## 4 Simulated Profile Retrievals Using Radiosonde Data
### Profile Retrievals Using OEM  
In Figure 6a,b, it is observed that when so-called exact S\({}_{a}\) and S\({}_{e}\) are applied, OEM solutions (solid red) for CrIS simulations are close to the shape of the a priori implying a minimal extraction of information from measurements due to higher regularization in terms of regularization theory. The results of O\({}_{3}\) retrieval are opposite to our previous observation on simulated H\({}_{2}\)O retrieval using of OEM [13], where solutions are highly oscillated under exact S\({}_{a}\) and S\({}_{e}\) due to very low regularization. This implies that OEM solutions perform inconsistently for different trace gas profile retrievals and for different case studies. This raises the fundamental question in applying OEM for satellite inverse problem when two similar profile retrieval problems will yield two different outputs when stochastically exact error covariances are used. This confirms that a method cannot reliably fulfill the scientific quest when the method is derived using fundamental flaw where error is treated as information. Satellite retrieval community treats OEM as \"accepted wisdom\" and it is widely applied to many satellite retrieval problems without scientifically verifying the outcome of this method, which may result in devastating consequences on science developments. This, to our understanding, knowingly or unknowingly motivates researchers to apply tweaking to S\({}_{a}\) and S\({}_{e}\), perhaps without examining the root cause e.g., [4; 45] to obtain some convincing output applying OEM. For example, there is no explanation for considering the 100% a priori error covariance in [45] and S\({}_{e}\) is not estimated using the error distribution of forward model and instrument, but it is estimated from the fit residual [4]. Such approaches yield seemingly better results but are not mathematically or physically justifiable and high chances for information loss. To test such tweaking, we explored more simulations with increased of 20%, 50% and 100% S\({}_{a}\) are shown in the same figure. This progressively yields profiles closer to the true profile, but still with significant oscillations. As compared to RTLS solution, OEM yields an inferior solution (see Figures 5 and 6). In Figure 6c,d, retrievals for TES are shown under identical setting as for CrIS (Figure 6a,b). However, interestingly, the O\({}_{3}\) profile retrieval outcome is even worse than that for CrIS that has a lower SNR (higher noise). Essentially, the solution closely follows a priori and information from (simulated) measurement is almost unused even after tweaking the S\({}_{a}\). Please note, however, that the ST retrieval has slightly improved for TES (bottom panel of Figure 6c,d) as compared to the same for CrIS. One may argue that this happens due to the functional property of the TES measurements that may have \"multiple solutions\". However, this is misleading since such inconsistency is not observed in the case of RTLS solutions shown in Figure 5c,d. This, therefore, accurately captures the inconsistencies in performance in OEM method when different instruments are simulated. In practice, one may tend to assign this non-performance to real instruments and further tweak S\({}_{e}\) and other associated errors, but we demonstrated the root cause is the selected inverse method and not a nominally performing instrument. This issue is discussed further in the following paragraph. The value of S\({}_{e}\) is not only dependent on the sensor error only, but it also depends on forward model and other associated realistic errors in operation. Therefore, S\({}_{e}\) is further increased (compared to reported sensor SNR of a particular instrument), perhaps based on trial and error, and the solutions achieve some meaningful numbers. The fundamental question is how to estimate the forward model error because it is almost impossible to develop a perfect forward for any science problem.