**7. Estimation of the c.d.f.**

10 Current Air Quality Issues

alternatively

the data,

2014

of the pollution levels.

concentrations.

June *PM*<sup>10</sup> *PM*<sup>10</sup> Est. Est. Est. Est. 2011 value value*<sup>a</sup>* Error*<sup>a</sup>* value*<sup>b</sup>* Error*<sup>b</sup>* 12th 22 20.970 -1.030 22.029 0.029 13th 22 22.821 0.821 22.940 0.940 14th 27 26.437 -0.563 23.178 -3.822 15th 25 24.867 -0.133 23.878 -1.122 16th 25 24.926 -0.074 24.473 -0.527 17th 30 25.466 -4.534 25.825 -4.175 Mean values 21.167 24.248 -0.919 23.720 -1.446 *a* Results obtained by using the periodic variogram model (5) *b* Results obtained by using the nonperiodic variogram model (6)

**Table 2.** Kriging estimations of a sequence of 6 missing values, from the 12th to the 17th of June 2011 and

been computed for the period ranging from the 1st to the 6th of January 2014, by using,

1. the available data, the variogram model (5) and the modified *GSLib* routine "KT3DP" which builds the searching neighborhood taking into account the periodicity exhibited by

2. the deseasonalized *PM*<sup>10</sup> observations, the variogram model (6) and the original *GSLib* routine "KT3D" which produces *PM*<sup>10</sup> predicted residuals at which the diurnal component of the day before has been added to obtain predictions of *PM*<sup>10</sup> daily

In Fig. 6, the time series of *PM*<sup>10</sup> daily concentrations measured from the 9th of December

PM observed values <sup>10</sup>

PM predicted values with periodic variogram model <sup>10</sup>

PM predicted values with nonperiodic variogram model <sup>10</sup>

9 11 13 15 17 19 21 23 25 27 29 31 2 4 6

days

**Figure 6.** Time plot of *PM*<sup>10</sup> predicted values and *PM*<sup>10</sup> daily concentrations (*µg*/*m*3), from the 1st to the 6th of January

2013 to the 6th of January 2014 is shown together with the predicted *PM*<sup>10</sup> values for the period ranging from the 1st to the 6th of January 2014. Note that the kriging procedure using the nonperiodic variogram model (6) related to *PM*<sup>10</sup> residuals has produced overestimates

Moreover, in Table 3 some results of the performance of the prediction procedure are presented. The mean value of the kriging standard error is lower if the periodic variogram model is used, compared with the case of kriging based on the nonperiodic variogram model.

corresponding errors for periodic and nonperiodic variogram models

0

50

100

PM ( / )

3 *g m*

> 10

150

200

For a given time series of *PM*10, it might be useful to estimate the probability that the variable under study exceeds a fixed limit, so that appropriate and prompt solutions might be adopted if necessary.

In this section, estimation of c.d.f. of *PM*<sup>10</sup> daily concentrations (*µg*/*m*3) has been conducted.

In particular, the c.d.f. of *PM*<sup>10</sup> at unsampled time points has been estimated by indicator kriging [17].

Six threshold values for *PM*<sup>10</sup> (22, 35, 50, 78, 98, and 108 *µg*/*m*3) have been properly chosen, and six indicator variables according to the fixed thresholds have been defined as follows


with *t* ∈ *T*. Note that indicator data are equal to 1 if the values of the variable under study are not greater than the considered threshold and they are equal to 0 otherwise. For each threshold, the temporal indicator variogram has been computed and modelled (Figs. 7, 8).

**Figure 7.** Indicator maps of *PM*<sup>10</sup> daily concentrations and their sample indicator variograms with the fitted models, for three threshold values. (a) Indicator map and (b) sample variogram indicator for the threshold *x*<sup>1</sup> = 22 *µg*/*m*3. (c) Indicator map and (d) sample variogram indicator for the threshold *x*<sup>2</sup> = 35 *µg*/*m*3. (e) Indicator map and (f) sample variogram indicator for the threshold *x*<sup>3</sup> = 50 *µg*/*m*<sup>3</sup>

12 Current Air Quality Issues

(a) *x*<sup>1</sup> = 22 *µg*/*m*<sup>3</sup> (b) *x*<sup>1</sup> = 22 *µg*/*m*<sup>3</sup>

(c) *x*<sup>2</sup> = 35 *µg*/*m*<sup>3</sup> (d) *x*<sup>2</sup> = 35 *µg*/*m*<sup>3</sup>

(e) *x*<sup>3</sup> = 50 *µg*/*m*<sup>3</sup> (f) *x*<sup>3</sup> = 50 *µg*/*m*<sup>3</sup>

**Figure 7.** Indicator maps of *PM*<sup>10</sup> daily concentrations and their sample indicator variograms with the fitted models, for three threshold values. (a) Indicator map and (b) sample variogram indicator for the threshold *x*<sup>1</sup> = 22 *µg*/*m*3. (c) Indicator map and (d) sample variogram indicator for the threshold *x*<sup>2</sup> = 35 *µg*/*m*3. (e) Indicator map and (f) sample

variogram indicator for the threshold *x*<sup>3</sup> = 50 *µg*/*m*<sup>3</sup>

**Figure 8.** Indicator maps of *PM*<sup>10</sup> daily concentrations and their sample indicator variograms with the fitted models, for three threshold values. (a) Indicator map and b) variogram for the threshold *x*<sup>4</sup> = 78 *µg*/*m*3. (c) Indicator map and (d) variogram for the threshold *x*<sup>5</sup> = 98 *µg*/*m*3. (e) Indicator map and (f) variogram for the threshold *x*<sup>6</sup> = 108 *µg*/*m*<sup>3</sup>

In particular the following models have been fitted


Thus, the c.d.f.s corresponding to six different unsampled time points, i.e. the days 1-6 of January 2014, have been estimated by using the "KT3DP" routine.

