## 4 Simulated Profile Retrievals Using Radiosonde Data
### Comparative Error Analysis between Both Sensors and Methods  
The error analysis of parameters' estimation from satellite measurement is always a debatable issue due to lack of high-quality and abundant in situ measurements. Even it is sometimes argued that in situ measurement is also not \"truth\" due to different error patterns of different measurement systems. The only way a proper error analysis can be performed is through a controlled experiment with simulated retrieval where the truth is known, and the performance of the inverse methods can be well analyzed. We have already discussed rigorously in our earlier publication [13] that operational validation is made using tuned reference data sets with many ambiguous mathematical constraints and after discarding a significant number of retrievals for the sake of in situ quality control (e.g., Chi-squares test). A most debated current practice to validate the prevalent stochastic-based profile retrieval is that it needs alteration of the reference dataset with an averaging kernel of the inverse model e.g., [4; 54; 49; 50; 51]. Our aim is not to present a gimmick (e.g., \"TES profiles agree within 5-10%, less than the variability in CO distributions obtained by both TES and the aircraft instruments [54]\") in peer-review literature. One can easily find in [54] that the a priori profile is inside of the posterior error bar, which implies that information is lost after using the measurement. In another published article, for validation purposes OMPS-LP measurements are required to be within 5\({}^{\circ}\) latitude and 10\({}^{\circ}\) longitude from the O\({}_{3}\) station, and within a 12 h time span around the sonde launch. For each sonde profile, all collocated OMPS-LP observations are again averaged before the comparison [4]. In our opinion, such approaches using repeated aggregation is excessive and will smooth out much of the existing error distribution especially for chemically active gases, such as O\({}_{3}\). We would rather present a comparative error analysis to evaluate the performance of inverse methods by considering simulated experiments for realistic situations (realistic O\({}_{3}\) profiles) instead of fine-tuning of end-numbers. Earlier the qualitative retrieval for realistic O\({}_{3}\) profiles are shown in Figure 7a,b using RTLS and difficulties of the problem are discussed, but when a number is assigned to this retrieval, the error can reach up to 2000% for realistic situation as shown in Figure 11.  
As seen in Figure 11a and discussed earlier in Section 3, for a complex profile with multiple inflection points and high gradient, retrieval errors can be very high for some points. In such extreme cases, however, LSDO from a straight-line IG (IG1) still achieves remarkably less error than LFDO. This potential improvement will be explored in our future work to get the best results in extremely difficult cases. For a comparatively smoother true profile (Figure 11b) point retrieval errors are within ~100% from any IG and both derivative operators. To conclude this particular discussion, we must exercise sufficient caution in reporting and interpreting errors, such as \"5-10%\" as these numbers are very subjective and depend on sub-selection and tweaking. Figure 11: CrIS retrieval error on individual points of two different profiles (**a**,**b**) employing two different IGs as shown in Figure 7a,b (ignored here for brevity) and two different regularizations (LFDO, red; LSDO, cyan). The solution from two IGs are separated by solid and dashed data. To give further comprehensive idea on the gross characteristics of the quality of the retrievals for all profiles employing different methods, we binned the data at 2 km interval, and calculated the root mean square error (RMSE) for each bin.