International Journal of Atmospheric Sciences

Volume 2013 (2013), Article ID 419694, 7 pages

http://dx.doi.org/10.1155/2013/419694

Research Article

Self-Organized Criticality: Emergent Complex Behavior in PM₁₀ Pollution

Shi Kai,1,2 Liu Chun-Qiong,1,2 and Li Si-Chuan²

¹Key Laboratory of Hunan Ecotourism, Jishou University, Jishou, Hunan 416000, China

²College of Biology and Environmental Sciences, Jishou University, Jishou, Hunan 416000, China

Received 14 February 2013; Revised 9 May 2013; Accepted 16 May 2013

Academic Editor: Prodromos Zanis

Copyright © 2013 Shi Kai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We analyze long-term time series of daily average PM₁₀ concentrations in Chengdu city. Detrended fluctuation analysis of the time series shows long range correlation at one-year temporal scale. Spectral analysis of the time series indicates 1/f noise behavior. The probability distribution functions of PM₁₀ concentrations fluctuation have a scale-invariant structure. Why do the complex structures of PM₁₀ concentrations evolution exhibit scale-invariant? We consider that these complex dynamical characteristics can be recognized as the footprint of self-organized criticality (SOC). Based on the theory of self-organized criticality, a simplified sandpile model for PM₁₀ pollution with a nondimensional formalism is put forward. Our model can give a good prediction of scale-invariant in PM₁₀ evolution. A qualitative explanation of the complex dynamics observed in PM₁₀ evolution is suggested. The work supports the proposal that PM₁₀ evolution acts as a SOC process on calm weather. New theory suggests one way to understand the origin of complex dynamical characteristics in PM₁₀ pollution.

1. Introduction

The adverse effects of PM₁₀ have been recognized in environmental sciences. Besides the reduction of visibility, the direct impact on human health via inhalation is an important issue [1]. It will be very useful to develop accurate PM₁₀ concentrations forecasting methods, which can help to put forward effective warning strategies to reduce impacts on public health during episodes or poor air quality [2]. In recent years, some PM₁₀ concentrations forecasting methods have been developed [3]. These methods mainly come from two approaches. One is to establish accurate atmospheric model based on meteorologic, physical, and chemical process. The other is to find inherent correlations based on the statistical analysis of the collected data. However, there are still some pending problems with predicting PM₁₀ concentrations. PM₁₀ evolution is highly complex events involving human factors as well as meteorologic, topographic, physical, and chemical conditions. Interrelationships between these processes and PM₁₀ concentrations are complex and nonlinear. The circumstances that determine high PM₁₀ concentrations are uncertain sometimes [4]. So the microscopic physical and chemical mechanisms that drive PM₁₀ temporal evolutions are not well understood. However, even if such microcosmic dynamical mechanisms have been illuminated, it is likely that the system would be highly nonlinear without any simple way to predict emergent behavior [5].

As asked by Nagel [6], “is there anything we can say about these systems from first principles without knowing about the ‘microscopic’ details of the problem?” It is obvious that the complexity theory and related methods are needed when confronted with these complex and nonlinear problem. So many researchers have investigated the nonlinear dynamics of air pollutants evolution with different aspects [7–9]. The relational studies have observed that air pollutants concentrations exhibited long range correlation, scale-invariant, and multifractal behavior. These progresses enhance our fundamental knowledge on the complex structure of PM₁₀ concentrations. Thus, one has to be very careful to employ the standard statistical methods in air pollution prediction (as, e.g., ARIMA and ARIMAX models). If one wants to make robust forecasting, the first important thing to do is to identify the structure of the process. If it is approximately linear, then a linear method (e.g., an autoregressive one) can be helpful. On the contrary, if it is nonlinear, completely different types should be used. In this case, predicting models can be validated by using the information on the persistence characteristics of air pollution time series.

However, why do the complex structures of PM₁₀ concentrations evolution exhibit scale-invariant?

