Temporal multiscaling characteristics of particulate matter PM10 and

Atmospheric Environment 169 (2017) 22e35

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Temporal multiscaling characteristics of particulate matter PM10 and ground-level ozone O3 concentrations in Caribbean region Thomas Plocoste a, *, Rudy Calif b, Sandra Jacoby-Koaly a -Pitre EA 4935 - LARGE (Laboratoire de Recherche en G eosciences et Energies), groupe A erosol, D epartement de Physique, Universit e des Antilles, Pointe-a 97157, Guadeloupe (F.W.I) b -Pitre EA 4935 - LARGE (Laboratoire de Recherche en G eosciences et Energies), groupe Energies, D epartement de Physique, Universit e des Antilles, Pointe-a 97157, Guadeloupe (F.W.I) a

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


First multifractal analysis for PM10 and O3 pollutants in Caribbean region.
Nonstationary and nonlinear nature of pollutants was analyzed in the fully developed turbulence framework.
Multifractal lognormal model was performed.
Intermittency parameter was estimated by a lognormal model.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 May 2017 Received in revised form 29 August 2017 Accepted 31 August 2017 Available online 6 September 2017

A good knowledge of the intermittency of atmospheric pollutants is crucial for air pollution management. We consider here particulate matter PM10 and ground-level ozone O3 time series in Guadeloupe archipelago which experiments a tropical and humid climate in the Caribbean zone. The aim of this paper is to study their scaling statistics in the framework of fully developed turbulence and Kolmogorov’s theory. Firstly, we estimate their Fourier power spectra and consider their scaling properties in the physical space. The power spectra computed follows a power law behavior for both considered pollutants. Thereafter we study the scaling behavior of PM10 and O3 time series. Contrary to numerous studies where the multifractal detrended fluctuation analysis is frequently applied, here, the classical structure function analysis is used to extract the scaling exponent or multifractal spectrum zðqÞ; this function provides a full characterization of a process at all intensities and all scales. The obtained results show that PM10 and O3 possess intermittent and multifractal properties. The singularity spectrum MS ðaÞ also confirms both pollutants multifractal features. The originality of this work comes from a statistical modeling performed on zðqÞ and MS ðaÞ by a lognormal model to compute the intermittency parameter m. By contrast with PM10 which mainly depends on puffs of Saharan dust (synoptic-scale), O3 is more intermittent due to variability of its local precursors. The results presented in this paper can help to better understand the mechanisms governing the dynamics of PM10 and O3 in Caribbean islands context. © 2017 Elsevier Ltd. All rights reserved.

Keywords: PM10 Ozone Spectral analysis Multifractal analysis Lognormal model Intermittency

* Corresponding author. E-mail addresses: [email protected] (T. Plocoste), [email protected] (R. Calif), [email protected] (S. Jacoby-Koaly). http://dx.doi.org/10.1016/j.atmosenv.2017.08.068 1352-2310/© 2017 Elsevier Ltd. All rights reserved.

T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

1. Introduction For the past few years, many megacities throughout the world have experienced air pollution episodes leading to negative impacts on human health (Alberini et al., 1997; Bernstein et al., 2004; Pope and Dockery, 2006; Kampa and Castanas, 2008; Matus et al., 2012). These pollutants are mainly from local origin. In Caribbean islands, anthropogenic pollution has been rising during the last 20 years because of the increase of urbanization and industrial development. In addition to the pollution of local origin (industrial area, transport activities, open landfill etc.), air quality over the Lesser Antilles is widely affected by multiple large-scale pollution. In Guadeloupe archipelago, major emitting sources of particulate matter (primary pollutant) with an aerodynamic diameter 10 mm or less (thereafter PM10) is African dust. Mineral dust is transported throughout the year to the Caribbean area from their African sources in the Sahara and in the Sahel (Petit et al., 2005). Generally PM10 quantities from saharan dust are more important between April and December (Prospero, 1981a,b). Some studies have shown the health impact of PM10 on human health in Guadeloupe archipelago (Cadelis et al., 2013, 2014). Ground-level ozone (thereafter O3 ) is a secondary pollutant formed by photochemical reactions involving nitrogen oxides (thereafter NOx), carbon monoxide (thereafter CO) and volatile organic compounds (thereafter VOCs) with solar energy (Windsor and Toumi, 2001). NOx, CO and VOCs mostly result from anthropogenic activities (Geng et al., 2015). As PM10, O3 is a major air pollutant that can cause serious damage to human health and ecological environment (McKee, 1993; Berman et al., 1995; Jalaludin et al., 2004). Because of their health impact linked to respiratory diseases, a good knowledge of process dynamics of these pollutants (e.g, variability) is crucial to quantify their harmfulness level. Numerous studies have relied on conventional statistical tools such as kurtosis to characterize the intermittent behavior of atmospheric pollutants (Windsor and Toumi, 2001; Chelani, 2009). Based on the concept of a fractal, in recent decades, the scaling behavior of the air pollution process has attracted increased attentions (Dong et al., 2017). Previously, numerous studies analyzed and modeled pollutant series as monofractal series (Lee et al., 2006; Yuval and Broday, 2010). This method is more suitable to model homogeneous time series because there is only one scaling component and the properties are constant (Stanley et al., 1996). To refine the investigation of the dynamics of more heterogeneous and complex series, it is necessary to analyze more scaling exponents, including through fractal and multifractal analyses, especially when the original series is composed of many interwoven fractal subsets (Pamuła and Grech, 2014). Nowadays, multifractality is regarded as the inherent property of complex and composite systems. Multifractal approach has already been widely used in the literature for different fields as finance (Grech, 2016), image analysis (Kestener et al., 2011), medicine (Pal et al., 2016), earthquakes (Fan and Lin, 2017) or wind farm (Calif et al., 2013a) just to mention a few. Multifractality theory is widely used to quantitatively delineate the nonlinear evolution of a complicated system and the multiscale characteristics of physical quantities. In tropical Atmospheric Boundary Layer (thereafter ABL), it is known that wind speed and solar radiation which are responsible of transport and production of atmospheric pollutants present multifractal properties (Calif and Schmitt, 2012; Calif et al., 2013b). It is the reason why we have decided to investigate the possible multifractal behavior of atmospheric pollutants. In literature, many authors use the MultiFractal Detrended Fluctuation Analysis (thereafter MF-DFA) for study multifractal properties of pollutants (Liu et al., 2015; Xue

