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Abstract. A specific statistical analysis is applied on surface ozone and NO2 measurements obtained at Patission air pollution monitoring station (Athens, Greece), during the period 1988-2008. The main conclusions obtained are that both air pollutants values do not obey the Gaussian distribution, but instead, follow the distribution of the seismic Gutenberg-Richter law. Moreover surface ozone and NO2 extremes follow the Generalized Pareto distribution.
The health effects of air pollution have been subject to intense study in recent years. Exposure to air pollutants such as nitrogen dioxide and ozone has been associated with increases in mortality and hospital admissions due to respiratory and cardiovascular disease. Effects have been seen at very low levels of exposure and it is unclear whether a threshold air pollutants concentration exists below which no effects on health are likely.
Given that air pollution may produce adverse human health effects, it is urgent to develop secure tools and models, which reliably predict pollutants concentrations (Sapkota et al., 2005; Efstathiou et al., 2005). Current predictive models are separated into two main categories: prognostic models, which are based on the physicochemical principles determining photochemical pollution and the diagnostic models, which are statistical descriptions of the observed air pollution and mainly based on classical statistical methods and neural network techniques (Gardner and Dorling, 1998; Perez and Trier, 2000; Windsor and Toumi, 2001; Varotsos et al., 2001).
Lu (2002) utilized three distributions (lognormal, Weibull and type V Pearson distribution) to simulate the PM10 concentration distribution in Taiwan areas and it was derived that the lognormal is the best distribution to represent the PM10 daily average concentration.
Recently, Varotsos et al. (2003) investigated the photochemical pollution of an urban and a rural station located in the greater area of the Athens basin, during the period (1987-2001). For the rural station, the investigation was also prolonged to the involved meteorological mechanisms. Furthermore, they examined the distribution of the mean monthly surface ozone concentration (SOC) at the urban station and they proposed an empirical tool, according to which the SOC distribution is of semi-log type. It was also derived that the empirical probability of exceeding or equalling an observed SOC value was equal to the theoretically derived value.
Very recently Reider et al. (2010) showed that a Gaussian distribution is inadequate to describe extremes in the daily mean values of the Arosa total ozone time series, because of a high frequency of low and high values, i.e. heavy tails. According to that analysis the Generalized Pareto Distribution (GPD) is one of the most commonly used distributions in the framework of extreme value theory.
The present work examines whether the surface NO2 and O3 concentrations which are considered as the most representative index for the atmospheric air quality, obey the Gaussian or other distribution, laying special emphasis on the extreme values (Rundle, 1989; Dubois, 1998; Varotsos et al., 2003; Reider et al., 2010).
Data and Analysis
In the current study, mean daily surface ozone and NO2 measurements, obtained at the Patission urban station of the National Air Pollution Monitoring Network, during the period 1988-2008, were employed. The Patission station was selected because it provides the longer, homogeneous and continued data-record (Varotsos et al., 2003).
3. Discussion and results
By analysing the data mentioned above, the mean daily values of each of surface ozone and NO2 were grouped into classes of equal length, in order to examine whether their concentrations obey the Gaussian distribution.
Figure 1. The percentage relative frequency histogram of the (a) surface ozone, (b) surface NO2 mean daily values, at Patission station, during the period 1988-2008. Smooth line presents the Normal distribution.
Figure 1 depicts the percentage relative frequency histogram of the surface O3 and NO2 mean daily values. In the same figure the smoothed line presents the Gaussian distribution. However, using the statistical best-fit tests Kolmogorov Smirnov, Chi-square and Anderson Darling the hypothesis whether surface ozone and NO2 concentration obey the Normal distribution was rejected, at the 95% confidence level.
Inspection of Figure 1ab shows that the extreme values (defined as those that are located out of the [μ€2σ, μ+2σ] interval, where μ and σ stand for the time series average and standard deviation, respectively) seem to be mainly responsible for the poor-flexibility of the normal distribution on surface ozone and NO2 values (Efstathiou, 2006).
Because the obtained distribution may be described by monotonously decreasing functions with heavy tails, the data were tested for meeting the requirements of an exponential distribution, a geometric distribution, a simple power law distribution, a generalized power law distribution (the Zipf-Mandelbrot distribution) and a lognormal distribution, respectively. The observed data showed a best fit to the simple power law and the Zipf-Mandelbrot distributions.
In order to define the above mentioned air pollutants distribution, the empirical probability P(X>x) of exceeding a fixed surface ozone or NO2 value x, was calculated and then plotted in semi-logarithmic graph, against the value x (see Figure 2).
