## 2 SiF Retrieval Method and Setup
### The GOME-2 Instrument  
The GOME-2 spectrometer aboard the Metop-A and Metop-B satellites (Stein  
is modelled using PCs from an analysis of measured reflectance spectra over fluorescence-free regions (see below). Convolution with the instrument's spectral response function and any other instrument effects causing spectral structures are implicitly described by the PCs. In the actual fluorescence retrieval, the surface albedo is described as a fourth-order polynomial in wavelength (\(n=4\)). The spectral shape of the fluorescence radiance across the retrieval window, which roughly corresponds to the second peak of the SiF spectrum, is parameterised with a Gaussian function located (\(\mu_{F}\)) at 737 nm and having a sigma-width (\(\sigma_{F}\)) of 34 nm following [PERSON] et al. [57].  
Finally, we replace the monochromatic solar irradiance \(E_{0}(\lambda)\) by the high-resolution solar irradiance reference spectrum from [PERSON] and [PERSON] [58] convolved with the GOME-2 instrument response function and scaled to the actual Earth-Sun distance (denoted as \(\overline{E}_{0}(\lambda_{i})\)). We use this spectrum instead of the actually measured irradiance spectrum \(E_{0}(\lambda_{i})\) because of radiometric degradation of the GOME-2 instrument, which largely cancels out in the calculation of the reflectance but would show up as a confounding trend in the SiF time series (for example, the SiF in 2013 would have been underestimated by more than 10% compared to 2007 if the Level-1b irradiance had been used). The use of \(\overline{E}_{0}(\lambda_{i})\) instead of \(E_{0}(\lambda_{i})\) is another difference compared with [PERSON] et al. [21]; in SiF NASA (v26) a (different) correction for radiometric degradation applied to the measured irradiance is now implemented [59].  
The forward model for the fluorescence retrieval is then completed and written as  
\[R(\lambda_{i};\mu,\mu_{0})\approx\left(\sum_{j=0}^{n}a_{j}\lambda_{i}^{j} \right)e^{-\sum_{k=1}^{n}b_{k}f_{k}(\lambda_{i})}+\frac{\pi I_{F,0}e^{-\frac{ 1}{2}\left(\frac{\lambda_{i}-\mu_{F}}{\sigma_{F}}\right)^{2}}}{\mu_{0} \overline{E}_{0}(\lambda_{i})}e^{-\frac{\mu^{-1}}{\mu^{-1}+\mu_{0}^{-1}}} \sum_{k=1}^{n}b_{k}f_{k}(\lambda_{i}) \tag{3}\]  
In our baseline retrieval setup, the number (\(m\)) of PCs (\(f_{k}\)) is 35. Thus, the total number of fit parameters is \(5+35+1\) or 41 (\(a_{j}+b_{k}+I_{F,0}\)). Extensive retrieval simulations were performed by [PERSON] et al. [21], which indicated that the accuracy and precision of the retrieved fluorescence peak value for a fit window running from 747 nm to 780 nm hardly changes when increasing the amount of PC from 25 to 35. In this paper, we will show results from real data experiments further investigating the effect of the number of PCs, which confirm the conclusion that at least 25 components are needed. In our real data experiments, we observed, however, a small improvement when using 35 compared to 25 components. Therefore this number has been chosen for the baseline retrieval setup. In SiF NASA (v26), \(m\) was reduced to twelve, because the fit window did not include the oxygen A band. The number of PCs is the second major difference between the two retrieval implementations. We remark that selection of the PCs can be further optimized by, for example, setting a lower threshold on the PC's explained variance [39] or using more sophisticated statistical criteria [36]. Finally, we note that by making additional simplifications, the forward model can be linearized to speed up the calculations [36; 39].