23

et al., 2015; Dong et al., 2017) or the Detrended Cross-Correlation Analysis (thereafter DCCA) to explore the multifractal properties between a pair of pollutant (He et al., 2016, 2017). The aim of this paper is to analyze multifractal properties of PM10 and O3 data with another method, the classical structure function approach from turbulence field. To our knowledge, there have been very few multiscaling analyses of PM10 and O3 pollutants in Caribbean region. This paper is organized as follows. Section 2 presents PM10 and O3 data used in this study. Section 3 describes the theoretical framework with spectral analysis and multifractal analysis using the structure function scaling exponent method for a time series. Section 4 comments on the results obtained and then discusses them in Section 5. Lastly, Section 6 presents the conclusion of this paper. 2. Description of data In Guadeloupe archipelago, located at the North of the Lesser Antilles, 16:25+ N 61:58 W, the hourly concentrations of PM10 and e es de Surveillance de O3 were released by Les Associations Agre  de l'air, a national organization that overseas air quality in Qualite each of the French administrative regions. Guadeloupe air quality network is managed by Gwad’Air agency (http://www.gwadair.fr/). The Air Quality Stations (thereafter AQS) are situated at BaieMahault (16:2561+ N 61:5903+ W, suburban areas), Abymes -Pitre (16:2611+ N 61:5269+ W, suburban areas) and Pointe-a (16:2422+ N 61:5414+ W, urban area). PM10 concentrations are measured using the Thermo Scientific Tapered Element Oscillating Microbalance (TEOM) models 1400ab and 1400-FDMS (Prospero et al., 2014) and O3 concentrations with the Thermo Scientific 49i by UV-absorption method. Measurements are made continuously and stored as 15 min averages used to calculate hourly data. In this -Pitre data (see AQS on Fig. 1 in red study we focused on Pointe-a square) which lies south of the major population centers in Guadeloupe (210 715 inhabitants). Contrary to PM10 measurement campaign which began in 2005 and ended in 2012, O3 measurement campaign also began in 2005 but continues until today. For each year, only long time series measured continuously are analyzed in order to not distorting the dynamics of PM10 and O3 over time. The lengths of time series measured and considered for this study are drawn up in Table 1. An example of pollutant data sequences is displayed in Fig. 2. Both signals exhibit huge fluctuations showing a strong variability of PM10 and O3 pollutants. To analyze the kind of stochastic processes, the adequate fully developped turbulence framework is applied to datasets. The following section is dedicated to description of the methods applied on pollutants time series. 3. Theoretical framework 3.1. Self-similarity and scale invariance The description of natural phenomena by the study of statistical scale-laws is not recent (Mandelbrot, 1982). Self-similarity of complex systems has been widely observed in nature and is the simplest form of scale invariance. A signal xðtÞ is self-similar if its statistical properties remain unchanged in the signal aH xðt=aÞ obtained from xðtÞ by dilation of a factor a > 0 for the time axis ðtÞ and a factor aH for the amplitude axis ðAÞ. The parameter H is called self-similarity parameter or Hurst parameter; it provides information on the degree of smoothing or variability of the signal. Hurst parameter is bounded between 0 and 1 (Zeng et al., 2013). For example, H ¼ 1=2 for a Brownian motion and H ¼ 1=3 for a nonintermittent turbulence (Kantelhardt, 2012; Calif et al., 2013b).

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T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

Fig. 1. Map of Guadeloupe archipelago with the locations of the Air Quality station (AQS) indicated by a red square and the French Met Office (MF) indicated by a purple triangle. The arrows indicate trade winds direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

For instance, Fig. 3 highlights the self-similar properties of PM10 pollutant. Fig. 3(b) and (c) are successively dilated portions of the signal represented in Fig. 3(a). A property of self similar signal is the presence of spectral power law in Fourier space for example. We also verified that O3 pollutant data present the same properties. 3.1.1. The existence of a power-law Commonly, a scale invariance can be detected by computing of power spectral density (thereafter PSD). The PSD Eðf Þ separates and measures the amount of variability occurring in different frequency bands. Scale invariance can be detected by computing Eðf Þ. For a scale-invariant process, the following power law is obtained over a range of frequency f in log-log representation:

Eðf Þ  f b

(1)

where f is frequency and b is the spectral exponent. It reveals the scale-free memory effect as a power-law dependence of the frequency distribution. PSD can characterize special kinds of signals in nature, for instance, 1=f noise. Their typical LogE versus LogF curves are shown in Fig. 4 (Hara et al., 1997; Shimizu and Hara, 1996). Of course, they all meet the power-law distribution and the corresponding power-

law exponents are respectively 0,1 and 2. In Fig. 4, it can be seen that the spectrum of white noise indicates the lack of memory in this process due to the presence of a plateau corresponding to a spectral exponent b ¼ 0. This means that data at time t has no relationship with the data at time t  t, so its power spectrum in double logarithmic diagram is a horizontal line (Zeng et al., 2013). However, Brown noise processes strong memory is highlighted in Fourier space by a 2 power-law. As mentioned by Schertzer et al. (1997), if a process shows a power-law distribution in spectrum, then its power-law exponent contains its nonstationary process characteristic (Schertzer and Lovejoy, 1987). Hence, b contains information about the degree of stationarity of the studied parameter (Mandelbrot, 1982; Schertzer and Lovejoy, 1987; Marshak et al., 1994):
b < 1, the process is stationary,
b > 1, the process is nonstationary,
1 < b < 3, the process is nonstationary with increments stationary. Consequently, depending on b value, the presence of long or short memory (temporal correlated structure) can be highlighted in the signal.

Table 1 Length of continuous time series (N) for the studied years (in hours). Year

PM10 (N)

O3 (N)

2005 2006 2008 2009 2010 2012 2014 2015

e 5622 5691 4679 e 4739 e e

8674 e 8214 8574 6337 5959 6251 8363

3.1.2. Spectral analysis Spectral analysis has been widely applied for pollution studies in China, United States or Europe (e.g., (Windsor and Toumi, 2001; Marr and Harley, 2002; Choi et al., 2008)). Few studies use this method for pollution data in Caribbean region. The expression of the PSD for a process xðtÞ is recalled here. A N point-long interval is used to construct the value at frequency domain point f, Xf (Bracewell, 1999):

T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

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Fig. 4. The LogE versus LogF diagram of three examples of 1=f noise with the spectral exponents b ¼ 0, 1 and 2. (b ¼ 0 corresponds to a White Gaussian noise and b ¼ 2 corresponds to a Brownian motion).

Xðf Þ ¼

N1 X

xj e2pijN f

(2)

j¼0

with f ¼ 0,1,…,N-1. The PSD is

2    Eðf Þ ¼ Xðf Þ

Fig. 2. An example of pollutant signals recorded continuously with a time sampling -Pitre: an extract of (a) PM10 during 5622 h and (b) O3 equal to 5 min at Pointe-a during 8214 h. Both signals show strong variability with huge fluctuations.

(3)

Nevertheless, the power spectrum Eðf Þ is second-order statistic (proportional to the square of the amplitude of a given frequency fluctuation) and its slope is not sufficient to fully specify a scaling process. Multifractal analysis is a natural generalization to study the scaling behavior of a nonlinear phenomenon, using qth order structure functions. 3.2. Multiscaling analysis and multifractal models We further investigate the possibility that time series generated by certain atmospheric systems may be members of a special class of complex processes, called multifractal, which require a large number of exponents to characterize their long memory properties (Liu et al., 2015).

Fig. 3. Figure ðbÞ shows a dilated portion from signal ðaÞ and Figure ðcÞ shows a dilated portion from signal ðbÞ. This illustrates the statistical self-similarity properties of PM10 data. O3 data present the same properties.

3.2.1. The structure function analysis Multifractal analysis was first introduced for the study of turbulence by Monin et al. (1971) and Mandelbrot (1972). Intermittency or multifractality is an important feature of the turbulent-like dynamical systems (Frisch, 1995). Numerous spatial or temporal freedoms exist simultaneously and interact with each other to transfer energy, momentum or other physical quantities (Gao et al., 2016). To characterize this multiscale interaction, structure function analysis is used to retrieve the scale invariance for high Reynolds turbulent flows (Kolmogorov, 1941). This analysis technique has been widely used in the turbulence research community (Anselmet et al., 1984; Frisch, 1995) and other research fields (Schmittbuhl et al., 1995; Calif et al., 2013b; Calif and Schmitt, 2014). Here, we may define intermittency as the property of having

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T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

large fluctuations at all scales with a correlated structure. Large fluctuations are much more frequent than what would be obtained for Gaussian processes (Frisch, 1995; Schertzer et al., 1997). This is typically studied considering the probability density function (PDF), ore more often using the moment of order q of these fluctuations, named “structure functions of order” q:

q   Sq ðtÞ ¼ Xðt þ tÞ  XðtÞ ;

(4)

E D Sq ðtÞ ¼ ðDXt Þq xtzðqÞ ;

(5)

where DXt ¼ jXðt þ tÞ  XðtÞj is the absolute value of time increments, t is a time scale and zðqÞ is the scale invariant exponent function. In the case of intermittency, the scaling moment function zðqÞ is nonlinear. We have here written the fluctuations in time, since below in this paper, we deal with time series analysis. The values of the function zðqÞ are estimated from the slope of Sq ðtÞ versus t in a log-log diagram for all moments q. The knowledge of the full ðq; zðqÞÞ curve for integer and non integer moments provides a full characterization of pollution data at all scales and all intensities. The parameter H ¼ zð1Þ is the Hurst exponent characterizing the non-conservation of the mean. Monofractal processes correspond to a linear function zðqÞ ¼ qH . The function zðqÞ defines the types of scaling behavior. This exponent function is valuable to characterize the statistics of the random process. If zðqÞ is linear, the statistical behavior is monoscaling and if zðqÞ is nonlinear and concave, the behavior is defined as multiscaling corresponding to a multifractal process. The concavity of this function is a characteristic of the intermittency, the more concave the curve is, the more intermittent is the process (Frisch, 1995; Schertzer et al., 1997). The spectral exponent b associated with the scale invariance of the power spectral density can be connected to the function of the exponents z by the following expression:

b ¼ 1 þ zð2Þ

(6)

