Modeling Based on the Analysis of Interval Data of Atmospheric Air Pollution Processes with Nitrogen Dioxide due to the Spread of Vehicle Exhaust Gases

Abstract

The article deals with the issue of modeling taking into consideration nitrogen dioxide pollution of the atmospheric surface layer caused by vehicle exhaust gases. The interval data analysis methods were suggested. The method of identifying the mathematical model of the distribution of nitrogen dioxide as an atmospheric air pollutant based on the analysis of data with known measurement errors was proposed and grounded for the first time. The obtained mathematical model in the form of a difference equation is characterized by the guaranteed accuracy of forecasting nitrogen dioxide concentrations in a specified area of the city. It also adequately takes into account traffic changes which significantly reduces the costs of environmental control and monitoring. The proposed new model identification method is more effective in terms of computational time complexity compared to the known method and it is based on taking into account measurement errors which in the final case provides predictive properties of the model with guaranteed accuracy.

1. Introduction

The problem of nitrogen dioxide pollution of the atmospheric surface layer in large cities is associated with the intensive of motor vehicles, the exhaust gases of which contain this atmospheric pollutant [1,2,3,4,5]. The concentration of this atmospheric pollutant in the exhaust gases from vehicles is quite high. In addition, nitrogen dioxide is very harmful for human because it belongs to the second type of danger. According to statistic data, traffic pollution in large cities is from 70 to 90 percent among all pollution [6,7,8,9]. Therefore, monitoring of this substance is a very relevant topic [9,10,11,12,13,14,15].

Air pollution monitoring processes are based on an extensive system of stations measuring the concentrations of the main atmospheric pollutants, analyzing these data and establishing the main indicators of atmospheric air quality [12,13,14,15,16]. The main focus of air pollution monitoring by European institutions is to the Air Quality Index as a research and communication tool for reporting the current state of air pollution to the public. The air quality index allows air quality monitoring in real time on the territory of those countries which implemented the data transfer protocols. The index uses data from more than 2000 air quality monitoring stations across the whole Europe that belong to the Copernicus atmospheric monitoring network [17] and evaluates air quality according to four indicators: solid particles, ground-level ozone, nitrogen dioxide and sulfur dioxide, each of which is evaluated accordingly to the standards approved by the European Union Directives. The Air Quality Index updates data every 6 h but it happens that data from the analyzers is not received in time. To solve this problem, the European Environment Agency uses an approximation method to model the necessary data. This approximation method contains differences for the indicator which is measured. The so-called differential method is used for nitrogen dioxide and solid particles where the value is obtained by simulating the Copernicus system with the addition or subtraction of a correction difference, which is the averaging of the difference between previous measurements and system-simulated values obtained at the same hour.

Recently, in Ukraine, an increasing number of public organizations and commercial projects appear which raise the issue of atmospheric air quality monitoring and the creation of separate IT products to improve air quality information. One of the currently active projects in Ukraine is the EcoInfo project, the purpose of which is to inform the public about the state of air quality in the location of a site user. However, despite the technical and visual attractiveness of this project, it does not show the methodology which is used to measure air quality. Therefore, such a system cannot be called a monitoring system and has nothing to do with air quality at the indicated points. The experience of the environmental organization “EcoCitizens”, namely their crowdsourcing project monitoring of the environment is no less interesting, the main idea of which was the accumulation and processing of a large array of environmental information using monitoring devices based on the Arduino or Raspberry Pi platforms. At the same time, it is worth noting that the development of air quality monitoring systems is just beginning at the regional level.

The website of the Ministry of Ecology and Natural Resources of Ukraine contains a list of software products in the field of air protection [18]. However, taking into consideration the description of their purpose, it becomes clear that they are intended for conducting calculations of air pollution and not for the monitoring process.

However, monitoring system deployment with a large number of sensors in high-traffic areas is a high-value task. Instead of it, a small number of sensors or mobile systems are typically used to selectively measure pollutant concentrations. Besides, mathematical models are used to quantify the concentrations of these gases with a given accuracy at any point of the atmospheric surface layer. Mostly these types of mathematical models are partial differential equations or their differential analogs [19,20,21,22]. In mathematical models in the form of differential equations, it is necessary, to take into account a significant number of factors, for example, the heterogeneity of the environment for the transport of air pollutants, physical processes that affect the transport of pollutants such as ventilation, thermal air flows, etc.

Mathematical models of the distribution of concentrations of atmospheric pollutants are difficult to set up because they are characterized by large values of errors in long-term prediction and require complex computational decision procedures. Therefore, it is advisable to use their difference analogs, the main idea of which is to determine a certain difference scheme by matching it with experimental data. To build mathematical models in the form of difference equations, it is necessary to solve the problems of their structural and parametric identification. The stated above methods are focused on cases of accurate presentation of observation results and do not take into account uncertainties in the form of errors, the values of which are often limited. In these cases, the methods of interval data analysis that is data with errors limited in value should be used. As a result, we obtain interval discrete dynamic models which describe the dynamics of pollution.

At the same time, setting the value of the measured concentrations, modern measuring systems simultaneously allow establishing the range of these measurements, based on the possible error bounds. Therefore, the results of pollutant concentration measurements can be presented in the form of intervals with numerically determined lower and upper bounds. This representation of variables is called an interval, and the analysis of this type of data is used to build mathematical models in the form of discrete equations [23,24,25,26,27], which specify the distribution of the pollutant in the atmospheric surface layer. It should be noted that the models of this type are so-called interval discrete models (IDM) [25,26,27,28].

As it is known, IDM construction procedures [25,26] consist of two aspects, namely structural and parametric identification. Both of these aspects in terms of interval data analysis are NP-complex computational problems. Therefore, the methods of evolutionary calculations should be used to solve NP-complex computational problems [29,30,31].

2. Materials and Methods

2.1. Mathematical Model of the Nitrogen Dioxide Distribution in the Atmospheric Surface Layer

As it was stated above the “classical” models of the distribution of air pollutants in the atmosphere are differential equations with partial derivatives [19,20,21,22]. Such equations are good descriptions of the transport of parts of chemical substances in the air. However, to set up them, it is necessary to measure diffusion coefficients in spatial coordinates, which is difficult because the environment is heterogeneous. Furthermore, it is necessary to use numerical techniques for such models [23,24,25,26,27,28]. That is why, the authors of this article suggest the usage of discrete analogs of the stated differential equations which are actually solutions and methods of building these discrete equations on the base of experimental data.

Let us formulate the assumptions which are based on the construction of a mathematical model of the distribution of nitrogen dioxide in the atmospheric surface layer.

Assumption 1.

