## 2 Materials and Methods
### Development of the Methodology  
In the present work, we tested the hypothesis that precipitation datasets at a higher resolution than that of the typical GRACE footprint have the information content needed to downscale GRACE TWSA observations. Precipitation is known to have a strong impact on TWS changes and the individual components of TWS, like soil moisture [27; 55]. Precipitation has also been shown to regulate shallow groundwater storage and, to a lower extent, deep groundwater storage [13]. Therefore, we evaluated the ability of a well-known and widely used remotely sensed daily precipitation product, i.e., GPM-IMERG at a spatial resolution of 0.1\({}^{\circ}\), to be used as an auxiliary variable to downscale GRACE TWSA. To account for the time lag and lead between changes in Total Water Storage and precipitation, a cross-correlation at various time lags was performed between TWSA and precipitation. The methodology follows the schematic diagram shown in Figure 2 and was applied at the national level in Greece, where downscaled time series of GRACE TWSA were constructed at the 0.1\({}^{\circ}\) spatial resolution; furthermore, spatial averages were evaluated at the Water District level. Initially, the GPM IMERG dataset was resampled to 1\({}^{\circ}\) to match GRACE TWSA spatial resolution, and their relationship was investigated. Previous works have highlighted the issue of time lags of changes among the various water budget components [15], therefore, a cross-correlation analysis between the changes of GRACE TWSA and GPM precipitation was conducted to determine the time lag(s) that could possibly contribute in the downscaling process. Therefore, the regression equation for each pixel at the coarse spatial resolution will be as follows:  
\[T_{1^{\circ}}^{t}=a_{1^{\circ}}+b_{1,1^{\circ}}P_{1^{\circ}}^{t}+b_{2,1^{ \circ}}P_{1^{\circ}}^{t-1}+\dots+b_{n,1^{\circ}}P_{1^{\circ}}^{t-n} \tag{1}\]  
where \(T_{1^{\circ}}^{t}\) (m) is the ensemble mean of GRACE TWSA at 1\({}^{\circ}\) and time \(t\), and \(P_{1^{\circ}}^{t}\) (m) is the GPM IMERG precipitation at time lags t resampled at 1\({}^{\circ}\), n denotes the number of time lags between TWSA and precipitation observations, \(a_{1^{\circ}}\) is the intercept of the regression equation and \(b_{1,1^{\circ}}\dots b_{n,1^{\circ}}\) are the coefficients of the auxiliary variables at a spatial resolution of 1\({}^{\circ}\). Figure 2: Workflow of the downscaling methodology. Applying Equation (1), an estimate of GRACE TWSA is obtained and the residuals \(\widetilde{T}_{1^{\circ}}^{t}\) were computed by subtracting the original GRACE TWSA data \(T_{1^{\circ}\ (\mathrm{o})}^{t}\) from the estimated \(T_{1^{\circ}}^{t}\) from Equation (1):  
\[\widetilde{T}_{1^{\circ}}^{t}=T_{1^{\circ}}^{t}-T_{1^{\circ}\ (\mathrm{o})}^{t} \tag{2}\]  
Subsequently, Equation (1) is applied at the finer spatial resolution, i.e., 0.1\({}^{\circ}\), to obtain spatially downscaled GRACE TWSA at improved resolution. The intercept and slope of Equation (1) were resampled to 0.1\({}^{\circ}\) using a bilinear interpolation, whereas precipitation observations were used at their native resolution of 0.1\({}^{\circ}\):  
\[T_{0.1^{\circ}}^{t}=a_{0.1^{\circ}}+b_{1,0.1^{\circ}}P_{0.1^{\circ}}^{t}+b_{ 2,0.1^{\circ}}P_{0.1^{\circ}}^{t-1}+\cdots+b_{n,0.1^{\circ}}P_{0.1^{\circ}}^{t -n} \tag{3}\]  
The final step focuses on correcting the downscaled GRACE TWSA using the residual correction approach. Analogous methodology of residual correction was presented previously while downscaling remotely sensed soil moisture [32] and has been shown to considerably improve the accuracy of the downscaled variable. Thus, the downscaled TWSA from Equation (3) was then further processed adding the residuals from Equation (2), resampled at 0.1\({}^{\circ}\) using a bilinear interpolation:  
\[\overline{T}_{0.1^{\circ}}^{t}=T_{0.1^{\circ}}^{t}+\widetilde{T}_{0.1^{\circ}} ^{t} \tag{4}\]  
where \(\overline{T}_{0.1^{\circ}}^{t}\) is the residually corrected TWSA at 0.1\({}^{\circ}\).