As an introduction to the concept, self-organized criticality (SOC) has been proposed by Bak et al. [10] to provide a framework of modeling such phenomena as persistent behavior, noise, and scale-invariant, which are widespread in nature. The Bak-Tang-Wiesenfeld (BTW) sandpile model is a classical numerical model in SOC theory. This concept is well illustrated with a model of sandpile. One considers a grain is dropped into a sandpile randomly and slowly. As we add new grains, the pile grows more and more until the pile reaches a critical slope in a statistically stationary state. At some critical point, the addition of other grains may cause either small avalanches or trigger a very large avalanche. Some phenomena, such as power-law temporal correlations and scale-invariant, emerged from SOC state in a dissipative system. The issue of determining whether evolution of PM₁₀ is governed by SOC is a difficult one. There are no unequivocal determining criteria. One approach is to compare characteristic measures of air pollution process to those obtained from a known SOC system [11, 12]. We consider that PM₁₀ pollution is a complex dynamical system, which will automatically adjust itself to a critical state characterized by power-law correlations in both space and time. This state is “critical” in the sense of an equilibrium critical point where there is no characteristic length or time scale that controls the behaviour of the system. So PM₁₀ pollution can be analogous to a variety of nonequilibrium relaxation processes in nature such as earthquakes and avalanches.

In this work, at first, long range correlation, noise, and scale-invariant of PM₁₀ concentrations measured at Chengdu are examined by detrended fluctuation analysis, power spectrum, and cumulative magnitude-frequency distribution, respectively. Then, in the SOC framework, we put forward the self-organized evolution theory of PM₁₀ pollution under calm condition. The SOC mechanism of PM₁₀ evolution is discussed. At last, a modified sandpile model of SOC, which describes the essential mechanisms of self-organized evolution of PM₁₀, is put forwarded to simulate the scaling characteristic of PM₁₀ concentrations and to illuminate the origin of scale-invariant in PM₁₀ pollution. This work provides new insight and approaches to research the nonlinear dynamics and emergent complex behavior in PM₁₀ evolution.

2. Materials and Methods

2.1. Data

Chengdu city is located in western Sichuan Basin of China. Sichuan Basin covers 260,000 km², generally at low altitudes of about 500 m. These lower lying areas are surrounded by mountains and a plateau higher than 4 km. The unique geographical environment directly affects meteorological condition of pollutant diffusion and increases the frequency of calm wind at Chengdu. At Chengdu city, the average annual frequency of calm wind, namely, wind speed being 0~0.2 m/s, is 46%. So the local pollution sources play a more significant role in air quality of Chengdu [13].

There are eight automatic monitoring stations at Chengdu city. The daily average concentrations of pollutants are made at each station. These concentrations are further averaged over the stations to provide the daily average values of pollutant to represent the daily average air quality of Chengdu city. In this work, the examined data set is daily average PM₁₀ concentrations data from 2001 to 2011 in Chengdu city, as shown in Figure 1. The length of data set is 4018. These data are provided by the Sichuan Environment Monitoring Center. These data are characterized by many large fluctuations with no obvious correlation that is difficult to interpret (see Figure 1).

Figure 1: The daily average PM₁₀ concentrations data from 2001 to 2011 in Chengdu city. The data lengths all are 4018 (days).

2.2. Detrended Fluctuation Analysis

Detrended fluctuation analysis (DFA), which was proposed by Peng et al. [14], can be used to demonstrate the scaling behavior associated with SOC systems. It can avoid spurious detection of correlations that are artifacts of nonstationary, which often affects the time series data. It is that, for sample size , the root mean square fluctuation of this integrated and detrended time series behaves as a power-law function of within the scaling region, data present scaling

A DFA exponent indicates a wholly stochastic process lacking correlation; indicates persistent long range power-law correlations; for , power-law anticorrelations are present such that large values are more likely to be followed by small values and vice versa. Although a DFA exponent between 0.5 and 1.0 does not absolutely prove the presence of SOC, it is a strong (in a statistical sense) indication of the presence of SOC.

