## 2 The definition of scale and relevant terminologies
### Characteristics of scales  
Although \"scale\" is a widely used term and has different meanings in various disciplines, in general, it can be thought of as having multi-dimensionality, complexity and variability. Firstly, scale has a multi-dimensional nature [11]. It can be expressed as the variability in space and the expansibility in time [1]. Consequently, the scale is divided into spatial scale and temporal scale, respectively, based on the different research targets or fields. As the observations of remote sensing are more specialized compared to conventional measurement methods, the scale notion can be extended to spectral scale, directional scale and polarization scale. The spectral scale expresses the ability to discriminate fine spectral differences and refers to the full width at half maximum (FWHM) of band. The directional scale describes the angular geometry of the sun-object-sensor system [11]. The polarization scale refers to the polarization direction of the signal. In general, the spatial and temporal scales are closely related. Then, within a specific discipline, the processes that operate over larger temporal scales may also operate over larger spatial scales due to the transport mechanisms between space and time [1]. [PERSON] and [PERSON] [5] demonstrated a scheme to classify hydrological processes according to typical space and time scales. [PERSON] _et al_. [20] used the ratio of spatial scale and temporal scale to discriminate the different processes in a space-time coordinate system. In reality, all sensors boarded on spacecrafts are a kind of \"hand-made\" equipment. The spatial resolution, the bandwidth and the spectral response function are not the same. The difference between sensors must be analyzed first. That is to say, both spatial and spectral scaling is necessary when comparing the retrieval products of different sensors. Even if the resolutions of the sensors are close to each other, i.e., MODIS and AVHRR, spectral scaling is needed. Such a point of view may improve our understanding of scale effects.  
Secondly, scale has complex hierarchies. It is the reflection of the level of the organization of nature, which results in the research targets varying with scales. For example, small-scale hydrological studies may mainly focus on the scale of vegetation and soil; meso-scale studies may focus on the response of the hydrology unit to the changes of land surface; while large-scale studies may be mainly about the interaction of the atmosphere and the land surface. These phenomena and processes occurring at different scales may interact with each other, consequently, many regional or global changes, such as pollution, the greenhouse effect and biodiversity may be rooted in local scale or small-scale environmental problems. Similarly, large-scale changes (such as global climate change and ocean circulation anomalies), in turn, will influence small-scale phenomena and processes. This shows that both large-scale and small-scale studies are equally important. Large-scale describes the abstract features or macro-structure, and small-scale characterizes the details. On the one hand, we can understand the macro-changes and the general trend through large-scale studies; on the other hand, we may find the mechanism of the development of the process and give reasonable explanations through small-scale studies.  
Finally, scale may also have variability, that is, the targets at different scales will show different characteristics. The isothermal surface would become non-isothermal. The spectral curve of emissivity may become smoother when the spectral scale is coarser. It has increased the difficulty of scale analysis.