exercise. As both rainfall and noise are normalised
to have zero mean, the standard deviation provides the
best parameter for testing for deviations between the
two processes. The F -test (Crow et al., 1960) is used
for this purpose. This test provides a measure for the
probability that two independent samples of size N have
the same variance. The estimators of the variances are
sr̂2(m) (Equation (3)) and sn̂2(m) [Equation (5)]. The test
statistic is the ratio
F =
sn̂2(m)
sr̂2(m)
(6)
which follows an F -distribution with N
2 − 1 degrees of
freedom. The null hypothesis is
H0 : sr̂2(m) = sn̂2(m)
(7)
The values of F for a specified confidence level can be
found from tables of the F -distribution in the literature.
When the computed F is too high (Crow et al., 1960),
we reject the null hypothesis at an appropriate confidence
level.
5. Spectrally homogeneous Indian monsoon rainfall
as example
Before presenting a classification of India into SHRs,
it is instructive to analyse HIM data and examine the
homogeneity among the spectra of the 14 sub-divisions
that constitute the region, in order to gain an appreciation
of the extent of spectral homogeneity that may already
be present in previously defined homogeneous-rainfall
regions.
To do this, the spectral mean square deviation of each
sub-divisional rainfall from HIM rainfall is calculated
using Equation (3). Similarly an ensemble of 14 white
noise deviations (of sample size 120) is then calculated
using Equation (5). The average σn2 over an ensemble
of 1000 such realisations is found to be 1.82, which
can be taken as a population statistic. To check the
null hypothesis that the two variances are the same,
we compute F = σn2 /sr̂2(m) , m = 1, . . . , 14, and apply the
F -test. Results are given in Table II. It is clear from the
table that the spectral mean square deviation of each subdivisional rainfall is appreciably lower than that for white
noise by a factor that varies from 0.76 to 0.39.
Based on Table II we can group sub-divisions into
sub-regions where sr̂2(m) is less than σn2 at specified
confidence level intervals. It is found that the four
sub-divisions Telangana, West Madhya Pradesh, East
Rajasthan and Vidarbha form a group with a rejection
probability >99.5%. We tentatively identify this group
as a candidate for a spectrally homogeneous sub-region,
and confirm that this is so by repeating the process above,
Copyright  2009 Royal Meteorological Society
Table II. F -test formulated from the spectral deviations of
sub-divisional rainfall in HIM region.
sr̂2(m) F = σn2 /sr̂2(m) Probability
(%)a
Sub-division
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Telangana
West Madhya Pradesh
East Rajasthan
Vidarbha
Konkan
East Madhya Pradesh
West Rajasthan
Marathwada
Haryana
Saurashtra
Gujarat
Madhya Maharashtra
North interior Karnataka
Punjab
0.715
0.735
0.771
0.860
0.947
1.077
1.082
1.148
1.160
1.168
1.176
1.231
1.293
1.381
2.55
2.48
2.36
2.12
1.92
1.70
1.69
1.59
1.57
1.56
1.55
1.48
1.41
1.32
>99.9
>99.9
>99.9
(99.5, 99.9)
(99.0, 99.5)
(97.5, 99.0)
(97.5, 99.0)
(95.0, 97.5)
(95.0, 97.5)
(95.0, 97.5)
(95.0, 97.5)
(90.0, 95.0)
(90.0, 95.0)
<90.0
a Probability for rejecting the hypothesis that the spectral standard
deviation of rainfall is the same as that of white noise.
replacing the parameters of the HIM by that of the subregion of four sub-divisions. We call this the ‘spectrally
homogeneous Indian monsoon region’ (SHIM). Figure 1
shows that the SHIM sub-region represents a contiguous
area embedded within the HIM region. We can therefore
assert, with very high confidence (>99.5%), that the four
sub-divisions mentioned constitute a strongly spectrally
homogeneous sub-region. We also see from Table II that
three sub-divisions are at a confidence level below 95%,
and one below 90%; so HIM is clearly not sufficiently
homogeneous in spectral space.
5.1. Periodicities in SHIM and HIM rainfall
We can now compare periodicities in SHIM and HIM
rainfall. For this purpose the time series of SHIM rainfall is prepared by area-weighted averaging over the time
series of the four sub-divisions constituting it. Figure 2
shows the result of the significance test for periodicities
in PSD against the classical reference spectrum [Equation (2)]. We observe here that six periods, respectively of
2.1, 2.3, 2.8, 7.5, 13.3 and 60.0 years, are at or above the
95% confidence line in SHIM, whereas only two periods,
at 2.3 and 2.8 years respectively, are so observed in HIM
rainfall (Azad et al., 2008). Also the 2.3-year period in
SHIM rainfall (Figure 2) is at 99.5% confidence, whereas
it is 99% in the case of HIM rainfall.