2.3. Spectral Analysis

The power spectrums have been used in investigating the noise behavior of some time series. If a time series spectrum obeys a power-law form

where is the frequency, it indicates the absence of a characteristic time scale, that is, a scaling behavior. Thus, fluctuations at all scales are related to each other and a fractal behavior may be assumed. Malamud and Turcotte [15] have shown theoretically that one should have . However, in practice this relation is only weakly satisfied.

2.4. Cumulative Magnitude-Frequency Distribution

In some natural phenomena, cumulative magnitude-frequency distributions exhibit power-law scaling. It is regarded as the typical “critical” dynamical behavior of SOC systems [16]. A power-law applied to a cumulative distribution has the relation

where is the cumulative number of events with size greater than the magnitude and is the scaling exponent.

2.5. Simplified Sandpile Model for PM₁₀ Pollution

In the study, a general sandpile model for PM₁₀ pollution has been established with a nondimensional formalism.

The model is defined on a square lattice of size with open boundaries in 2D. The boundaries of the lattice are open. So sands are allowed to leave the system through the boundaries. A given amount of PM₁₀ pollutants is associated with each site .

Driving mechanism: at a given time, as a consequence of pollution source emission at some site , amount of PM₁₀ pollutants changes as follows:

Redistribution and relaxation mechanism: when the amounts of PM₁₀ pollutants at some site reach some threshold magnitude , the site becomes unstable or critical and it relaxes by a toppling. The redistribution and relaxation rule is

This rule will be circulating running in accordance with the above method until a new stable configuration is reached, namely, all . Avalanche size () is measured as the total number of toppling during an avalanche.

These rules represent the movements and transformation process of PM₁₀ under calm condition. When PM₁₀ pollutants diffuse, the site of pollution source will reserve partial pollutants, which are set to one-fifth of its original value. At the same time, owing to precipitation, adsorption, and chemical action, some PM₁₀ pollutants will be lost during transportation and diffusion. In our model, we presume that 4% of PM₁₀ pollutants will be lost when they topple to the four adjacent neighbor sites. So the model is local and nonconservative.

Temporal degradation mechanism: PM₁₀ pollutants will decay with time owing to self-purification of atmospheric environment. We simplify this process and presume that degradation of PM₁₀ follows the first level of decaying kinetics. So when a new stable configuration is reached after each relaxation rule, PM₁₀ pollutants at all sites will decay to of the original level as follows:

After all lattice sites are stable, another grain of sand is added.

If PM₁₀ evolution is an example of a SOC process, the avalanche size distribution will follow power-law distribution. In a nondimensional formalism, we select and referring to the classical BTW sandpile model.

3. Results and Discussion

3.1. Long Range Correlation as Detected by DFA Method

Figure 2 shows the DFA for daily average PM₁₀ concentrations data of Chengdu from 2001 to 2011. The observed relationship shows obviously two different period regimes, with a critical time scale of about one year. Fitting by the least square method the plot, for , , while , . For one-year periods (), it indicates high persistence. For example, there is a tendency for increase in PM₁₀ concentration to be followed by another increased tendency in PM₁₀ concentration at a different time in a power-law fashion. This suggests that the correlations between the fluctuations in PM₁₀ concentrations do not obey the classical Markov-type stochastic behavior (exponential decrease with time) but display more slowly decaying correlations. Over longer time periods, , is close to 0.5 and indicates that the fluctuations in PM₁₀ concentrations behave like the stochastic process at a large temporal scale. This phenomenon perhaps reflects an influence of the annual climate cycle.

Figure 2: DFA of the study data. The dots are values of against the corresponding . The solid lines are power law , with and , respectively. The vertical line indicates that is about one year.

