Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> Scaling methods for measurements <s> Preface. 1. Introduction. 2. Upscaling. 3. Downscaling. 4. A simple DSS for upscaling and downscaling. Appendix: Random Variables and Stochastic Functions. Glossary. References. Index. <s> BIB001 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> Scaling methods for measurements <s> Abstract Land surface temperature (LST) is a key parameter in numerous environmental studies. Surface heterogeneity induces uncertainty in pixel-wise LST. Spatial scaling may account for the uncertainty, however, different approaches lead to differences in scaled values. Satellite-retrieved LST may be representative of the pixel-wise LST and useful for scaling analysis, but the limited accuracy of retrieved values adds uncertainty into the scaled values. Based on the Stefan–Boltzmann (S–B) law, this study proposed scaling approaches for LST over flat and relief areas to explore the combined uncertainties in scaling using satellite-retrieved data. To take advantage of simultaneous, multi-resolution observations at coincident nadirs by the Advanced Spaceborne Thermal Emission Reflection Radiometer (ASTER) and the MODerate-resolution Imaging Spectroradiometer (MODIS), LST products from these two sensors were examined for part of the Loess Plateau in China. 90-m ASTER LST data were scaled up to 1 km using the proposed approaches, and variation in the LST was generally reduced after scaling. Amongst the sources of uncertainties, surface heterogeneity (emissivity) and different scaling approaches resulted in very minor differences, with a maximum difference of 0.2 K for the upscaled LST. Terrain features, taken as an areal weighting factor, had negligible effects on the upscaled value. Limited accuracy of the retrieved LST was the major uncertainty. The overall LST increased 0.6 K on average with correction for terrain-induced angular effect and 0.4 K for both angular and adjacency effects over the study area. Accounting for terrain correction in scaling is necessary for rugged areas. With terrain correction, the upscaled ASTER LST achieved an agreement of − 0.1 ± 1.87 K and a root mean square error (RMSE) of 1.87 K overall with the 1-km MODIS LST rectified by Wan et al.'s approach [Wan, Z., Zhang, Y., Zhang Q., Li, Z.-L. (2002b), Validation of the land-surface temperature products retrieved from Terra Moderate Resolution Imaging Spectroradiometer data. Remote Sensing of Environment, 83 , 163–180]. Refining the rectification approach resulted in a better agreement of − 0.2 ± 1.57 K and a RMSE of 1.58 K. <s> BIB002
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 BIB002 . These methods were all developed on the basis of the 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. Bierkens et al. BIB001 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.