This expression recalls that spectral analysis is a second order moment analysis. Another way to characterize a multifractal series is the singularity spectrum MS ðaÞ. This singularity spectrum was first introduced in the 1980s to characterize the multifractality of turbulence and chaotic systems (Benzi et al., 1984; Parisi and Frisch, 1985). MS ðaÞ is related to a via a Legendre transform (Feder, 1988; Peitgen et al., 2006; Liu et al., 2014):

a¼

dzðqÞ ; dq

(7)

MS ðaÞ ¼ minðqa  zðqÞ þ 1Þ q

specific a0 , which corresponds to the peak of the multifractal spectrum (Telesca et al., 2003). In more concrete terms, the bigger the Da and the larger the MSmax , the stronger is the degree of multifractality. Furthermore, the difference of the fractal dimensions between the minimum probability and maximum probability subsets DMS ðDMS ¼ MS ðamin Þ  MS ðamax ÞÞ describes the proportion of the number of elements at the maximum and minimum in the subset, which refers to the proportion of the large and small peaks of signals (Liu et al., 2014). 3.2.2. Multifractals models In literature, there are two models frequently used to model zðqÞ, the lognormal model (Frisch, 1995; Schertzer et al., 1997; Medina et al., 2015; Schmitt and Huang, 2016) and the log-stable model or universal multifractal (Schertzer and Lovejoy, 1987; Kida, 1991). For a lognormal model we consider a classical fit for which the function is quadratic. Knowing the values zð0Þ ¼ 0 and zð1Þ ¼ H, the following equation is written:

zðqÞ ¼ qH 

m 2

 q2  q

(9)

where the parameter m is chosen as verifying m ¼ 2H  zð2Þ ¼ 2zð1Þ  zð2Þ and is called intermittency parameter. The larger m is, more the time series displays intermittent behavior. The expression of the log-stable model is (Schertzer and Lovejoy, 1987):

zðqÞ ¼ qH 

C1 ðqa  qÞ ða  1Þ

(10)

where H ¼ zð1Þ the Hurst parameter defines the degree of smoothness or roughness of the field. The parameter C1 is the fractal co-dimension of the set giving the dominant contribution to the mean ðq ¼ 1Þ and bounded between 0 and d (d is the dimension space). It measures the inhomogeneity mean or the mean intermittency characterizing the sparseness of the field. The larger C1 is, vy more the mean field is inhomogeneous. The multifractal Le parameter a is bounded between 0 and 2, where a ¼ 0 corresponds to the monofractal case and a ¼ 2 corresponds to the multifractal lognormal case. The parameter a measures the degree of multifractality, i.e., how fast the inhomogeneity increases with the order of the moments (Seuront et al., 1996). In this study, we consider the lognormal model which provides a reasonable fit up to q ¼ 5 (Calif and Schmitt, 2014). The question of the best model is not the topic of the present paper, therefore we consider here the lognormal fit as convenient for the structure function analysis. Next sections describe the results achieved by applying the methods described previously. These results are then discussed.

(8) 4. Results

where a is the singularity exponent and MS ðaÞ is the singularity spectrum. In this frame for a monofractal process, a ¼ H and MS ðaÞ ¼ 1. The curve a  MS ðaÞ is a single-humped function for multifractal, and reduces to a point for monofractal. The shape and the extension of MS ðaÞ curve contain significant information about the distribution characteristics of the studied data. The broader measured a and MS ðaÞ are, the more intermittent the field is (Frisch, 1995). Da and DMS are two very important parameters describing the complexity of the multifractal spectrum (Liu et al., 2014). The spectrum width Da is defined as Da ¼ amax  amin . Da describes the inhomogeneity of the distribution of probability measured on the overall fractal structure, which has been assimilated as the degree of multifractality and intermittency (Macek et al., 2011; Liu et al., 2014). MS ðaÞ takes a maximum value MSmax ðMSmax ¼ MS ða0 ÞÞ at a

4.1. Spectral analysis of PM10 and O3 time series In order to detect the presence of scaling in the pollutants series considered in this study, we applied a data spectral analysis in the Fourier space. For that, a logarithmic average technique of the spectra is performed to avoid fluctuations and have a better visibility on scaling ranges. More precisely, Fig. 5 shows the spectra EPM10 ðf Þ of PM10 for the considered years. Overall, PM10 spectra EPM10 ðf Þ show one scaling regime between low and middle frequencies 106 ⩽ f ⩽ 104 Hz which corresponds to timescales 104 ð104 sz3 hÞ⩽T⩽106 s ð106 sz12 daysÞ with a spectral slope bPM10 between 1.26 and 1.37 (see Table 2). The scaling behavior of PM10 series has been observed in previous studies by Windsor and

