## 4 Principle Laws
### Spatial Dependence  
Spatial dependence is a rule used by nearly all spatiotemporal fusion algorithms. According to [PERSON]'s first law of geography, everything is related but nearby things are more related [74]. Satellite images observe the spectral information of objects on land surface. It is very reasonable to assume that nearby objects have similar spectral characteristics if they are from the same land cover. The pixels of these objects are named as similar pixels in the STARFM method and other STARFM-like methods (Figure 5a) [8; 11; 26]. Similar pixels exhibit spatial dependence, for example, in a forest region, neighboring pixels are more likely from the same tree species and have similar structural attributes, such as tree height and stem density. As a result, these pixels have higher similarity in spectral characteristics, and higher consistency in temporal change across time. In Geostatistics, for similar pixels with a spatial distance h, the spatial dependence can be estimated as covariance C(h) [75]:  
\[C(h)=\frac{1}{n(h)}\sum_{d_{ij}=h}\left[(F_{1}(x_{i},y_{i},b)-\overline{F})(F_ {1}(x_{j},y_{j},b)-\overline{F})\right] \tag{5}\]  
where \(n(h)\) is the number of similar pixel pairs with distance \(h\), \(d_{ij}\) is the spatial distance between \(i\)th and \(j\)th similar pixels, and \(\overline{F}\) is the average value of these similar pixels. Normally, the covariance is different at each spatial distance as depicted by the curve in Figure 5b. The covariance decreases with distance until a saturated point is reached.  
In STARFM and all STARFM-like algorithms, they use the spatial dependence property to reduce the uncertainty and block effect of fusion results, but they do not use covariance in Equation (5) to describe the spatial dependence. For simplification, these algorithms use the inverse distance (1/h) as the alternative of the covariance function. The initial fused result of an individual pixel has a lot of uncertainties due to the error in geo-referencing between two types of images and image noises, however, combining fused results of all similar pixels can average out abnormal prediction of an individual pixel. The spatial dependence is used to weight these similar pixels when they are combined [8; 11]. For methods such as FSDAF, most of the calculations are implemented within each  
Figure 5: (a) Diagram of similar pixels and (b) shape of normal covariance functions.  
coarse pixel, which may lead to the block effect in the fused image, i.e., showing the footprint of coarse pixels in the fused fine image. To mitigate this block effect, predictions of similar pixels from a moving window covering several coarse pixels are combined using spatial dependence as the weight [26].  
In unmixing-based methods, the spatial dependence property of satellite images is used to solve fine-scale endmember values. To successfully estimate the unknown variables in the linear equations (Equation (3)), the number of coarse pixels must be equal or larger than the number of land cover classes. Assuming nearby coarse pixels should have high similarity in endmember values due to spatial dependence, then, a given coarse pixel and its spatially neighboring coarse pixels are employed to compose the equation system in Equation (3) to estimate its fine-scale endmember values [43].