## 2 Method
### Definition of the Modelled Geophysical Gravity Signals-South America
#### 2.2.3 Soil Moisture Variations over SA  
We employed the GLDAS product [25] to predict the temporal variations in soil moisture and the effects on the gravity field over South America. GLDAS reports the total water mass in the first 2 m of soil; it is a global model, with several spatial and temporal resolutions. We employed monthly models with 0.25\({}^{\circ}\) of spatial resolution.  
For the scopes of the sensitivity analysis, the GLDAS model was analyzed together with a database of hydrologic basins outlines in order to obtain soil moisture time variations for each hydrologic basin. This allowed us to express the sensitivity in terms of the smallest basin, which can be monitored. The outlines and names of hydrologic basins are derived from the watershed database of the USGS (HydroSHEDS [26]), which maps the principal drainage systems worldwide from the analysis of SRTM products at 500-m resolution. The watersheds for South America are shown in the following Figure 3b in grey, while three basins, which hereinafter are analyzed in detail, are reported with three different colors (green--Orinoco; yellow--Amazon; red--Parana).  
From these databases, we calculated the gravity variations at 250 km above ground level employing a tesseroid discretization.  
Once we obtained the 4-D gravity and mass models for the basins, we extracted amplitude and long-period trends by fitting every gravity/moisture time-series at a given longitude and latitude with the following relation (Equation (1)):  
\[y(t)=k+mt+\sum_{i=1}^{2}a_{i}\cos\biggl{(}i\frac{2\pi t}{365.25}\biggr{)}+b_ {i}\sin\biggl{(}i\frac{2\pi t}{365.25}\biggr{)} \tag{1}\]  
Figure 3a,b show the amplitude of the seasonal component and the long-period trend of the gravity effect of the GLDAS. The side-plots report the mass variations as functions of longitude and latitude. We see that seasonal effects are clearly larger compared to long-period variations (about a factor 100) with respect to both the mass and gravity variations. The largest variations are mostly related to the three basins considered; the Amazon in particular is associated with both strong seasonal signals and a long-term depletion signal. Parana and Orinoco show important seasonal variations but generally less pronounced long-period trends. This is also more evident when observing the mass variations of the single basins (last three plots of Figure 3).  
Figure 3: (**a**) Amplitude signal of the GLDAS for South America. (**b**) Long period (LP) gravity signal of the GLDAS model; colored lines report the outlines of the Paraná (red), Amazon (yellow), and Orinoco (green) basins. Plots on the side and bottom report for both the maps the seasonal and LP hydrologic mass changes as functions of longitude and latitude. Central plot: long-period signal according to GLDAS for the Amazon basin; top right: seasonal annual amplitude of the Amazon. Bottom plot reports the time-series of the mass variations of the Orinoco, Amazon, and Paraná basins. Blue curve = observed mass variations for GLDAS; red curve = fitted mass variations of the basin with Equation (1).