**2. Data set and method**

Series of (daily) global solar radiation data were studied and all data set is within the period (Jul 97 – Apr 2011). Data were provided by Technological Institute SIMEPAR-Brazil from 18 meteorological stations in Paraná State (PR) (Fig. 1) located in Southern Brazil (Fig. 2). The coordinates of each station are in Table 1. Data cover a region with 199,314.9 km2 and the climate is classified as subtropical. Mean temperature oscillates between 4 °C and 21 °C. The weather conditions in Paraná are usually associated with incursions of polar air toward the Equator and areas of convective clouds associated with extra tropical cold fronts. It also shows areas of local instability associated with mesoscale convective complexes, squall lines, convective clouds, heavy precipitation and lightning.

Fig. 1. Meteorological stations in Paraná state. Adapted from: <http://www.simepar.br>.

more exponents (theoretically infinite) to characterize their scaling properties. In this work multifractality in time series data of global solar radiation is studied by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA) proposed in (Kantelhardt *et al*., 2002). It is a modified version of DFA to detect multifractal properties of time series and provides a systematic tool to identify and quantify the multiple scaling exponents in the data. This method was applied in several cases, in particular in climatological data series as presented in (Kantelhardt *et al*. 2003; Alvarez-Ramirez *et al*., 2008; Pedron, 2010; Zhang,

Series of (daily) global solar radiation data were studied and all data set is within the period (Jul 97 – Apr 2011). Data were provided by Technological Institute SIMEPAR-Brazil from 18 meteorological stations in Paraná State (PR) (Fig. 1) located in Southern Brazil (Fig. 2). The coordinates of each station are in Table 1. Data cover a region with 199,314.9 km2 and the climate is classified as subtropical. Mean temperature oscillates between 4 °C and 21 °C. The weather conditions in Paraná are usually associated with incursions of polar air toward the Equator and areas of convective clouds associated with extra tropical cold fronts. It also shows areas of local instability associated with mesoscale convective complexes, squall lines,

Fig. 1. Meteorological stations in Paraná state. Adapted from: <http://www.simepar.br>.

2010).

**2. Data set and method** 

convective clouds, heavy precipitation and lightning.

Fig. 2. The Paraná (PR) state location in Southern Brazil.

segment as

Data series provide the incidence of direct solar radiation measured on the surface. There is no correction carried out regarding the presence of clouds.

The MF-DFA method is a generalization of the standard DFA, being based on identification of the scaling of the *mth*-order moments of the time series which may be non-stationary. The modified MF-DFA procedure consists of a sequence of steps and detailed information about computation can be found in Kantelhardt et al. (2002). The steps are essentially identical to

the conventional DFA procedure. First we construct the profile *X i*( ) as () ' *i k k Xi x* .The index *i* counts the data points in the record, i.e., *i=1,2,..., N*. For eliminating the periodic seasonal trends daily differences ' *iii x xx* were computed, where *<sup>i</sup> x* represents the average value of radiation for each calendar date *i*. The profile is then divided into ( /) *N int N s <sup>s</sup>* non-overlapping segments of length *s,* where *s* represents time intervals measured in days*.* To accommodate the fact that some of the data points may be left out, the procedure is repeated from other end of the data set. The local trend is determined by using the least-squared fit to each segment ν and we obtain the detrended time series () () () *X i Xi p i <sup>s</sup>* where *p* ( )*i* is the polynomial fit to the νth segment. In this work it is used as linear fit (DFA1). The variance of the detrended time series is calculated for each

$$F^2(\nu, s) = \frac{1}{s} \left\{ \sum\_{i=1}^s X\_s^2 \left[ (\nu \text{-1})s + \text{i} \right] \right\}. \tag{1}$$

Averaging over all segments the *mth* order fluctuation can be obtained and

$$F\_m(\mathbf{s}) = \left\{ \frac{1}{2N\_s} \sum\_{\nu=1}^{2N\_s} \|\,^2F^2(\nu, \mathbf{s})\|\,^{m/2} \right\} \Big|\Big/\_{m}^{\nu}.\tag{2}$$

