Spectral Observations of PM10 Fluctuations in the Hilbert Space

During the last 20 years, many megacities have experienced air pollution leading to negative impacts on human health. In the Caribbean region, air quality is widely affected by African dust which causes several diseases, particularly, respiratory diseases. This is why it is crucial to improve the understanding of PM10 fluctuations in order to elaborate strategies and construct tools to predict dust events. A first step consists to characterize the dynamical properties of PM10 fluctuations, for instance, to highlight possible scaling in PM10 density power spectrum. For that, the scale-invariant properties of PM10 daily time series during 6 years are investigated through the theoretical Hilbert frame. Thereafter, the Hilbert spectrum in time-frequency domain is considered. The choice of theoretical frame must be relevant. A comparative analysis is also provided between the results achieved in the Hilbert and Fourier spaces.


Introduction
Generally, the concentration of air pollutants varies and is impacted by the local pollutant emission levels and meteorological and topographical conditions [1,2]. Particulate matter (PM) is a complex mixture of elemental and organic carbon, ammonium, nitrates, sulfates, mineral dust, trace elements, and water [3]. PM with an aerodynamic diameter of <10 μm, i.e., PM10, are well known for their impact on human health [4]. Many studies have highlighted that exposure to PM increases the number of hospital admissions for cardiovascular disease, acute bronchitis, asthma attacks, respiratory disease, and congestive heart failure [5][6][7][8]. In the Caribbean area, one of the main emitters of PM10 is from large-scale sources, i.e., African dust [9]. Knowledge of the dynamics of PM10 process is crucial to elaborate strategies and construct tools to predict dust events. The time-frequency distribution of a signal provides information about how the spectral content of a signal evolves with time, thus providing an ideal tool to dissect, analyze, and interpret nonstationary signals [10]. Contrary to classical methods, the need of a timefrequency representation (TFR) is stemmed from the inadequacy of either time domain or frequency domain analysis to fully describe the nature of nonstationary signals [10]. In literature, there are numerous methods to obtain energy density as a function of time and frequency simultaneously as the short-time Fourier transform (STFT), Hilbert-Huang transform (HHT), and wavelet transform (WT) [10][11][12].
In this study, the scaling properties of PM10 data are firstly analyzed, and then the TFR is investigated. In order to highlight the performance of the Hilbert space, an analysis of PM10 data was also performed in the Fourier space.
This chapter is organized as follows. Section 2 presents PM10 data analyzed in this study. Section 3 describes the methods applied in order to investigate PM10 dynamics. Section 4 comments on the results obtained and then discusses them.

Experimental data
Guadeloupe archipelago is a French West Indies island located in the middle of the Caribbean basin, i.e., 16.25°N latitude and 61.58°W longitude, which experiences a tropical and humid climate [13,14]. The time series analyzed here belong to Guadeloupe air quality network which is managed by the Gwad'Air agency (http:// www.gwadair.fr/). PM10 concentrations are measured at Pointe-à-Pitre (16.2422°N 61.5414°W) using the Thermo Scientific tapered element oscillating microbalance (TEOM) models 1400ab and 1400-FDMS. Hourly PM10 concentrations were sampled during the period from 1 January 2005 to 31 December 2010. We processed these data into daily average concentrations. In total, there are 2150 daily averaged data points available continuously for 6 years. Figure 1 displays PM10 daily signal illustrating huge fluctuations and thus indicating a strong variability. These strong oscillations observed in the middle of each year are attributed to PM10 related to dust outbreaks coming from the African coast from May to September [9]. For the rest of the year, PM10 is mainly generated by anthropogenic pollution [15].

Scaling analysis (1D representation)
The description of natural phenomena by the study of statistical scale laws is not recent [16]. Self-similarity of complex systems has been widely observed in nature and is the simplest form of scale invariance. A scale invariance can be detected by computing of power spectral density (PSD). The PSD separates and measures the amount of variability occurring in different frequency bands. In this study, PSD are estimated through the Fourier and Hilbert spaces.