Azad et al. (2008) also report that seven out of the 14
sub-divisions constituting the HIM region show a spectral
dip around a frequency of 0.25 per year. In the SHIM subregion we find all four sub-divisions showing a spectral
dip, three of them in the frequency band 0.2–0.31 per
year (Figure 3) and the fourth (East Rajasthan) over
the slightly lower frequency band 0.13–0.26 per year.
We can thus confirm that while the HIM region is not
spectrally homogeneous, a subset of it that we have called
the SHIM sub-region is spectrally homogeneous to a high
degree of confidence.
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PERIODICITIES IN INDIAN MONSOON RAINFALL
Figure 1. SHIM sub-region within the HIM region.
6.
Spectrally homogeneous clusters
Following the procedure of Section 4.1, we first identify
spectrally homogeneous sub-regions in each of the homogeneous regions of Parthasarathy et al. (1993). Those
which are left out of these sub-regions are then reconsidered with neighbouring sub-divisions to determine
whether they form additional SHRs. By this process we
identify 10 SHRs in India, as listed below, and shown in
Figure 4:
Figure 2. The estimated SHIM spectrum obtained from the Welch
technique compared with the reference spectra obtained from PSD of
the AR1 process at different confidence levels.
In summary, it is seen that HIM rainfall time series
over the time period 1871–1990 shows one 99% significant period at 2.3 years and a 95% significant period at
2.8 years, tested against the classical reference spectrum
of Equation (2). However, by introducing the spectrally
homogeneous sub-region SHIM, we have a confidence
level of 99.5% for the 2.3-year period, and five other
significant peaks at or above 95% confidence at 2.1, 2.8,
7.5, 13.3 and 60.6 years.
Copyright  2009 Royal Meteorological Society
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NA, SA, HWB, BPL;
GWB, BPT;
ORS;
EUP, WUP, EMP;
WMP;
HAR, PUN, WRA, ERA, GUJ, SAU;
KNK, MMH, MTW, VDB, TEL, CKA, NIK;
CAP, RAY, SIK;
KER;
TN.
We find that there are four sub-divisions (West Madhya
Pradesh, Orissa, Kerala and Tamil Nadu), none of which
is spectrally homogeneous with any other sub-division.
To assess how robust this classification into SHRs
is, we present here a second method of identifying
homogeneous zones based on spectral data. This is done
by the technique of hierarchical clustering.
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S. AZAD et al.
Figure 3. Estimated PSD of four sub-divisions constituting the SHIM sub-region.
Hierarchical clustering of a given set of M entities
(Dunham 2002) requires an M × M distance matrix
whose ij element is a suitably defined distance between
the corresponding entities i and j . In the present case
we select, for any two sub-divisions i and j , a distance
measure sij between their spectra the departure of the
spectral cross-correlation coefficient cij from unity. That
is, cij is evaluated between the power spectra of the
rainfall in the two sub-divisions,
sij ≡ 1 − cij = 1−
r̂ (i) (ωκ ) − r̂ (i) (ωκ ) r̂ (j ) (ωκ ) − r̂ (j ) (ωκ )
E
σr̂ (i) .σr̂ (j )
Figure 4. The 10 SHRs in India as defined by the spectral deviation method.
Copyright  2009 Royal Meteorological Society
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PERIODICITIES IN INDIAN MONSOON RAINFALL
Figure 5. Dendrogram representing the hierarchical clustering of the 29 monsoon regions. The y-axis represents the separation metric as defined
in Section 6. The x-axis has abbreviations of the 29 monsoon regions (Table I).
Here E is the expectation, r̂ (i) (ωκ ) and r̂ (j ) (ωκ ) are
the spectra of the rainfall in sub-divisions i and j
respectively, and σr̂ (i) and σr̂ (j ) are the respective standard
deviations; bars denote average over ωκ .
If i and j have identical spectra sij = 0, and if they
are perfectly anti-correlated sij = 2. So sij can be called
the separation metric between sub-divisions i and j ,
1 ≤ i < j ≤ 26. Hierarchical clustering, as defined in
Dunham (2002), can be implemented using so-called
(1) single, (2) complete or (3) average linkage functions.
To define these functions, let Si , i = 1 to M, denote
the sub-divisions and Rα , α = 1 to N , the collection of
SHRs. A linkage function Lκλ is a separation metric that
measures how close any two clusters Rκ , Rλ are.