We present the results of nitrogen dioxide concentration measurements in the form of intervals of possible values of this concentration:

$$[z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}],i=0,\dots ,I,j=0,\dots ,J,h=0,\dots ,H,k=0,\dots ,K$$

(1)

where $[z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}]$—respectively, the lower and upper bounds of the interval of possible nitrogen dioxide concentrations at a point with discretely specified spatial coordinates $i=0,\dots ,I,j=0,\dots ,J,h=0,\dots ,H$ and time discrete $k=0,\dots ,K$.

It should be noted that received interval values of the measured concentration are determined by absolute ${\Delta }_{i,j,h,k}$ or relative ${\delta }_{i,j,h,k}$ metrological and methodical errors of the measuring system. In this case, the lower and upper bound of the interval of possible values of the concentration of nitrogen dioxide is calculated by the following formula:

$$z_{i,j,h,k}^{-}=z_{i,j,h,k}^{}-z_{i,j,h,k}^{}\cdot {\delta }_{i,j,h,k}^{}, i=0,\dots ,I, j=0,\dots ,J, h=0,\dots ,H, k=0,\dots ,K$$

(2)

$$z_{i,j,h,k}^{-}=z_{i,j,h,k}^{}+z_{i,j,h,k}^{}\cdot {\delta }_{i,j,h,k}^{}, i=0,\dots ,I, j=0,\dots ,J, h=0,\dots ,H, k=0,\dots ,K$$

(3)

where $z_{i,j,h,k}^{}$—the measured value of nitrogen dioxide concentration.

In general, measurement errors may depend on the point in space and time of measurement. In our case, we assume that these errors are constant for all measurements. Mainly for the measurement of concentrations of chemical substances in the atmosphere, such relative errors depending on measurement methods and measuring device may vary from 5% to 50%.

Assumption 2.

As it was stated, to model the distribution of nitrogen dioxide concentration in the atmospheric surface layer, we use a discrete model in the following general form [23,24]:

$$\begin{array}{l}{v}_{i,j,h,k}={\overrightarrow{f}}^T(v_{i-d,j-d,h-d,k-d},v_{i-d+1,j-d,h-d,k-d,\dots ,}{v}_{j-d+1,d,d,d,\dots ,}{v}_{i-1,j-1,h-1,k-1,\dots ,} \\ v_{i,j,h-1,k})\cdot \overrightarrow{g}, i=d,\dots ,I, j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K\end{array}$$

(4)

where $v_{i,j,h,k}$—the calculated concentration of nitrogen dioxide at a point with discrete spatial coordinates $i=d,\dots ,I, j=d,\dots ,J, h=d,\dots ,H$ and at discrete time intervals $k=d,\dots ,K$. $d$—the order of DM (4), which is known to be equivalent to the order of a differential equation—the analog of a discrete model; $\overrightarrow{g}$—the vector of unknown parameters of the model; ${\overrightarrow{f}}^T(•)$—a vector of basic functions, which are used to transform the values of the modeled characteristic at discrete points in space and for certain discrete periods.

The stated above elements form structural elements

$${\overrightarrow{f}}^T(v_{i-d,j-d,h-d,k-d},v_{i-d+1,j-d,h-d,k-d,\dots ,}{v}_{j-d+1,d,d,d,\dots ,}{v}_{i-1,j-1,h-1,k-1,\dots ,} v_{i,j,h-1,k})$$

of the discrete model.

Let us indicate the vector to simplify further consideration:

$$\overrightarrow{V}={(v_{i-d,j-d,h-d,k-d},v_{i-d+1,j-d,h-d,k-d},\dots ,v_{j-d+1,d,d,d},\dots ,v_{i-1,j-1,h-1,k-1},\dots ,v_{i,j,h-1,k})}^T$$

Assumption 3.

To identify DM (4) based on the interval data (1), we use the criterion of ensuring the accuracy of the mathematical model within the accuracy of the measurement experiment, that is the following conditions:

$$[{\widehat{v}}_{i,j,h,k}^{-};{\widehat{v}}_{i,j,h,k}^{+}]\subset [z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}] \forall i=0,\dots ,I,\forall j=0,\dots ,J, \forall h=0,\dots ,H, \forall k=0,\dots ,K$$

(5)

where $[{\widehat{v}}_{i,j,h,k}^{-};{\widehat{v}}_{i,j,h,k}^{+}]$—the interval estimates of the modeling concentration of nitrogen dioxide based on model (4).

These conditions define the accuracy of the mathematical model despite the case when scholastic models of such type are examined. As it is known, [23,24,25,26,27], such an approach does not require statistical research during the verification of the predictive properties of the mathematical model.

Let us introduce the mathematical operator $Q({\lambda }_s)$, which generates a mathematical model based on the transformation of its structure ${\lambda }_s=\{f_1^s(\overrightarrow{V})\cdot g_1^s; f_2^s(\overrightarrow{V})\cdot g_2^s;\dots ;f_{m_s}^s(\overrightarrow{V})\cdot g_{m_s}^s\}$ into an additive convolution, i.e.,:

$$Q({\lambda }_s)=f_1^s(\overrightarrow{V})\cdot g_1^s+f_2^s(\overrightarrow{V})\cdot g_2^s+\dots +f_{m_s}^s(\overrightarrow{V})\cdot g_{m_s}^s$$

(6)

In Equation (6), s—means a certain set of structural elements, on the base of which we build the s—model in the form of (6), that is, as a convolution of this set of elements. The notation ${\lambda }_s$ means s is that structure.

To use the “classical” model of the distribution of air pollutants in the atmosphere in the form of the differential equation, it is necessary to calculate it by specifying boundary conditions. In the case of the use of a discrete equation, it is necessary to specify initial conditions. In our case, taking into account available interval data (1), the initial conditions should be specified in such way:

$$\begin{array}{l}[{\widehat{v}}_{0,0,0,0}^{-};{\widehat{v}}_{0,0,0,0}^{+}]\subseteq [z_{0,0,0,0}^{-};z_{0,0,0,0}^{+}],\dots ,[{\widehat{v}}_{1,d-1,1,0}^{-};{\widehat{v}}_{1,d-1,1,0}^{+}]\subseteq [z_{1,d-1,1,0}^{-};z_{1,d-1,1,0}^{+}],\dots ,\\ [{\widehat{v}}_{d-1,d-1,d-1,d-1}^{-};{\widehat{v}}_{d-1,d-1,d-1,d-1}^{+}]\subseteq [z_{d-1,d-1,d-1,d-1}^{-};z_{d-1,d-1,d-1,d-1}^{+}].\end{array}$$

(7)

The stated above conditions (7) mean that depending on the order $d$ of the difference scheme it is necessary to conduct a series of measurements in a certain point of space and in a certain time period to set a difference scheme for concrete environmental conditions. It should be noted that these conditions fully characterize exclusively measured values of the concentration of a chemical substance in the atmosphere at certain points. For example, traffic may be characterized by measured values of nitrogen dioxide concentration. We assume that the increased concentration of nitrogen dioxide in the air at the measurement points near the road surface is associated with an increase in the intensity of motor vehicle traffic.