Fourier analysis
In order to investigate the scaling properties of PM10 data, classically the discrete Fourier transform of the times series considered is computed. The expression of Fourier transform X(f) for a process x(t) is recalled here. An N point-long interval is used to construct the value at frequency domain point f, X f [17]: Thus, the analytical expression of Consequently the power spectral density E(f) is estimated by computing the following expression:

Hilbert analysis
To determine the scale invariance of a given time series in a joint amplitudefrequency space, the Hilbert-Huang transform [19,20] is performed. HHT can be summarized in two steps: (i) empirical mode decomposition (EMD) and (ii) Hilbert spectral analysis (HSA). Empirical mode decomposition is a powerful tool to separate a nonlinear and nonstationary time series into a sum of intrinsic mode functions (IMF) without a priori basis as required by traditional Fourier-based method [19][20][21]. An IMF must satisfy the following two conditions: (i) the difference between the number of local extrema and the number of zero-crossings must be zero or one, and (ii) the local maxima and the envelope defined by the local minima are close to zero. Therefore, the original signal x(t) is decomposed into a sum of n1 IMF modes with the residual r n (t): To obtain a physically significant IMF, this selection process must be stopped by a certain criterion. For more details, EMD decomposition is widely described in the literature [19][20][21][22][23].
To characterize the time-frequency energy distribution from the original signal x (t), HSA is applied on each obtained IMF component C m (t) to extract the instantaneous amplitude and frequency [19,24]. The Hilbert transform is defined by: with P the Cauchy principal value [24,25]. We can specify an analytical signal z for each IMF mode C m (t) with describes an amplitude and represents the phase function of IMF modes. Consequently, the instantaneous frequency ω m t ð Þ is defined from the phase φ m t ð Þ by Thus, the original signal x(t) can be expressed as where Re is a part real [19,20,26]. Due to the simultaneous representation of frequency modulation and amplitude modulation, the HHT can be considered as a generalization of the Fourier transform [19,20]. The energy in a time-frequency space is designated as the Hilbert spectrum where T is the total data length. The Hilbert spectrum H(ω,t) gives a measure of amplitude from each frequency and time, while the marginal spectrum h(ω) gives a measure of the total amplitude from each frequency [27]. As a result, the marginal spectrum can be compared to the Fourier spectrum [19,20].
In conclusion, for a scale-invariant process, the Fourier E(f) and the Hilbert h(ω) spectral densities obtained follow a power law over a range of frequencies: where f and ω are the frequencies and β f and β h are the spectral exponents, respectively, in the Fourier and Hilbert spaces. It reveals the scale-free memory effect as a power law dependence of the frequency distribution. Consequently, β f and β h contain information about the degree of stationarity of the studied parameter [16,28,29]: • If β f or β h <1, the process is stationary.
• If β f or β h >1, the process is nonstationary.
• If 1< β f or β h <3, the process is nonstationary with increments stationary.

Spectrogram
The spectrogram (SPEC) of a signal x(t) is defined as the squared magnitudes of the STFT as shown in Eq. (12) [12]: is a window (e.g., Hanning, rectangular, Hamming), t is time, and f is frequency.
As depicted in Eq. (13), SPEC roughly describes the energy density of the signal at point (t,f) [12]: The SPEC has been applied successfully in various research fields [12, [35][36][37]. The main advantages of SPEC are an easily understanding interpretation, and it allows a fast computation. However, the main drawback of SPEC is the same as that of the STFT [12]. Indeed, there is a trade-off between time and frequency resolution.

Hilbert spectrum
The Hilbert spectrum (HS) is a joint time-frequency representation introduced by [19]. It is important to notice that the two important tools (i.e., EMD and HS) for exploratory analysis of the data are provided by HSA method. This approach was applied successfully in various research fields as fault diagnosis for rolling bearing [11], turbulence [38], environment [34,39], and geophysics [40], to cite a few.

Scaling properties
In order to identify the presence of scaling in PM10 time series, the PSD is estimated in the Hilbert and Fourier spaces. Figure 2 depicts the power spectral density provided by the Hilbert transform and the Fourier transform. On this figure, we try to detect a power law behavior of the form h ω ð Þ $ ω Àβ h and E f ð Þ $ f Àβ f where β h and β f are, respectively, the spectral exponents in the Hilbert and Fourier spaces. On the frequency range 2:09 Â 10 À7 ⩽f ⩽4:57 Â 10 À5 Hz which corresponds to time scales 6:1 hours⩽T⩽55:4 days, a power law behavior is clearly noticed in the Hilbert space with an estimated spectral exponent β h = 1.02 AE 0.10. β h is equal to 1 power law scaling observed in the mesoscale range [41]. In the Fourier space, this power law is not significant. This is due to the existence of intermittent dust events with huge fluctuations in PM10 data (see Figure 1). Indeed, the Fourier transform is a linear asymptotic approach which requires high-order harmonic components to mimic nonlinear and nonstationary process [42]. Thus, the highorder harmonics may lead an artificial energy transfer flux from a large scale (low frequency) to a small scale (high frequency) in the Fourier space. Consequently, the Fourier-based spectrum may be contaminated by this artificial energy flux [42]. The artificial energy transfer may give a less steep power spectrum as we observed in Figure 2. By contrast, combined with the EMD method, HSA has very local abilities both in physical and spectral domains and does not require any higherorder harmonic components to simulate the nonlinear and nonstationary events. As a consequence, HSA method may provide a more accurate scaling exponent and singularity spectrum [42].
According to [43], wind speed dominates the amount of pollutant dispersion in the atmospheric boundary layer. In addition, this meteorological parameter could also transport PM10 from large-scale sources, i.e., African dust [9]. To complete our results, we used hourly wind speed measurements provided by the French weather office (Météo France Guadeloupe) located at Abymes (16.2630°N 61.5147°W). PM10 and wind speed measurements are very close, i.e., ≈8.1 km of distance, and performed at the center of the island under the same atmospheric conditions [2]. Figure 3 illustrates the PSD provided by the Hilbert transform and the Fourier transform for wind speed data. This time, a power law behavior is observed in both spaces on the same frequency range 3:54 Â 10 À7 ⩽f ⩽1:36 Â 10 À4 Hz which corresponds to time scales 2:1 hours⩽T⩽32:7 days. Contrary to PM10 which is a passive scalar, wind speed is a vector quantity. The estimated spectral exponents are identical with, respectively, 0.89 AE 0.06 and 0.90 AE 0.12 in the Hilbert and Fourier spaces. As PM10, spectral exponent values are also close to À1. For wind speed, at low frequencies, a spectrum close to the 1 power law is likely occurs close to a rough surface, due to a strong interaction between the mean flow vorticity and the fluctuating vorticity [44,45].

