3.2.3. GWR Downscaling

Since the GWR model was first proposed by Brunsdon et al. [36] in 1996, it has been extensively applied in research on spatial heterogeneity [17–20,37,38]. The basic idea of GWR is that the relationship between variables varies with changes in spatial location; thus, a regression model can be established by estimating the parameters of the correlated variables and explanatory variables at each given location in the study area. Figure 3 shows the spatial distributions of the intercept, NDVI regression coefficient, and local R<sup>2</sup> obtained via GWR, and these values are in accordance with the definition. These parameters exhibit significant spatial variations. The intercept ranges from -78.7 to 178.4, the NDVI coefficient ranges from -123.3 to 203.4, and the local R2 ranges from 0.01 to 0.92.

**Figure 3.** The spatial distributions of the (**a**) intercept, (**b**) slope of the NDVI and, (**c**) local R<sup>2</sup> in July 2016.

In this study, a GWR regression model was established based on the NDVI and the IMERG precipitation data as follows:

$$\mathcal{Y}\_{\dot{\jmath}} = \beta\_0 \{ u\_{\dot{\jmath}}, v\_{\dot{\jmath}} \} + \sum\_{i=1}^p \beta\_i \{ u\_{\dot{\jmath}}, v\_{\dot{\jmath}} \} X\_{\dot{\jmath}\dot{\jmath}} + \varepsilon\_{\dot{\jmath}} \tag{4}$$

where *Yj* is the IMERG precipitation at point *j*; *Xij* is the NDVI at point *i* in the vicinity of point *j*; β0 - *uj*, *vj* and β*<sup>i</sup>* - *uj*, *vj* represent the intercept and slope, respectively, at point *j*; - *uj*, *vj* represents the two-dimensional coordinates of point *j*; and ε*<sup>j</sup>* is the residual error. Unlike traditional global regression models, Equation (4) is based on the assumption that the shorter the distance between the observation point and point *j* is, the greater the influence on point *j* will be, with the coefficient acting as a damping function that depends on the distance from point *j*. This damping function can be obtained in accordance with Equation (5):

$$\hat{\beta}(\boldsymbol{u}\_{\boldsymbol{\uprho}}, \boldsymbol{v}\_{\boldsymbol{\uprho}}) = \left(\boldsymbol{X}^{T} \{\mathcal{W}(\boldsymbol{u}\_{\boldsymbol{\uprho}}, \boldsymbol{v}\_{\boldsymbol{\uprho}})\} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{T} \mathcal{W}(\boldsymbol{u}\_{\boldsymbol{\uprho}}, \boldsymbol{v}\_{\boldsymbol{\uprho}}) \boldsymbol{Y} \tag{5}$$

where βˆ *uj*, *vj* represents the coefficient of point *j*; *X* and *Y* are the independent and dependent variables, respectively; and *W*- *uj*, *vj* is a weight matrix. This matrix ensures that the shorter the distance between points *i* and *j* is, the greater the weight, and the elements of the matrix can be obtained as follows:

$$\begin{aligned} w\_{ij} &= \left[1 - \left(d\_{ij}/b\right)^2\right]^2 \text{ when } d\_{ij} \le b\\ w\_{ij} &= 0 \text{ when } d\_{ij} > b \end{aligned} \tag{6}$$

where *dij* is the distance of point *j* from the nearby observation point *i,* and *b* is a fixed threshold defined in terms of a distance metric.

In detail, the following procedures were applied for GWR-based downscaling (Figure 2).

(1) To effectively establish the precipitation-NDVI model, anomalous NDVI areas corresponding to snow and water bodies were removed from the high-spatial-resolution NDVI data [17,39].

(2) After the removal of outliers, the 1-km NDVI data were aggregated to a resolution of 10 km by means of pixel averaging. Then, a GWR model of the 10-km IMERG data and the 10-km NDVI data was established with the NDVI as the independent variable and the IMERG precipitation data as the dependent variable. By introducing 1-km and 10-km grid point coordinates into the GWR model, the constants and corresponding coefficients for the 1-km and 10-km NDVI were obtained, as shown in Equation (4).

(3) The 10-km NDVI data were entered into the regression model to obtain NDVI-based precipitation predictions with a 10 km resolution (Predicted Precipitation 10 km in Figure 2).

(4) The residual errors between the values predicted by the 10-km resolution model and the original IMERG precipitation values were calculated (Residuals 10 km in Figure 2). The 10-km residuals were then transformed into 1-km residuals through spline interpolation (Residuals 1 km in Figure 2).

(5) The 1-km NDVI data after anomalous data removal were used to force the regression model, thus obtaining 1-km model-predicted precipitation values. Spline interpolation was then applied to fill in the values missing after outlier removal to obtain downscaled data (DS\_CIMERG in Figure 2).

(6) The 1-km model-predicted precipitation values and 1-km residual data were summed to obtain post-residual-corrected 1-km downscaled precipitation data.
