## 3 Retrieval Adaptations
### Noise Model  
The measurement error covariance matrix [23] has to account not only for the measurement noise but for the total error including also the forward model error. The measurement noise of the instrument is well known from laboratory measurements and in-flight estimates. In theoretical studies (as in part 1, [23]), it is often assumed for convenience, that the measurement noise dominates and that other error components can be neglected, i.e., the noise model is approximated by the measurement noise. Especially when analyzing measured data, unknown inaccuracies of the forward model can violate this assumption and lead to larger fit residuals and unrealistic results (and error estimates) because the optimal estimation retrieval puts too much trust in the measurement. This may happen, e.g., due to imperfect knowledge of the instrumental line shape function (ILS), unconsidered spectroscopic effects such as Raman scattering, inaccuracies of the spectroscopic data bases, approximations of the radiative transfer model, or imperfect meteorology. Ideally, one would reduce the fit residuals to the instrument's noise level by improving the forward model, but this is often not possible. A potential solution is to fit parts of the residuum by empirical orthogonal functions (EOF) computed from a representative set of measurements [21].Another approach is to adjust the noise model so that it accounts for measurement noise plus forward model error (e.g., [16; 17; 27]) and a variant of this approach is also used by us. Most forward model errors can be interpreted to result from inaccuracies of the computed (effective) atmospheric transmittance. However, the largest scene-to-scene variability of the simulated radiance is due to changes of, e.g., albedo and solar zenith angle. Therefore, it is reasonable to assume forward model errors to be approximately proportional to the continuum signal \(\mathrm{I}_{\mathrm{cont}}\) which we obtain from up to nine spectral pixels at the fit windows' lower wavelength length ends. We model the root mean square residual to continuum signal ratio RSR by  
\[RSR=\sqrt{NSR^{2}+\delta F^{2}}, \tag{1}\]  
where NSR represents the root mean square of the spectral 1\(\sigma\) radiance noise (as reported in the OCO-2 L1b data) to continuum signal ratio and \(\delta F\) the relative forward model error. In order to estimate the free parameter \(\delta F\), we analyzed a representative set of pre-filtered soundings (Figure 3) with a modified FOCAL setup for which we (quadratically) added 2% of the continuum radiance to the measurement noise. This overestimation of the expected total error effects that the retrieval usually converges towards values being not very far away from the a priori, i.e., values being more or less realistic. Additionally, we switched off the SIF retrieval (which is basically identical to a zero level offset retrieval in the SIF fit window) and switched on the retrieval of zero level offsets in all four fit windows. If the instrument noise would dominate the total error, RSR and NSR would (statistically) lie on a 1:1 line. After the removal of outliers (Figure 4, gray dots), this is basically the case for the SIF fit window with forward model errors estimated to be about 0.5% of the (continuum) signal (Figure 4). The forward model error within the other fit windows is estimated to be between 2.5% and 3.2% (Figure 4). This means, the total error in dark scenes (large NSR) is still dominated by the instrumental noise but in bright scenes (small NSR), the forward model error dominates. Outliers are removed as follows: The data set is grouped in 35 NSR bins. Only bins with more than 500 samples are further considered. Within each bin, RSR should follow a \(\chi^{2}\)-distribution with as many degrees of freedom as spectral pixels of the fit window. The number of spectral pixels is always large enough to approximate the \(\chi^{2}\)-distribution with a Gaussian distribution.