In the following, applying linear regression analysis between the logarithm of the probability P(X>x) and the surface ozone value x, the slope derived α equalled €0.026, with coefficient of determination R2 = 0.98. The latter denotes that the slope derived above is statistically significant, at 95% confidence level yielding to the relationship below:
P(X>x) ~ 10αx (1)
According to the equation (1), surface ozone values do not obey the Gaussian distribution but instead follow the distribution of the Gutenberg-Richter law (Goldstein et al., 2004; Rundle, 1989). This result was confirmed by using the statistical best-fit tests Kolmogorov-Smirnov, Chi-square and Anderson Darling at 95% confidence level.
Similarly, applying linear regression analysis between the logarithm of the probability P(X>x) and the fixed NO2 x value, the derived slope αÎ„ equalled €0.008, with coefficient of determination R2 = 0.99, which is statistically significant, at 95% confidence level, according to the above mentioned non parametric statistical tests. Thus the following relationship can be written:
P(X>x) ~ 10αÎ„x (2)
According to the equation (2), ΝΟ2 values follow the distribution of the Gutenberg-Richter law, similarly to surface ozone values.
Figure 2. Semi-log plot of the probability P(X>x) as a function of the fixed (a) surface ozone or (b) NO2 value x. Least-square fit for the surface ozone values (y = €0.026x + 0.21 and R2 = 0.98) and for the NO2 values (y = €0.008x + 0.44 and R2 = 0.99).
In the literature many seemingly unrelated parameters show power law distributions, e.g., the size distributions of earthquakes measured by the Gutenberg-Richter scale (Gutenberg and Richter, 1994) the growth of breast, colorectal, and renal cell carcinomas. To explain some of these phenomena, Bak et al. (1998) hypothesized the self-organized criticality (SOC) according to which complex systems evolve toward a state of maintained criticality at which they are able to propagate perturbations on all of the possible length and size scales, often described as avalanches. An alternative explanation may be that evolving systems in general, are inclined to show highly optimized tolerance (HOT), where systems are tuned, through selection, to highly structured and efficiently operating states within a given environment (Carlson and Doyle, 1999). This is partly achieved by generating barriers that reduce the effects of cascading failures caused by perturbations encountered frequently during the evolutionary process. However, rare disturbances may lead to dramatic consequences. The results of such failures show power law distributions with exponents close to unity (Carlson and Doyle, 1999). In this context, the complex system of the air pollution extremes would be caused by a cascading failure in a system showing HOT. However, the avalanche dynamics of the air pollution system usually associated with SOC and HOT are rapid, compared with the driving processes. That is, the system dynamics are slow but will eventually result in a state where a rare single event may, at a single time, cause a large effect i.e., a large number of aberrations. This makes the suggested models poorly compatible with the observations. On the other hand, a variety of stochastic growth processes have been shown to converge to power law distributions (Marsili and Zhang 1998). One such process is multiplicative fluctuations (Pietronero et al. 2001).
As mentioned above, Reider et al. (2010) showed that the Generalized Pareto Distribution is one of the most commonly used distributions in the framework of extreme value theory. In the current study GPD was used in order to model the mean daily surface ozone and NO2 values above an extreme, during 1988€2008. Figure 3 depicts the empirical and the predicted cumulative distribution function of both air pollutants extreme values x > μ+2σ (over 58 μgr/m3 and 242 μgr/m3 for surface ozone and NO2, respectively). Studying that figure and using the statistical best-fit tests Kolmogorov Smirnov, Chi-square and Anderson Darling, the hypothesis whether surface ozone and ΝΟ2 extremes follow the GPD was accepted at 95% confidence level.
Figure 3. Comparison between the empirical and the predicted probability P(X≤x) where x is a fixed extreme value of (a) surface ozone or (b) NO2. Least-square fit for the surface ozone extremes (y = 1.01x and R2 = 1) and for the NO2 extremes (y = 1x and R2 = 1).
The presented results point to an important aspect of air pollution episodes, the power law distribution of surface ozone and NO2 extremes in Athens, during 1988-2008. Both pollutants concentration seemed not to obey Gaussian distribution and the poor-flexibility was mainly due to the extremes values. It was also derived that surface ozone and NO2 values follow the distribution of the Gutenberg-Richter law, while the surface ozone and ΝΟ2 extremes follow the Generalized Pareto distribution.