Abstract
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.
Keywords
- PM10 data
- empirical mode decomposition
- Hilbert spectral analysis
- time-frequency representation
- Fourier space
1. 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 time-frequency 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.
2. 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].
3. Methods
3.1 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.
3.1.1 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
Thus, the analytical expression of
Consequently the power spectral density
3.1.2 Hilbert analysis
To determine the scale invariance of a given time series in a joint amplitude-frequency 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
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
with
where
Thus, the original signal
where
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 with
where
In conclusion, for a scale-invariant process, the Fourier
where
3.2 Time-frequency representation (2D representation)
3.2.1 Spectrogram
The spectrogram (SPEC) of a signal
where
As depicted in Eq. (13), SPEC roughly describes the energy density of the signal at point (
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.
3.2.2 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.
4. Results
4.1 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
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
4.2 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
5. 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
Acknowledgments
The authors would like to thank Guadeloupe air quality network (Gwad’Air) and the French Met Office (Météo France Guadeloupe) for providing air quality and meteorological data.
Nomenclature
Fourier spectral density frequency (Hz) spectral exponent instantaneous amplitude intrinsic mode function component Hilbert spectrum instantaneous frequency (Hz) scale index total length of a sequence particulate matter signal (μg/m3) residual of the intrinsic mode function phase function of the intrinsic mode function
Abbreviations
PM10 | particulate matter with an aerodynamic diameter 10 μm or less |
PSD | power spectral density |
SPEC | spectrogram |
EMD | empirical mode decomposition |
IMF | intrinsic mode function |
HSA | Hilbert spectral analysis |
HHT | Hilbert-Huang transform |
TFR | time-frequency representation |
HS | Hilbert spectrum |
STFT | short-time Fourier transform |
WT | wavelet transform |
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