3.2. Noise as Detected by Spectral Analysis

On the power spectrum plot shown in Figure 3, we can see that the power spectral density obeys two different power laws in the high-frequency and low-frequency regimes. We have power-law fits for and for year⁻¹. In shorter period, spectral analysis shows that the fluctuations in PM₁₀ concentrations are characterized by noise and self-affine type fractal behaviors, which are similar to the results of DFA methods. For and values, we note that they are higher than the ones estimated by the relation . It shows that this relation is only weakly satisfied.

Figure 3: The power spectrum plot for the study data. The black lines are power law , with and corresponding to high-frequency and low-frequency regimes, respectively.

3.3. The Magnitude-Frequency Distribution of PM₁₀ Concentrations

Figure 4 shows the number () of PM₁₀ events from 2001 to 2011, with size greater than some PM₁₀ concentration () on a double logarithmic plot. Fitting by the least square method, the scaling exponent () is 3.58 according to (1). The scale invariance region starts from 0.128 mg/m³ and ends at 0.518 mg/m³. In the PM₁₀ concentrations regions, a typical scale of pollution events does not exist. There is inherent dynamical connection among the fluctuations in PM₁₀ concentrations. In smaller PM₁₀ concentrations regions, the power-law breaks down obviously. We consider that low monitoring frequency of PM₁₀ concentrations results in the low-size tail of the frequency distribution. A similar phenomenon in rainfall has been reported by Peters and Christensen [17].

Figure 4: The number () of PM₁₀ events from 2001 to 2011, with size greater than some PM₁₀ concentration () on a double logarithmic plot.

In order to investigate the robustness of the scale invariance in PM₁₀ concentrations, the same analysis is performed at the different time intervals, shown in Figure 5 (from 2006 to 2011) and Figure 6 (from 2010 to 2011). These observed statistical distributions all exhibit power laws and the scaling exponents are almost the same. The robustness to the time intervals is the critical behavior of SOC system.

Figure 5: The number () of PM₁₀ events from 2006 to 2011, with size greater than some PM₁₀ concentration () on a double logarithmic plot.

Figure 6: The number () of PM₁₀ events from 2010 to 2011, with size greater than some PM₁₀ concentration (We have found the value ) on a double logarithmic plot.

3.4. Simulation Results of Sandpile Model

The SOC state is stationary in the sense that over long timescales, the average height neither grows nor decays. The average height can be calculated according to the relation

The simulations are performed for 50 × 50 lattice sizes. The average height is plotted against the number of avalanches up to 50000 in Figure 7 for . It can be seen that a constant average height is achieved and it remains constant over a large number of avalanches. The inset figure is a closeup on the average height curve, which looks like very low amplitude “ripples” propagating along the mean field. This phenomenon indicates that the SOC state is reached.

Figure 7: Plot of average height against the number of avalanches. A closeup on the average height curve is shown in the inset.

When reaching the nonequilibrium steady state, extensive data collection has been made for in runs of 10⁷ avalanches. The simulated result of avalanche size distribution is shown in Figure 8. We have found the value when for the exponent of power-law relation , quite close to the exact value in Figure 4. We emphasize that the parameter is closely related to characteristics of PM₁₀ pollutants in atmospheric environment. The emerged scale-invariant in the avalanches size distribution demands no external parameter tuning and should be seen as an evidence of SOC.

Figure 8: Avalanche size distribution for the PM₁₀ pollution sand model when . It follows the power-law relation , with an exponent of (solid line), which is consistent with that of the size distribution of PM₁₀ pollution events in Figure 4.

3.5. Possible Explanation of PM₁₀ Evolution SOC

There are no unequivocal determining criteria to ascertain whether evolution of some natural phenomenon is governed by SOC. One accustomed approach is to compare characteristic measures of some natural phenomenon to those obtained from a known SOC system.

