Multifractal analysis of rainfall dynamics in Piracicaba, São Paulo, Brazil.

Transcription

1 Multifractal analysis of rainfall dynamics in Piracicaba, São Paulo, Brazil. Sílvio Fernando Alves Xavier Júnior 1 Tatijana Stosic 2 Borko Stosic 2 Wilson Rosa de Oliveira Júnior 2 1 Introduction Over the last several decades the multifractal theory [1] initially developed to model velocity fluctuations in turbulent flow [2] has been extensively used to describe and model temporal and spatial distribution of rainfall and stream flow [3]-[8]. By analyzing correlations within the data on different scales, the multifractal approach thus makes it possible to assess the full range of precipitation dynamics, which is crucial for understanding and modeling various hydrological phenomena such as floods, droughts, runoff, soil erosion, pollution transport, water infiltration etc [9],[10]. Global historical rainfall series obtained by traditional rain gauge measurements are mostly available at coarse time scales (daily or above resolution). The lack of high resolution rainfall data, due to their costly and time consuming acquisition, can be overcome by i) developing efficient downscaling models that produce synthetic data at a higher temporal and spatial resolution based on low resolution observations, or ii) by downscaling the output of the regional climatic models and global circulation models. While a decrease in water discharge can be explained by anthropogenic intervention (in the 3 1 early 80's a series of reservoirs were built in order to export approximately 31 m s from the Piracicaba river basin to the metropolitan region of Sao Paulo city), the increase in precipitation is the result of complex interactions between human and natural factors and it is not yet completely understood [11]. In order to contribute to a better understanding of hydrological processes in this basin and possible relation with natural and anthropogenic factors, we apply multifractal analysis on daily rainfall time series. We compare the results of multifractal analysis for the periods before and after 1970, when the positive trend in 1 UFRPE Depto. de Estatística e Informática. silvioxj@gmail.com 1 UFRPE Programa de pós-graduação em Biometria e Estatística Aplicada. wilson.rosa@gmail.com 2 UFRPE Programa de pós-graduação em Biometria e Estatística Aplicada. tatstosic@gmail.com 2 UFRPE Programa de pós-graduação em Biometria e Estatística Aplicada. borkostosic@gmail.com 1

2 precipitation was found to be much stronger [11],[12]. In the following section we describe the data and present the multifractal analysis. The subsequent section deals with the results of our analysis, and finally the conclusions are drawn. 2 Material and Methods Rainfall data The data used in this work are provided by the Agro-meteorological Sector of the Department of Exact Sciences of the College of Agriculture Luiz de Queiroz, University of Sao Paulo (Escola Superior de Agricultura Luiz de Queiroz-ESALQ, USP), Brazil. The data are collected at an agro-meteorological station located in the city of Piracicaba (latitude S, longitude ' ' W and altitude 546m), state of Sao Paulo, southeast Brazil, and they are available at The climate in the region is sub-tropical C with a wet season from October to March and a dry season from April to September. Eighty percent of the total annual rainfall of mm occurs in the wet season with an average monthly precipitation of more than 300mm, while average monthly precipitation in the dry season is less than 29mm. The average temperature in Piracicaba ranges from 19C in the winter to 24C in the summer [13]. We analyze daily rainfall records collected during the period , with a total of observations. Multifractal analysis Multifractal time series are characterized by a hierarchy of scaling exponents that describe different scaling behavior of many subsets of the series [1]. The standard multifractal analysis based upon partition function multifractal formalism [1] is not appropriate for non-stationary time series, for which the improved methods were developed, as wavelet transform modulus maxima (WTMM) method [14] and multifractal detrended fluctuation analysis (MF-DFA) method [15]. In this work we use MF-DFA method which produces slightly more reliable results than WTMM method [16],[17], and is characterized by simplicity of the implementation algorithm. This method can systematically detect non-stationarities and overcome trends at all timescales, and was successfully applied in various phenomena such as hydrological processes [5], physiological signals [18], geophysical data [19],, and financial time series [20]. 2

3 The MF-DFA procedure is briefly described as follows. The original temporal series is integrated to produce y( k) x( i) x, k 1,, x ( i), i 1,, x 1 i 1 x( i) k i 1, where is the average. ext, the integrated series y (k) is divided into n nonoverlapping segments of length n and in each segment the linear (or higher order polynomial) least square fit (representing local trend) is estimated. The integrated series y (k) is then detrended by subtracting the local trend y i (k) (ordinates of straight line or higher order polynomial segment) from the data in each segment and a q th order fluctuation function is calculated as F ( n) q 1 in n n i1 1 n k( i1) n1 y( k) y ( k) i 2 q / 2 1/ q (1) where, in general, q can take any real value except zero. Repeating this calculation for all box sizes provides the relationship between fluctuation function F q (n) and box size n, where typically F q (n) increases with n as power law h( q) F q ( n) ~ n. The scaling exponent (q) h is obtained as the slope of the regression (least square line fitting) of log F q ( n) versus log n. For stationary time series, h (2) is identical to well-known Hurst exponent H [1], and therefore h (q) is called the generalized Hurst exponent. For monofractal time series h (q) is independent of q (a constant), while for multifractal time series, for which small and large fluctuations scale differently, h (q) is a decreasing function of q [15]. 3 Results and Discussion In order to make sure that seasonal periodicity does not affect power law behavior we apply MF-DFA on the deseasonalized (normalized) series x( t) x( t) X ( t), (2) where x (t) is the mean daily precipitation calculated for each calendar date by averaging over all years in the record, and is the standard deviation of x (t) for each calendar date [5]. To verify if the multifractality is due to broad probability density function for the values 3