Then, using the operator $Q({\lambda }_s)$ and taking into account the initial conditions (7), we obtain discrete models—candidates evaluated in the process of structural identification in the form of an interval discrete model (IDM):

$$[{\stackrel{⌢}{v}}_{i,j,h,k}({\lambda }_s,[\overrightarrow{\widehat{V}}])]=[f_1^s([\overrightarrow{\widehat{V}}])]\cdot {\widehat{g}}_1^s+[f_2^s([\overrightarrow{\widehat{V}}])]\cdot {\widehat{g}}_2^s+\dots +[f_{m_s}^s([\overrightarrow{\widehat{V}}])]\cdot {\widehat{g}}_{m_s}^s$$

(8)

where

$$\begin{array}{c}{f}_i^s([\overrightarrow{\widehat{V}}])=f_i^s([{\widehat{v}}_{i-d,j-d,h-d,k-d}],\dots ,[{\widehat{v}}_{i,j-d,h,k}],\dots ,[{\widehat{v}}_{i,j,h-1,k}],{\overrightarrow{u}}_{i,j,h,0},\dots ,{\overrightarrow{u}}_{i,j,h,k}) \_\\ [\overrightarrow{\widehat{V}}]=[{\widehat{v}}_{i-d,j-d,h-d,k-d}],\dots ,[{\widehat{v}}_{i,j-d,h,k}],\dots ,[{\widehat{v}}_{i,j,h-1,k}]\end{array}$$

an interval vector with components that mean interval estimates of the modeled characteristic given in the form of initial conditions or predicted at previous points in space and time.

Structural identification gives a possibility to define a general form of IDM that is searching difference scheme. However, to use this difference scheme, it is necessary to calculate its parameters on the base of the experimental data presented in the form of an interval (1).

The vector of known parameters of the IDM (8) ${\widehat{\overrightarrow{g}}}^s$ which is obtained in the form of a solution of the interval system of nonlinear algebraic equations (ISNAE), fulfilling the conditions (4) [24]:

$$\{\begin{array}{c}\begin{array}{c}\begin{array}{l}[{\widehat{v}}_{0,0,0,0}^{-};{\widehat{v}}_{0,0,0,0}^{+}]\subseteq [z_{0,0,0,0}^{-};z_{0,0,0,0}^{+}],\dots ,\\ [{\widehat{v}}_{d-1,d-1,d-1,d-1}^{-};{\widehat{v}}_{d-1,d-1,d-1,d-1}^{+}]\subseteq [z_{d-1,d-1,d-1,d-1}^{-};z_{d-1,d-1,d-1,d-1}^{+}];\end{array}\\ \\ z_{i,j,h,k}^{-}\le [f_1{}^s(\overrightarrow{\widehat{V}})]\cdot {\widehat{g}}_1^s+[f_2{}^s(\overrightarrow{\widehat{V}})]\cdot {\widehat{g}}_2^s+\dots +[f_{m_s}^s(\overrightarrow{\widehat{V}})]\cdot {\widehat{g}}_{m_s}^s\le z_{i,j,h,k}^{+};\end{array}\\ i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K.\end{array}$$

(9)

Therefore, after the transformations, we obtained the general form of the problem of parametric identification of the IDM in the form of ISNAE for each s—separate candidate model. If ISNAE (9) turns out to be incompatible with the current structure ${\lambda }_s$, then we build a new structure, and on its basis—a new ISNAE (9) and again check its compatibility. Compatibility of ISNAE (9) means that the intervals of values of $[{\stackrel{⌢}{v}}_{i,j,h,k}({\lambda }_s,[\overrightarrow{\widehat{V}}])]$ the predicted characteristic in all discrete belong to the intervals $[z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}], i=0,\dots ,I, j=0,\dots ,J, h=0,\dots ,H, k=0,\dots ,K$ obtained from the results of observations. It is worth noting that guaranteed predictive properties of IDM are insured in such a way unlike stochastic analogs of such mathematical models.

Then we shall consider the mathematical formulation of the construction problem of IDM.

As it is known [24,25], the interval data analysis procedures, in this case, involve finding at least one solution ${\widehat{\overrightarrow{g}}}_l^s$ ISNAE (9) or its interval estimation. Such procedures are interactive and consist of some evolution of the values of the initially specified or subsequently calculated components of the current vector ${\widehat{\overrightarrow{g}}}_l^s$ IDM parameters. At the same time, the “quality” indicator is used to assess the quality of the evolution $\delta ({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)$. The function $\delta ({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)$ quantitatively reflects the degree of closeness of the solution to the solution that transforms ISNAE (9) into a compatible system. The closer the predicted corridor, built based on the given proximity of the parameter vector to the experimental one, the higher the quality of the proximity. In the works [23,24,25,26,27], the expression for this function is substantiated based on the following considerations. If the predicted and experimental intervals do not cross with each other, the closeness quality is defined as the difference between the centers of the most distant intervals:

$$\begin{array}{l}\delta ({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)=\underset{i=d,\dots ,I,j=d,\dots ,J,h=d,\dots ,H,k=d,\dots ,K}{\max }\{|mid(f_1^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l1}^s+f_2^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l2}^s+\dots \\ +f_m^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{lm}^s)-mid([z_{i,j,h,k}])|\}\end{array}$$

(10)

if

$$[{\widehat{v}}_{i,j,h,k}]\cap [z_{i,j,h,k}]=Ø, \exists i=d,\dots ,I,\exists j=d,\dots ,J,\exists h=d,\dots ,H,\exists k=d,\dots ,K$$

If there is a crossing of the predicted and experimental corridors, then the quality is defined as the smallest value of the intersection width:

$$\begin{array}{l}\delta ({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)=\underset{i=d,\dots ,I,j=d,\dots ,J,h=d,\dots ,H,k=d,\dots ,K}{\max }\{wid(f_1^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l1}^s+f_2^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l2}^s+\dots \\ +f_m^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{lm}^s)-wid(f_1^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l1}^s+\\ +f_2^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{l2}^s+\dots +f_m^s([\overrightarrow{\widehat{V}}])\cdot {\widehat{g}}_{lm}^s)\cap [z_{i,j,h,k}]\}\end{array}$$

(11)

if

$$[{\widehat{v}}_{i,j,h,k}]\cap [z_{i,j,h,k}]\ne Ø, \forall i=d,\dots ,I,\forall j=d,\dots ,J,\forall h=d,\dots ,H,\forall k=d,\dots ,K$$

where $mid(•),wid(•)$. are operations for determining the center and width of the intervals, respectively.

