## 5 Overview of general scaling methods
### Scaling methods for measurements  
Scaling methods for measurements are easier to deal with because the measurements recorded by remote sensors usually capture the radiance emitted or reflected by the surface. In such cases, the Area-Weighted Scaling Methods (AWM) may be applicable in a flat region. Otherwise, the influence of slope angle should be taken into account [22]. These methods were all developed on the basis of the  
\begin{table}
\begin{tabular}{c c c c} \hline WTM & It can investigate features of interest in the data set at an appropriate scale and find the length scale of the variability. & - The dimension of data set must be the exponent of 2. & [16] \\  & The manner of WTM is dependent on the mother wavelet. & [60] \\ \hline LVM & The principle is easy to be understood. & - It is unrealistic to assume the pixel value of a coarse resolution image is simply the average of finer resolution pixels within the corresponding coarse pixel. & [61] \\  & & - It is dependent on the global variance in the image and the values of local variance cannot be directly compared between different images & [19] \\ \hline SVM & - It can be used to judge whether the geographical scale is large enough to detect the length scales of the landscape. & The second order stationarity hypothesis should be satisfied. & [19] \\  & - The loss of image spatial variability at a given spatial resolution can be estimated. & [63] \\ \hline FM & - It has a theoretical basis that many curves or surfaces in the world may show the statistical self-similar property. & No agreement has been reached on the definition of fractal dimension which can be used to determine the characteristic scale. & [13] \\  & - The more irregular an object, the bigger the fractal dimension. The turning points of fractal dimension may contain some important information. & [67] \\ \hline \end{tabular}
\end{table}
Table 2: Cont.  
law of conservation of energy or matter. If the retrieval models at both local and large scale are known in priori, the scaling methods mentioned above would work well. However, if the retrieval model at one scale, either local scale or large scale is known, the retrieval model derived from one scale cannot be guaranteed to be used at the other scale. Therefore, AWM alone is insufficient; correction items to the area-weighted measurements need to be added in order to scale products correctly. This process could be thought of as finding the representative measurements corresponding to a large scale. The aim of these methods is to get the scale invariant result at large scale when the local scale retrieval model is used. [PERSON] _et al_. [15] proposed a scaling method, Finding Representative Parameters Method (FRPM), and gave an example of finding the representative conductivity for blocks in a numerical model of groundwater flow. Through the comparison of model output at both local scale and large scale, the representative conductivity can be thought of as the sum of two items. One is the arithmetic block average of parameters. Another is the block covariance of parameters. Through numerical analysis and comparison, we may give the analytical solution to the representative measurements in remote sensing. Then, the correction items may be the combined effects of the nonlinearity of the retrieval model and the heterogeneity of measurements.