T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

27

Toumi (2001); Varotsos et al. (2005); Kai et al. (2013); Dong et al. (2017). We obtained smaller timescale values than those found previously in other studies carried out in large megalopolis: z 14 months ⩽T in Windsor and Toumi (2001), 4 h ⩽T⩽ 9 months in Varotsos et al. (2005) and 1 day ⩽T⩽ 1 year in (Kai et al., 2013). For the spectral exponent, bPM10 ¼ 1:021 in Kai et al. (2013) is smaller our study values. As regards O3 , Fig. 6 exhibits the spectra EO3 ðf Þ. O3 spectra EO3 ðf Þ also show one scaling behavior between low and middle frequencies 107 ⩽ f ⩽ 104 Hz which corresponding to timescales 104 ð104 sz3 hÞ⩽T⩽107 s ð107 sz4 monthsÞ with a spectral slope bO3 between 1.11 and 1.27 (see Table 2). The scaling behavior of O3 series has been observed in previous studies by Windsor and Toumi (2001); Varotsos et al. (2005); Hsu et al. (2011). As for PM10, we obtained smaller values than those found previously in other studies carried out in large megalopolis for timescales z14 months⩽T in Windsor and Toumi (2001). When we compare EO3 ðf Þ and EPM10 ðf Þ for all years, we can clearly see that EO3 ðf Þ always shows two characteristic peaks at the same frequencies (e.g. 2005 in Fig. 6): the first (the largest) at z1:2105 Hz (timescales z1:4 days) and the second (the smallest) at z2:4105 Hz (timescales z2:8 days). These peaks are characteristic of cyclical events such as diurnal cycle. The first results obtained from PSD for PM10 and O3 data show that pollutants fluctuations obey to a power-law behavior which means long-term correlated structure in data. All values of bPM10 and b03 > 1, meaning that PM10 and O3 processes are nonstationary with increments stationary. However, Fourier spectrum is a secondorder statistics providing information on medium level fluctuations. For a full description of PM10 and O3 at all intensities and all scales, multifractal analysis is used below. 4.2. Multifractal analysis of PM10 and O3 time series 4.2.1. Structure function analysis The structure function analysis is applied to both series. Fig. 7 presents the multiscaling analysis DPM10t versus t for PM10 data, for moment orders q ¼ 1 to 4 with increment 0.13 in log-log plot for the studied years. The scaling range is from 1 h to 10 h, which is a scaling range narrower than the one found from the Fourier analysis. The slope for each structure function of order q is then estimated for all moments between q ¼ 0 and q ¼ 4. The same protocol was applied to O3 data. The knowledge of the full ðq; zðqÞÞ curve for integer and non integer moments provides a full characterization of PM10 and O3 fluctuations at all scales and intensities. Figs. 8 and 9 present the empirical structure functions zPM10 ðqÞ and z03 ðqÞ for each year compared to the linear model zðqÞ ¼ qH which corresponds to monofractal processes. All the curves obtained for zPM10 ðqÞ and z03 ðqÞ are nonlinear and concave. The concavity of scaling exponents curves highlights the intermittent and multiscaling features of PM10 and O3 fluctuations. This has already been observed in China (Gao et al., 2016) or Europe (Windsor and Toumi, 2001). In Fig. 8, 2006 shows the curve with the highest concavity (more intermittent) and 2009 has the lowest (less intermittent) for zPM10 ðqÞ. Overall, zPM10 ðqÞ values obtained in our study are approximately in the same value range for small (q < 2) and large fluctuations (q > 2) as those obtained by Gao et al. (2016) which use the spatial structure function analysis. For z03 ðqÞ (see Fig. 9) it is less obvious to determine the degree of concavity from the observation of the curves. However, 2005, 2008 and 2014 seem to roughly have greater degrees of concavity. The second moment zð2Þ is linked to the spectral exponent b through bc ¼ 1 þ zð2Þ. Tables 3 and 4 provide the values of H, bc , zð2Þ and their all-years average values for PM10 and O3 .

Fig. 5. The mean Fourier power spectra EPM10 ðf Þ of PM10 data in log-log plot, showing a power-law with slightly different slope between years (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In Table 3, we can see that 0:154 < HPM10 < 0:206 with an average of 0.181. For O3 (see Table 4), 0:130 < HO3 < 0:280 with an average of 0.230. This indicates that PM10 and O3 processes are

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T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35 Table 2 Values of spectral exponent for PM10 and 03 data for the studied years. Year

