J. Ind. Geophys. Union ( October 2013 ) v.17,no.4,pp:335-339 Studies on the trend and chaotic behaviour of Tamil Nadu rainfall P. Indira 1, S. Stephen Rajkumar Inbanathan 2 1 Research and Development Centre, Bharathiar University, Coimbatore, India 2 Post-Graduate Research Department, The American College, Madurai-625 002, India Email: indiraindhu2006@gmail.com Abstract This study is aimed at in finding the trend or nonlinearity that may exist in the rainfall pattern of the State of Tamil Nadu. Based on the observed rainfall data for a period of 111 years (1901-2011), a trend analysis with Hurst exponent and non-linearity analysis with Lyapunov exponent is employed. The interpretation of the Hurst exponent of Tamil Nadu rainfall indicates that the Southwest Monsoon rainfall shows a persistent behaviour, whereas the annual rainfall shows an anti-persistence behaviour. The Northeast Monsoon, which is the most important season for the State of Tamil Nadu shows a randomness or non-linearity in its value. The non-linearity in the Northeast Monsoon has been verified through Lyapunov Exponent. The Northeast Monsoon shows a largest positive Lyapunov exponent value than the Southwest and Annual rainfall. This indicates that the Northeast Monsoon of Tamil Nadu is chaotic. The chaotic nature of the Northeast Monsoon is also verified using surrogate data method. INTRODUCTION Tamil Nadu mostly depends on monsoon rains, and thereby is prone to droughts when the monsoons fail. It has distinct periods of rainfall, which are the advancing monsoon period (second half of May), Southwest Monsoon (from June to September) with strong southwest winds, the Northeast Monsoon (from October to December) with dominant northeast winds. The normal Annual rainfall of the State is about 945 mm (37.2 in) of which 48% is through the Northeast monsoon, and 32% through the Southwest monsoon. Since the State is entirely dependent on rains for recharging its water resources, monsoon failures lead to acute water scarcity and severe drought. The Southwest Monsoon (SWM) has been studied by the researchers throughout the world to understand its dynamics, predictability, cloud physics and tele-connection aspects; on the other hand the Northeast Monsoon (NEM) has not been studied extensively. However, interesting results, on the pulsatory behaviour of NEM, its discordant relationship with SWM etc., have been documented in the literature based on the limited studies. In analysis the behaviour of the NEM, the prevalence of strong winds over India during March May is favourable for an above normal NEM rainfall activity over the southern peninsular India (Raj and Geetha, 2008), while the same is unfavourable for the ensuing SWM rainfall over India. An important historical trend that is in favour of a good Northeast Monsoon is its discordant relationship with the summer monsoon. Studies that have analysed 110 years of rainfall data have revealed that mostly, when the SWM had been below normal or had failed the NEM had strengthened. In 29 years when the SWM was below normal, the NEM was above normal (Samuel Selvaraj and Raajalakshmi Aditya, 2011). Likewise, in 34 years when the SWM was above normal, rainfall from NEM was below normal (Dhar and Rakhecha, 1983). ONSET AND WITHDRAWAL OF NORTH EAST MONSOON During October, the monsoon trough is positioned over northern parts of India in a NW-SE orientation at the surface and the lower troposphere starts a rapid shift southwards (Khole and De, 2003). The low level winds reverse to north easterlies and Southwest Monsoon withdraws from these regions. However, over the southern peninsula, the rainfall continues in October also with a sharp increase of rainfall over coastal Tamil Nadu (sometimes) in the second half of October which is taken as NEM onset. The onset
P. Indira, S. Stephen Rajkumar Inbanathan Season Table 1. The average Annual rainfall pattern of Tamil Nadu Months Normal Rainfall in mm Percentage of annual Rainfall Southwest Monsoon June-September 322.00 32.96% Northeast Monsoon October-December 470.00 48.10% Average Rainfall January-December 977.00 100.00% of NEM is clearly delineated and well defined over coastal Tamil Nadu. The average Annual rainfall pattern for Tamil Nadu is shown in Table1. The presence of non-linearity in the NEM is analysed in the present paper using the mathematical tools of Hurst Exponent and the Lyapunov exponent. DATA AND METHODOLOGY The data for the Tamil Nadu rainfall for a period of 1901-2011 was procured from regional meteorological centre, Chennai. The data is categorised into Southwest, Northeast and Annual rainfall. We then execute the Hurst exponent for the three categories. Through the H value, the given time series data is identified as persistence, anti-persistant or chaotic. If the time series data shows persistence behaviour, then autoregressive process or moving average or