## 4 Simulated Profile Retrievals Using Radiosonde Data
### Comparative Retrievals for CrIS  
The current mainstream approach for parameter estimation from space-based measurement is based on constraining by climatological data as a priori and some additional ad hoc procedures on top of OEM. The main concern is how reliable climatological daily/monthly/yearly averaged state space parameter for the actual state when satellite measurements are conducted in under a minute for a highly dynamic atmosphere. Moreover, several ad hoc methods including different operational constraints are available in literature and it is not always comprehensible or straightforward to account for such constraints in simulated retrieval. Thus, a comparative study between the original inverse methods of deterministic and stochastic (with some tweaking of errors covariance) is discussed here. RTLS solutions in Figure 5a,b can retrieve state parameters from CrIS simulated measurement unambiguously (with associated error and dependent on available information in measurement). These are significantly less error than those from OEM solutions in Figure 6a, under so-called exact error covariances. Recall that (Figure 5a,b, two profiles) LSDO performs slightly better than LFDO for most altitudes. For all 277 profiles (not two), we show results along the same line in Figure 7a,b but are presented in a different way that is convenient than line-plots to show multiple profiles. As is seen (Figure 7b), LSDO concentration follows a more systematic distribution, with high density at the core and less scatter confirming the observation seen from only two profiles (Figure 5) but for all profiles now. Please note that, since no error covariance is involved in RTLS, a single profile retrieval or collective retrieval of any number of profiles has no effect on the results. It can be concluded that the method is reliable and appropriate to be applied globally. Figure 7c-f show OEM retrievals. Although error is known in this simulated study, to understand the effect of tweaking of a priori covariances on solutions, three additional irrationals a priori covariances of 20%, 50% and 100% are considered, which are shown in Figure 7d-f, respectively. The high systematic error observed in Figure 7c gradually decreases with increasing values of a priori covariances because of reducing regularization strength (in terms of deterministic interpretation). Please note that Figure 7c,f are the two extreme cases of S\({}_{\text{z}}\), and Figure 7d,e are in between. While Figure 7c shows less scattered as well as less saturated (restrained boundary) points, Figure 7f shows significantly higher scatter and saturated points. However, the 1-to-1 ratio is stronger in Figure 7f as compared to Figure 7c. The peak-density reduces progressively between Figure 7c,f as those points contribute to the scatter, likely because of posterior error is higher than a priori error for high S\({}_{\text{z}}\) values. Another interesting observation in Figure 7c is the presence of horizontal stripes. These stripes originate when posterior resembles the a priori, meaning that the algorithm did not yield any solution without further decreasing the regularization strength as shown in Figure 7d-f. It can be concluded from this study that optimum regularization for all iterations, by tweaking a priori error covariance, is almost impossible. The tweaking of error covariance or additional method on top of the OEM may improve the solution to some degree, but it poses more ambiguities and leaves us with several unanswered questions. Figure 8a shows the retrieval errors of ST using various schemes. The RMSE of retrieval using RTLS for all profile is 0.06 K, whereas the same for OEM with stochastically correct a priori error is 0.6 K. The ST error in OEM is one order magnitude higher than that of RTLS retrieval.