## 3 Mechanism analyses of scale effects
### Effects of scale on the measurements, retrieval models and products  
The mechanism analyses of scale issues undoubtedly involve several questions. The first one is what the main causes of scale issues are in remote sensing, which has already been analyzed above. The following one is what the effects of scale on the measurements, models and products are. With measurements, what we are more concerned about may be the mean, variance and correlation lengths. [PERSON] _et al_. [5] and [PERSON] _et al_. [35] have already demonstrated the effect of the measurement scale on the apparent variance and the apparent integral scale (apparent correlation length). Here, the \"apparent\" means the statistical properties that appear in the data. The apparent variance decreases with increasing resolution, while the apparent correlation length, the average distance (or time) over which a property is correlated, always increases with increasing resolution. To the retrieval models, the effects of scale may be difficult to analyze. We may select suitable models at the corresponding scale; for example, the radiative transfer model of vegetation at the leaf scale [36] or canopy scale [37]. Whether the change of scale has an effect on the retrieval model is dependent on the status of the measurements and the form of the retrieval model. In the case of three-dimensional structures, the retrieval models may need to be remodeled. [PERSON] and [PERSON] [38] used a clumping index to correct the radiation attenuation coefficient at a shoot scale and proposed the forest reflectance models for coniferous forests. In the case of big-leaf assumptions, if measurements are homogeneous, the scale has no effects on the retrieval model no matter if the model is linear or non-linear. If the measurements are heterogeneous, there is a scale effect on the retrieval model. There is only no effect when the retrieval model is linear and does not vary with the land cover types, which means the model has a uniform form. Otherwise, the retrieval model may need to be simplified or re-parameterized. [PERSON] [4] proposed a spatialisation model to overcome the scale effects by using the lower and upper bounds of the retrieval model. [PERSON] _et al_. [21] addressed the problem of how the scale, or spatial resolution of reflectance measurements impacts retrievals of LAI and developed a physically radiative transfer formulation with explicit spatial resolution dependents. To the products, the effects of scale have already been widely discussed. There is conflicting conclusions in the literature as to whether products are scale dependent or scale free. The main reason is that the scaling effects are usually dependent on the real application. If the retrieval model that is used is linear, there may be no scale effects, yet when the retrieval model has a uniform form for all the land covers, for example, if simply mapping the reflected solar radiation to the surface albedo, we can argue that the reflected solar radiation parameterization is scale invariant. Otherwise, the nonlinearity of albedo with topography and spectral dependence of albedo and the reflected solar radiation would be scale dependent. This demonstrates that a different parameterization and different assumptions related to the retrieval model can lead to different conclusions for the same physical process [39]. [PERSON] [4] used a convex hull to judge whether products are overestimated or not. The concave retrieval model overestimates the arithmetic average of values derived independently from fine resolution, while the convex retrieval model underestimates them. [PERSON] _et al_. [34] pointed out that the magnitude of scaling effects increases rapidly with pixel size until the size is larger than the typical length scale of data for the univariate retrieval model. For the bivariate transfer function, the scaling effects are the combined effects of several components, which may add up or compensate for each other. The effects of scale on specific products are theoretically and practically analyzed.