## Material and Method
### Methodology
#### 2.3.1 Model Building  
Soil moisture downscaling was realized by the downscaling factor; therefore, the downscaling factor was key to the soil moisture scale. The expression for the downscaling factor is shown below [31]:  
\[F=\frac{SMI_{high}}{SMI_{low}} \tag{1}\]  
where \(F\) is the scale factor, \(SMI_{high}\) is the soil moisture index at a high resolution, and \(SMI_{low}\) is soil moisture at a low resolution. According to previous studies, the latent heat flux index was used as the _SMI_, which was constructed mainly using temperature, vegetation index, and latent heat flux. The expression of the original _SMI_ was as follows [20]:  
\[SMI_{high}=1-\frac{(1-\alpha NDVI_{PROBA-V})(T_{MODIS}-T_{min})}{(1-\alpha NDVI _{PROBA-V})(T_{MODIS\_max}-T_{MODIS\_min})+(T_{e}-T_{MODIS\_min})} \tag{2}\]\[\alpha=1-\frac{Q-Q_{min}}{Q_{max}-Q_{min}} \tag{3}\]  
where \(NDVI_{PROBA-V}\) is the \(NDVI\) based on \(PROBA-V\) data, \(T_{MODIS}\) is the surface temperature corresponding to \(MODIS\) data, \(T_{MODIS\_min}\) is the minimum surface temperature based on \(MODIS\) in the study area, \(T_{MODIS\_max}\) is the maximum surface temperature of \(MODIS\) in the study area, \(T_{e}\) is the maximum surface temperature under complete vegetation cover, \(Q_{max}\) is the maximum latent heat flux on the surface, and \(Q_{min}\) is the minimum latent heat flux on the surface. \(SMI\) relates to soil moisture through latent heat flux. As can be seen from formulas (2) and (3), the surface temperature, vegetation index, and latent heat flux must be obtained to achieve downscaling. As the latent heat flux could not be measured directly, additional inversion experiments were required. Additionally, the \(NDVI\) description of the area with high vegetation coverage was insufficient [20]. Therefore, this study introduced the \(TVDI\) to enhance \(SMI\) performance, which is expressed as:  
\[F=\frac{1-TVDI_{high}}{1-TVDI_{low}} \tag{4}\]  
where \(TVDI_{high}\) denotes the \(TVDI\) in high resolution (0.3 km) using \(PROBA-V\) and \(MODIS\) data, and \(TVDI_{high}\) is the \(TVDI\) at low resolution (9 km) using SMAP and \(PROBA-V\). The \(TVDI\) mainly comprises surface temperature and coefficients \(a\), \(b\), and \(EVI\), among which \(EVI\) and surface temperature are relatively easy to obtain. According to the scatter plot (Figure 2) of surface temperature and the vegetation index, the simulation parameters \(a\) and \(b\) could be estimated using the surface temperature and vegetation coverage with the following expression [32]:  
\[TVDI_{high}=\frac{T_{MODIS}-T_{MODIS\_min}}{a_{1}+b_{1}\times EVI_{PROBA-V}-T_ {min}} \tag{5}\]  
\[TVDI_{low}=\frac{T_{SMAP}-T_{SMAP\_min}}{a_{2}+b_{2}\times EVI_{PROBA-V}-T_ {min}} \tag{6}\]  
In formulas (5) and (6), \(T_{min}\) is the minimum surface temperature in the study area, \(a_{1}\) and \(b_{1}\) are simulation parameters for dry edges, and \(a_{2}\) and \(b_{2}\) are simulation parameters for wet edges. The expression is as follows [33]:  
\[Dry\;egde:\;T_{NDVI\_max}=a_{1}+b_{1}\times Fr \tag{7}\]  
\[Wet\;edge:\;T_{NDVI\_min}=a_{2}+b_{2}\times Fr \tag{8}\]  
\[Fr=\left(\frac{NDVI-NDVI_{min}}{NDVI_{max}-NDVI_{min}}\right)^{2} \tag{9}\]  
where \(Fr\) is the vegetation fraction, \(T_{NDVI\_max}\) is the maximum surface temperature corresponding to the \(Fr\) value, and \(T_{NDVI\_min}\) is the minimum surface temperature corresponding to the \(Fr\) value. \(NDVI_{max}\) and \(NDVI_{min}\) correspond to the maximum and minimum of the \(NDVI\) values, respectively. Figure 2 shows that under conditions of identical vegetation coverage, soil moisture and surface temperature were inversely related [34]. Based on this principle, \(a_{1}\), \(a_{2}\), \(b_{1}\), and \(b_{2}\) can be obtained by extracting the vegetation coverage and surface temperature from satellite images and constructing their characteristic space. Parameters \(a_{1}\) and \(b_{1}\) were obtained by fitting the dry edge, and parameters \(a_{2}\) and \(b_{2}\) were obtained by fitting the wet edge.