## 2 Materials and Methods
### Methodology
#### 2.3.2 Ist-Cube Model
Figure 3: Methodological flowchart for the (**a**) annual composite and reconstruct and (**b**) merging the IST-cube in the spatial domain. MSOMB = minimum possible side length of the minimum bounding box. (**c**) Merging the IST-cube in the temporal domain. where \(\omega_{shape}\) ranges between 0 and 1, as decided by the user. In general, \(\omega_{shape}\) is a small number that reduces the influence of the shape while spectral has a large weight to scale-up its effect [64]. In this study, the value of \(\omega_{shape}\) was set as 0.1 which is a default value [65; 66]. The weight of the sum of \(H_{color}\), \(H_{EVI}\), and \(H_{NDVI}\) is determined by 1-\(\omega_{shape}\). Here, \(H_{shape}\) determines the shape difference of the IST-cube before and after the merger. In the same manner, \(H_{color}\) denotes the spectral difference, as measured by the difference in the standard deviation of all bands before and after merging. \(H_{EVI}\) and \(H_{NDVI}\) indicate the differences in the vegetation function or greenness, as measured by the difference in the standard deviation of the EVI and NDVI before and after merging. \(H_{EVI}\) and \(H_{NDVI}\) were calculated as follows:  
\[H_{EVI}=\omega_{EVI}\Big{[}n_{cb}\cdot\Delta T_{cb}\cdot h_{EVI}^{cb}-\Big{(} n_{cb1}\cdot\Delta T_{cb1}\cdot h_{EVI}^{cb1}+n_{cb2}\cdot\Delta T_{cb2} \cdot h_{EVI}^{cb2}\Big{)}\Big{]}\text{ and} \tag{4}\]  
\[H_{NDVI}=\omega_{NDVI}\Big{[}n_{cb}\cdot\Delta T_{cb}\cdot h_{NDVI}^{cb}- \Big{(}n_{cb1}\cdot\Delta T_{cb1}\cdot h_{NDVI}^{cb1}+n_{cb2}\cdot\Delta T_{ cb2}\cdot h_{NDVI}^{cb2}\Big{)}\Big{]}, \tag{5}\]  
where \(h_{EVI}\) and \(h_{NDVI}\) are the variance in the EVI and NDVI of the IST-cube, respectively, and \(\omega_{EVI}\) and \(\omega_{NDVI}\) are the weights of EVI and NDVI, respectively. For the purpose of forest change detection, this study set \(\omega_{EVI}\) and \(\omega_{NDVI}\) as five times the weight of other bands and set \(\omega_{EVI}\) and \(\omega_{NDVI}\) as 1.5625 in order to maintain the range of spatiotemporal heterogeneity criterion consistent with the original model. \(\sigma_{EVI}^{i}\) and \(\sigma_{NDVI}^{i}\) indicate the standard deviation of the EVI and NDVI at time \(i\), respectively, as follows:  
\[h_{EVI}=\frac{\sum_{i=1}^{T}\sigma_{EVI}^{i}}{T}\text{ and} \tag{6}\]  
\[h_{NDVI}=\frac{\sum_{i=1}^{T}\sigma_{NDVI}^{i}}{T}, \tag{7}\]  
where \(n\) is the number of pixels in the spatial domain of the IST-cube at time \(i\), \(p_{jEVI}\) and \(p_{jNDVI}\) are the EVI and NDVI values of the jth pixel, respectively, and \(\overline{p}_{EVI}\) and \(\overline{p}_{NDVI}\) are the average EVI and NDVI values of \(n\) pixels, respectively. Next, \(\sigma_{EVI}^{i}\) and \(\sigma_{NDVI}^{i}\) were calculated as follows:  
\[\sigma_{EVI}^{i}=\sqrt{\frac{\sum_{j=1}^{n}\left(p_{jEVI}-\overline{p}_{EVI} \right)^{2}}{n}}\text{ and} \tag{8}\]  
\[\sigma_{NDVI}^{i}=\sqrt{\frac{\sum_{j=1}^{n}\left(p_{jNDVI}-\overline{p}_{NDVI} \right)^{2}}{n}}, \tag{9}\]  
where \(H_{t}\) refers to the average spectral variance in the IST-cubes in the temporal domain and is normalized as follows:  
\[H_{t}=\frac{\sum_{i=1}^{n}\left(\sum_{k=1}^{m}\omega_{k}\cdot\sigma_{ik}^{i}+ \omega_{EVI}\cdot\sigma_{IEVI}^{i}+\omega_{NDVI}\cdot\sigma_{INDVI}^{i}\right) }{n}, \tag{10}\]  
where \(m\) represents the number of bands in the Landsat image, \(n\) is the number of pixels at the bottom surface of an IST-cube, and \(\sigma_{EVI}^{i}\) and \(\sigma_{NDVI}^{i}\) refer to the EVI and NDVI standard deviations in the temporal pixel column formed by pixels of all time point layers in the \(i\)-th pixel of the spatial domain, respectively. The normalization of \(\sigma_{EVI}^{i}\) and \(\sigma_{NDVI}^{i}\) was calculated as follows:  
Figure 4:0  
Figure 4:1Similarly, \(\sigma^{i}_{EVI}\)_and_\(\sigma^{i}_{NDVI}\), indicate the EVI and NDVI standard deviations of the temporal pixel column formed by the pixels of all time point layers in the \(i\)-th pixel of the spatial domain, respectively.