## 4 Discussion  
In this study, we investigated daily ground-level PM\({}_{2.5}\) estimates at 1 km spatial resolution in the Beijing metropolitan area using the LMEM by incorporating the MAIAC AOT data and ground-based PM\({}_{2.5}\) measurements. The results show that both model fitting and cross validation can generate higher R\({}^{2}\) values for monitoring sites. We also analyzed the model performance for each year in the five-year study period, indicating that both Model-I and Model-II give higher model fitting R\({}^{2}\) values and cross validation R\({}^{2}\) values relative to a linear regression model based on the same data inputs (Figure S1), and can be hugely improved when considering that daily variations of PM\({}_{2.5}\)-AOT relationship are correlated to mixing height, relative humidity, PM\({}_{2.5}\) composition, and PM\({}_{2.5}\) vertical profile. Both Model-I and Model-II have higher R\({}^{2}\) values, indicating that the PM\({}_{2.5}\) 24 h average and the PM\({}_{2.5}\) period average between 10:00 a.m. and 2:00 p.m. can be used to fit LMEM models with high confidence. Overall, Model-I has slightly better performance as demonstrated by its lower RMSPE and MPE values and similar R\({}^{2}\) values to Model-II. AOT represents the amount of aerosols in the vertical column of atmosphere from the ground to satellite sensors (not only limited to the vertical column segment close to the monitoring sites). The PM\({}_{2.5}\) 24 h average captures more of the aerosols which have subsided down to ground level at monitoring sites than does the PM\({}_{2.5}\) period average between 10:00 a.m. and 2:00 p.m. This might explain why the LMEM model fitted with the PM\({}_{2.5}\) 24 h average performs a little better than the LMEM model fitted with the PM\({}_{2.5}\) period average. The LMEM in our study generated higher model fitting R\({}^{2}\) values of 0.90-0.94 and cross validation R\({}^{2}\) values of 0.87-0.93 than those in previous studies. For example, the GWR model applied to the  
Figure 10: Seasonal mean PM\({}_{2.5}\) concentrations (\(\upmu\)g/m\({}^{3}\)) calculated from daily PM\({}_{2.5}\) estimations during 2013–2017. whole of China with an overall cross validation R\({}^{2}\) value of 0.64 [31] and to the Pearl River Delta region with an R\({}^{2}\) value of 0.74 [21]; the empirical nonlinear model applied for the Xian metropolitan area with an R\({}^{2}\) value of 0.67 [32]; the new LMEM model incorporating VIIRS night light data applied to the Beijing-Tianjin-Hebei Region with an R\({}^{2}\) value of 0.75; the improved LMEM model applied to three megalopolises in China, namely, Beijing-Tianjin-Hebei, with an R\({}^{2}\) value of 0.77, to the Yangtze River Delta region with an R\({}^{2}\) value of 0.80, and to the Pearl River Delta region with an R\({}^{2}\) value of 0.80 [23]; the observation-based model using MAIAC AOT applied to the Beijing-Tianjin region with an R\({}^{2}\) value of 0.70, to the Yangtze River Delta region with an R\({}^{2}\) value of 0.77, and to the Pearl River Delta region with an R\({}^{2}\) value of 0.83 [33]; and the LMEM models using MOD-3K AOT applied to Beijing with R\({}^{2}\) values of 0.796 and 0.81 [24; 25]. The model performance for Beijing in this study is also comparable with those studies conducted in the U.S. with MAIAC AOT data as inputs. For example, model fitting over the southeastern U.S. achieved an R\({}^{2}\) value of 0.83, while cross validation achieved an R\({}^{2}\) value of 0.67 [14]; annual model fitting gets an R\({}^{2}\) value of 0.71-0.85, while cross validation gets an R\({}^{2}\) value of 0.62-0.78 during 2001-2010 [15], and cross validation in the New England area achieved an R\({}^{2}\) value of 0.89 [34].