## 3 Methods
### Evaluation Strategies  
To determine the spatial representativeness of the 500 m product used in this study, variogram estimation was performed at nine scales for each site. When estimating the variogram, the common spatial step was one MODIS pixel, and according to the result of [41], three scales were added between 1 \(\times\) 1 and 2 \(\times\) 2 MODIS pixels; that is, 21 \(\times\) 21, 23 \(\times\) 23, and 29 \(\times\) 29 TM pixels. The 2.5 \(\times\) 2.5 MODIS pixels were also added to keep an intensive estimation. The estimating scales used in this study are shown in Table 2.  
To make the analysis representative, for each site, not only was the tower-located MODIS pixel analyzed, the research area was enlarged to 9 \(\times\) 9 pixels with the tower-located MODIS pixel as the central pixel. In this research area, for every MODIS pixel, the semivariogram parameters (nugget, sill, range) and the statistical value, including mean and standard deviation, were calculated at the nine scales illustrated in Table 2.  
The spatial representativeness was evaluated according to the calculated parameters and values. The sill value represents the magnitude of spatial variability. In this study, the sill values of 9 \(\times\) 9 MODIS pixel in every research area were compiled (for every scale, sill values were compiled as a group; thus, we obtained nine groups of data), and the paired t-test was implemented every adjacent scale data pair (e.g., 15 \(\times\) 15 and 21 \(\times\) 21) to find whether the land surface at adjacent scales was significantly different. Statistical significance was determined by the \(p\)-value. If the \(p\)-value was zero, it indicated that the difference of the land surface heterogeneity at adjacent scales was not significant; the central MODIS pixel was able to represent an area determined by the larger scale.  
\(R_{CV}\) depicts spatial variation in the landscape. In this study, \(R_{CV}\) was calculated between each pair of scales in Table 2, then all the data from 109 sites and 81 pixels were compiled and the histogram was plotted. The scale in which the \(R_{CV}\) value is smallest indicates that the central pixel has the similar spatial representativeness in these adjacent scales.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Scale** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline TM pixels & 15 \(\times\) 15 & 21 \(\times\) 21 & 23 \(\times\) 23 & 29 \(\times\) 29 & 31 \(\times\) 31 & 39 \(\times\) 39 & 47 \(\times\) 47 & 61 \(\times\) 61 & 75 \(\times\) 75 \\ Scale in meters & 450 & 630 & 690 Taking the calibrated TM data as the albedo reference, we aggregated the TM value at different scales and compared the aggregated 30 m albedo data directly with the 500 m albedo product, and two aggregation methods were used. One was simple average, and the other considered the point spread function. [PERSON] (2010) and [PERSON] et al. (2010) used the convolution of a Gaussian function to characterize the optical PSF of the MODIS instrument, assuming that the central area of the pixel made a greater contribution to the signal. In this paper, we adopt the Gaussian function, as was done by [PERSON] (2010) and [PERSON] et al. (2010), but set an asymmetric Gaussian point spread function. The PSF was defined below.  
\[\mathrm{PSF}(x,y)=\frac{1}{2\pi a^{2}}exp^{-0.5(x^{2}/a^{2}+y^{2}/a^{2})} \tag{3}\]  
where  
\[\mathrm{a}=\frac{\mathrm{FWHM}}{2.355} \tag{4}\]  
The FWHM is the full width at half maximum of the PSF. In this study, we set it as the sensor spatial resolution as [PERSON] and [PERSON] (2010), which also represents the spatial representativeness of the pixel.  
The determination coefficient was used to judge the satisfactorily of the consistency between the two datasets. The scale where the highest determination coefficient appeared was considered that the MODIS data has the best spatial representativeness.