Therefore, if in the process of identification of the IDM its structure is determined in some way ${\lambda }_s$, then for this fixed structure the task of parametric identification of the IDM is to find a solution for the following optimization problem:

$$\delta ({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)\stackrel{{\widehat{\overrightarrow{g}}}_l^s}{\to }\min , {\widehat{g}}_{li}\in [g_{li}^{low},g_{li}^{up}], i=1,\dots ,n, l=1,\dots ,S$$

(12)

where $g_{li}^{low},g_{li}^{up}$ the lowest and highest value of each IDM parameter.

However, this problem is very difficult from a computational point of view. The complexity of the problem is connected with the complexity of the objective function, which for various conditions requires calculations at each iteration of the method according to the expression (10) or (11). It is worth noting that the objective function (10) or (11) is given algorithmically, it is discrete and it does not have analytical representation, which complicates the calculation and analysis of the optimization problem (12). In addition, the algorithm for solving the optimization problem (12) becomes more complicated even for simple cases of the mathematical model in the form of difference operator. At the same time, the criterion for minimization of standard deviation is used in the vast majority of problems for both structural and parametric identification of mathematical models. On the other hand, mainly even setting the interval problem statement in the sense of solving the ISNAE (9), to find at least one difference operator is sufficient. Then conditions (7) can be rewritten in the following form:

$$\begin{array}{l}{\widehat{\overline{v}}}_{0,0,0,0}^{}\in [z_{0,0,0,0}^{-};z_{0,0,0,0}^{+}],\dots ,\\ {\widehat{\overline{v}}}_{d-1,d-1,d-1,d-1}^{}\in [z_{d-1,d-1,d-1,d-1}^{-};z_{d-1,d-1,d-1,d-1}^{+}]\end{array}$$

(13)

where ${\widehat{\overline{v}}}_{0,0,0,0}^{},\dots ,{\widehat{\overline{v}}}_{d-1,d-1,d-1,d-1}^{}$ are some initial values.

Then ISNAE (9) is

$$\{\begin{array}{c}\begin{array}{c}\begin{array}{l}{\widehat{\overline{v}}}_{0,0,0,0}^{}\in [z_{0,0,0,0}^{-};z_{0,0,0,0}^{+}],\dots ,\\ {\widehat{\overline{v}}}_{d-1,d-1,d-1,d-1}^{}\in [z_{d-1,d-1,d-1,d-1}^{-};z_{d-1,d-1,d-1,d-1}^{+}];\end{array}\\ \\ z_{i,j,h,k}^{-}\le f_1{}^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_1^s+f_2{}^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_2^s+\dots +f_{m_s}^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_{m_s}^s\le z_{i,j,h,k}^{+};\end{array}\\ i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K.\end{array}$$

(14)

In this case one of the interval models is

$${\widehat{\overline{v}}}_{i,j,h,k}({\lambda }_s,\overrightarrow{\widehat{V}})=f_1^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_1^s+f_2^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_2^s+f_{m_s}^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_{m_s}^s$$

(15)

Statement 1.

If at least one solution of ISNAE (9) exists, then it is equivalent to the solution to the following problem:

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s,{\alpha }_{i,j,h,k})\stackrel{{\widehat{\overrightarrow{g}}}_l^s,{\alpha }_{i,j,h,k}^{}}{\to }\min ,{\alpha }_{i,j,h,k}\in [0,1],\\ i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K\end{array}$$

(16)

where

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)={\displaystyle \sum _{i=d}^I{\displaystyle \sum _{j=d}^J{\displaystyle \sum _{h=d}^H{\displaystyle \sum _{k=d}^K(}}}}{f}_1^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_{l1}^s+f_2^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_{l2}^s+\dots \\ +f_m^s(\overrightarrow{\widehat{V}})\cdot {\widehat{g}}_{lm}^s-({\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+}){)}^2\end{array}$$

(17)

In Equation (17) $\overrightarrow{\widehat{V}}$—vector with components that means given in the form of initial conditions, or predicted at previous points in space and time interval estimates of the simulated characteristic.

Statement 2.

At least one solution to the problem (16) satisfies the condition

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s,{\alpha }_{i,j,h,k})=0,\\ i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K\end{array}$$

(18)

As we can see, Statement 2 is a sufficient condition for Statement 1. Therefore, proving Statement 2, we will simultaneously show the validity of a very important Statement 1.

Proof of Statement 2.

Let us rewrite Equation (17) taking into account Equation (15) for ${\widehat{\overline{v}}}_{i,j,h,k}({\lambda }_s,\overrightarrow{\widehat{V}})$:

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)={\displaystyle \sum _{i=d}^I{\displaystyle \sum _{j=d}^J{\displaystyle \sum _{h=d}^H{\displaystyle \sum _{k=d}^K(}}}}{\widehat{\overline{v}}}_{i,j,h,k}({\lambda }_s,\overrightarrow{\widehat{V}})-\\ ({\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+}){)}^2\end{array}$$

(19)

If the solution of ISNAE (14) exists, then it means that equivalent conditions (5) are fulfilled, which in this case is:

$$\begin{array}{l}{\widehat{\overline{v}}}_{i,j,h,k}^{}\in [z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}] \\ \forall i=0,\dots ,I,\forall j=0,\dots ,J, \forall h=0,\dots ,H, \forall k=0,\dots ,K\end{array}$$

(20)

Then:

$$\begin{array}{l}{\widehat{\overline{v}}}_{i,j,h,k}^{}={\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+}, {\alpha }_{i,j,h,k}\in [0,1]\\ i=0,\dots ,I,j=0,\dots ,J, h=0,\dots ,H, k=0,\dots ,K\end{array}$$

(21)

since ${\widehat{\overline{v}}}_{i,j,h,k}^{}$ is always a linear combination of bounds of the intervals $[z_{i,j,h,k}^{-};z_{i,j,h,k}^{+}],i=0,\dots ,I,j=0,\dots ,J, h=0,\dots ,H, k=0,\dots ,K$.

Partial conditions arise from these conditions (21):

$$\begin{array}{l}{\widehat{\overline{v}}}_{i,j,h,k}({\lambda }_s,\overrightarrow{\widehat{V}})={\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+}, {\alpha }_{i,j,h,k}\in [0,1]\\ i=d,\dots ,I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K\end{array}$$

(22)

Substituting Equation (19) instead ${\widehat{\overline{v}}}_{i,j,h,k}({\lambda }_s,\overrightarrow{\widehat{V}})$ of Equation (21), we will get:

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s)={\displaystyle \sum _{i=d}^I{\displaystyle \sum _{j=d}^J{\displaystyle \sum _{h=d}^H{\displaystyle \sum _{k=d}^K({\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+})}}}}-\\ ({\alpha }_{i,j,h,k}\cdot z_{i,j,h,k}^{-}+(1-{\alpha }_{i,j,h,k})\cdot z_{i,j,h,k}^{+}){)}^2=0\end{array}$$

(23)

It confirms the truth of Statement 2 and accordingly Statement 1. □

The advantage of using the objective function in the form (17) is the fact that it has analytical representation and it is quadratic at least relatively ${\alpha }_{i,j,h,k}\in [0,1],i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K$.

