Periodicities in Indian monsoon rainfall over ... - Wiley Online Library

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 Ni˜no-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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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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the spectral representation is a periodogram defined as N−1 2 2 −iωk t r(t)e (1) rˆ (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 rˆ (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 sr2ˆ (m) =

1

N −1 2

k

(ˆr (m) (ωk ) − rˆ (ωk ))2

(3)

where rˆ (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 sn2ˆ (m) =

1

N −1 2

k

(nˆ (m) (ωk ) − n(ω ˆ k )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 Int. J. Climatol. 30: 2289–2298 (2010)

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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 sr2ˆ (m) (Equation (3)) and sn2ˆ (m) [Equation (5)]. The test statistic is the ratio F =

sn2ˆ (m) sr2ˆ (m)

(6)

which follows an F -distribution with N 2 − 1 degrees of freedom. The null hypothesis is H0 : sr2ˆ (m) = sn2ˆ (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 /sr2ˆ (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 sr2ˆ (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. sr2ˆ(m) F = σn2 /sr2ˆ(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) 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. Int. J. Climatol. 30: 2289–2298 (2010)

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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. Int. J. Climatol. 30: 2289–2298 (2010)

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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− rˆ (i) (ωκ ) − rˆ (i) (ωκ ) rˆ (j ) (ωκ ) − rˆ (j ) (ωκ ) E σrˆ (i) .σrˆ (j )

Figure 4. The 10 SHRs in India as defined by the spectral deviation method. Copyright  2009 Royal Meteorological Society

Int. J. Climatol. 30: 2289–2298 (2010)

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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, rˆ (i) (ωκ ) and rˆ (j ) (ωκ ) are the spectra of the rainfall in sub-divisions i and j respectively, and σrˆ (i) and σrˆ (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; Int. J. Climatol. 30: 2289–2298 (2010)

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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)

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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 Ni˜no-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. References Allen MR, Smith LA. 1996. Monte Carlo: detection irregular oscillations in the presence of colored noise. Journal of Climate 9: 3373–3404. Annamalai H. 1995. Intrinsic problem in the seasonal prediction of the Indian summer monsoon rainfall. Meteorology and Atmospheric Physics 55: 61–76. Azad S, Narasimha R, Sett SK. 2007. Multiresolution analysis for separating closely spaced frequencies with an application to Indian monsoon rainfall data. International Journal of Wavelets, Multiresolution and Information Processing 5(5): 735–752. Azad S, Narasimha R, Sett SK. 2008. A wavelet based significance test for periodicities in Indian monsoon rainfall data. International Journal of Wavelets, Multiresolution and Information Processing 6(2): 291–304. Bhalme HN, Rahalkar SS, Sikder AB. 1987. Tropical quasi-biennial oscillation of 10 mb wind and Indian monsoon rainfall-implication for forecasting. International Journal of Climatology 7: 345–353. Bhattacharyya S, Narasimha R. 2005. Possible association between Indian monsoon rainfall and solar activity. Geophysical Research Letters 32: L05813. Bhattacharyya S, Narasimha R. 2007. Regional differentiation in multidecadal connections between Indian monsoon rainfall and solar activity. Journal Geophysical Research 112: D24103. Blackman RB, Tukey JW. 1958. The Measurement of Power Spectra from the Point of View of Communications Engineering. Dover Publications: New York; 190. Broomhead DS, King G. 1986. Extracting qualitative dynamics from experimental data. Physica D 20: 217–236. Crow EL, Davis FK, Maxfield MW. 1960. Statistics Manual: with Examples Taken from Ordnance Development. Dover Publications: New York. Dunham MH. 2002. Data Mining Introductory and Advanced Topics: (Chapter 5). Prentice Hall: Upper Saddle River; New Jersey. Int. J. Climatol. 30: 2289–2298 (2010)

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