## 3 Methodology and Data Sources
### Model Formulation
#### 3.5.2 Evaluating the Contribution of SOB-Caused Inflow PM\({}_{2.5}\) Concentration  
Similar to the above mass balance equation, the differential equation of the average PM\({}_{2.5}\) concentration in the receptor city sub-box can also be obtained as follows [47]:  
\[\frac{\mathrm{d}C_{rc}}{\mathrm{d}t}=\frac{q_{rc}}{H_{rc}}+R_{rc}-\frac{v_{d}} {H_{rc}}C_{rc}+\frac{1}{T}(C_{n}-C_{rc}) \tag{7}\]  
The terms on the right-hand side of Equation (7) stand for changes in PM\({}_{2.5}\) concentration due to local emissions, chemical reactions, deposition, and horizontal transport. \(C_{rc}\), \(q_{rc}\), \(R_{rc}\), \(H_{rc}\), and \(u_{rc}\) are, respectively, the average PM\({}_{2.5}\) concentration, local PMemission rate, PM\({}_{2.5}\) chemical reaction rate, PBLH, and wind speed of the receptor city. According to the average 25 km diameter of the urban geographic space of the receptor city Changchun, it can be estimated that the residence time of the air mass is generally greater than 1 h at normal wind speeds. Therefore, when simulating the hourly average PM\({}_{2.5}\) concentration of the city, \(T\) is the moving time in the recipient city of the air mass with the PM\({}_{2.5\text{-inflow}}\) concentration. Subjected to a given initial condition, \(\left.C_{rc}(t)\right|_{t=0}=C_{rc}(0)\), the hourly average PM\({}_{2.5}\) concentrations of the receptor city (\(\overline{C_{rc}}\)) predicted by the box model can be written as:  
\[ \begin{array}{c}\overline{C_{rc}}=\frac{1}{T}\int_{0}^{T}\left[\begin{array} []{c}\frac{A}{B}\left(1-e^{-Bt}\right)+C_{rc}(0)e^{-Bt}\end{array}\right] \text{d}t\\ =\frac{A}{B}+\frac{e^{-Bt}-1}{T}\left(\frac{A}{B^{2}}-\frac{C_{rc}(0)}{B}\right) \end{array}  \tag{8}\]  
with  
\[A=\frac{q_{rc}}{H_{rc}}+R_{rc}+\frac{C_{n}}{T}\]  
\[B=\frac{v_{d}}{H_{rc}}+\frac{1}{T}\]  
The solution of Equation (8) is obtained for the hourly average PM\({}_{2.5}\) concentration of the receptor city:  
\[ \begin{array}{c}\overline{C_{rc}}=\Bigg{[}\frac{1}{v_{d}+\frac{H_{rc}}{T}}- \frac{1}{\overline{H_{rc}}\left(\frac{v_{d}}{H_{rc}}+\frac{1}{T}\right)}\Bigg{]} q_{rc}+\\ \Bigg{[}\frac{\frac{1}{\overline{H_{rc}}}-\frac{1}{\overline{H}}-\frac{1}{ \overline{H}\left(\frac{v_{d}}{H_{rc}}+\frac{1}{T}\right)^{2}}}{\Bigg{]}}R_{rc }+\\ \frac{1}{\frac{1}{\overline{H_{rc}}}+1}-\frac{1}{\overline{r}^{2}\left(\frac{ v_{d}}{H_{rc}}+\frac{1}{T}\right)^{2}}\Bigg{]}C_{n}+\\ \frac{1}{\overline{T}\left(\frac{v_{d}}{H_{rc}}+\frac{1}{T}\right)}C_{rc}(0) \end{array}  \tag{9}\]  
Equation (9) stands for the hourly average concentration of PM\({}_{2.5}\) of the receptor city, which has a multivariate linear relationship with the local primary emission rate (\(q_{rc}\)), chemical reaction rate (\(R_{rc}\)), horizontal transport concentration (\(C_{n}\)), and initial local concentrations (\(C_{rc}(0)\)) of PM\({}_{2.5}\). Equation (9) is written as \(\overline{C_{rc}}=\gamma q_{rc}+\delta R_{rc}+\alpha C_{n}+\beta C_{rc}(0)\), with the coefficients \(\gamma\), \(\delta\), \(\delta\), and \(\beta\) depending upon meteorological properties in urban PBLH and wind speed (\(H_{rc}\), \(n_{rc}\)), as well as the specific properties of PM\({}_{2.5}\) (\(v_{d}\)). Since the residence time of the air mass is generally greater than 1 h, the contribution of PM\({}_{2.5\text{-inflow}}\) will be estimated in \(m\) periods (\(\tau_{j}\), \(j=1\), 2,..., \(m\)) by hourly time steps. Each period \(\tau_{j}\) is determined by the following formula, in which \(\tau_{1}\), \(\tau_{2}\),..., \(\tau_{(m-1)}\) is one hour and \(\tau_{m}\) is usually less than one hour. \[\tau_{j}=\left\{ \begin{array}{c}3600,\\ \frac{L-3600(m-1)}{u_{rc}(m)}\end{array} ,\begin{array}{c}j= \begin{array}{ c}1,\,2,\ldots,\,m-1\\ \frac{L-3600(m-1)}{u_{rc}(m)}\end{array} , \begin{array}{c}j=\begin{array}{ c}m-1\\ \end{array}\end{array} \right. \tag{10}\]  
where \(L\) is the average diameter of the urban geographic space; \(j\) is the \(j\)th hour since the air mass flowed into the city; and \(u_{rc}(j)\) is the average urban wind speed during the \(j\)th hour.