For *m=2* the standard DFA procedure is retrieved. If the original series are long-range power-law correlated, the fluctuation function will vary as ( ) ( ) *h m Fs s <sup>m</sup>* (for large values of *s*). Note that *h(m)* is the generalized Hurst exponent and *h(2)* is the usual exponent H previously mentioned. A multifractal description can also be obtained from considering partitions functions ( ) ( 1) 1 ( ) *Ns <sup>m</sup> <sup>m</sup> Zs X x s m ss* where ( ) *m* is the Renyi exponent (Barabasi & Vicsek, 1991). A linear scaling of ( ) *m* with *m* is characteristic of a monofractal

data set, whereas a nonlinear scaling is indicative of multifractal behaviour. The exponent *h(m)* is related to the Renyi exponent ( ) *m* by

$$
\pi \left( m \right) = m \ln \left( m \right) - 1 \,\,. \tag{3}
$$

It is also possible to verify the multifractality degree by defining the ratio *H H* where *H* is the slope of the function ( ) *m versus m* for *m* 0 and *H* is the equivalent for *m* 0 . By definition such relation is equal to unity in monofractal signals and a deviation from this value indicates multifractal properties.

#### **3. Results and discussion**

The mean global solar radiation at each station is showed in (Fig. 3).

Fig. 3. Latitude dependence of the mean global solar radiation in the Paraná state.

1 <sup>1</sup> ( ) [ ( , )] <sup>2</sup> *Ns <sup>m</sup> <sup>m</sup> <sup>m</sup>*

For *m=2* the standard DFA procedure is retrieved. If the original series are long-range power-law correlated, the fluctuation function will vary as ( ) ( ) *h m Fs s <sup>m</sup>* (for large values of *s*). Note that *h(m)* is the generalized Hurst exponent and *h(2)* is the usual exponent H previously mentioned. A multifractal description can also be obtained from considering

 

*s Fs F s N* 

( 1)

where

data set, whereas a nonlinear scaling is indicative of multifractal behaviour. The exponent

It is also possible to verify the multifractality degree by defining the ratio *H H* where

By definition such relation is equal to unity in monofractal signals and a deviation from this

 

*Ns <sup>m</sup> <sup>m</sup> Zs X x s m ss*

( ) *m* by

partitions functions ( )

( )

(Barabasi & Vicsek, 1991). A linear scaling of

*h(m)* is related to the Renyi exponent

*H* is the slope of the function

**3. Results and discussion** 

value indicates multifractal properties.

1

The mean global solar radiation at each station is showed in (Fig. 3).

Fig. 3. Latitude dependence of the mean global solar radiation in the Paraná state.

<sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup>

. (2)

() ()1 *m mh m* . (3)

( ) *m versus m* for *m* 0 and *H* is the equivalent for *m* 0 .

( ) *m* with *m* is characteristic of a monofractal

( ) *m* is the Renyi exponent

In general, the mean intensity of solar radiation measured on the surface (global solar radiation) at each station decreases with increasing latitude, it is expected (Table 1). The lowest values were recorded for the stations closest to the coast, Curitiba and Guaratuba, not necessarily being of higher latitude. Local characteristics of clouds can affect actual radiation on the surface.

The typical distribution of radiation during the year is shown in Fig. 4. This seasonal behaviour implies a natural correlation, the consequent periodic trends are eliminated being the daily differences, previously discussed in the DFA method. On the other hand, the phenomena ElNiño/LaNiña affect the region, particularly in precipitation and temperatures. The radiation itself is not affected by the phenomenon, but a greater or lesser distribution of clouds would affect the radiation values measured directly on the surface. In a first approach, it is assumed that the presence of clouds have a random effect in the data stream and does not represent an actual correlation.

Fig. 4. Distribution of mean global solar radiation during the year, typical of the South Hemisphere.

Fig. 5 shows the global solar radiation series and the profile of cumulative series *X i*( ) for the Curitiba station. To perform the DFA method the lower and upper limits for *s* values for the time windows were chosen as 4 and *N/*4, where *N* corresponds to the number of records. The result for Hurst exponent is presented in Fig. 6 for both original and shuffled series. For randomic data series is expected *H=0.5*.

Fig. 5. a) Solar radiation measured on the surface at Curitiba station. b) Cumulative series of daily deviations for the same station.

Fig. 6. a) Hurst exponent for Curitiba station. b) The exponent H for the shuffled series.