bPM10

b03

2005 2006 2008 2009 2010 2012 2014 2015

e 1.29 1.26 1.35 e 1.37 e e

1.18 e 1.11 1.20 1.26 1.27 1.23 1.18

very variable. Variabilities of both pollutants are in the same range. Furthermore, for all the studied years, 0 < H < 0:5 which signifies that we have a high variability of pollutant values and the fluctuations of PM10 and O3 time series are negatively correlated in a power law fashion (anti-persistence) (Carbone et al., 2004). It is important to note that values of HPM10 and HO3 are independent of the length of time series (see Table 1). This shows that our study samples are representative of the dynamics of both pollutants. In other studies where they used DFA and MF-DFA analysis, PM10 and O3 fluctuations were found positively correlated (H > 0:5) which means persistency in time series (Varotsos et al., 2005; Liu et al., 2015; Dong et al., 2017). For the second moment, 0:290 < zPM10 ð2Þ < 0:381 with an average of 0.347 and 0:210 < zO3 ð2Þ < 0:480 with an average of 0.382. As a consequence by linking to the spectral exponent, we found 1:30 < bcPM10 < 1:38 and 1:21 < bcO3 < 1:48. Previously in Table 2 with spectral analysis, we found 1:26 < bPM10 < 1:37 and 1:11 < bO3 < 1:27. When we compare year by year, bcPM10 gives the closest results to spectral analysis. bcO3 always overestimates the values of the spectral exponent. We obtain an average bcPM10 -bcO3 of 1.342e1.382 for multiscaling analysis and an average bPM10 -bO3 equal to 1.28e1.21 for spectral analysis. Overall, these results appear to be satisfactory when looking at previous values estimated by Fourier analysis. Another representation to highlight a multifractal series is the singularity spectrum MS ðaÞ. Fig. 10 presents the corresponding singularity spectrum MsðaÞ of PM10 and O3 series for moments order 0⩽q⩽4, respectively for 2012 and 2014 as an example. For the considered years, all singularity spectra exhibit the same shape. Visually, the measured a and MsðaÞ confirm the existence of the multifractality for both pollutants; both singularity spectra possess a certain wide highlighting the multifractal data behavior. The multifractality through the singularity spectrum has been observed in recent studies for PM10 by Liu et al. (2015); Gao et al. (2016); nez-Hornero et al. (2010); Dong et al. (2017) and for O3 by Jime Pavon-Dominguez et al. (2013). Da and DMS describe the complexity of the multifractal spectrum. Table 5 gives Da and DMS values for both pollutants. For PM10, 0:07 < DaPM10 < 0:27 with 2006 associated to the highest value and 2009 to the lowest. Concerning O3 , 0:10 < DaO3 < 0:24 with 2014 presenting the highest value and 2009e2012 the lowest values. For all years, we found a smaller degree of multifractality than those obtained previously in megalopolis by Liu et al. (2015); Gao et al. (2016); Dong et al. (2017) for nez-Hornero et al. (2010); Pavon-Dominguez PM10 and by Jime et al. (2013) for O3. The results obtained with DaPM10 and DaO3 on the degree of multifractality show that O3 seems to be more intermittent than PM10 because on the whole, DaO3 is higher than DaPM10 . The shape of MsðaÞ curve obviously looks like a hook to the right with DMS always negative for both pollutants, which indicates that the frequency of PM10 and O3 values at the lower values is more than the higher values one for the studied year. This result was previously found by Liu et al. (2015) for PM10 data.

4.2.2. Lognormal model To predict the scaling moment function zðqÞ, we use a lognormal model (see section 3.2.2) to fit the scaling exponents curves zPM10 ðqÞ and z03 ðqÞ. Following this model, the intermittency parameter m can be estimated by m ¼ 2H  zð2Þ ¼ 2zð1Þ  zð2Þ. This parameter characterize the intensity of the intermittency or multifractality. H is the so-called Hurst exponent characterizing the signal variability; more H is higher and more the data values are smooth; the weaker H is and more the data values are rough. In Figs. 8 and 9, the lognormal provides a good fit for all scaling exponents zPM10 ðqÞ and z03 ðqÞ. Table 6 presents the intermittency values for each pollutant. By analyzing intermittency annual values, we note that mPM10 is higher for 2006 (0.029) and lower for 2009 (0.005). For mO3 , 2014 has the higher intermittency value (0.100) following by 2005 (0.093) and 2008 (0.090). mO3 presents the lowest values (0.050) for 2009 and 2012. These results confirm those obtained in section 4.2.1 with the concavity of zPM10 ðqÞ-z03 ðqÞ and the width of DaPM10 -DaO3 . O3 is more intermittent than PM10 because mO3  mPM10 for all the considered years. It is the same in average, with mO3 equal to 0.078, and mPM10 equal to 0.016. The lognormal model is also used to predict the degree of multifractality or intermittency via the singularity spectrum curve MS ðaÞ (see Fig. 10). As scaling exponents curves, MS ðaÞ curves are well fitted by the lognormal model. Overall, our results highlight that O3 has higher multifractality degree than PM10. 5. Discussion In this article, in order to study the scaling invariance of PM10 and O3 times series, we firstly used spectral analysis. The difference of scaling invariance behavior observed between our study and previous ones made in megacities can be caused by local features such as pollution origin and level, city configuration and meteorological parameters. For example, due to many local pollution sources in megacities, PM10 and O3 concentrations are high and events such as smog episodes can occur (Moussiopoulos et al., 1995; Beeson et al., 1998; Ma et al., 2012). In contrast, for Guadeloupe, major emitting source of PM10 are due to Saharan dust which is a episodic phenomenon coming from synoptic-scale (Prospero et al., 2014). Thereafter, for analyzing the scaling invariance behavior of both pollutants and its nature, we applied the classical structure function analysis to extract the scaling exponent zðqÞ. For all the considered years, we found a high variability (H < 0:5) in PM10 and O3 time series. In megacities, there is a small variability of pollutants concentrations with H > 0:5 for PM10 and O3 (Varotsos et al., 2005; Liu et al., 2015; Dong et al., 2017). Besides, as shown in nez-Hornero et al., 2010; Pavon-Dominguez previous studies (Jime et al., 2013; Liu et al., 2015; Gao et al., 2016; Dong et al., 2017), PM10 and O3 exhibit intermittent and multifractal behavior. With 0:07 < DaPM10 < 0:27 and 0:10 < DaO3 < 0:24, a weaker degree of multifractality was found in Guadeloupe for both pollutants contrary to those obtained previously. Pollution origin and level can be key parameters to explain this difference in multifractal degree. In Europe and China for example, because of heavy traffic, exhaust from motor vehicles has been regarded as the major source of particulate matter in the ABL (Künzli et al., 2000; He et al., 2016). In our case, road traffic is less important and high levels of PM10 is caused by dust originated from African coasts (Euphrasie-Clotilde et al., 2017). Furthermore, city configuration may also a crucial feature in atmospheric pollution behavior. Indeed, in megacities, there are many high-rise buildings influencing the wind circulation and inducing the pollutants accumulation in the surface layer. In Guadeloupe, most of buildings are four stories high or less (Plocoste

