## 2 The definition of scale and relevant terminologies  
### Notions of scales  
In the field of scientific research, scale is often one of most involved terminologies. The scale can refer both to the magnitude of a study (e.g., its geographic extent) and also to the degree of detail (e.g., its level of geographic resolution) [1]. There are several terms that share the same meaning with observation scale that describe the detail, such as resolution, grain, support and so on. With remote sensing, scale might be resolution and can be thought of as the smallest objects being distinguished by sensors. For ecology, scale is likely to be grain, which is the measured size of patches. To environmental research, scale could be support, the largest area or time interval in which the parameter of interest is homogeneous [15]. To cartography, scale may be defined simply as the ratio between distance on the map and on the ground. There is much confusion and abuse concerning the term \"scale\" and it is necessary to identify the meaning of this word. [PERSON] and [PERSON] [13] and [PERSON] _et al_. [3] pointed out that at least four meanings of scale can be identified within the spatial domain. They are the observation scale, the operational scale, the geographic scale and the cartographic scale. [PERSON] and [PERSON] [5] and [PERSON] _et al_. [15] proposed another two meanings of scale: the modeling scale and the policy scale, respectively. From analysis, modeling and demonstrating perspective, scale needs to be divided into at least such six types. Here are the definitions used by the authors [3,5,13,15] with some modifications combined. Table 1 shows a comparison of the six meanings of scale used in various fields of scientific research. \begin{table}
\begin{tabular}{c c c} \hline
**Meaning** & **Description** & **Remarks** \\ \hline Observation & The measurement units at which & Referring to the description of resolution, time interval, spectral range, solid angle or polarization direction. \\ scale & data is measured or sampled & In order to better reveal the process, the modeling scale \\ \hline Modeling & The scale at which the model is & In order to better reveal the process, the modeling scale should be coincided with both the observation scale and the operational scale. \\ scale & built or derived & In order to better reveal the process, the modeling scale should be coincided with both the observation scale and the operational scale. \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the six meanings of scale used in the field of scientific research. 1. Observation scale can be called a \"measurement scale\". It depends on the method or the characteristics of the instrument and can be thought of as measurement units (i.e., intervals or areas or volumes) at which data is measured or sampled. To remote sensing, the measurement scale refers to the description of resolution, time interval, spectral range, solid angle or polarization direction. As the limitation of data collection and storage capacity, the smaller measurement scale usually corresponds to the smaller geographic scale and vice versa. 2. Modeling scale is the scale at which the model is built or derived in order to give reliable output. Both the measurement scale and the operational scale may influence the modeling scale. Observations sampled at a measurement scale are used as input for models, so the measurement scale must coincide with the modeling scale. If the measurement scale is smaller or larger than the modeling scale, it should be scaled. Again, a model needs to reveal the process; the modeling scale should also coincide with the operational scale. Similarly, it also needs to be scaled. 3. Operational scale refers to the scale at which a certain process is supposed to operate. It can also be called the \"scale of action\". For example, thunderstorms may happen in an area of dozens of square kilometers. The operational scale of thunderstorms may be dozens of kilometers.