To motivate comparisons between PM₁₀ pollution and SOC sandpile, we firstly take a qualitative description of the complexity in PM₁₀ pollution system which could give rise to SOC dynamics. PM₁₀ pollution system contains many components such as pollutant sources, atmospheric pollutants components, solar radiation, wind speed, temperature, atmospheric self-purification, topographical feature, and other meteorologic factors. Each component has a certain influence on the average PM₁₀ concentration each day. When all the components are considered together, they interact and correlate with each other on vastly different timescales. One group of comparatively fast radical chemical reactions relaxes on timescales of fractions of seconds up to hours, while another group the rather slow processes (e.g., the movement of PM₁₀ pollutants) relaxes on timescales of days up to years or even longer. Thus, PM₁₀ pollution system is a complex system composed of a series of interconnected components. These components have some complex pattern of influence on the daily average PM₁₀ concentration, which result in the fluctuations of PM₁₀ concentrations in long term.

We make an analogy between the sandpile and PM₁₀ evolution. PM₁₀ pollutants form mainly as a result of first and (or) secondary pollutants produced from the emission of air pollution sources. The driving force is the continuously PM₁₀ pollutants emission in atmospheric environment, which serve as the grains continuously dropped on a pile. We can define the superposition of local PM₁₀ pollutants concentration which represents the chain of forces in sandpile. When the amounts of microscopic condensed PM₁₀ pollutants reach some threshold magnitude, the pollutants masses can be transported on microscopic scales by diffusion or convection. They reach a new location, where the local PM₁₀ pollutants concentration is lower and can be diluted. If the local PM₁₀ pollutants concentration in the neighborhood is also high, the amount of condensed pollutants will increase. Once the system reaches some critical point, any small perturbation, in principle, can trigger a chain reaction like the avalanches in atmospheric system. The normal atmospheric environmental capability serves as the critical state. If the local PM₁₀ pollutants concentration is higher than the same critical value, pollutants are assembled and precipitated in the atmosphere. Thus, the system will adapt itself by removing these dissidents to maintain the critical state just as the sandpile adapts itself by avalanching to reach its constant angle of repose. Therefore, we can define the fluctuations of PM₁₀ concentrations as avalanches events in a SOC sandpile. At the critical state, long range correlation, noise, and scale-invariant of PM₁₀ concentrations will emerge from the dissipative system. It is important to note that PM₁₀ pollution system is “tuned” to a critical state solely by its own internal dynamics rather than external dynamics. The high correspondence of the simulated results to observations supports that PM₁₀ evolution acts as a SOC process on calm weather. And SOC is a useful framework to explain the nonlinear evolution of PM₁₀ concentrations.

The micromechanisms of PM₁₀ concentrations evolution are very complex. Some microscopic physical and chemical mechanisms are still uncertain. For example, how do photochemical reaction rate change with first and (or) secondary pollutants? How do the components of pollutants affect mass transport and chemical reaction at gas and solid two phases? Based on this traditional “reductionism” science perspective, the origin of robust scale-invariant in PM₁₀ concentrations evolution can be quite hard to understand. However, when we turn sight to “holism” science perspective, the satisfactory understanding is achieved to this problem. Considering the similarities between sandpile system and PM₁₀ evolution, a simplified sandpile model for PM₁₀ pollution with a nondimensional formalism is put forward. This model mechanism only includes the emission of PM₁₀ pollutants, the movements and transformation of PM₁₀, and temporal degradation process of PM₁₀. The high correspondence of the results to observations indicates that the model provides an effective parameterization of the key physical process that governs PM₁₀ concentrations evolution.

It is important to note that the power system organizes itself to an operating point near to, but not at, a critical value. This could make the system quite robust in different time intervals. One consequence is that the measured frequency of occurrence of small events can be used to estimate the frequency of occurrence of large events. For example, the recurrence interval for serious air pollution can be estimated from the frequency of smaller air pollution.