Time-frequency domain
The TFR in the Fourier and Hilbert spaces are, respectively, illustrated in Figures 4 and 5. Both figures show a color gradient from strong energy (in red) to weak energy (in blue). This highlights the energy activity related to PM10 concentrations during the study period. Such an approach gives the possibility of tracking the evolution of PM10 data spectral content in time, which is typically represented by variations of the amplitudes and frequencies of the components from which the signal is composed [46]. On Figure 4, strong energies are observed throughout the years with slight fluctuations on the frequency range 0⩽f ⩽1 Â 10 À6 Hz. For f >4 Â 10 À6 Hz, strong energies are also noticed in the middle of each year and at the beginning of 2010 with more fluctuations. In Figure 5, energy distributions are more localized. On the frequency range 0⩽f ⩽1 Â 10 À6 Hz, we can observe the influence of small-scale event on energy behavior. As noticed, this energy may be weak or null. As an example, the impact of a general strike in early 2009 that paralyzed Guadeloupean archipelago at least 2 months is highlighted by zero energy due to the lack of PM10 sources, i.e., industrial activity and road traffic. For f >1 Â 10 À6 Hz, one can see more precisely energy variation related to dust events from mesoscale to large scale. Contrary to SPEC, HS clearly illustrates localized energy fluctuations due to small-scale event. In fact, the STFT makes an assumption that any signal as piecewise stationary and uses suitable window function to produce the short-time spectral characteristics of the signal. However, in   [48]. However, the SPEC does not detect energy fluctuation related to anthropogenic pollution, i.e., local sources. This shows HS is a robust method in timefrequency domain. Indeed, based on the EMD method, this TFR is fully data adaptive, and the signal decomposition is performed without any predefined basis functions. These results confirm the superiority of HS over STFT in TFR.

Conclusion
In this paper, we investigated scaling and time-frequency properties of PM10 data in Hilbert frame. The performances obtained in the Hilbert space are compared with those achieved in the Fourier space. Firstly, with the Hilbert spectral analysis (HSA), a power law behavior is clearly observed on the frequency range 2:09 Â 10 À7 ⩽f ⩽4:57 Â10 À5 Hz which corresponds to time scales 6:1 hours⩽T⩽55:4 days with an estimated spectral exponent β h = 1.02 AE 0.10. As HSA methodology has a very local ability in both physical and spectral spaces, the influence of intermittent dust events with huge fluctuations is included in the amplitude-frequency space which is not the case in Fourier spectrum. Thereafter, PM10 data are illustrated in time-frequency representations with the Hilbert spectrum and spectrogram. The results provide the evidence that HS-based TFR performs better than SPEC. The higher resolution in TFR offers better fluctuations of PM10 energy for f < 1μ Hz. This is due to the fact that it is impossible to increase the TF resolution at the desired level in SPEC. The major asset of HS is that the time resolution can be as precise as the sampling period and the frequency resolution depends on the choice up to the Nyquist limit. In addition, contrary to SPEC which introduces a noticeable amount of cross-spectral energy terms during the use of window function with overlapping, HS is fully adaptive to datasets due to the decomposition of the signals. These first results suggest a substantial possibility to perform a profound dynamical analysis of PM10 concentrations for the Caribbean area in order to quantify the origin and the threshold pollution.