auto regressive moving average methods are applied for their analysis. If chaos is present in the system, the next step is to check for magnitude of chaos. The Lyapunov exponent is employed to analysis weather the data has low dimensional chaos or high dimensional chaos. Generally, Lyapunov exponent is the best method of identification of presence of chaos but this method alone does not prove the chaotic behaviour, so surrogate data method is also used to confirm the presence of chaos in NEM rainfall of Tamil Nadu. HURST RESCALED RANGE ANAYSIS We provide below a concise summary of Hurst s rescaled range method below. To calculate the Hurst exponent, one must estimate the dependence of the rescaled range on the time span of n observation (Mandelbrot and Wallis, 1969). A time series of full length N is divided into a number of shorter time series of length n = N, N/2, N/4... The average rescaled range is then calculated for each value of n. For a (partial) time series of length n, X1, X2...X n, the rescaled range is calculated as follows: 1. A time series (X k ) K 1,N is divided into d sub-series of length m. for each sub-series n=1,.d; 2. Find the mean E n and the standard deviation S n. 3. Normalize the data (X in ) by subtracting the sub-series mean: Z in = X in E n, i=1,...,m 4. Create a cumulative time series: i Y in = S Z jn, i=1...,m; j=1 5. Find the range R in = max Y jn min Y jm j=l,m j=1,m 6. Rescale the range R n /S n 7. Calculate the mean value of the rescaled range for all sub-series of length m 1 d (R/S) m = S R n /S n d m=1 Hurst found that (R/S) increments by power-law as time increases, which indicates (R/S) t = c.t H H can be estimated as the slope of log-log plot of (R/S) t versus t. H describes the correlation between the past and future in the time series. For independent random processes with finite variances, the H value is 0.5. When H > 0.5, the time series is persistent, which means that an increasing trend in the past is indicative of an increasing trend in the future. Conversely, as a general rule, a decreasing trend in the past signifies a persistent decrease in the future. When H < 0.5, the time series is anti-persistent, which means that an increasing trend in the past implies a decreasing trend in the future and viceversa. If H is more or less equal to 0.5 it indicates that the time series is random. The values of the Hurst exponent were tabulated in Table 2. 336
Studies on the trend and chaotic behaviour of Tamil Nadu rainfall Table 2. Hurst and Lyapunov Exponent Values of Tamil Nadu rainfall Season Hurst Exponent Lyapunov Exponent Southwest Monsoon 0.59 2.013 Northeast Monsoon 0.53 2.717 Annual 0.4343 2.158 Figure 1. Hurst Exponent (R/S) graph for Northeast, Southwest and Annual rainfall of TamilNadu LYAPUNOV EXPONENT The Lyapunov exponent gives the quantitative value for a non-linear dynamical system. A positive largest Lyapunov exponent indicates chaos. It is thus useful to study the mean exponential rate of divergence of two initially close orbits using the formula (Dechert and Gencay, 1992). This number, called the Lyapunov exponent "λ", is useful for distinguishing among the various types of systems. It works for discrete as well as continuous systems. λ < 0 Negative Lyapunov exponents are characteristic of dissipative or non-conservative system λ = 0 A Lyapunov exponent of zero indicates that the system is in steady state mode or conservative. λ > 0 A large positive Lyapunov exponent indicates the system is unstable and chaotic The values obtained from the qualitative and quantitative analysis of the Tamil Nadu Rainfall pattern are tabulated in Table 2. SURROGATE DATA ANALYSIS The method of surrogate data (Theiler, Eubank and Physica, 1992) is an approach that makes use of the substitute data generated in accordance with the probabilistic structure underlying the original data. This means that the surrogate data possess some of the properties, such as the mean, the standard deviation, the cumulative distribution function, the power spectrum, but are otherwise postulated as random. It is generated according to a specific null hypothesis. Here, the null hypothesis is that the rainfall time 337