T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

29

Fig. 6. The mean Fourier power spectra EO3 ðf Þ of ground-level ozone data in log-log plot, showing a power-law with slightly different slope between years (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 7. The scaling of the PM10 structure function SPM10 ðqÞ, for q ¼ 1, 2, 3 and 4.

et al., 2014) and their density is much less important. Its almost flat topography in the center of the island facilitate pollutants dispersion and reduce risk of stagnation which increases pollutants concentrations in the ABL. The permanent trade winds increase the pollutants dispersion. In case of sustained trade winds, pollution should be swept toward the Caribbean Sea (see Fig. 1). In addition, weather conditions can also play a significant role in multifractal properties. In literature, previous studies have already shown that pollutants concentrations are closely linked to meteorological parameters (Bloomfield et al., 1996; Zhang et al., 2015; He, 2017); wind speed dominates the amount of pollutants dispersion in the ABL while temperature (due to solar radiation) contributes to transformation of pollutants. To flesh out our discussion on both pollutants with meteorological parameters evolution, we used te o France measurements from The French Met Office (Me Guadeloupe, thereafter MF in purple triangle on Fig. 1) located at Abymes (16:2630+ N 61:5147+ W) for the same period. Data provided by MF are hourly averaged data. A previous study made by vignon (2003) has already demonstrated that MF weather Bre

station data is representative for our study area. As for pollutants time series, the methodologies described in section 3 were applied to meteorological parameters time series. Figs. 11 and 12 show the spectra EW ðf Þ and ESR ðf Þ of wind speed and solar radiation. Between low and middle frequencies, Fig. 11 shows one scaling regime whereas Fig. 12 presents two scaling regimes with a higher slope for middle frequencies. On both spectra, we find the same peaks characteristic of cyclical events as on Fig. 6 at z1:2105 Hz (timescalesz1:4 days) and z2:4105 Hz (timescalesz2:8 days). We note that ESR ðf Þ peaks have higher intensities than EO3 ðf Þ and EW ðf Þ. This is due to the fact that O3 production and wind speed are closely linked to solar radiation. Let us recall here that O3 is a secondary pollutant which forms in two steps by the photolysis of nitrogen dioxide (thereafter NO2 ) due to solar energy (Jenkin and Clemitshaw, 2000; Seguel et al., 2012):
Step 1: NO2 þ hvðl < 420nmÞ/NO þ Oð3 PÞ
Step 2: Oð3 PÞ þ O2 ðþMÞ/O3 ðþMÞ

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31

Fig. 9. Measured scaling exponent zO3 ðqÞ with vertical error bar for each data point (blue line) compared to the linear model zðqÞ ¼ qH (red line with square) and the lognormal model (green line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 3 Values of scaling and multifractal parameters for PM10 data.

2006 2008 2009 2012 Average

HPM10 ¼ zPM10 ð1Þ

zPM10 ð2Þ

bcPM10

0.195 0.154 0.170 0.206 0.181

0.361 0.290 0.335 0.381 0.342

1.361 1.290 1.335 1.381 1.342

Table 4 Values of scaling and multifractal parameters for O3 data.

2005 2008 2009 2010 2012 2014 2015 Average

Fig. 8. Measured scaling exponent zPM10 ðqÞ with vertical error bar for each data point (blue line) compared to the linear model zðqÞ ¼ qH (red line with square) and the lognormal model (green line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

HO3 ¼ zO3 ð1Þ

zO3 ð2Þ

bcO3

0.257 0.250 0.130 0.280 0.150 0.280 0.260 0.230

0.421 0.410 0.210 0.480 0.250 0.460 0.440 0.382

1.421 1.410 1.210 1.480 1.250 1.460 1.440 1.382

O3 needs solar radiation for its formation. In regards to wind parameter, the thermal turbulence generated by solar radiation intensity in the ABL strongly influences the local wind speed for a vignon, 2003; Stull, 2012). It is the continental wind regime (Bre reason why we found the same characteristic frequencies in solar radiation and wind spectra. As mentioned previously, wind speed and solar radiation govern PM10-O3 dispersion and O3 formation. Both meteorological parameters also exhibit multifractal behavior. This multifractal nature is shown in Figs. 13 and 14 with the scaling exponent zðqÞ and the singularity spectrum MS ðaÞ in the insets. Fig. 13 shows the concavity and the nonlinearity of zW ðqÞ for one scaling regime (see Fig. 11) whereas Fig. 14 highlights the same properties for zSR ðqÞ in two scaling regimes (lower and middle frequencies in Fig. 12). With the shape of zSRM and zSRL , we observe that zSR ðqÞ is more concave for middle frequencies than for lower frequencies. In the insets, MS ðaÞ highlights the multifractal behavior of these both meteorological parameters. Here, we put

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T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35 Table 6 Intermittency parameter m with its average value for PM10 and O3 data. Year

mPM10

mO3

2005 2006 2008 2009 2010 2012 2014 2015 Average

e 0.029 0.019 0.005 e 0.012 e e 0.016

0.093 e 0.090 0.050 0.080 0.050 0.100 0.080 0.078

Contrary to previous studies, our study also quantifies pollutants intermittency through the intermittency parameter m. O3 revealed more intermittent than PM10 because mO3 is greater than or equal to mPM10 . Furthermore, mO3 values seems to be less variable than mPM10 . This can be explained by the origin of both pollutants. Indeed, O3 is a pollutant produced by photochemical reactions involving NOx and VOcs emitted by local pollution sources (constant sources throughout the year), while PM10 mainly depends on puffs of dust coming from Sahara mainly between April and December (variable source during the year) (Prospero, 1981a,b). One must notice that it is the first study quantifying the intermittency of PM10 and O3 in Caribbean region where PM10 are mainly advected with Saharan dust.

