## 2 Materials and Methods
### The BRIDGE DIAL Data Analysis Technique
#### 2.2.5 Evaluation of Uncertainty Sources  
The main sources of uncertainty that affected our CO\({}_{2}\) flux measurements are reported below [39,41]:  
1. Systematic uncertainty of the carbon dioxide concentration measurement. Taking into account the experience [21,41], the systematic uncertainty of the BILLI system was dominated by inaccuracies in wavelength setting [39], and this led to errors in differential absorption cross section and, as a consequence, in CO\({}_{2}\) mixing ratio. In this regard, in order to control the transmitted wavelength before each atmospheric measurement, a photo-acoustic cell filled with pure CO\({}_{2}\) (at atmospheric pressure and temperature) was implemented close to the laser exit. This allowed setting the ON/OFF wavelengths used in this study with an accuracy better than the laser linewidth [39], with an overall uncertainty in the wavelength setting equal to \(\pm\) 0.02 cm\({}^{-1}\) (half laser linewidth). For these reasons, the systematic uncertainty of the CO\({}_{2}\) concentration measurement was below 5.5% [41,42].
2. Statistical uncertainty of the carbon dioxide concentration measurement. Starting from the calculation of the standard deviation of the signals corresponding to every ADC channel, and implementing the standard uncertainty propagation technique [48], it has been possible to evaluate the statistical uncertainty. This uncertainty varies as a function of distances from the system, considering typical atmospheric and plume conditions encountered during field campaigns. Typical values at 2.5 km and over 4 km from the system were estimated to be equal to 2% and slightly above 5%, respectively [41,42].
3. Uncertainty linked to plume transport velocity. Mean and standard deviation of the wind velocity were calculated, also in this case by the standard uncertainty propagation technique. Then, the corresponding relative uncertainty was evaluated [48]. The assessed values ranged from 3% (at Stromboli [41]) to 19% (at Pozzuoli [21]).
4. Uncertainty in the characterization of the integration area. Due to difficulties in accurately assessing the surface where there is an excess of carbon dioxide concentration, this parameter  
Figure 4: (a) In-plume CO\({}_{2}\) concentration map acquired on 26 June 2015 at Stromboli. Carbon dioxide concentration varies as a function of increasing altitude (on the left of the map) and elevation (on the right of the map). The superimposed grey points represent the vertical wind profile retrieved by BILLI in the same dataset. Here, the mean value of the wind velocity was found about 1.3 m/s. The x-axis error bar is the weighted average standard uncertainty of wind velocity, equal to 0.52 m/s. Instead, the error bar on the y-axis represents the altitude halfwidth between adjacent regions (A, B, C, and D) [40]). This figure has been obtained by combining Figure 4 and Figure 8 reported in [40]. (b) Time-series of CO\({}_{2}\) fluxes from Stromboli volcano from 24 to 29 June 2015, rearranged from Figure 5 reported in [41]. The BILLI based fluxes (red circles) were obtained using the procedure detailed here [39,41]. For comparison, independent CO\({}_{2}\) flux estimates, obtained by multiplying the in-plume CO\({}_{2}\)/SO\({}_{2}\) ratio (from Multi-GAS) by the SO\({}_{2}\) flux (from UV cameras) are also presented (green squares) [41].  
represented the main source of uncertainty for the calculation of volcanic CO\({}_{2}\) fluxes, with an order of magnitude estimated to be about 25% [41].  
If all uncertainty sources are statistically independent, all the components have to be quadratically summed [48] in order to obtain the total uncertainty of volcanic CO\({}_{2}\) fluxes, which varies from 20% to 30% (dominated by the area uncertainty [41; 42]). This value is consistent with the results obtained during the field campaigns of Pozzuoli (2014) [21], Stromboli (2015) [41], and Mt. Etna (2016) [42].