## 2 Materials and Methods
### Diffusive Flux Calculation  
The CO\({}_{2}\) and CH\({}_{4}\) diffusive fluxes at the water-air interface (\(\Phi\)CO\({}_{2}\) and \(\Phi\)CH\({}_{4}\), respectively) were determined on the basis of the measured dissolved gas concentrations (C\({}_{i,w}\), in mol/L), according to the thin boundary layer (TBL) model [61] as follows:  
\[\Phi_{i}=\beta k_{i}\big{(}C_{i,w}-C_{i,\sigma q}\big{)} \tag{1}\]  
where \(C_{i,\sigma q}\) is the dissolved gas concentration calculated assuming equilibrium with the atmosphere (based on gas solubilities as a function of temperature and salinity [62]), \(\beta\) is the chemical enhancement applicable for CO\({}_{2}\) only (see below) and \(k_{i}\) is the gas transfer velocity (in cm h\({}^{-1}\)). The \(k_{i}\) values were estimated as follows:  
\[k_{i}=k_{600,i}\bigg{(}\frac{Sc_{i}}{600}\bigg{)}^{x} \tag{2}\]  
where (i) \(k_{600,i}\) is the \(k_{i}\) value for each gas normalized to 600, (ii) \(x\) is dependent upon the roughness of the water surface (\(-0.67\) or \(-0.5\) for wind speed <3 m s\({}^{-1}\) or >3 m s\({}^{-1}\), respectively [63]) and (iii) \(Sc_{i}\) is the Schmidt number. The \(Sc_{i}\) values were calculated for each gas based on temperature, according to the fourth order polynomial fits proposed by [PERSON] [62]. The \(k_{600,i}\) values were calculated from local wind speed using the following empirical relationships [63; 64]:  
\[k_{600,i}=0.23{U_{10}}^{2}+0.1U_{10} \tag{3}\]  
\[k_{600,i}=0.72U_{10} \tag{4}\]  
where \(U_{10}\) is the wind speed at a height of 10 m. The latter was calculated from average wind speed recorded during the sampling campaigns (Weather Underground company, San Francisco, CA, USA; [[https://www.wunderground.com](https://www.wunderground.com)]([https://www.wunderground.com](https://www.wunderground.com))) as described in [PERSON] and Wanninkhof [63] and references therein.  
The chemical enhancement factor \(\beta\) for the determination of \(\Phi\)CO\({}_{2}\) values was computed according to Hoover and Berkshire [65] and Wanninkhof and Knox [66] as follows:  
\[\beta=\frac{\tau}{(\tau-1)+\frac{tanh\big{[}(r\tau D^{-1})^{0.5}Dk_{\rm CO_{2} }^{-1}\big{]}}{\big{[}\big{(}r\tau D^{-1}\big{)}^{0.5}Dk_{\rm CO_{2}}^{-1} \big{]}}} \tag{5}\]  
where: (i) \(D\) is the molecular diffusivity (in cm\({}^{2}\)/s), calculated as \(D=14.6836\times 10^{-5}\times\big{[}\frac{(273.15+t(^{\circ}C))}{217.2056}-1 \big{]}^{1.997}\)[67] ; (ii) \(r\) (in s\({}^{-1}\)) is the combined rate constant for the hydration of CO\({}_{2}\), calculated as \(r=r_{1}+r_{2}K_{w}^{*}a_{H}^{-1}\), where \(r_{1}\) (in s\({}^{-1}\)) and \(r_{2}\) (in Lmol\({}^{-1}\)s\({}^{-1}\)) are the CO\({}_{2}\) hydration rate and the CO\({}_{2}\) hydroxylation rate constants, respectively [68], \(K_{w}*\) is the equilibrium constant for water and \(a_{H}\) is the activity coefficient for the hydrogen ion; (iii) \(\tau=1+\frac{a_{H}^{2}}{\left(K_{1}^{\prime}K_{2}^{\prime}+K_{1}^{\prime}a_{H} \right)}\), where \(K^{\prime}{}_{1}\) and \(K^{\prime}{}_{2}\) are the first and second equilibrium constants for carbonic acid, respectively [69].  
The overall diffusive flux in terms of CO\({}_{2}\) equivalents (\(\Phi CO_{2\alpha\eta}\)) was calculated as:  
\[\Phi CO_{2 eq}=\Phi CO_{2 eq}+GWP_{CH4}\times\Phi CH_{4} \tag{6}\]  
where \(GWP_{CH4}\) is the global warming potential value of CH\({}_{4}\) over a 100-year time horizon (i.e., 28 [10]).