P. Indira, S. Stephen Rajkumar Inbanathan Table 3. Lyapunov Exponent value for Original and Surrogate rainfall data of Tamil Nadu Season Lyapunov Exponent Southwest Monsoon Original rainfall data Surrogate rainfall data Southwest Monsoon 2.013 1.352 Northeast Monsoon 2.717 0.686 Annual 2.158 0.938 series is a linear process, and the goal is to reject the hypothesis that the original data have come from a linear process. The rejection of the null hypothesis can be made based on some discriminating statistics. Since the primary interest is to identify chaos in the time series, it would be desirable to use any of the statistics used for the identification of chaos, such as the correlation dimension, the Lyapunov exponent, the Kolmogorov entropy, the prediction accuracy, etc. If the discriminating statistics obtained for the surrogate data are significantly different from those of the original time series, then the null hypothesis can be rejected, and original time series may be considered to have come from a nonlinear process. On the other hand, if the discriminating statistics obtained for the original data and surrogate data are not significantly different, then the null hypothesis cannot be rejected, and the original time series is considered to have come from a linear process (Bellie Sivakumar et al., 1999) In the present study the Surrogate data was generated using mat lab programme and Lyapunov exponent was carried out for the surrogate data generated for the Southwest Northeast and Annual rainfall of Tamil Nadu. The Lyapunov exponent value for both surrogate data and the original data is listed in Table 3. RESULT AND DISCUSSION Table 3 illustrates the difference in the Lyapunov exponent that was calculated for the Tamil Nadu rainfall data. From it, we infer that the Lyapunov value for the original data and that of the surrogate data is different for all three cases and this is more explicit in the Northeast rainfall data, which we assumed to be chaotic in nature from the results of Hurst exponent. The surrogate results also suggest that Northeast monsoon rainfall is a nonlinear system. The value 2.717 for northeast monsoon rainfall shows that a low dimensional chaos is present in the Northeast monsoon rainfall. CONCLUSION The Northeast monsoon rainfall in Tamil Nadu is an interesting weather phenomenon that is chaotic but challenging for the monsoon meteorologists. The surrogate data method has ensured the Northeast monsoon to be chaotic and the Lyapunov exponent value for the Northeast monsoon rainfall was estimated to be 2.7, the Lyapunov value for the Annual rainfall and the Southwest monsoon rainfall was calculated to be 2.15 and 2.01, respectively. The Hurst exponent was found to be persistent for the Southwest monsoon. The Southwest monsoon was studied by many researches and it showed a decreasing trend. This trend on Southwest monsoon will be persistent as indicated by the Hurst exponent value. Similarly, the Annual rainfall with H value 0.43 will have an anti- persistent nature. The Northeast monsoon with H=0.53 exhibits a chaotic nature. Thus, the improved methods of observation and the analysis of the data through the sophisticated mathematical techniques are essential to understand the chaotic nature of the Northeast Monsoon rainfall. REFERENCES Bellie Sivakumar, Shie-Yui Lion, Chih-Young Liaw, and Kok-Kwang Phoon, 1999. Singapore rainfall behavior: Chaotic? Journal of Hydrologic Engineering, 4, 1, January, Paper No. 14421. Dechert, W.D and Gencay. R., 1992. Lyapunov exponents as a nonparametric diagnostic for stability analysis, Journal of Applied Econometrics, 7, S41-S60. 338
Studies on the trend and chaotic behaviour of Tamil Nadu rainfall Dhar, O.N. and Rakhecha, P.R., 1983. Foreshadowing Northeast monsoon rainfall over Tamil Nadu, India, Monthly Weather Review, v.111, pp:109-112. Khole, M. and De U.S., 2003. A study on northeast monsoon rainfall over India, Mausam, v.54, no.2, pp:419-426. Mandelbrot, B.B. and Wallis, J.R., 1969. Robustness of the rescaled range R/S in the measurement of non-cyclic long-run statistical dependence, Water Resources, v.5, pp:967-988. Raj, Y.E.A. and Geetha, B., 2008. Relation between southern oscillation index and Indian northeast monsoon as revealed in antecedent and concurrent modes, Mausam, v.59, no.1, pp:15-34. Samuel Selvaraj, R. and Raajalakshmi Aditya, 2011. Study on correlation between southwest and northeast monsoon rainfall over Tamil Nadu, Universal Journal of Environmental Research and Technology, v.1, no.4, pp:578-581. Theiler, J., Eubank, S., Longtin, A., Galdrikian, D. and Farmer, J.D., 1992. Testing for non-linearity in time series: the method of surrogate data, Physica, D 58 (1992), 77, Reprinted in: E. Ott, T. Sauer, J.A. Yorke (Eds.), 1994, Coping with Chaos, Wiley, New York. Manuscript received: Feb, 2013, accepted: July, 2013 P. INDIRA has obtained her M.Sc., M.Phil. degrees in Physics from Madurai Kamaraj University. She is presently pursuing Ph.D. from Bharathiar University. She has good teaching experience in physics and is presently working in Prince Dr. K. Vasudevan College of Engineering and Technology, Chennai. Her current research interests include atmospheric physics and weather prediction. Dr. S. Stephen Rajkumar Inbanathan is working as Assistant Professor of Physics at the American College, Madurai. He received his Ph.D. from Banaras Hindu University, Varanasi. He was a visiting faculty at Hebrew University, Jerusalem, Israel and visiting associate professor of Inter University Consortium on Astronomy and Astrophysics, Pune. 339