## 5 Overview of general scaling methods
### The scaling methods for products  
Compared to the scaling methods of measurements and retrieval models, the scaling methods for products are more widely studied in research. These scaling methods provide a few systematic approaches to characterize the land surface heterogeneity and compensate for scaling effects. Here, we just emphasize the spatial domain. Other scaling methods applied in different domains can be realized by certain approaches. For example, scaling in the temporal domain may consider the general diurnal patterns of meteorological variables [69] or assume that the evaporative fraction (EF) is constant throughout the day [70]. Scaling in the spectral domain and the directional domain would rely on experimental regression or look-up tables (LUT) to solve the scale problem, respectively. Since land surface is very heterogeneous, up-scaling is probably more important than downscaling in product validation. Therefore, we may be more concerned about the up-scaling method. The downscaling method resorts to relevant literature [18]. In general, the simple up-scaling method can use statistical algorithms to estimate the spatial means at a large scale by either area-weighted products of homogeneous patches or the integration over the probability density function of products [42]. In the following, other extensively used scaling methods in the spatial domain, such as Empirical Regression Method (ERM), Taylor Series Expansion Method (TSEM), Contextual Parameters Method (CPM), Statistical Fractal and Self-similar Method (SFSM) will be analyzed in detail. ERM is simply used to empirically calibrate the relationship of products between fine and coarse scales [71; 72; 73]. The goal is to relate the products at different scales by regression. [PERSON] _et al_. [53] simply used empirical relationships between spectral vegetation indices and surface estimations of LAI, which are in-situ measurements from the auxiliary sites, to estimate coarse-scale LAI. [PERSON] _et al_. [74] used a multivariate ordinary least squares (OLS) algorithm which uses an iteratively re-weighted least squares (IRLS) algorithm to build an empirical relationship and upscale the field LAI data to the corresponding satellite products. Although the scaling method is simple, the relationship of regression is usually site, time, model and scale dependent. If any one factor is changed, the relationship may need to be recalibrated. Consequently, it would gradually be replaced by other scaling methods. TSEM is based on [PERSON]'s theorem of linearizing the retrieval model around the arithmetic average of measurements [16; 33; 34]. It characterizes the scaling effects as the combined effects of degree of non-linearity of the model and the heterogeneity of the land surface. The key to using this method is to know how to estimate the variance and covariance within the pixel at coarse resolution. [PERSON] and [PERSON] [33] proposed a novel method based only on the mean values of measurements at pixel scale to parameterize the variance and covariance terms by the half ellipse relationship between the normalized standard deviation of radiance and the normalized average radiance over the pixel. They used this relationship to study the effects of subgrid scale heterogeneity of soil wetness and temperature on grid-scale evaporation [75]. [PERSON] _et al_. [34] used the local dispersion variance and covariance which are calculated by semivariogram and cross semivariogram from concurrent high spatial resolution images or a spatial sampling per type of landscape of the high spatial resolution data to quantify the spatial variability within the moderate resolution pixel. Afterwards, [PERSON] _et al_. [76] proposed a spatio-temporal model of the variogram to estimate the intra-pixel spatial heterogeneity. It is a novel approach to predict the variogram at a date at which the high spatial resolution scene is not availableby the seasonal cycle of phenological variability. Although its validity is argued by [PERSON] [31], it is still useful for product scaling.