## 2 SiF Retrieval Method and Setup
### PC Analysis of Atmospheric Absorption Optical Thickness  
The assumption is made that the same set of empirical basis functions can be used to describe the atmospheric absorption optical thickness both over fluorescent target scenes and over non-fluorescent reference scenes. In the baseline retrieval setup, we construct a set of non-fluorescent reference spectra from a year of observations over the Saharan desert. The PCs for the two-way slant absorption optical thickness are determined from this set of reference spectra in the following way.  
When there is no fluorescence, the second term in Equation (3) cancels. A number of sub-windows are selected in which the absorption optical thickness becomes very small (712-713 nm, 748-757 nm and 775-783 nm) so that the top-of-atmosphere reflectance approaches the surface reflectance. We fit a second-order polynomial to the measured reflectance in these sub-windows to determine the surface albedo across the entire retrieval window (the surface reflectivity for non-vegetated surfaces shows much less spectral variation across the retrieval window, so that a polynomial of second-order can be used here). The two-way slant absorption optical thickness for a particular ground pixel is then given by \[\tau_{\rm s}^{\downarrow\uparrow}(\lambda)=-\ln\left(\frac{R^{ref}(\lambda_{i};\mu, \mu_{0})}{\sum_{j=0}^{2}c_{j}\lambda_{i}^{j}}\right) \tag{4}\]  
Finally, PCs are determined from an analysis of slant absorption optical thicknesses for this set of non-fluorescent reference scenes. The set of reference spectra is constructed from observations over the Saharan desert (here: the latitude-longitude box defined by corners 16N, 8W and 30N, 29E). GOME-2A spectra are included only if they are completely free of vegetation according to the 1 km resolution USGS Global Land Cover Characterization database (version 2; [[https://ita.cr.usgs.gov/GLCC](https://ita.cr.usgs.gov/GLCC)]([https://ita.cr.usgs.gov/GLCC](https://ita.cr.usgs.gov/GLCC))) and if the pixel's effective cloud fraction in the near-infrared is below 0.4. The effective cloud fraction is taken from the FRESCO product [60; 61] as included in the GOME-2 radiance product. We may make different choices for the region of vegetation-free reference areas. Initial experiments indicated that the Saharan desert works well in our retrieval. In Section 6.3 we return to the question of PC reference area.  
For SiF retrievals in a particular month, the set of reference spectra is collected from the twelve months preceding and including this month. For example, SiF retrievals for the month of June 2015 use PCs determined from reference spectra in the time frame of July 2014 to June 2015. For SiF retrievals in 2007, which is the first year of the GOME-2A time series, reference spectra are taken from that year.  
The number of reference spectra is very large (in the order of 30,000 spectra) and calculating the PCs using the covariance method is problematic. Instead, we calculate PCs sequentially in order of decreasing explained variance using the Non-linear Iterative Partial Least-Squares algorithm [62; 63].  
The selection of reference spectra is a third difference with SiF NASA (v26), for which observations mainly over cloudy ocean, and to some extent also desert and snow covered land are used, but for a single day only.