The H exponent for all studied data series presented value in the range 0.53 to 0.76 with mean value 0.65 (Table 1). Remembering that 0.5 *H* is related with correlation and persistence the values for it in solar radiation is not surprising. It is reasonable to suppose that the incidence of solar radiation in the next day is not too different from the previous day. No correlation, in a statistical sense, it was found with the exponent H and latitude, altitude and average solar radiation, respectively.


**1.6**

**2.0**

**2.4**

**Log (F(s))**

Fig. 6. a) Hurst exponent for Curitiba station. b) The exponent H for the shuffled series.

The H exponent for all studied data series presented value in the range 0.53 to 0.76 with mean value 0.65 (Table 1). Remembering that 0.5 *H* is related with correlation and persistence the values for it in solar radiation is not surprising. It is reasonable to suppose that the incidence of solar radiation in the next day is not too different from the previous day. No correlation, in a statistical sense, it was found with the exponent H and latitude,

**2.8**

**3.2**

**b**)

**Cumulative series (W/m2**

Fig. 5. a) Solar radiation measured on the surface at Curitiba station. b) Cumulative series of

**)**

**0 1000 2000 3000 4000 5000**

**Time (days) - Jul 97/Apr 2011**

**0.5 1.0 1.5 2.0 2.5 3.0 3.5**

**Log (s)**

**H= 0.51**

**0 1000 2000 3000 4000 5000**

**Time (days) - Jul 97/Apr 2011**

**0.5 1.0 1.5 2.0 2.5 3.0 3.5**

**Log (S)**

altitude and average solar radiation, respectively.

**H= 0.72**

daily deviations for the same station.

**1.5**

**2.0**

**2.5**

**Log (F(s))**

**3.0**

**3.5 a**)

**Solar radiation (W/m2**

**)**


Table 1. Location of the meteorological stations, mean value of global solar radiation, Hurst exponent H and the relation *H H* , the multifractal degree.

The values grouped by frequency and intensity of Hurst exponent occurrences are presented in Fig. 7. Our results present contrasting value with those presented by (Harrouni &, Guessoum, 2009). They indicate high degree of anti-persistence. When the method is applied to temperature data series, the H value obtained is closer to a universal value, however this may depend of geographical position. It is expected that global solar radiation has similar performance regarding the persistence, in this case depending on the distribution of clouds or other elements in the local atmosphere.

Fig. 7. Frequency of occurrence of Hurst exponents H.

In Table 1 it is also possible to observe the ratio *H H* which indicates the deviation from monofractal behaviour. Monofractal signals are characterized by unity in this relation. In this sense, a representative graphic is presented in Fig. 8 were the perfomance of the exponent ( ) *m* with different values of *m* is showed*.* In general the values for all stations indicate weak multifractality (Table 1). In this sense radiation time series present stationarity. Intrinsic correlations of time series represent the behavior of global solar radiation and, despite the presence of clouds, it does not demand the need for multiple Hurst exponents to describe the time series. Note that negative values of *m* emphasize on the parts with small fluctuations. For positive values of *m* the focus is on the parts with large fluctuations. Small deviations of the ratio above or below from unity are related with this characteristic.

Fig. 8. The multifractal exponent versus several moments *m*. Different slopes indicate multifractality. At the left is the result for original series from Curitiba station. In the right is the result for the shuffled series.

As described in (Kantelhardt *et al*., 2002) two different types of multifractality in time series can be identified. In the first case the multifractality of a time series which can be due to the shape of probability density function and the second case, the multifractality which can also be due to different long-range correlations for small and large fluctuations. It is possible to distinguish between these two types of multifractality. To achieve this the corresponding randomly shuffled series was analyzed. The correlations are destroyed by shuffling procedure but dependence of broad distribution function remains, and consequently the multiplicity of exponents is maintained.

Applying the method in the shuffled series of all stations, the ratio *H H* approaches unity, indicating a tendency toward monofractal nature. This demonstrates that the multifractality present in the series of solar radiation are due to different long-range correlations of the small and large fluctuations. In Fig. 8 the right graph presents the result for the shuffled series from Curitiba station.

It is interesting to note here the role of clouds or the presence of other particles or aerosols in the atmosphere. The method could be applied to time series related with solar radiation at the top of the atmosphere, for each location, and the series of solar radiation measured on the surface. In this case it would be possible to obtain information about the behaviour of clouds or other elements in the atmosphere at each location. On the other hand, variations of radiation incident on the planet due to periodic variations in solar activity could be detected in longer series.
