## 2 Methods and Data  
### Spatial Downscaling Based on ATPRK  
ATPRK was employed as a standard spatial downscaling method in this study because of its great potential for spatial downscaling. It is a multivariate version of ATPK proposed initially to predict fine-scale attribute values from areal data [28]. It is a hybrid method combining regression-based modeling with ATPK-based residual correction [2] and was named by [PERSON] et al. [21; 29]. Its implementation requires coarse resolution data and fine resolution auxiliary variables associated with the target attribute.  
The whole procedure for ATPRK is illustrated in Figure 1. In this section, the main theoretical background and application procedures of ATPRK are briefly provided. Suppose that the attribute values of coarse resolution data (\(z^{\mathrm{c}}(\mathbf{v})\)) are available in the study area of interest, where \(\mathbf{v}\) denotes a coarse resolution pixel. In theory, ATPRK regards the target variable as a random variable that is decomposed into TC and RC as:  
\[z^{\mathrm{c}}(\mathbf{v})=T^{\mathrm{c}}(\mathbf{v})+R^{\mathrm{c}}(\mathbf{ v}), \tag{1}\]  
where \(T\) and \(R\) are the TC and RC at a coarse resolution, respectively.  
By applying the same decomposition to the attribute at a fine resolution, the attribute value at a fine resolution location \(\mathbf{u}\) (\(\mathbf{z}^{\mathrm{F}}(\mathbf{u})\)) within a coarse resolution pixel is predicted as the sum of TC and RC predicted at a fine resolution:  
\[\mathbf{z}^{\mathrm{F}}(\mathbf{u})=\mathbf{T}^{\mathrm{F}}(\mathbf{u})+ \mathbf{\dot{R}}^{\mathrm{F}}(\mathbf{u}), \tag{2}\]  
Figure 1: Workflow of ATPRK for spatial downscaling with fine resolution auxiliary variables.  
where \(\hat{T}^{\text{F}}\) and \(\hat{R}^{\text{F}}\) are the TC and RC predicted at a fine resolution, respectively.  
The TC at a fine resolution is predicted using regression modeling between the target attribute and the auxiliary variables. Prior to predicting the TC at a fine resolution, the quantitative relationships between the target attribute and auxiliary variables are first derived at a coarse resolution since the target attribute is available only at a coarse resolution. When there are \(N\) auxiliary variables at a fine resolution in the study area (\(y^{\text{F}}\)), the fine resolution auxiliary variables to the coarse resolution are first upscaled to the coarse resolution. Regression modeling is then conducted to quantify the relationship between the target attribute and the upscaled auxiliary variables. Finally, the TC at a fine resolution (\(\hat{T}^{\text{F}}(\mathbf{u})\)) is predicted using the relationship modeled at a coarse resolution and the auxiliary variables at a fine resolution, under the assumption that the quantitative relationships between the target attribute and auxiliary variables remain unchanged across spatial resolution [2]:  
\[\hat{T}^{\text{F}}(\mathbf{u})=f({y_{i}}^{\text{F}}(\mathbf{u})),\ i=1,\cdots,N, \tag{3}\]  
where \(f(\cdot)\) denotes the regression function applied at a coarse resolution and \(y_{i}^{\text{F}}(\mathbf{u})\) is the \(i\)th auxiliary variable at a fine resolution.  
As regression modeling is conducted at a coarse resolution, the RC at a fine resolution (\(\hat{R}^{\text{F}}(\mathbf{u})\)) in Equation (2) has to be predicted from the coarse resolution RC via another estimation procedure. ATPK has great potential for area-to-point predictions because of its ability to explicitly account for scale differences between input data and output results [2]. This study employed ATPK to predict the RC at a fine resolution. The theory and detailed explanations of ATPK can be referred to in [PERSON] [28] and some previous studies [7; 29].