## 2 Materials and Methods
### LMEM Model Fitting and Validation  
[PERSON] et al. [13] developed the LMEM algorithm to estimate PM\({}_{2.5}\) from MODIS MOD04 AOT at 10 km resolution; it takes into account the day-to-day variability of the PM\({}_{2.5}\)-AOT relationship based on mixing height, relative humidity, PM\({}_{2.5}\) composition, and PM\({}_{2.5}\) vertical profile with the assumption that the PM\({}_{2.5}\)-AOT relationship varies largely day to day but minimally spatially on a given day at the study area. The LMEM calculates day-specific random intercepts and slopes for the PM\({}_{2.5}\)-AOT relationship and incorporates both fixed-effects terms and random-effects terms. The LMEM equation is  
\[PM_{ij}=\left(\alpha+\mu_{j}\right)+\left(\beta+v_{j}\right)\times AOT_{ij}+s_{i }+\epsilon_{ij} \tag{1}\]  
where \(PM_{ij}\) is the daily average PM\({}_{2.5}\) concentration at site \(i\) on day \(j\); \(AOT_{ij}\) is the daily average AOT value corresponding to site \(i\) on day \(j\); \(\alpha\) and \(\mu_{j}\) are the fixed and random intercepts on day \(j\), respectively; \(\beta\) and \(v_{j}\) are the fixed and random slopes on day \(j\), respectively; \(s_{i}\) is the random intercept of site \(i\); and \(\epsilon_{ij}\) is the error term at site \(i\) on day \(j\). In this model, the fixed parameter \(\beta\) for AOT represents the average PM\({}_{2.5}\)-AOT relationship for the entire space and time in the specific urban area, while the random parameter \(v_{j}\) represents daily variation of the PM\({}_{2.5}\)-AOT relationship. We trained (fitting and validating) the LMEM using a large number of PM\({}_{2.5}\)-AOT sample data pairs for PM\({}_{2.5}\) estimation from 2013 to 2017. As an AOT pixel value represents the average aerosol level in a given 1 km \(\times\) 1 km area while a ground-based PM\({}_{2.5}\) measurement only represents the aerosol level at a point, theoretically, there may be systematic bias between AOT-derived PM\({}_{2.5}\) and ground-measured PM\({}_{2.5}\). The site bias may also vary by surface cover type, topography, distance to pollution source, and other anthropogenic and natural environmental conditions. It should be noted that site random effect (\(s_{i}\)) was not considered when estimating PM\({}_{2.5}\) concentrations for our study period because AOT values are unbiased representative of the corresponding grid cells; in other words, PM\({}_{2.5}\) values derived from AOT reflect the overall PM\({}_{2.5}\) levels in the grid cell, but the unadjusted predicted PM\({}_{2.5}\) levels with site effects maybe be more representative of the average population exposures to PM\({}_{2.5}\)[13]. After the model training, the PM\({}_{2.5}\) estimates over our study area for the whole time period were generated using LMEM and 1 km MAIAC AOT. Further, the model performance using 1 km AOT data and 3 km AOT data was compared using the 1 km MAIAC AOT and 3 km MOD-3K AOT data sets for 2014 as inputs. In addition, as the PM\({}_{2.5}\) 24 h average and the PM\({}_{2.5}\) period average may differ, we also established different LMEM models and compared the model performance with these two PM\({}_{2.5}\) data sets. Tenfold cross-validation (CV) was applied to test the potential model overfitting through randomly splitting the entire samples (PM\({}_{2.5}\)-AOT pairs) into 10 subsets with each subset containing approximately 10% of the whole sample dataset. One sample subset was used for model testing, while the remaining nine sample subsets were used for model fitting. This procedure was repeated 10 times until every subset was tested.