Thus, if in the process of identifying the IDM in some way determined, its structure ${\lambda }_s$ then for this fixed structure the task of parametric identification of IDM is to find the solution of such an optimization problem:

$$\begin{array}{l}\mathrm{\Lambda }({\lambda }_s,{\widehat{\overrightarrow{g}}}_l^s,{\alpha }_{i,j,h,k})\stackrel{{\widehat{\overrightarrow{g}}}_l^s,{\alpha }_{i,j,h,k}^{}}{\to }\min ,{\alpha }_{i,j,h,k}\in [0,1],\\ i=d,\dots I,j=d,\dots ,J, h=d,\dots ,H, k=d,\dots ,K,\\ {\widehat{g}}_{li}\in [g_{li}^{low},g_{li}^{up}], i=1,\dots ,n, l=1,\dots ,S\end{array}$$

(24)

2.2. Characteristics of Parametric Identification IDM Method

The given problem is an NP-complex problem, since it cannot be solved in a predetermined number of iterations. General approaches to the solution of problems of this type are heuristic methods, in particular random search methods. However, recently, methods based on swarm intelligence became widely used as one of the most effective tools for the solution of complex optimization tasks using elements of self-organization and adaptation. In swarm algorithms, the interaction between individuals of the population occurs locally which ensures the detection of local extrema and, accordingly, the faster finding of the extremum of the global objective function [32,33,34,35,36,37,38]. In addition, another advantage of swarm methods and algorithms is the smaller number of algorithm parameters. The investigation of known swarm algorithms and other algorithms of random search, for instance, genetic ones is presented in the scientific work [24]. While the objective function (19) in optimization problem (24) because of its non-linear and recursive character, has many extreme properties, and also this problem can not be solved by a finite amount of steps then optimization problem (24) is NP computationally complex. Therefore, the comparison with other heuristic methods, for instance, presented in the works [34,35,37] requires computing experiments beyond the framework of this work. However, the analysis of the known swarm algorithms, presented in the work [24], showed that the artificial bee colony algorithm is the most appropriate in the context of solving the problem of parametric IDM identification. Moreover, none of the known computing algorithms, besides the artificial bee colony algorithm, does not solve the problem of the structural identification of IDM, where a general view of difference scheme are solutions [24,25,26]. Therefore, in general, computing scheme of structural and parameter identification of IDM, the artificial bee colony algorithm has advantages over other heuristic computing algorithms of complex optimization problems.

Taking into consideration the stated above information, the method of identifying the mathematical model of the distribution of atmospheric pollutants of motor vehicles will be built on the principles of behavioral models of bee colonies [24].

The main phases of the method are described below.

Initialization phase. Initialization of the initial ${\stackrel{⌢}{\overrightarrow{g}}}_l$ estimates of the vector of IDDM parameter values from $n$ variables $g_{li},i=1,\dots ,n$, which should be optimized to minimize the objective function in the problem (24):

$$g_{li}=g_{li}^{low}+rand(0,1)\ast (g_{li}^{up}-g_{li}^{low}),i=1,\dots ,n,l=1,\dots ,S$$

(25)

where $rand(0,1)$—a random real number from the range [0, 1], which is generated according to the law of uniform distribution.

Also, at this stage, the main parameters of the search method are set, namely LIMIT; S; $[I_{\min };I_{\max }]$; current iteration number (mcn = 0), total number of iterations (MCN).

Worker bees phase. In the context of the optimization problem (24), the search for new solutions, that is the calculation of current $g_{li}^{mcn}$ estimates of the vector of IDDM parameter values is carried out using the equation:

$$g_{li}^{mcn}=g_{li}^{}+{\Phi }_{li}^{}\ast (g_{li}^{}-g_{pi}^{}),i=1,\dots ,n,p\ne l=1,\dots ,S$$

(26)

where $g_{li}^{mcn}$ estimates of the vector of IDDM parameter values at the current iteration; ${\overrightarrow{g}}_p^{}$ a randomly selected vector of parameter values from $p\ne l=1,\dots ,S$; $i=1,\dots ,n$ the index of a randomly selected parameter; ${\Phi }_{li}^{}$ a random number from the range [–1,1].

After calculation the current estimates of the vector of IDDM parameter values ${\overrightarrow{g}}_l^{mcn}$ reassignment of estimates of parameter value vectors between previously defined and current ones is carried out using (17):

$${\stackrel{⌢}{\overrightarrow{g}}}_l=\left\{{\stackrel{⌢}{\overrightarrow{g}}}_l, if \delta ({\stackrel{⌢}{\overrightarrow{g}}}_l)\le \delta (\left({\stackrel{⌢}{\overrightarrow{g}}}_l^{mcn}\right)\right\} \mathrm{or} {\stackrel{⌢}{\overrightarrow{g}}}_{}^{}{}_l=\left\{{\stackrel{⌢}{\overrightarrow{g}}}_{}^{mcn}{}_l,if \delta ({\stackrel{⌢}{\overrightarrow{g}}}_l)>\delta (\left({\stackrel{⌢}{\overrightarrow{g}}}_l^{mcn}\right)\right\}.$$

(27)

Explorer bees phase. In the context of the optimization problem (24), at this stage, there is an opportunity to investigate in more detail the area of the solution space of this problem, namely to determine such vectors of parameter values, around which it is necessary to conduct a more detailed study of the objective function. To do this, we calculate the probability of feasibility of the study for each of the previously established vectors of parameter values (with the condition to prior normalization of the values of the objective function $\delta ({\stackrel{⌢}{\overrightarrow{g}}}_l)$ of the interval [0, 1]):

$$P_l^{}=\frac{1-\delta ({\stackrel{⌢}{\overrightarrow{g}}}_l)}{{\displaystyle \sum _{l=1}^S(1-\delta ({\stackrel{⌢}{\overrightarrow{g}}}_l))}}$$

(28)

Based on computing probabilities, we choose a neighborhood of a certain vector of parameter values ${\stackrel{⌢}{\overrightarrow{g}}}_l$, that contains $m_l$ points whose coordinates are calculated according to Equation (26). At the same time, the number of points around each vector is calculated according to the following formula: $m_l=P_l\cdot S$.

