## 2 Materials and Methods
### Methods
#### 2.2.4 Suitable Grid Scale Evaluation Method  
The selection of grid scale plays a crucial role in both the accuracy of population spatialization results and the representation of population distribution heterogeneity. To evaluate the population estimation results across different grid scales, this study employs two distinct sets of evaluation indicators. The first set consists of three indicators previously discussed: the coefficient of determination (R\({}^{2}\)), mean relative error (MRE), and root mean square error (RMSE), which are used to evaluate the accuracy of the estimation results. The second set incorporates landscape ecology indicators, namely standard deviation of population density (SDPD), Shannon-Wiener diversity index (SWDI), and Simpson diversity index (SIDI), to evaluate the heterogeneity of population spatial distribution [40]. Notably, higher values of SDPD, SWDI, and SIDI indicate superior population density differentiation and, consequently, a more appropriate grid scale selection. (1) Standard deviation of population density (SDPD)  
The standard deviation is a statistical metric that measures the dispersion degree of a data series. By treating the pixel values of population density grid data as numerical sequences, the standard deviation of population density effectively reveals the variation in population density across different grid scales. (2) Shannon-Wiener Diversity Index (SWDI)  
The Shannon-Wiener diversity index is a biodiversity metric grounded in information theory principles. This index comprehensively integrates both species richness and evenness to quantify ecosystem diversity levels, in which higher values indicate greater species abundance within an ecological system. In this study, we adapt the Shannon-Wiener diversity index to assess population distribution heterogeneity, calculated as follows:  
\[SWDI_{k}=-\sum_{i=1}^{m}[T_{ik}\times\ln(T_{ik})] \tag{12}\]  
where \(SWDI_{k}\) indicates the Shannon-Wiener diversity index for different grid scales \(k\) (\(k\) = 100 m, 200 m, 300 m, and 500 m); \(m\) represents the number of different population density types; and \(T_{ik}\) denotes the ratio of population density type \(i\) at grid scale \(k\) relative to the total amount of all density types. \(SWDI_{k}\) is always greater than or equal to zero, with no upper limit. When \(SWDI_{k}\) = 0, it indicates that there is only one population density type in the study area. As the number of population density types increases, or as the proportions of various density types become more balanced, the \(SWDI_{k}\) value increases,indicating greater variation in population distribution and improved differentiation across the region. (3) Simpson Diversity Index (SIDI)  
The Simpson diversity index is an indicator for assessing species diversity within an ecosystem. It quantifies the probability that two randomly selected individuals from a community belong to the same species. A higher index value indicates a greater species richness, more equitable distribution among species, and overall enhanced community diversity. The calculation formula is as follows:  
\[SIDI_{k}=1-\sum_{i=1}^{\text{m}}T_{ik}^{2} \tag{13}\]  
where \(SIDI_{k}\) represents the Simpson diversity index of population spatial distribution at scale \(k\). The SWDI and SIDI were originally designed as landscape pattern assessment metrics using discrete data types. However, our multi-scale population density simulation data consist of continuous variables, necessitating a discretization process for the population density data. For this purpose, the minimum scale of 100 m resolution data was selected as the baseline. With an interval of 1 person/10,000 m\({}^{2}\), the data were divided into 447 categories, and each category represents a population density value. Keeping the number of categories constant, the continuous population density data at other scales were classified into corresponding categories based on their unit population density values, resulting in discrete population density data at different scales.