## 2 Materials and Methods
### Methods
#### 2.2.2 Relativity Correction  
Based on the Mie scattering theory [11; 48], the near-ground extinction coefficient of aerosols \(\sigma_{\text{a}}(\lambda)\) can also be expressed as:  
\[\sigma_{\text{a}}(\lambda)=\frac{3\langle\text{Q}_{\text{ext}}\rangle}{4\text {r}_{\text{eff}}\rho}.\text{PM}_{\text{x}} \tag{3}\]  
where \(\langle\text{Q}_{\text{ext}}\rangle\) denotes the scattering efficiency of normalized particles, r\({}_{\text{eff}}\) denotes the effective radius of the particle, \(\rho\) denotes the average density of particles, PMx denotes the mass concentration of aerosol, and all parameters are influenced by relative humidity. Assuming that the particles, particle spectrum, and density of the aerosol remain relatively constant, \(\langle\text{Q}_{\text{ext}}\rangle\), r\({}_{\text{eff}}\), and \(\rho\) can be viewed as constants so that there is a certain positive relationship between \(\sigma_{\text{a}}(\lambda)\) and PMx. According to the average mass extinction rate defined by [PERSON] [11].  
\[\text{E}_{\text{ext}}=\frac{\sigma_{\text{a}}(\lambda)}{\text{PM}_{\text{x}}} \approx\text{E}_{\text{ext}}(\text{RH}) \tag{4}\]  
Since the fine particulate matter is the main contributor to aerosol extinction, here, PMx refers to particulate matter with a particle radius less than 2.5 \(\upmu\)m, and \(\text{E}_{\text{ext}}(\text{RH})\) describes the average state of the extinction ability of particles with different properties. To quantitatively describe the hygroscopic growth characteristics of \(\text{E}_{\text{ext}}(\text{RH})\), the extinction hygroscopic growth factor of the average mass extinction rate f\((\text{RH})\) can be expressed as  
\[\text{f}(\text{RH})=\frac{\text{E}_{\text{ext}}(\text{RH})}{\text{E}_{\text{ dry}}} \tag{5}\]  
where \(\text{E}_{\text{dry}}\) denotes the average mass extinction rate under dry conditions (RH = 40%). According to the [PERSON]'s research [52], \(\text{E}_{\text{ext}}(\text{RH})\) can be represented by Equation (6),  
\[\text{E}_{\text{ext}}(\text{RH})=\text{a}+\text{b}\times\left(\frac{\text{RH} }{100}\right)^{\text{c}} \tag{6}\]where \(\rm a\), \(\rm b\), and \(\rm c\) are coefficients fitted with RH variation, depending on the aerosol physicochemical properties. According to the above equations, the expression of \(\rm PM_{2.5}\) can be obtained as follows:  
\[\rm PM_{2.5}=\frac{\sigma_{a}(\lambda)}{E_{ext}(RH)}=\frac{\rm AOD}{\rm H_{a}} \over\left(a+b\times\left(\frac{RH}{100}\right)^{c}\right)\times E_{dry} \tag{7}\]  
Since fine particles can stay in the atmosphere for days or even weeks and can be well mixed in the atmosphere and transported over long distances to other regions, the spatial distribution of the fitted coefficients \(\rm a\), \(\rm b\), and \(\rm c\) can be obtained by interpolation, assuming that the aerosol properties vary smoothly on a regional scale. Subsequently, assuming that the regional relative humidity varies smoothly, the distribution of relative humidity in the study area can be obtained by IDW interpolation. Finally, the \(\rm PM_{2.5}\) concentration value of each pixel can be calculated based on the AOD, \(\rm H_{a}\), relative humidity, \(\rm a\), \(\rm b\), and \(\rm c\) in the study area.