For each day of interest, the c.d.f. has been estimated by solving as many kriging systems as the number of threshold values considered. For each threshold, the corresponding indicator variogram model has been used for the kriging procedure.

Figure 9 shows the c.d.f.s estimated at days 1-6 of January 2014. It is clear that the probability of not exceeding a fixed threshold increases gradually from the 1st to the 6th of January 2014. For example, the estimated probability that *PM*<sup>10</sup> concentrations, on the 1st of January 2014, do not exceed 22 *µg*/*m*<sup>3</sup> is lower than the estimated probability on the 3rd or the 6th of the considered month.

**Figure 9.** C.d.f.s estimated for *PM*<sup>10</sup> daily concentrations (*µg*/*m*3) at days 1-6 of January 2014


**Table 4.** Estimated values for c.d.f. at days 1-6 of January 2014, for fixed thresholds

Moreover, note that it is almost sure that *PM*<sup>10</sup> concentrations do not exceed the cutoff 78 *µg*/*m*<sup>3</sup> at days the 5th and 6th (Table 4).

The probability that the variable under study doesn't exceed the threshold fixed by the National Law (50 *µg*/*m*3) is high (equal to 60%) for the last day of interest. In fact, the 6th of January is a non-working day (Epiphany) characterized by low traffic and low industrial emissions.

The local government could use these results in order to carry out environmental policies for the control of high levels of *PM*10, since it is well known that high concentrations of this pollutant are dangerous for the human health.

Indeed, the estimation of the c.d.f. is a very powerful tool since any action of environmental protection might be adopted in advance by taking into account the actual likelihood of dangerous *PM*<sup>10</sup> exceeding. For example, decisions about traffic limitation in high traffic urban area might be supported by the knowledge of the probability that a hazardous pollutant exceeds the level of attention.

## **8. Conclusions**

14 Current Air Quality Issues

considered month.

In particular the following models have been fitted

• *γI*<sup>1</sup> (*ht*; 22) = 0.185 *Exp*(|*ht*|; 10) + 0.023 *Cos*(|*ht*|; 365),

• *γI*<sup>2</sup> (*ht*; 35) = 0.15 *Exp*(|*ht*|; 10) + 0.07 *Cos*(|*ht*|; 365),

• *γI*<sup>3</sup> (*ht*; 50) = 0.102 *Exp*(|*ht*|; 10) + 0.036 *Cos*(|*ht*|; 365),

• *γI*<sup>4</sup> (*ht*; 78) = 0.039 *Exp*(|*ht*|; 10) + 0.0004 *Cos*(|*ht*|; 365),

• *γI*<sup>5</sup> (*ht*; 98) = 0.013 *Exp*(|*ht*|; 10) + 0.001 *Cos*(|*ht*|; 365),

• *γI*<sup>6</sup> (*ht*; 108) = 0.007 *Exp*(|*ht*|; 10) + 0.0004 *Cos*(|*ht*|; 365).

variogram model has been used for the kriging procedure.

January 2014, have been estimated by using the "KT3DP" routine.

**Figure 9.** C.d.f.s estimated for *PM*<sup>10</sup> daily concentrations (*µg*/*m*3) at days 1-6 of January 2014

Thus, the c.d.f.s corresponding to six different unsampled time points, i.e. the days 1-6 of

For each day of interest, the c.d.f. has been estimated by solving as many kriging systems as the number of threshold values considered. For each threshold, the corresponding indicator

Figure 9 shows the c.d.f.s estimated at days 1-6 of January 2014. It is clear that the probability of not exceeding a fixed threshold increases gradually from the 1st to the 6th of January 2014. For example, the estimated probability that *PM*<sup>10</sup> concentrations, on the 1st of January 2014, do not exceed 22 *µg*/*m*<sup>3</sup> is lower than the estimated probability on the 3rd or the 6th of the

> In this paper, *PM*<sup>10</sup> time series analysis, by using geostatistical techniques, has been discussed and the importance of appropriate tools of Geostatistics to study the temporal evolution of this environmental phenomena has been highlighted.

> The seasonal behavior of *PM*<sup>10</sup> levels has been evaluated through the variogram, that is the basic tool of Geostatistics. Moreover, estimation and prediction problems in the analysis of the time series of this pollutant, characterized by a periodic behavior, have been solved through kriging geostatistical techniques.

> The computational aspects have been performed through the use of "KT3D" for the observed values and "KT3DP" for the residuals obtained after removing the periodic component.

> Finally, the indicator approach and its capability for assessing the probability that *PM*<sup>10</sup> exceeds the specific threshold values have been demonstrated.

> The results obtained in this paper by applying geostatistical techniques to analyze *PM*<sup>10</sup> time series could be useful to support national policies for environmental and health protection. Governments' activity must be oriented to control that the concentrations of the analyzed pollutant don't exceed specific thresholds according to national or international directives, since it has been demonstrated that particulate matter is dangerous for the human health.
