## 2 Materials and Methods
### Methodology
#### 2.3.2 Ist-Cube Model  
The ST-cube model proposed by [PERSON] [31] has been used and improved in this study. We combined the ST-cube and vegetation indices to develop IST-cube. The essence of this model is to merge IST-cubes or pixels. Before merging, the model successively performs a merger judgment with adjacent IST-cubes, finally merging with the adjacent IST-cube that meets the merger conditions and has the smallest heterogeneity difference. It has been demonstrated that the NDVI provides important information that can be used to estimate vegetation photosynthesis activity and detect vegetation phenology [54; 55]. At the same time, previous studies have established that the EVI can be used to estimate the gross primary production (GPP) [56; 57; 58]. The EVI has been widely used to monitor the conditions of plants [59; 60], which can be calculated using blue and red reflectance bands, as well as near infrared. Both indices have a range between -1 and 1 [61]. Furthermore, previous studies have experimentally demonstrated that using multiple indices yields better performance than the use of a single spectral index [25; 62]. Consequently, we chose the parameters and threshold reported in a previous study [31] while adding the EVI and NDVI. Figure 3 presents these details. A cell cube is the smallest computing unit in the IST-cube model. In this study, a cell cube is defined as a pixel, as in the original model [31], which is determined by the feature that raster data are composed of pixels in a matrix structure. Figure 3a shows that this model reconstructs satellite images into cell cubes before the IST-cube merging. The first step of the IST-cube model (Figure 3b) merges the IST-cube or pixel in the spatial domain every year. This merging method is similar to the region growing method [63]. During this merge, five indicators [i.e., the perimeter of the IST-cube, minimum possible side length of the minimum bounding box (MSOMB) of the IST-cube, standard deviation of the band value, standard deviation of the NDVI, and standard deviation of the EVI] were included in the decision on whether to merge. As shown in Figure 3c, the second step of the IST-cube model merges the IST-cube with adjacent years in the temporal domain. In this study, a two-step procedure (first merge in the spatial domain, followed by merge in the temporal domain) was used for IST-cube merging. Two IST-cubes must satisfy two conditions to make a merge decision in the temporal domain: 1. they must be adjacent in the temporal domain and 2. they must have at least one pixel with the same X, Y coordinates, as shown in Figure 4, where the red, yellow, and blue IST-cubes represent individual cubes resulting from merging in the spatial domain. Figure 4a shows the red and blue IST-cubes in a time merge judgment because they satisfy two conditions required for a merge decision in the temporal domain. Similarly, Figure 4b displays the result after the red and blue IST-cubes merge into one IST-cube, which is represented by the red IST-cube. The red and yellow IST-cubes then form a time merge judgment. Finally, the yellow and red IST-cubes merge. Figure 4 illustrates that two different IST-cubes can be combined into a single cube by merging with other adjacent IST-cubes in the temporal domain. Based on a previous study [31], we set the spatial heterogeneity value, \(S_{h}\), as the sum of the weighted combination of the spectra, shape, EVI, and NDVI:  
\[S_{h}=\omega_{shape}\cdot H_{shape}+\Big{(}1-\omega_{shape}\Big{)}\cdot(H_{ color}+H_{EVI}+H_{NDVI}), \tag{3}\]  
Figure 4: Schematic merging of the spatial results for the IST-cube in the temporal domain. Red, yellow, and blue cubes represent individual cubes resulting from merging in the spatial domain. (**a**) Red and blue IST-cubes in the merge judgment in the temporal domain; (**b**) red and yellow IST-cubes in the merge judgment in the temporal domain; and (**c**) final merge into a single cube.