## 3 Simulated Theoretical Smooth Profile Retrieval
### Profile Retrievals from Simulated CrIS Measurements  
The sensor specifications of the CrIS hyper-spectral sounder onboard Suomi National Polar-orbiting Partnership (NPP) is considered in this study since this instrument will continue to be flown in a series of Joint Polar Satellite System (JPSS) missions until at least 2038. For real data from CrIS, channel radiance is given by a convolution of the instrument line shape (ILS) function with the monochromatic radiance from the simulation at the entrance to the interferometer [38]. For the simulated retrieval study, a simplified \"sinc\" function is considered to produce equivalent CrIS measurements by convolving with the simulated spectrum. For example, the simulated measurement is calculated by the spectra for O\({}_{3}\) at a resolution of 0.06 cm\({}^{-1}\) using the GENSPECT LBL model, for the US 1976 standard atmospheric temperature and O\({}_{3}\) profiles, and ST of 300 K and surface emissivity of one and convolved with the \"sinc\" function. To achieve a more realistic condition, we applied our \"select channel\" algorithm [16] to discard the channels, which are contaminated by the radiance from other interfering trace gases (e.g., CO\({}_{2}\), N\({}_{2}\)O, CH\({}_{4}\), SO\({}_{2}\), NO\({}_{2}\), NH\({}_{3}\), HNO\({}_{3}\), OCS, HOCI, H\({}_{2}\)O\({}_{2}\), H\({}_{2}\)O and H\({}_{2}\)S). For example, \"select channel\" algorithm congregate 169 channels out of 302 channels for a selected window (900-1090 cm\({}^{-1}\)). Monte Carlo noise realizations were added to calculated spectrum of 1% (signal to noise ratio, SNR = 100), which is conservative because reported CrIS SNR is more than 150 for longwave IR (LWIR) channels e.g., [39; 40], to produce equivalent realistic measurement. During retrieval, we also consider the surface temperature (ST) as an additional retrieved parameter and the first guess is arbitrarily set to 275 K.  
Retrievals adding Monte Carlo noise in the simulated spectrum have been made using RTLS for three different true profiles (TP), which are TP1 (realistic), and TP2 and TP3 (extreme case full-sinusoidal profiles). TP1 is a 1976 US standard O\({}_{3}\) profile for the earth's atmosphere. To improve confidence on the outcome, two initial guess (IG) profiles, one is a constant (IG1, green) and the other is a realistic (0.8 times of TP1, IG2, blue), are considered as shown in Figure 1. Only TP1 is solved from IG2 (Figure 1a). We have purposefully done this simulated experiment using two unrealistic sinusoidal profiles to understand the inverse properties of RT function and it does not violate any limits from the point of RT physics. It is obvious that parameters can go beyond the boundary for a specific iteration when Newtonian iterative optimization is used in a nonlinear problem. Also, the solution of sinusoidal profiles will give us an additional advantage to understand and analyze altitudinal information content for such measurements. The simulated retrieval result confirms as shown in Figure 1a that CrIS measurement can retrieve good O\({}_{3}\) realistic profiles up to \(\sim\)30 km using RTLS method without any a priori information from two different IGs (IG1 and IG2) for TP1. However, a higher deviation in solution is observed for TP2 at \(\sim\)10 km and at same time the oscillation for TP3 at \(\sim\)10 km is lower as compared to that of TP2. This implies that there is a weak space for such measurement at 10-15 km and the severity of weak space depends on the shape of the profile (see Figure 1b). The profile retrievals of TP1 above 30 km are noisy even when solved from IG2, which confirms that the information available from CrIS measurement is low. On the contrary, the profile retrieval of TP2 is improved above 30 km and some oscillation is observed for TP3 at that level. This implies that the information of retrieval is not only dependent on CrIS measurements but also significantly dependent on the shape of the profiles resulting in different information contents.