while f(αmax) decreasing. That means the number of event for the
largest index has increased, but that of the smallest has decreased.
Generally, The B value shows growth trend, the α~f(α) curve show
developing tendency from right to left. In 14 years, except for July
2004 to June 2005, the B are all negative, indicating that the events
of larger NO2 indices are almost dominated.
Figure 4d shows that the multifractal parameters for APIs in
each year are similar to that of PM10.
Now, it is demonstrated that some methods, such as iterated
function systems, are useful in time series predicting based on the
fractal scaling feature of the data. Thus, a natural step forward in
this line of research would be the use of the different fractal
characteristics of API to develop new predicting methods.
The differences among these four multifractal spectra indicate
that heterogeneity (disordered state) characterizes NO2 and PM10
indices dynamics owing to airflow, pollutant emissions and other
related factors; while the atmospheric buffer capability results in
SO2 indices being characterized by a dynamical change from
heterogeneity (disordered state) toward homogeneity (ordered
state). As to APIs, they are determined by air quality indices,
various pollutants and their effects on human health, so there are
more complex temporal evolution dynamics and physical
mechanism, therefore, APIs display the “richest” signal in structure
in these four series.
3.3. Cumulative frequency–size distribution
Previous studies indicate that the atmosphere share all
characteristic features of SOC systems, which is an excellent
example of an SOC process (Vattay and Harnos, 1994; Peters and
Christensen, 2006). The movements and transformation of air
pollutants occur in the atmosphere. Thus, the air pollutants should
also share these generic dynamical properties. From the complex
point of view, the power–law scaling and long–term memory can
be recognized as the footprint of SOC behaviors. Cumulative
frequency–size distributions associated with many natural systems
exhibit power law scaling. This is the typical “critical” dynamical
behavior found in the SOC systems (Matsoukas et al., 2000; Liu et
al., 2014b). A power law applied to a cumulative distribution has
the relation N=cr–λ, where N is the cumulative number of events
per unit time with size greater than or equal to the magnitude (r),
is the scaling exponent, c is a constant.
Figure 5 shows the number density (N) of air pollution events
(of SO2, NO2 and PM10), with size greater than or equal to some
pollution index value (r), respectively. Note the similarity to the
Gutenberg–Richter law in the earthquakes study (Turcotte and
Malamud, 2004). We found that these pollution indexes in
Shanghai exhibit different power law behavior. For NO2 and PM10
indexes, the plots exhibit curvature, showing obviously two
different scaling regions with the scaling exponent of 0.024 (0.031)
and 0.025 (0.008), respectively, while for SO2 indexes, only one
scaling region appears with the scaling exponent of 0.085.
Therefore, the results show that the SOC of SO2 series is fully
developed (continuous high persistence at all the scales that we
have analyzed), while the PM10 and NO2 series seem to be only
partially developed (reduced organization for NO2 series and
increased organization for PM10 at larger scales). We note that the
power law breaks down in smaller pollution index magnitude
regions. We think that low monitoring frequency of pollution
indexes series result in the low–size tail of the frequency
distribution. Shi et al. (2010, 2013) and Peters and Christensen
(2006) have found a similar phenomenon in water pH, PM10 and
rainfall, respectively.
Figure 5. The number density (N) of SO2, NO2 and PM10 pollution events,
with size greater than or equal to some pollution index value (r).
We think that the fluctuation of pollution index values do not
look very different from avalanches from the point of view of SOC.
In order to further explain it, we show an analogy between the air
pollutants and the sand–pile. In the complex atmospheric
environment, air pollution is affected by several variables driving
the change of APC, such as the movements and transformation of
air pollutants, precipitation and climate condition, interact and
correlate with each other and so on. All the variables driving the
change of air pollutants interact with each other and these inter–
relationships are complex, nonlinear. We define the measurement
value of air pollution as an avalanche event, and the magnitudes of
various pollution index values as avalanche sizes in a granular pile.
And the superposition of local APC represents the chain of forces in
the sand–pile. When the amounts of microscopic condensed air
pollutants reach some threshold magnitude, the air pollutant
masses can be transported on microscopic scales by diffusion or
convection. They reach a new location, where the local APC is
lower, and can be diluted. If the local APC in the neighborhood is
also high, the amount of condensed air pollutants masses will
increase. Once the system reaches some critical point by its own
internal tuning, any small fluctuation, in principle, can trigger a
chain reaction like the avalanches in the sand–pile. It is important
to note that the 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 the three kinds of pollutants evolution acts as a SOC process
on calm weather. And SOC is a useful framework to explain the
nonlinear evolution of the three kinds of pollutant concentrations.
Therefore, we may interpret the air pollution, which goes through
very large fluctuations, as consequences of SOC processes. So the
power–law scaling in these pollution indexes series is equivalent to
that of avalanche size evolution.
In Figure 5, we found two different scaling regions for NO2 and
PM10 series, and only one scaling region for SO2 series. For NO2 and
PM10 pollution, atmospheric motion and pollution discharge
change greatly. So we infer that the SOC of air pollutants (NO2 and
PM10) and SOC of airflow may independently affect the temporal
variation of NO2 and PM10 in different regimes respectively, and it
is these compound mechanisms that result in the appearance of
two different power–law regions for NO2 and PM10 pollution.
However, for SO2 pollution, owing to the effect of atmospheric
buffer capability, atmospheric motion and pollution discharge have
small fluctuation and the independent effect of airflow SOC
becomes small. Thus the SOC of NO2 and PM10 pollution play a
major role in the temporal variation of air pollution. At the critical
state, high persistence, and scale–invariant of the three kinds of
Liu