## 2 Data and Methods
### Identifying the Effect of GNSS-IWV and CAPE on Extreme Rainfall Formation  
Understanding the conditions leading to extreme rainfall events is difficult, because of complex, interfering atmospheric processes in the eastern central Andes. Previous research indicates that extreme rainfall in the south-central Andes is often caused by deep convective storms [5; 10; 46]. An analysis requires the investigation of the dominant climatic variables leading to extreme rainfall events with reliable data. In this study, we analyse the joint effect of GNSS-IWV and CAPE on extreme rainfall generation. First, we analyse the rainfall distribution for each GNSS-IWV station (TUCU and CATA, see Figure 1A) in the south-central Andes using the complementary cumulative distribution function (CCDF) and by comparing the best-fit parameters. Second, we investigate the seasonal behaviour and the fluctuations in the GNSS-IWV and CAPE in conjunction with rainfall for both stations (TUCU and CATA) on the daily  
Figure 1: (**A**) Topographic overview (from [45] data) of the study region in the central Andes in north-western Argentina with the outline of the internally-drained central Andes in white (see inset for location in South America with the internally-drained central Andes shown in red). Black lines are international borders. White stars show the GNSS stations used in this study: San Miguel de Tucumán (TUCU, \(n=15\) years) and San Fernando del Valle de Catamarca (CATA, \(n=6\) years). (**B**) Annual mean rainfall derived from TRMM 3B42 (1999–2013) shows rainfall distribution and the location of the GNSS station locations (white stars). scale. We use the wavelet coherence analysis to confirm the seasonal agreement between rainfall, GNSS-IWV, and CAPE. Third, we analyse the relation between the daily mean of GNSS-IWV and rainfall for both station locations (TUCU and CATA). We perform a curve fitting process to model the rainfall as a function of the GNSS-IWV. After fitting several different models to the data, we found that the data were best fitted by an exponential model Equation (2) based on model fitting statistics (root mean squared error, R-squared value, statistic test for the \(F\)-test on the regression model, and \(p\)-value). \[rainfall=\exp(\alpha*\text{GNSS-IWV}) \tag{2}\]  
where \(\alpha\) is the regression coefficient for GNSS-IWV. We take the natural logarithm of both sides of the equation to linearize it:  
\[\ln rainfall=\alpha*\text{GNSS-IWV} \tag{3}\]  
Fourth, we analyse the correlation between the daily mean of rainfall and CAPE at both station locations (TUCU and CATA) using a power-law relationship as described in previous research based on parcel theory [4; 26; 27]. Based on parcel theory, we expect the rainfall intensity to be commensurate to \(\sqrt{CAPE}\) (\(\beta=0.5\)), if CAPE is efficiently transferred to parcel kinetic energy [26; 27]. \[rainfall=\text{CAPE}^{\beta} \tag{4}\]  
where \(\beta\) is the regression coefficient for CAPE. After data linearisation, the following equation is substituted to describe the parcel theory:  
\[\ln rainfall=\beta*\ln\text{CAPE} \tag{5}\]  
Fifth, we generate the regression model Equation (6), which shows a log-linear relationship between the daily mean rainfall and both variables at both station locations. \[\ln rainfall=c+\alpha*\text{GNSS-IWV}+\beta*\ln\text{CAPE} \tag{6}\]  
where \(\alpha\) and \(\beta\) are the regression coefficients for GNSS-IWV and CAPE, respectively. Our approach and the formulations are based on previous studies for extreme rainfall [4; 26; 27]. Sixth, we compare the contribution of each variable to the rainfall extreme events in the presence of each other by joint regression of both variables using quantile regression [47]. We then test the goodness of our model based on the goodness of fit criterion for a certain quantile [47].