Next, the objective function is calculated according to Equation (17) and a pair is selected between existing and current vectors of parameter values based on Equation (27) for each newly created vector.

Scout bees phase. In this phase, the new vectors of model parameter values are randomly generated. It means that the parameters of the mathematical model of the distribution of automobile atmospheric pollutants are calculated using Equation (25) at this stage.

In the context of the artificial bee colony algorithm, it means exhausting the current sources of nectar (vectors of parameter values) when the limit counter exceeds the limit value LIMIT, and in the context of solving the parametric identification problem (24), this mechanism means the possibility of avoiding local minima.

3. Results and Discussion

As it follows from the stated above, IDM built on the base of the interval data received from experimental measurements is used for the modeling of chemical substances in the atmosphere. An example is the distribution of nitrogen dioxide concentration caused by atmospheric pollutants of automobile exhaust gases in the given central area of the city of Ternopil on the ring road with S. Krushelnytska street, Zbarazka street, Brodivska street and Halytska street joined. This area is characterized by changeable traffic intensity at different time intervals. It should be noted that the model does not take into account the time factor directly. It means that it is necessary to assess spacial distribution of air pollution on a specified area, to make control measurements entering new initial conditions in the form (5) for the mathematical model. Therefore, other factors, for instance traffic intensity will be taken into account under new initial conditions.

Directly, information measuring aeromobile complex on the base of Sniffer4D Hyper-local Air Quality Analyzer with installed sensors for measuring nitrogen dioxide and metereological characteristics was used for identification (setting up) mathematical model in the form of IDM [23]. It is worth noting that the relative error for the installed nitrogen dioxide sensor was 15%. The module was installed on mobile system board—quadrocopter DJI M100. Measuring height was stable, it was h = 1.5 m on the area of 32.4328 m on 32.4328 m (1051.883 m²) with discretization of a given area at 20 points for each coordinate. Measurements were made during three minutes during intensive traffic at 5:00 p.m. of a working day. The characteristics of the area and conditions of measurements were the next: average coordinates of a square area (in a global system of coordinates according to navigation system of aeromobile complex): Longitude: 25.5; Latitude: 49.5; the temperature was −2.6 °C; the relative humidity was 69.8%; atmospheric pressure was 97,473 Pa; wind speed was 2 m/s. The picture received from program module Sniffer4D Mapper is presented in Figure 1.

The results of measurements in the form of immediate nitrogen dioxide concentrations $z_{i,j,h,k}^{}$,$i=0\dots ,19$, $j=0\dots ,19$, $h=const$, $k=const$ presented in

Table 1. Measured instantaneous concentrations of nitrogen dioxide NO₂ (μg/m³) at the given spatial discrete of the highway ring of S. Krushelnytska street, Zbarazka street, Brodivska street and Halytska street at 5:00 p.m.

i/j	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	45.13	45.6	45.13	44.66	44.66	44.19	44.19	44.66	45.13	45.6	45.6	45.13	47.01	47.01	47.01	47.48	47.48	48.89	48.42	47.48
1	48.42	48.42	48.89	49.84	50.31	49.84	49.36	49.36	49.36	49.36	49.84	49.84	50.31	49.84	49.84	50.78	49.84	48.89	48.89	48.42
2	49.84	49.36	49.36	50.78	49.84	49.84	49.84	49.36	48.89	49.84	50.31	50.78	51.25	51.25	51.72	52.19	51.25	50.78	51.25	50.78
3	48.89	47.48	47.01	47.01	47.01	46.54	47.95	47.01	46.07	47.48	47.95	47.95	48.42	47.95	47.95	48.89	48.42	47.48	47.48	46.54
4	46.07	46.07	45.6	44.66	45.13	44.66	45.13	44.66	44.19	45.13	44.19	44.19	45.13	44.66	44.19	45.13	45.13	44.66	45.6	45.6
5	46.54	46.54	47.01	47.48	48.89	48.89	48.42	48.89	48.42	49.36	49.36	49.84	51.72	51.72	51.72	52.66	52.66	53.13	53.13	55.01
6	56.89	58.3	58.77	59.71	60.65	61.12	60.65	61.59	61.59	61.59	62.06	61.59	61.12	62.06	61.59	61.59	62.53	62.06	61.59	62.53
7	62.53	63.47	63.47	63.94	65.35	65.35	65.35	65.82	65.82	64.88	64.41	63.47	63	63	62.06	61.59	62.06	60.65	60.18	61.12
8	60.65	60.18	60.65	60.18	60.65	61.12	60.65	61.59	61.59	61.12	62.06	62.53	61.59	62.06	63	63.47	64.88	65.35	67.23	67.7
9	67.23	68.17	69.58	70.05	70.05	70.99	71.46	72.4	73.34	73.81	75.22	76.16	77.1	78.51	79.92	80.39	81.33	82.27	83.22	84.63
10	85.57	85.57	87.92	88.39	88.86	89.8	92.15	93.09	94.03	96.38	97.32	99.2	99.67	100.14	101.08	102.02	101.08	102.02	101.55	101.55
11	101.08	100.61	100.14	100.14	100.14	99.67	100.61	99.67	98.26	97.79	96.38	95.91	94.97	94.03	94.5	94.03	93.09	92.15	90.74	88.39
12	88.39	86.51	86.04	86.04	85.1	84.16	84.63	83.69	83.22	84.16	82.74	81.33	80.86	79.92	80.39	78.98	78.04	78.51	77.57	76.63
13	77.1	76.63	76.16	76.16	75.69	75.22	75.69	74.75	74.75	74.75	74.28	73.81	74.28	73.34	72.4	71.93	71.46	71.46	70.52	69.58
14	69.58	70.05	69.11	69.11	68.17	67.23	67.23	66.29	66.29	65.35	64.88	65.35	66.76	66.29	67.23	68.17	67.7	67.23	67.23	66.76
15	66.76	66.76	65.82	66.76	65.35	64.88	64.88	64.41	63	63.47	63.47	62.53	63	62.53	62.53	63	62.53	62.06	63	62.06
16	61.12	60.65	59.71	60.18	59.71	59.24	60.18	59.71	59.24	59.71	59.24	58.77	58.3	57.83	57.83	57.83	57.36	56.89	57.83	56.89
17	57.36	58.3	57.36	57.36	58.3	58.3	61.12	61.59	62.06	63.94	63.94	64.41	65.35	65.82	65.82	65.82	65.82	65.82	66.76	66.76
18	66.76	67.7	66.76	66.76	66.29	65.82	64.41	64.41	63.47	63.94	63.47	62.06	62.53	61.12	61.12	60.65	61.12	58.77	59.24	58.77
19	58.77	58.3	57.36	56.89	57.36	56.89	56.42	57.83	56.42	55.48	55.48	55.48	55.95	55.48	55.01	54.07	54.07	53.13	52.19	52.19

. It is worth noting that $h=const$, because the height is stable (1.5 m) and time discrete coordinate $k=const$, since time is also considered to be constant.