We consider three widely used linkage types. The
first, called single linkage, takes as the separation metric
between Rκ , Rλ the quantity
Lκλ = min s (iκ , jλ )
1 ≤ iκ ≤ MK
1 ≤ jλ ≤ Mλ
where Mκ , Mλ are the number of sub-divisions in Rκ , Rλ ,
respectively. That is, in this linkage type, Lκλ is taken as
the minimum separation between any sub-division in Rκ
and any sub-division in Rλ . This metric is clearly too
lenient. The second, complete linkage, goes to the other
extreme and picks the maximum separation instead; i.e.
Lκλ = max s(iκ , jλ )
1 ≤ iκ ≤ Mκ
1 ≤ jλ ≤ Mλ
The third takes the average value among all s (iκ , jλ );
Lκλ = ave s(iκ , jλ )
1 ≤ iκ ≤ Mκ
1 ≤ jλ ≤ Mλ
Copyright  2009 Royal Meteorological Society
Figure 6. Sub-divisional rainfall spectra in SHR7.
We have investigated the hierarchical clusters that
emerge from each of these three linkage functions.
Figure 5 presents the results for complete linking, in
a cluster diagram (generally called ‘dendrogram’). This
diagram plots the separation metric as ordinate, with
points on the abscissa denoting the sub-divisions as
marked. Each horizontal bar in the diagram has two
descending limbs linking the sub-clusters below the bar.
The height of any horizontal bar in the diagram denotes
the separation between the sub-clusters below the bar. For
example, the separation metric between KNK (Konkan)
and MTW (Marathwada) is 0.33, and that between the
KNK-MTW and MMH-NIK sub-clusters is 0.54. By
drawing across the whole diagram a horizontal line marking a threshold for the maximum acceptable separation
between clusters, we can quickly identify all the clusters which are closer to each other than the selected
threshold. For example the following constitute the set
of clusters (numbering 11 in all) if we take the threshold
as 0.83:
1. KNM, MTW, MMH, NIK, VDA, TEL, CKA;
2. HAR, PUN, EMP, WMP;
3. WRA, ERA, GUJ, SAU;
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S. AZAD et al.
Table III. Ninety percent significant periodicities present in the rainfall of the seven sub-divisions constituting SHR7.
Sub-division
1.
2.
3.
4.
5.
6.
7.
Konkan
Madhya Maharashtra
Marathwada
Vidarbha
Telangana
Coastal Karnataka
North interior Karnataka
Periodicities (year)
2.14
2.31
2.31
2.31
2.31
2.31
3.0
3.0
2.66
20.2
5.71, 5.99
7.51
2.79
14.99
10.91
10.0
7.51
23.98
59.88
14.99
2.26
2.60, 2.66
59.88
Note: Numbers in bold numerals refer to the singletons discussed in para 2 of section 7.
NA, SA;
EUP, WUP;
CAP, RAY;
TN, SIK;
MWB, ORS;
GWB, BPT;
KER;
BPL.
Remarkably the first nine multi-member clusters all have
geographically contiguous sub-divisions. Furthermore,
the first cluster is identical with SHR7.
The application of this technique for defining other
clusters of meteorological interest will be described
elsewhere (Vignesh and Narasimha, forthcoming). It is
enough to note here that SHR7 emerges as the largest
single spectrally homogeneous region in India by two
independent methods, and many of the other clusters
are very similar in composition. We therefore proceed
to present an analysis of the SHRs defined in Section 5.
7.
Discussion and Conclusions
To demonstrate explicitly spectral similarity, the estimated PSD functions of the sub-divisions constituting
SHR7 are shown in Figure 6 and the significant periodicities in these sub-divisions above 90% confidence level
[using Equation (2)] are listed in Table III. It is found that
peaks in any frequency band for these sub-divisions are
in near-coincidence with those from other sub-divisions,
which is clearly different from the seven white noise
realisations, each of the same sample size as rainfall
(Figure 7). The similarity in the sub-divisional spectra
in SHR7, including the presence of the spectral dip, is
visibly evident in Figure 6. The seven sub-divisions constituting SHR7 come from what are often thought to
be different homogeneous regions by other criteria, as
they cut across the hilly Western Ghats, from the coast
(Konkan and Coastal Karnataka) to the ‘rain-shadow’
areas of North interior Karnataka and Madhya Maharashtra beyond the Western Ghats (Figure 4). Comparison
of Figures 6 and 7 shows the power of the concept of
spectral homogeneity, and suggests that similar dynamical forcing must operate in the region, even though the
Copyright  2009 Royal Meteorological Society
Figure 7. White noise spectra of seven realisations, each of sample size
120.
3500
3000
Mean rainfall (mm)
4.
5.
6.
7.
8.
9.
10.
11.