4. Conclusion

Based on DFA, power spectrum, and statistical analysis, we have identified long range correlation, noise, and scale-invariant of PM₁₀ concentrations measured at Chengdu. These statistics seem consistent with avalanche sizes in a running sandpile known to be SOC. In order to explain the origin of scale-invariant in PM₁₀ evolution, according to the characteristics of PM₁₀ evolution on calm weather, a simplified sandpile model for PM₁₀ pollution with a nondimensional formalism is put forward. The simulated result is consistent with the actual monitored data very well. The work supports the proposal that PM₁₀ evolution acts as a SOC process on calm weather. Far from being equilibrium forms, PM₁₀ pollutants will evolve into a nonequilibrium and critical state. At the critical state, long range correlation, noise, and scale-invariant of PM₁₀ concentrations will emerge from the dissipative system. This insight will inspire new research into the macroeffect of air pollution processes and improvement of modeling of PM₁₀ pollution.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (no. 41105118) and Hunan Provincial Natural Science Foundation of China (13JJB012).

References

A. Nel, “Air pollution-related illness: effects of particles,” Science, vol. 308, no. 5723, pp. 804–806, 2005.

A. B. Chelani, “Change detection using CUSUM and modified CUSUM method in air pollutant concentrations at traffic site in Delhi,” Stochastic Environmental Research and Risk Assessment, vol. 25, no. 6, pp. 827–834, 2011.

A. Zolghadri and F. Cazaurang, “Adaptive nonlinear state-space modelling for the prediction of daily mean PM₁₀ concentrations,” Environmental Modelling and Software, vol. 21, no. 6, pp. 885–894, 2006.

B. Sivakumar and R. Berndtsson, “Modeling and prediction of complex environmental systems,” Stochastic Environmental Research and Risk Assessment, vol. 23, no. 7, pp. 861–862, 2009.

C. Lee and S. Lin, “Chaos in air pollutant concentration (APC) time series,” Aerosol and Air Quality Research, vol. 8, no. 4, pp. 381–391, 2008.

S. R. Nagel, “Instabilities in a sandpile,” Reviews of Modern Physics, vol. 64, no. 1, pp. 321–325, 1992.

C.-K. Lee, D.-S. Ho, C.-C. Yu, and C.-C. Wang, “Fractal analysis of temporal variation of air pollutant concentration by box counting,” Environmental Modelling and Software, vol. 18, no. 3, pp. 243–251, 2003.

C. Varotsos, J. Ondov, and M. Efstathiou, “Scaling properties of air pollution in Athens, Greece and Baltimore, Maryland,” Atmospheric Environment, vol. 39, no. 22, pp. 4041–4047, 2005.

C. Lee, L. Juang, C. Wang et al., “Scaling characteristics in ozone concentration time series (OCTS),” Chemosphere, vol. 62, no. 6, pp. 934–946, 2006.

P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality: an explanation of the 1/f noise,” Physical Review Letters, vol. 59, no. 4, pp. 381–384, 1987.

K. Shi, C. Q. Liu, N. S. Ai, and X. H. Zhang, “Using three methods to investigate time-scaling properties in air pollution indexes time series,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 693–707, 2008.

K. Shi and C. Liu, “Self-organized criticality of air pollution,” Atmospheric Environment, vol. 43, no. 21, pp. 3301–3304, 2009.

D. Chang, Y. Song, and B. Liu, “Visibility trends in six megacities in China 1973–2007,” Atmospheric Research, vol. 94, no. 2, pp. 161–167, 2009.

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, “Mosaic organization of DNA nucleotides,” Physical Review E, vol. 49, no. 2, pp. 1685–1689, 1994.

B. D. Malamud and D. L. Turcotte, “Self-affine time series: measures of weak and strong persistence,” Journal of Statistical Planning and Inference, vol. 80, no. 1, pp. 173–196, 1999.

D. L. Turcotte and B. D. Malamud, “Landslides, forest fires, and earthquakes: examples of self-organized critical behavior,” Physica A, vol. 340, no. 4, pp. 580–589, 2004.

O. Peters and K. Christensen, “Rain viewed as relaxational events,” Journal of Hydrology, vol. 328, no. 1-2, pp. 46–55, 2006.