To obtain intervals of possible nitrogen dioxide concentration, $i=0\dots ,19$, $j=0\dots ,19$, $h=const$, $k=const$ according to Equation (1), the relative error of the measured sensor of nitrogen dioxide 15% was taken into consideration.

In order to build IDM that predicts nitrogen dioxide concentrations in the specified area, it is necessary to solve the problem of identification of this model using the method of identification based on the behavioral model of the bee colony. Taking into account recent investigations made on the basis of structural identification the following initial structure of IMD was chosen for modeling on the base of experimental data:

$${\widehat{v}}_{i,j,h,k}=g_1+g_2\cdot {\widehat{v}}_{i,j-1,h,k}+g_3\cdot {\widehat{v}}_{i-1,j,h,k}+g_4\cdot {\widehat{v}}_{i-1,j-1,h,k}+g_5\cdot {\widehat{v}}_{i,j-2,h,k}+g_6\cdot {\widehat{v}}_{i,j-3,h,k}$$

(29)

Therefore, in the difference scheme (29) six parameters are unknown $g_1,g_2,g_3,g_4,g_5,g_6$, which should be calculated using the interval data. As it is known, it is necessary to enter initial conditions for setting the difference scheme (7) that is, to define interval concentrations of nitrogen dioxide at specified discrete. Interval data from Table 1 should be used. Hence, a difference scheme is of the third order according to discrete coordinate j (availability of the element ${\widehat{v}}_{i,j-3,h,k}$) and the first order according to discrete coordinate i, then for its setting, it is necessary to enter the next general initial conditions:

$$\begin{array}{l}{\widehat{v}}_{i,j,h,k}\in [z_{i,j,h,k}{}^{-}; z_{i,j,h,k}{}^{+}], i=0, \dots ,19, j=0,\dots ,2,h=const,k=const;\\ i=0,j=3,\dots ,19,h=const,k=const.\end{array}$$

(30)

As it can be seen in (30), initial conditions for the difference scheme (29) should be calculated using the first line from Table 1 and three first columns with numbers j = 0, j = 1, j = 2.

Measured instantaneous concentrations of nitrogen dioxide NO₂, on the base of which the intervals of these concentrations in the form of initial conditions (30) are calculated, they are red presented in Table 1.

Having used the stated above method of parameter identification of IDM we receive the following model:

$$\begin{array}{l}{\widehat{v}}_{i,j,h,k}=0.7952+0.3128{\widehat{v}}_{i,j-1,h,k}-0.1935{\widehat{v}}_{i-1,j,h,k}+0.17{\widehat{v}}_{i-1,j-1,h,k}+\\ +0.7149{\widehat{v}}_{i,j-2,h,k}-0.0171{\widehat{v}}_{i,j-3,h,k}\end{array}$$

(31)

Using the obtained IDM in the form of the difefrence scheme (31) with indicated initial conditions (30), concentrations of nitrogen dioxide were calculated and the graph was built. The results was shown in Figure 2.

Figure 2 shows the results of predictive nitrogen dioxide concentrations on the base of the model (31) (blue color line) and their comparison with experimental interval data (green color lines).

As we can see, the obtained model ensures the calculation of the concentration of nitrogen dioxide with a specified accuracy that meets the requirements of the “guarantee” (5) that is, such a model can be considered adequate.

Altogether, the assessment of the predictive properties of the obtained model and setting bounds of its usage requires additional investigations. In the given example, the conditions of the usage of the obtained model for other traffic intensity and other meteorological conditions will be widened. Therefore, the investigated area will be the same that is the highway ring of S. Krushelnytska street, Zbarazka street, Brodivska street and Halytska street in Ternopil city.

For a verification model, the measurements were made the next day at 1:00 p.m. at the same height of 1.5 m with the discretization of the area in 20 points for each coordinate. Excellent conditions besides time and day were: temperature: −3.2 °C; relative air humidity—72.4%; atmospheric pressure—97,473 Pa; north wind—3 m/s. There was moderate traffic during the time of measurements. All stated differences were indicated under initial conditions $\begin{array}{l}{\widehat{v}}_{i,j,h,k}\in [z_{i,j,h,k}{}^{-}; z_{i,j,h,k}{}^{+}], i=0, \dots ,19, j=0,\dots ,2,h=const,k=const;\\ i=0,j=3,\dots ,19,h=const,k=const.\end{array}$ for the difference scheme (31), calculated on the base of the data of red color presented in

Table 2. Instantaneous concentrations of nitrogen dioxide NO₂ (μg/m³) at specified discrete (the highway ring of S. Krushelnytska street, Zbarazka street, Brodivska street and Halytska street) at 1:00 p.m.