2500
2000
1500
1000
500
0
1
2
3
4
5
6
7
Sub-divisions
Figure 8. Mean of the rainfall of seven sub-divisions constituting
SHR7: 1 = Coastal Karnataka, 2 = Konkan; 3 = Madhya Maharashtra;
4 = North interior Karnataka; 5 = Marathwada; 6 = Telangana;
7 = Vidarbha.
magnitude of rainfall varies across SHR7 between 600
and 3200 mm (Figure 8).
To assess the difference between SHR7 and white noise
spectra, we define the null hypothesis that the numbers of
singletons (= unrepeated peaks) above 90% significance
level in the seven sub-divisional rainfall time series in
SHR7 are the same as in seven realisations of white noise.
We find on an average there are 22.15 unrepeated peaks
or singletons in white noise above 90% confidence level,
Int. J. Climatol. 30: 2289–2298 (2010)
2297
PERIODICITIES IN INDIAN MONSOON RAINFALL
Table IV. Significant periodicities (year) obtained by the methods of PSD and PSD + MRA in SHRs.
SHR
1
2
3
4
5
6
7
8
9
10
PSD
95% significance
MRA + PSD
99% significance
2.3, 6.0
3.3
2.8, 3.3, 3.6, 60.0
2.1, 10.9
2.1, 2.3, 13.3, 60.0
2.3, 2.8, 3.3
2.3
7.5
3.4
2.3, 3.7, 10.9
3.5, 4.5, 6.0, 8.6, 10.8, 20.0, 30.3
3.3, 4.0, 10.0, 30.3, 60.0
2.8, 3.3, 3.6, 4.1, 8.6, 13.3, 24.0, 60.0
2.8, 3.3, 5.5, 6.6, 7.5, 8.6, 10.9, 24.0, 60.0
2.8, 3.5, 7.5, 8.6, 13.3, 30.3, 60.0
2.8, 3.3, 4.0, 4.8, 8.6, 12.5, 17.8, 30.3
3.0, 5.7, 10.9, 13.3, 24.0, 30.3, 60.6
3.3, 4.6, 7.5, 13.3, 24.0, 60.0
3.4, 4.5, 6.0, 9.2, 12.0, 17.2
3.4, 3.7, 4.6, 5.7, 10.9, 20.0, 42.0
with a standard deviation of 3.86. From Table III we see
that there are 10 singletons in SHR7 spectra. Using the
z-test (Crow et al., 1960) with 6 degrees of freedom,
the null hypothesis is rejected at a confidence level of
99.99%.
The significance levels of periodicities obtained in the
10 SHRs are assessed using the method of direct PSD
and hybrid technique of MRA + PSD proposed in Azad
et al. (2008). Results are listed in Table IV.
Following are the major conclusions from the present
work:
The homogeneous-rainfall regions identified in earlier
works by various authors based on various criteria are
in general spectrally heterogeneous. Regions that do not
have similar spectra cannot be dynamically similar either.
The few significant periodicities found in earlier work
are therefore at least in part attributable to the use of
(spectrally) mixed samples, but in part also to use of less
powerful analysis techniques. We have sought to overcome these problems here by using spectral homogeneity
as a criterion for classifying the country into distinct
spectrally homogeneous regions. We have two different
procedures for identifying spectral homogeneity. In the
first we define a measure of the spectral deviation of subdivisional rainfall anomalies from the regional characteristic, and identify clusters within which the deviations are
small to a high degree of confidence assessed by a Monte
Carlo-type test using equivalent ensembles of white noise
realisations. In the second, a separation metric is defined
in correlation space, and an objective, automated procedure is used for defining spectrally homogeneous clusters.
Remarkably, both methods identify a cluster of seven
sub-divisions (called here SHR7) as the largest spectrally homogeneous region in India. In general, the new
regions so identified cut across those defined earlier by
other workers. For example, SHR7 has six sub-divisions
from the HIM region and one from the PENSI region,
and spans the northern west coast to the northern part
of the peninsula. SHR7 is of particular interest as all
the constituent sub-divisions show a spectral dip around
the frequency of 0.25 per year, and most of the spectral peaks in the different sub-divisions nearly coincide
with each other. We have not pursued here in detail the
Copyright  2009 Royal Meteorological Society
possible dynamical mechanisms for the computed spectral structure, but conjecture that the spectral dip is due to
the ENSO (EI Niño-Sourthern Oscillation) phenomenon,
which has similar time scales and is anti-correlated with
monsoon rainfall.
The present findings have strong implications for
rainfall prediction, which will be considered separately.
Acknowledgements
The authors are grateful to the Centre for Atmospheric
and Oceanic Sciences of the Indian Institute of Science for their continued hospitality. R. N. is grateful to DRDO for financial support through project
no. DRDO/RN/4124.
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