## 2 Materials and Methods
### Methodology
#### 2.2.1 The Retrieval of the Bidirectional Reflectance Distribution Factor (BRDF) Characterization  
The key process of the proposed method is the retrieval of the BRDF characterization. For a stable surface, the BRDF characterization is very constant, so it can be retrieved through time-series analysis using large amount of historical data, such as the method proposed in [16,18]. Although the method in [16,18] has been proved effective in some scenarios [16], some problems still remain; therefore, some key improvements are subsequently proposed in this study to retrieve better BRDF and the following atmospheric correction in arid aeras. Three improvements are made as follows:  
1. Instead of being applied to all land surfaces, the proposed method is only for the arid areas with bright and stable surfaces; the stability of the land surface enables use of much longer temporal range to retrieve more data to fit the BRDF characterization and subsequently improve the fitting accuracy of the BRDF characterization. Therefore, instead of using 1 km MODIS data within only one year, the proposed method uses a multi-year time-series dataset composed of 500 m MODIS data from both Terra and Aqua, which has higher resolution and a longer temporal period; thus, higher resolution means more detail and a multi-year time series dataset retrieves more \"clearest\" observations which captures better the BRDF characterization. Whether the AOD's resolution of 500 m from MODIS data is enough for atmospheric correction of MHSR data needs to be clarified before moving on; the method is only applied to the arid areas, such as desert and the Gobi, where the spatial variation of aerosol is usually very small, so 500 m is a suitable resolution of AOD retrieval, which can also greatly improve the algorithm's efficiency. 2. Due to the path becoming longer with the increasing of view zenith angle, the TOA reflectance increases; however, the view zenith angle (VZA) is not taken into consideration while choosing the clearest observations and most of the clearest observations concentrate on observations with smaller VZA, which may cause large error. Thus, the proposed method chooses the clearest observations for every 10 degrees' interval of VZA from 0 to 40 degrees (0-10, 11-20, 21-30, and 31-40), which will obtain observations with an even distribution of VZAs and subsequently improve the accuracy of the simulated BRDF. In addition, VZAs of MHSR data are seldom larger than 40 degree. [PERSON] et al. [17] undertook an accuracy evaluation of different BRDF models at a desert site, which showed that the Staylor-Suttles BRDF model [19] had the best fitting  
Figure 1: Flowchart of the new aerosol optical depth (AOD) retrieval method. accuracy and the kernel model [20] was the second best. Therefore, a comparison between the Staylor-Suttles and the kernel BRDF models is carried out at four different AERONET sites (Fowlers_Gap, Mezaira, and Tamarasset_INM, and Dalanzadgad) based on the method of [PERSON] et al. [21]. In this comparison, the reflectance of long time series from different models is normalized. The normalized result is shown in Figure 2. The information of the four AERONET sites is listed in Table 2.
3. The TOA reflectance observed from MODIS in the blue band (459-479 nm) over the four different AERONET sites were normalized by the Staylor-Suttles and the kernel BRDF models in 2018 and the mean/standard deviation pairs (Staylor-Suttles, kernel) for the four sites are (1.0069/0.0852, 1.0072/0.0936), (1.0047/0.0613, 1.0025/0.1090), (1.0015/0.0430, 1.0014/0.0464), and (1.0163/0.0563, 1.0077/0.0580) respectively. The comparison between the Staylor-Suttles and the kernel BRDF models at the red band is also carried out at the four AERONET sites, and the mean/standard deviation pairs (Staylor-Suttles, kernel) for the four sites at the red band are (1.0009/0.0307, 1.0005/0.0247), (1.0004/0.0240, 1.0004/0.0135), (1.0005/0.0227, 1.0005/0.0313), and (1.0026/0.0505, 1.0116/0.0336), respectively.