Spatial analysis of tails of air pollution PDFs in Europe

TL;DR

The paper tackles the problem of characterizing the tails of outdoor air-pollution PDFs across Europe, focusing on extreme concentration events. It adopts a superstatistical framework and fits $q$-exponential tails $f_{q,\lambda}$ to 3544 European monitoring sites, while extracting a long time scale $T$ via local kurtosis to capture dynamics. The main findings show heavy-tailed distributions with spatially varying parameters $(q,\lambda)$ and region-specific time scales, revealing pronounced East–West differences and pollutant-dependent patterns. These results enable spatially resolved risk assessment and policy design by combining tail statistics with dynamical information, and the authors provide data and code for replication.

Abstract

Outdoor air pollution is estimated to cause a huge number of premature deaths worldwide, it catalyses many diseases on a variety of time scales, and it has a detrimental effect on the environment. In light of these impacts it is necessary to obtain a better understanding of the dynamics and statistics of measured air pollution concentrations, including temporal fluctuations of observed concentrations and spatial heterogeneities. Here we present an extensive analysis for measured data from Europe. The observed probability density functions (PDFs) of air pollution concentrations depend very much on the spatial location and on the pollutant substance. We analyse a large number of time series data from 3544 different European monitoring sites and show that the PDFs of nitric oxide ($NO$), nitrogen dioxide ($NO_{2}$) and particulate matter ($PM_{10}$ and $PM_{2.5}$) concentrations generically exhibit heavy tails. These are asymptotically well approximated by $q$-exponential distributions with a given entropic index $q$ and width parameter $λ$. We observe that the power-law parameter $q$ and the width parameter $λ$ vary widely for the different spatial locations. We present the results of our data analysis in the form of a map that shows which parameters $q$ and $λ$ are most relevant in a given region. A variety of interesting spatial patterns is observed that correlate to properties of the geographical region. We also present results on typical time scales associated with the dynamical behaviour.

Figures (5)

• Figure 1: (a) The white dots show the positions of all European measuring stations considered. A red dot singles out an example of a station at Illmitz, Austria. For such a given example location, the time series data of the four measured concentrations are shown in panels b and d. The corresponding tails of the PDFs (as shown in c, e) are then automatically analysed with a fitting program code that extracts the parameters $(q, \lambda)$ for the best-fitting $q$-exponential. This is done for all 3544 sites
• Figure 2: Measured histogram of $NO$ concentrations, recorded at the example site Barnsley Gawber, UK, together with the best log-normal (blue), Weibull (orange), Gamma (brown) and $q$-exponential (purple) fits, together with their respective log-likelihood values. Also displayed are the optimum values of the $q$ and $\lambda$ parameters for the $q$-exponential distribution. The $q$-exponential fits the measured data best, as indicated by the highest log-likelihood value. Similar plots can be produced for all 3544 sites
• Figure 3: Best-fitting parameters of $q$-exponentials at the various measuring stations. There is an increasing trend of $q$ versus $\log \lambda$ for $NO$ (a) and $NO_2$ (b), whereas a disk-shaped pattern is observed for $PM_{2.5}$ (c) and $PM_{10}$ (d). The colors encode the area type where the measurements were done. There are predominant patches of a single color in the parameter space for $NO$ and $NO_2$, where for $PM_{2.5}$ and $PM_{10}$ the color pattern looks more mixed. Deep green points correspond to cleaner rural areas.
• Figure 4: Spatial distribution of best-fitting parameters $(q,\lambda)$ characterizing the measured PDFs of $NO_x$ and $PM_x$ pollutants across Europe. The color codes are directly indicated in the individual figures. A large value of $q$ indicates heavy tails in the distribution. A small value of $\lambda$ indicates heavy average pollution. The pollutant characteristics across Europe is quite inhomogeneous
• Figure 5: The long time scales $T$ describing the scale of typical changes of the temporal mean and variance of the measured time series for $NO$ (a), $NO_2$ (b), $PM_{2.5}$ (c) and $PM_{10}$ (d). The color coding is explained at the bottom of each figure