A Brief multifractal analysis of rainfall dynamics in Piracicaba, São Paulo, Brazil

The aim of this study was to evaluate the statistical properties of a daily rainfall time series recorded in Piracicaba river basin, state of Sao Paulo, Brazil in the period 1917-2016. We apply Multifractal Detrended Fluctuation Analysis (MF-DFA) on deseasonalized data and calculate generalized scaling exponents, h(q). We find that h(q) is a decreasing function of q, indicating multifractal behavior of precipitation dynamics. We also perform calculations on two subseries corresponding to the periods before and after 1970, where the second period is characterised by a significant increase of positive trend in rainfall intensity. We find no significant change in behaviour between these two subseries and the entire series, indicating that the dynamics of the underlying process did not change. Our results agree with previous studies on multifractal properties of rainfall time series in other regions of the world supporting the hypotheses of the existence of multiplicative cascade processes in the atmosphere.


Introduction
Rainfall is one of the most remarkable hydrometeorological variables studied because of its non-homogeneous behaviour in event and intensity, leading to water runoff, drought with adverse environmental and social consequences (Walther et al., 2002).Rainfall time series often exhibit strong variability in time and space (Yu et al., 2014).
By analysing the data correlations on different scales, the multifractal technique thus makes it possible to evaluate the full range of precipitation dynamics; which is essential for understanding and modeling several hydrological phenomena such as droughts, floods, runoff, soil erosion, pollution transport, water infiltration etc. (Berne, Delrieu, Creutin, & Obled, 2004;Moriasi & Starks, 2010;Zhang, Ni, Ni, Li, & Zhou, 2016).
Piracicaba river basin is situated in the Southeastern region of Brazil.This region is experiencing an accelerated urban and agro-Acta Scientiarum.Technology, v. 40, e35116, 2018 industrial development is leading to quantitative and qualitative degradation of hydric resources (Lara et al., 2001;Martinelli, Gat, De Camargo, Lara, & Ometto, 2004;Coltri, Ferreira, Freitas, & Demetrio, 2009).In the early 80's a series of reservoirs were constructed to export approximately 31 m³ s -1 from the Piracicaba river basin to the metropolitan region of São Paulo city.Earlier analysis performed on records of precipitation and streamflow showed a positive trend for precipitation for the entire basin and a negative trend for stream flow for some locations within basin.While a decrease in water discharge can be explained by anthropogenic intervention, the increase in precipitation is due to the result of complex interactions between human and natural factors, and it is not yet completely understood (Moraes et al., 1998;Oki & Kanae, 2006;Haddeland et al., 2014).
Global historical rainfall series obtained by traditional rain gauge measurements are mostly available at coarse time scales (daily or higher resolution).This facilitates the manipulation, storage and dissemination of data, but can lead to the lack of information.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.
Our objective in this paper focuses on the examination of the multifractal features of a longterm daily rainfall time series recorded in the Piracicaba river basin.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 precipitation was found to be much stronger (Liebmann et al., 2004;Moraes et al., 1998).
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.

Rainfall data
The data used in this work are provided by the Agrometeorological 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 agrometeorological station located in the city of Piracicaba (latitude 22 42 S, longitude 47 38 W and altitude 546m), state of São Paulo, southeast Brazil, and they are available at http://www.lce.esalq.usp.br/base.html.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 1200-1500 mm occurs in the rainy season with an average monthly precipitation of more than 300 mm, while average monthly precipitation in the dry season is less than 29 mm.The average temperature in Piracicaba ranges from 19°C in the winter to 24°C in the summer (Lara et al., 2001).We analyze daily rainfall records collected during the period 01-jan-1917 to 31-dec-2016, with a total of 36864 observations.
The MF-DFA procedure is briefly described as follows.The original temporal series ( ) , = 1, … , is integrated to produce ( ) = ∑ ( ) − , = 1, … , , where = ∑ ( )is the average.Next, the integrated series ( ) is divided into non-overlapping segments of length and in each segment, the linear (or higher order polynomial) least square fit (representing local trend) is estimated.The integrated series ( ) is then detrended by subtracting the local trend ( ) (ordinates of straight line or higher order polynomial segment) from the data in each segment and a -th order fluctuation function is calculated as where, in general, q can take any real value except zero.Repeating this calculation for all box sizes provides the relationship between fluctuation function ( ) and box size n, where typically ( ) increases with n as power law ( )~ ( ) .The scaling exponent h(q) is obtained as the slope of the regression (least square line fitting) of log ( ) versus log .For stationary time series, h( 2) is identical to well-known Hurst exponent H (Feder, 1988), and therefore h(q) is called the generalised 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 (Kantelhardt et al., 2002).

Results and discussion
In order to make sure that seasonal periodicity does not affect power law behaviour we apply MF-DFA on the deseasonalized (normalised) series where ( ) 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 (Kantelhardt et al., 2006).To verify if the multifractality is due to broad probability density function for the values 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.In order to apply the MF-DFA technique, we wrote a C programming language using the Microsoft Visual Studio Compiler 2008.
The results of MF-DFA analysis are presented on Figure 1 and Figure 2. Figure 1 shows log ( ) versus log plots for −10 ≤ ≤ 10 we can see linearity within scaling region 10<n<100 days, with decreasing slopes (representing generalized Hurst exponents h(q)) from negative to positive values, indicating multifractal behaviour.Figure 1 also displays the linear fit (red line) for q = 2, using the Ordinary Least Squares Method (OLSM).We obtained an adjusted value of = 0.999, indicating an optimal fit. Figure 2 shows scaling exponents h(q) for original and shuffled series.After shuffling the range of h(q) decreases but does not become constant, meaning that both large probability density function and different long-term correlations for small and large fluctuations contribute to observed multifractality (Kantelhardt et al., 2002).It is seen from Figure 2 that h(q) is a decreasing function of q, indicating multifractal behavior of precipitation dynamics, which is in agreement with the results obtained for other locations (Kantelhardt et al., 2006), supporting the hypotheses of the existence of multiplicative cascade processes in the atmosphere (Lovejoy & Schertzer, 2010;De Souto Araújo, Stosic, & Stosic, 2014).These empirical findings provide the base for evaluation of the applicability of universal multifractal model (Schertzer & Lovejoy, 1987) for generating synthetic rainfall data of higher resolution in this location, which may be used to improve the estimation of extreme values (Douglas & Barros, 2003).
Motivated by the results of Moraes et al. (1998) that reveal a strong positive trend in precipitation for the entire basin after 1970, we repeat MF-DFA analysis for two subseries, corresponding to the periods before and after 1970, and obtained an almost identical ℎ( ) spectrum.These findings indicate that although the intensity of the rainfall increased, the dynamics of the underlying process did not change.

Conclusion
In this work, the dynamics of a long-term daily rainfall time series for one station located in Piracicaba River basin were analyzed via Multifractal Detrended Fluctuation Analysis (MF-DFA).Numerical results show that these time series have multifractal behavior which stems from both the broad probability density function and from different long-term correlations for small and large fluctuations The current findings should be taken into account when formulating and/or validating new simulation models, in the sense that any future "microscopic" statistical models should reflect the nature of long-term correlations observed in real data, thus leading to more reliable predictions.

Figure 1 .
Figure 1.Fluctuation function ( ) versus box size on double logarithmic scale, for different values of from -10 to 10 with a step of 1 (from bottom to top).

Figure 2 .
Figure 2. Generalised Hurst exponent ℎ( ) before and after shuffling of the data.