6. Conclusion

Fig. 10. PM10 and O3 singularity spectrum Ms ðaÞ respectively for 2012 and 2014 with multifractal process (blue circle), monofractal process (red asterisk) and lognormal model (green line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 5 Values of singularity spectrum quantities with their all-years average values for PM10 and O3 data.

2005 2006 2008 2009 2010 2012 2014 2015 Average

DaPM10

DaO3

DMSPM10

DMSO3

e 0.27 0.17 0.07 e 0.13 e e 0.16

0.21 e 0.21 0.10 0.19 0.10 0.24 0.18 0.18

e 0.68 0.44 0.19 e 0.50 e e 0.45

0.34 e 0.35 0.17 0.34 0.14 0.39 0.25 0.28

into evidence the multifractal nature of the meteorological parameters which allow the formation or the advection of both pollutants. Being closely related, the multifractal nature of wind speed and solar radiation will influence the scaling invariance nature of PM10 and O3 .

In this paper, we analyzed the scaling and the intermittency properties of PM10 and O3 time series in Caribbean region through structure function analysis at all scales and all intensities. A sharp description of the dynamics of these pollutants, here, is provided in the fully developed turbulence. We firstly applied a spectral analysis to PM10 and O3 time series in the Fourier space. One scaling regime is found between low and middle frequencies for PM10 and O3 spectra. We obtained smaller timescales values than those found previously in other studies carried out in large megalopolis where pollution levels are higher than those of Guadeloupe. Topography and climatic conditions which are key parameters for atmospheric pollutants production and dispersion also differ. Secondly, we analyzed the scaling invariance behavior of both pollutants and his nature with the structure function scaling exponent zðqÞ. With the Hurst parameter H ¼ zð1Þ, we found a high variability in PM10 and O3 time series contrary to other studies (0 < H < 0:5). The structure functions analysis highlighted the concavity and the nonlinearity of scaling exponents zPM10 ðqÞ and z03 ðqÞ due to multifractal nature of both pollutants. Another representation was used and confirmed the multifractal behavior of PM10 and O3 time series: the singularity spectrum MS ðaÞ. Our results showed the existence of multifractality in both times series, which indicate a new level of complexity distinguished by the wide range of necessary fractal dimensions to characterize the dynamics of PM10 and O3 . For the first time in Caribbean region, a study of the intermittency has been also carried out on PM10 and O3 times series. The scale invariance enable the function zðqÞ to be estimated. A statistical modeling is performed by a lognormal model to quantify the curvatures of zðqÞ and MS ðaÞ. By using this model, the degree of multifractality or intermittency was calculated with the intermittency parameter m. For each year, the obtained results with mPM10 -mO3 confirmed those obtained visually by the concavity of zPM10 ðqÞ-z03 ðqÞ and those calculated by DaPM10 -DaO3 with MS ðaÞ. In

T. Plocoste et al. / Atmospheric Environment 169 (2017) 22e35

Fig. 11. The mean Fourier power spectrum EW ðf Þ of wind data between 2005 and 2015 in log-log plot, illustrating one scaling regime (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

33

Fig. 13. Scaling exponent zW ðqÞ and the corresponding multifractal spectra Ms ðaÞ (in the insets) with vertical error bar for each data point (blue line) compared to the linear model zðqÞ ¼ qH (red line with square) and the lognormal model (green line) for wind speed. Ms ðaÞ (blue line for multifractal process and red asterisk for monofractal process) is also fit with the lognormal model (green line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. The mean Fourier power spectrum ESR ðf Þ of solar radiation data between 2008 and 2015 in log-log plot, illustrating two scaling regimes for lower frequencies (black line) and middle frequencies (red dashed line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

other words, the singularity spectrum MS ðaÞ validated the structure function scaling exponent zðqÞ method for our database. Overall, we found that the degree of multifractality is higher for O3 than PM10. The results presented in this paper may aid in attaining a full understanding of complex PM10 and O3 processes dynamics in Caribbean region. A good knowledge of pollutant sequence dynamics is fundamental for management of air pollution. The results obtained from multifractal analysis and modeling should allow the construction of forecasting tools. Indeed, two forecast models can be developped (Medina et al., 2017): i) on the scaling properties of times series considered with a fractional brownian motion; ii) on the intermittency in the increments of times series considered with a multifractal lognormal fit. In future works, multifractal parameters obtained for both pollutants will be used to elaborate these forecast models, in order to evaluate their performance compared with others models. To better understand O3 behavior in Caribbean

Fig. 14. Scaling exponent zSR ðqÞ for lower (zSRL , red circle) and middle frequencies (zSRM , blue diamond) and the corresponding multifractal spectra Ms ðaÞ (in the insets with red circle and blue diamond for multifractal process and red asterisk and purple star for monofractal process) compared to the linear model zðqÞ ¼ qH (red line with square for lower frequencies and pink line with star for middle frequencies) for solar radiation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

islands, a study of correlations between this pollutant, its precursors and meteorological parameters will be performed by using stochastic methods. Disclosure statement No potential conflict of interest was reported by the authors.

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