**3. Methods**

To quantitatively evaluate the spatial representativeness of the MODIS daily albedo product, 30 m albedo data were derived from Landsat TM data and then calibrated using field observation data. Semivariagrams are calculated with calibrated 30 m albedo data for different scales, in which sill value is considered as the key index to measure the magnitude of field homogeneity, i.e., sill values of adjacent scales differ from each other significantly, indicating that the field homogeneity changed significantly between these two scales. Relative coefficient of variation (*RCV*) is also employed to determine the land surface homogeneity: *RCV* tends to be 0, which indicates that the adjacent scales have similar homogeneity, the MODIS pixel represents the larger scale. The last index used is the determination coefficient, 30 m albedo data is aggregated to different scales and compared with MODIS data. The scale in which it has the highest determination coefficient indicates the MODIS pixel represents the spatial scale best.

#### *3.1. Variogram Model Parameters from TM Data*

When using tower observation to validate the remote sensing albedo product, the spatial representativeness of the observation footprint was investigated on semivariogram models [37,39,47,54]. In this study, the method of deriving variogram functions to analyze surface albedo with TM data was used [36]. The variogram estimator γ(*h*) was used to obtain the half-average squared difference between albedo values within a certain distance. According to Román et al. [47], the isotropic spherical variogram model [55] was used to fit the variogram model parameters—the range (a), the sill (c), and the nugge<sup>t</sup> effect (*c*0), as below:

$$\gamma\_{sph}(h) = \begin{cases} \ c\_0 + c \left( 1.5 \frac{h}{a} - 0.5 \left( \frac{h}{a} \right)^3 \right) for \ 0 \le h \le a\\ \ c\_0 + c \text{ } for \ h > a \end{cases} \tag{1}$$

The range is the distance from a point beyond which there is no further correlation of the albedo associated with that point. It is the average patch size of the landscape in landscape ecology, which represents a region that differs from its surroundings, but is not necessarily inter-homogeneous. The sill is the maximum semivariance, and is the ordinate value of the range at which the variogram levels off to an asymptote. The non-zero value of the variogram when h = 0 is called the nugge<sup>t</sup> effect. It depends on the variance associated with small-scale variation, measurement errors, or their combination [56]. The range, the sill, and the nugge<sup>t</sup> effect all reveal the spatial variation of the land surface and the scale effect associated with remote sensing data [47,57]. It has been suggested that the land surface is homogeneous (representative) when the sill value is less than 0.001 [37]. In this paper, the semivariogram model was used as well. The 30 m Landsat albedo was first re-projected to a sinusoidal projection and the Landsat pixel located at the center of the MODIS 500 m pixel was determined. The semivariogram was calculated from Landsat data. The model parameters were fitted according to the spherical model.

#### *3.2. Relative Coefficient of Variation*

The indices deduced from the parameters of the semivariogram model and the statistical values were also used in this study. The relative coefficient of variation ( *RCV*), the scale requirement index, the relative proportion of structural variation, and the relative strength of spatial correlation, derived from the semivariogram model, were first used by Román et al. [47] to depict spatial variation. When the measurement site was spatially representative, the overall variation between the internal components of the measurement site (scale 1) and the adjacent landscape (scale 2) should have been similar in magnitude, and the *RCV* should have approached zero. The *RCV* was also calculated to check the spatial variation in the landscape. To calculate it, the coefficient of variation (cv) was first calculated as the ratio of the standard deviation to the mean. The *RCV* is given below:

$$R\_{CV} = \frac{\text{CV}\_{\text{scale}2} - \text{CV}\_{\text{scale}1}}{\text{CV}\_{\text{scale}1}} \tag{2}$$