4 log F q () of precipitation time series, or due to different long range correlations for small and large fluctuations, we also apply the MF-DFA on the shuffled series. The results of MF-DFA analysis are presented on Fig1 and Fig2. Fig.1 shows log ( n) F q versus log n plots for 2 q 4 we can see linearity within scaling region 10 n 100 days, with decreasing slopes (representing generalized Hurst exponents h (q) ) from negative to positive q values, indicating multifractal behavior log n Figure 1. Fluctuation function F q (n) versus box size n on double logarithmic scale, for different values of q from -2 to 4 with a step of 0.1 (from bottom, to top). Fig.2 shows scaling exponents h(q) for original and shuffled series. After suffling the range of h(q) decreases but does not become constant, meaning that both broad probability density function and different long term correlations for small and large fluctuations contribute to observed multifractality [15]. 4

5 h(q) Original series Shuffled series q Figure 2. Generalized Hurst exponent h (q) before and after shuffling of the data. Motivated by the results of [11] that reveal strong positive trend in precipitation for the entire basin after 1970, we repeat MF-DFA analysis for two sub series, corresponding to the periods before and after 1970, and obtained almost identical h (q) spectrum. These findings indicate that although the intensity of the rainfall increased, the dynamics of the underlying process did not change. 4 Conclusion In this work we examine the multifractal properties of a long-term daily rainfall time series recorded in Piracicaba river basin, located in the southeastern region of Brazil. We apply Multifractal Detrended Fluctuation Analysis (MF-DFA) on deseasonalized data and calculate generalized scaling exponents h (q). Our results show that h (q) is a decreasing function of q, indicating multifractal behavior of precipitation dynamics, which is in agreament with the results obtained for other locations [5], supporting the hypotheses of the existence of multiplicative cascade processes in the atmosphere [21]. These empirical findings provide the base for evaluation of applicability of Universal multifractal model [7] for generating synthetic rainfall data of higher resolution in this location, which may be used to improve the estimation of extreme values [22]. Besides the multifractal analysis on the entire series, we also perform calculations on two sub series corresponding to the periods before and after 1970, where the second period is characterized with significant increase of positive trend in 5

6 rainfall intensity. We find no significant change of behavior between these two subseries and the entire series, indicating that the dynamics of the underlying process did not change. We also find that shuffling of the series decreases the amplitude of the generalized Hurst exponent h (q), but does not turn it constant, meaning that the observed multifractality stems from both the broad probability density function, and from different long term correlations for small and large fluctuations [15]. The current findings provide some additional insight into the phenomenon under study, and should be taken into account when constructing and/or validating new simulation models. 5 References [1] FEDER, J. Fractals. Plenum Press, ew York Y, USA [2] BEZI, R.; PALADI, G.; PARISI, G.; VULPIAI, A. On the multifractal nature of fully developed turbulence and chaotic systems. Journal of Physics A: Mathematical and General 17: [3] GARCIA-MARI, A.P.; JIMEEZ-HORERO, F.J.; AYUSO, J.L Applying multifractality and self-organized criticality theory to describe temporal rainfall regimes in Andalusia (southern Spain). Hydrological Processes 22: [4] GUPTA, V.K.; WAYMIRE, E. A statistical analysis of mesoscale rainfall as a random cascade. Journal of Applied Meteorology 32: [5] KATELHARDT, J.W.; KOSCIELY-BUDE, E.; RYBSKIi, D.; BRAU, P.; BUDE, A.; HAVLI, S. Long-term persistence and multifractality of precipitation and river runoff records. Journal of Geophysical Research 111: D [6] LABAT, J.; MAGI, A.; ABABOU, R. Rainfall- runoff relations for karstic springs: Multifractal analysis. Journal of Hydrology 256: [7] SCHERTZER, D.; LOVEJOY, S. Physical modelling and analysis of rain and clouds by scaling multiplicative process. Journal of Geophysical Research 92: [8] SVESSO, D.; OLSSO, J.; BERDTSSO, R. Multifractal properties of daily rainfall in two different climates. Water Resource Research 32:

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