## 3 Methodology and Data Sources
### Model Formulation
#### 3.5.1 Estimating SOB-Caused Inflow PM\({}_{2.5}\) Concentration  
By examining the interrelationships among the sub-boxes of HWFGs, the mass balance equation of the change in total PM\({}_{2.5}\) suspended in the atmosphere per unit of time in each sub-box (\(\Delta M_{i}\)) is established as follows:  
\[\Delta M_{i}=M_{i}^{Emis}+M_{i}^{Chem}-M_{i}^{Depo}+M_{i}^{Trans} \tag{1}\]  
where the terms on the right side of Equation (1) stand for the primary emission (\(M_{i}^{Emis}\)), chemical reaction (\(M_{i}^{Chem}\)), deposition (\(M_{i}^{Depo}\)), and horizontal transport (\(M_{i}^{Trans}\)) mass of PM\({}_{2.5}\) per unit of time. The model assumes that the pollutant is completely mixed in each sub-box and pollutant concentration is uniform in the whole volume of air, so the change in total PM\({}_{2.5}\) per unit of time in sub-box \(i\) can be expressed as:  
\[\Delta M_{i}=\frac{\Delta C_{i}}{\Delta t}\cdot S_{i}H_{i} \tag{2}\]  
where \(\Delta C_{i}\) is the change in average PM\({}_{2.5}\) concentration in sub-box \(i\) (\(\text{\mu g/m}^{3}\)); \(S_{i}\) is the plane size of the sub-box; \(H_{i}\) is the height of the sub-box and is obtained by averaging the PBLH of the four endpoints at the top of the sub-box; and \(\Delta t\) is a specified period (s). Figure 4: Schematic diagram of two-stage spatiotemporal multi-box modeling based on hourly wind-field grids. The PM\({}_{2.5}\) primary emission of SOB per unit of time can be described from the equation below:  
\[M_{i}^{Emi}\!\!=S_{i}^{c}\gamma_{i}^{c}\ \rho^{c}\eta^{c}\delta^{c}+S_{i}^{r} \gamma_{i}^{r}\ \rho^{r}\eta^{r}\delta^{r}\eta^{r}\delta^{r}\eta^{r} \tag{3}\]  
where \(S_{i}^{c}\) is the corn straw burning area in sub-box \(i\) (ha); \(\gamma_{i}^{c}\) is the corn yield per unit area in sub-box \(i\) (kg/ha); \(\rho^{c}\) is the corn straw-to-grain ratio (%); \(\eta^{c}\) is the combustion efficiency of corn straw (%); \(\delta^{c}\) is the dry matter rate of corn straw (%); \(\delta^{c}\) is the average PM\({}_{2.5}\) emission rate of burning corn straw (%); \(\delta^{c}\) is the rice straw burning area in sub-box \(i\) (ha); \(Y_{i}^{r}\) is the rice yield per unit area in sub-box \(i\) (kg/ha); \(\rho^{r}\) is the rice straw-to-grain ratio (%); \(\eta^{r}\) is the combustion efficiency of rice straw (%); \(\delta^{r}\) is the dry matter rate of rice straw (%); and \(\delta^{r}\) is the average PM\({}_{2.5}\) emission rate of burning rice straw (\(\mu\)g/(kg-s)). For particulates, both wet deposition and dry deposition are major removal mechanisms. Because there was little precipitation along the HTPs during PEs according to the prepared meteorological data, dry deposition was the main factor in removing particulate matter during the study period. The PM\({}_{2.5}\) deposition rate is given by  
\[M_{i}^{Depo}\!\!=C_{i}\cdot S_{i}v_{d} \tag{4}\]  
where \(C_{i}\) is the average PM\({}_{2.5}\) concentration in sub-box \(i\) (\(\mu\)g/m\({}^{3}\)) and \(v_{d}\) is the PM\({}_{2.5}\) dry deposition rate in the atmosphere (m/s). The model assumes that advection is dominant along the wind direction, and turbulent dispersion is negligible along the wind direction and in vertical and crosswind directions.