i/j	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	20.22	20.69	20.22	19.75	20.69	19.75	19.75	21.16	20.22	19.75	21.16	19.75	19.28	19.75	19.75	20.69	20.22	20.22	20.69	21.16
1	20.69	21.16	22.10	22.57	23.51	24.45	25.86	26.33	26.33	27.27	27.74	27.74	27.74	29.15	28.68	28.68	29.15	28.68	29.62	29.15
2	28.68	29.15	28.68	28.21	29.15	29.62	28.68	29.15	28.68	28.68	29.15	28.68	28.21	28.68	27.74	27.27	28.21	27.74	27.74	27.74
3	27.74	28.21	27.27	27.27	28.68	27.74	27.74	28.21	28.21	27.74	27.74	27.74	27.27	28.21	27.27	27.27	28.21	27.27	26.80	27.27
4	26.80	27.27	28.21	29.15	30.56	30.56	30.56	31.97	32.44	32.44	34.32	34.79	35.26	36.2	35.26	35.26	36.2	36.2	37.14	37.61
5	38.08	39.02	39.02	40.43	40.9	41.37	41.84	42.78	42.78	42.78	43.72	44.19	44.19	44.66	44.66	44.19	44.66	45.13	44.19	45.13
6	43.25	42.78	43.72	42.78	41.84	42.78	41.84	41.37	41.84	40.43	41.37	41.84	40.9	41.84	42.31	42.31	44.66	44.66	46.07	47.48
7	48.42	49.84	50.78	50.78	51.25	52.19	52.19	52.19	53.6	53.6	53.6	53.6	53.13	54.07	53.13	52.66	54.07	53.13	52.66	53.13
8	53.13	53.13	52.66	52.19	52.19	52.19	51.72	51.72	52.19	52.19	52.19	52.66	51.72	51.25	50.78	49.84	50.31	49.36	48.42	49.36
9	48.42	47.48	47.95	48.89	48.89	48.89	48.42	48.89	49.84	49.36	49.84	50.31	50.31	51.25	52.19	51.72	51.72	52.66	54.07	55.95
10	55.95	56.89	59.71	60.18	61.12	64.88	66.29	68.64	71.46	72.87	76.16	78.98	78.98	79.92	82.74	82.74	83.22	84.63	84.16	84.16
11	84.16	84.16	85.1	84.16	83.69	84.63	83.69	83.22	83.22	83.69	82.74	82.27	82.27	81.33	80.86	79.92	79.45	79.45	78.51	78.04
12	78.51	77.57	77.57	75.69	75.22	75.22	74.75	74.28	74.28	73.34	72.87	73.34	71.93	71.46	70.05	68.64	68.17	68.64	65.82	64.88
13	64.88	63.94	63	63	62.06	61.12	60.65	60.65	61.59	60.65	59.71	60.18	60.18	59.24	60.18	59.71	59.24	59.24	58.77	58.77
14	59.24	58.77	58.3	58.3	57.36	57.36	56.89	56.42	55.01	54.54	54.54	54.54	53.13	52.19	52.66	52.19	50.31	50.78	50.78	48.89
15	48.89	48.42	47.48	47.48	46.54	46.07	46.54	45.6	46.07	47.48	47.01	46.07	46.54	45.13	46.07	45.13	44.19	45.13	44.19	43.72
16	43.72	43.72	42.31	41.84	41.37	40.9	40.9	40.43	40.43	39.96	39.49	38.55	39.49	39.02	38.55	38.55	38.08	39.49	38.55	37.61
17	39.02	39.02	38.55	38.08	39.02	38.08	38.08	38.55	39.02	39.96	39.02	38.55	39.49	39.02	39.02	39.02	39.49	39.02	39.02	38.55
18	39.02	39.96	39.49	39.49	39.96	39.96	39.96	40.9	40.43	39.96	40.9	40.9	41.37	41.84	41.84	42.78	42.31	41.84	42.31	42.78
19	42.78	44.19	45.13	45.6	47.95	48.42	48.89	51.25	51.25	51.25	52.66	52.19	51.72	51.72	51.72	52.19	51.72	51.72	52.66	52.19

.

The results of the prediction of the distribution of nitrogen dioxide concentrations on the base of IDM (31) and specified initial conditions $\begin{array}{l}{\widehat{v}}_{i,j,h,k}\in [z_{i,j,h,k}{}^{-}; z_{i,j,h,k}{}^{+}], i=0, \dots ,19, j=0,\dots ,2,h=const,k=const;\\ i=0,j=3,\dots ,19,h=const,k=const.\end{array}$, were calculated from the first line and three first columns from Table 2, shown in Figure 3. The comparison of the results of the prediction with measured intervals of nitrogen dioxide concentrations is shown in Figure 3.

As it can be seen, predictive concentrations for all discrete belong to intervals obtained experimentally. Therefore, under the stated above conditions, the obtained IDM has sufficient predictive properties for the prediction the limit of concentrations of nitrogen dioxide depending on the change of conditions particularly the change of traffic. In this case, the difference equation is set on the base of characteristic concentrations (initial conditions) for the indicated moment.

Hence, the implementation of the suggested method of parameter identification of IDM gives possibilities for the investigation of nitrogen dioxide concentrations caused by heavy traffic in different parts of the city. Besides, the quantity of measurements, which should be made is defined only by points for the setting of difference equation. For instance, in our case, 77 measurements are enough instead of 400 points of measurements (first line and three first columns in Table 1 and Table 2), to enter initial conditions for the mathematical model in the form of expression (31). It should be noted that mathematical models for different parts of the city will be different.

The statement of this problem in Equation (12) and the Equation (24) was examined during the investigation of the computational method itself for constructing the model. For both cases, the optimization method based on the behavioral model of the artificial bee colony was used. However, the formulation of the parametric identification problem in Equation (24), which was proposed for the first time, has an analytically defined objective function, unlike the case (12), where the objective function is algorithmically defined. In making computational experiment, the computational complexity of implementing the parametric identification method was investigated. The results are shown in

Table 3. Results of the comparison of finding a solution to optimization problems (12) and (24) using the method based on the ABC.

	Finding Solutions of the Optimization Problem (24)		Finding Solutions of the Optimization Problem (12)
	Number of Search Iterations	Total Search Time in Seconds	Number of Search Iterations	Total Search Time in Seconds
Try 1	500	560	706	732
Try 2	604	627	1590	1563
Try 3	1208	1239	5468	4973
Try 4	1745	1758	2072	1886

.

As it is shown in Table 3, during all four estimation experiments, the time complexity of implementing method of parametric identification of IDM based on ABC is lower in case of the usage the first proposed objective function (19) instead of (10) or (11). Therefore, the method of parametric identification of the IDM based on the optimization problem (24) is more effective.

4. Conclusions

The task of modeling the processes of pollution of the atmospheric surface layer caused by vehicle exhaust gases is one of the most important in monitoring the environment of cities. In particular, the main component of these gases is nitrogen dioxide in high concentration, which is dangerous for human health. Measuring and monitoring of city pollution caused by nitrogen dioxide is a high-cost task. Therefore, mathematical modeling methods were suggested to solve this task. Nitrogen dioxide concentrations within the city depend on many factors, in particular on the traffic intensity. The choice for modeling this distribution of discrete equations as an analog of differential equations with partial derivatives has certain advantages. Firstly, it gives the possibility for identifying the model itself using experimental data. Secondly, it gives the possibility for adjusting each model to specific conditions using partial measurements of nitrogen dioxide concentrations in a limited area of a city. In our case, it consists of 77 points instead of the necessary 400. Such an approach gives the possibility to take into account traffic intensity and changing meteorological conditions.

In the text of the paper, we described the shortcomings of classical chemistry transport models. In addition to the problem of measurements, namely measurements, and not calculations of diffusion coefficients in a heterogeneous environment and the setting of boundary conditions and the intensity function of a distributed emission source, there is a need to verify this model based on the results of nitrogen dioxide concentration measurements. Therefore, we did not compare the results obtained from different models, but the simulation results with the actually measured concentrations. To do this, IDM identification was first carried out with some measurement results, and then the simulation results were compared with the measurement results under other conditions.

The proposed new model of identification method is more effective in terms of computational time complexity compared with the known methods based on taking measurement errors into account, which in the final case provides predictive properties of the model with guaranteed accuracy.

Studies in a specific area of the city showed that the distribution of nitrogen dioxide obtained on the base of the interval discrete model in the form of the difference equation reflects the regularities of a place of research and adequately takes into account changes in traffic, which significantly reduces the costs for air quality control and environmental monitoring.