## 4 Simulated Profile Retrievals Using Radiosonde Data
### Comparative Error Analysis between Both Sensors and Methods
The RMSE values have been calculated in terms of percentage of error as \((\delta x/x)\) for individual points as the values of the O\({}_{3}\) profile varies by more than two orders of magnitude along the profile that makes it difficult to show and interpret in its original value. Please note that this inter-comparison study described below uses IG2 only (the straight-line IG is excluded) because OEM mechanism does not permit an unrealistic a priori and requires a close-to-truth profile as a priori. Also note that a priori and IG are not necessarily the same, i.e., any a prior may serve as the IG, but the IG does not require a priori knowledge and it could be far from the truth or of a different shape from true profile. We have assumed that the IG and a priori are identical for this study. As mentioned earlier, an exact solution is not possible using a few pieces of information from measurements. The reconstruction of retrieved profiles using RTLS is dependent on the available DFR and the distance from the IG profile, and so is OEM by definition. Thus, the retrieval error is dependent on the IG error and we propose an innovative way to describe the inter-comparison study employing information gain (\(G_{inf}\)) after measurement as the percentage of error reduction (PER) from IG error as shown in Figure 12. The G\({}_{\text{inf}}\) is calculated as follows:  
\[G_{inf}=\mathbf{\epsilon}_{ig}-\mathbf{\epsilon}_{rtv};\quad\textit{where}\ \ \mathbf{\epsilon}_{r}=rms\{(\mathbf{x}_{r}-\mathbf{x}_{true})/\mathbf{x}_{true}\}\|_{2 km} \tag{12}\]  
where, \(\mathbf{\epsilon}_{ig}\) and \(\mathbf{\epsilon}_{rtv}\) are the percentage of altitudinal RMSE for IG and retrievals from RTLS or OEM respectively. \(\mathbf{\epsilon}_{r}\) is the percentage of the altitudinal RMSE for the reference r, which can be IG, RTLS and OEM. The simplest way to understand information gain from Figure 12 is by looking at the values to the left and right of the zero-point on the abscissa. Any point with less than zero indicates information loss (when compared to IG error) and vice versa. The legends for RTLS1 (LFDO), RTLS2 (LSDO) and OEM (S\({}_{a}\) = exact, S\({}_{e}\) = exact) have been already mentioned and are the same for two figures. Only OEM\({}_{\text{tw}}\) is different between Figure 12a,b, corresponding to Figures 7d and 9f, respectively. For RTLS, the performances are satisfactory and similar for both the instruments, with LSDO overly outperform LFDO and there is no information loss (retrieval error is always lesser than IG error). The information gain above \(\sim\)20 km is lower than that of the bottom part. This is because IG error is already low above \(\sim\)20 km (cf., Figure 4a) and there is little scope for gaining further information. On the contrary, OEM results drastically vary between the two sensors. In the case of CrIS in Figure 12a the loss of information is very high, and we have restricted the abscissa to \(\pm\)100% for retaining the clarity of presentation. Generally, for CrIS (Figure 12a), it is observed that there is virtually no information gain using OEM (solid red line) and the results worsen with tweaking (dashed red line). The posterior error higher than a priori error using OEM is not reporting here the first time, it is often reported in  
Figure 12: Information gain at different altitudes in comparison to percentage of IG error for different retrievals. (**a**) CrIS; (**b**) TES. published literature e.g., [9; 48; 54]. On the other hand, for TES in Figure 12b, there is no update of solutions using OEM and the results are very good after tweaking and is comparable to that of LFDO. It is possible to cherry-pick such random success or at times obtain good results for a given sensor by trial-and-error or tweaking, but this has no objective basis.