## 3 Simulated Theoretical Smooth Profile Retrieval
### Information Content Analysis Using Subspace
We calculate the rate of change of information along the altitude grids by calculating the angle between two adjacent column (measurements) vectors of the Jacobian as:  
\[\theta=\cos^{-1}\frac{\mathbf{K}(:,i)^{T}\mathbf{K}(:,j)}{\|\mathbf{K}(:,i)\|\|\mathbf{K}(:,j)\|} \tag{8}\]  
where \(i,j\) are the adjacent column of the Jacobian. We have also verified our results using \"subspace\" routine available in MATLAB. The information is calculated as  
\[\mathrm{H_{sb}}=\sin\theta \tag{9}\]  
We have calculated values of \(\mathrm{H_{sb}}\) for the original Jacobian (\(\mathbf{K}\)) for three different profiles as \(\mathrm{H_{sb}(org)}\) and inverted matrix of RTLS solution at the last iteration ([\(\mathbf{K}\); \(\alpha\)L1; \(\alpha\)L1]) from IG1, which is the regularized Jacobian for the retrieved profiles as \(\mathrm{H_{sb}(reg)}\). The calculated subspace information for two different sensors are shown in Figure 3.  
As mentioned earlier, a \(\mathrm{H_{df}}\) value of 0.2 does not necessarily mean that there is 80% error. Take for example, in the case of CrIS (Figure 1), retrievals both for TP2 and TP3 are in weak space around ~10 km, but the \(\mathrm{H_{df}}\) values are highest along the altitude and ~0.17. This cannot explain the true retrieval quality of an inversion. Contrary to \(\mathrm{H_{df}}\) values, RTLS-based \(\mathrm{H_{sb}(org)}\) values (Figure 3a) clearly demonstrate lesser information contents along the altitude around ~10 km for both TP2 and TP3, which correspond to the weak-space retrievals as is seen in Figure 1. As a consequence of the weak space, RTLS inherently regularizes the original Jacobian to a higher strength resulting in \(\mathrm{H_{sb}(reg)}\) value close to 1. This ensures a reduction in the propagation of the random error, but the regularization error is relatively higher than other altitudinal grids and resulting in a change in the shape of the retrieved profile. Although the values of \(\mathrm{H_{sb}(org)}\) are not directly related to the quality of retrieval, the consequences of the high values of \(\mathrm{H_{sb}(org)}\) for TP2 and TP3 around ~20 km and for TP1 around  
Figure 3: 2D distribution of information content for simulated retrievals for three different profile shapes: TP1 (blue), TP2 (red) and TP3 (green) for (**a**) CrIS and (**b**) TES. Dashed lines: \(\mathrm{H_{sb}}\) original \(\mathrm{H_{sb}(org)}\); Dashed-dotted lines: \(\mathrm{H_{sb}}\) regularized \(\mathrm{H_{sb}(reg)}\). -35 km (Figure 3a) are not explored here. Similar weak space behavior is also noticed for TES simulated retrievals (Figures 2 and 3b) but in lesser magnitude due to the value of H\({}_{\text{sb}}\)(org) being higher than for CrIS and it is around \(\sim\)15 km. Therefore, we would like to emphasize that only a straightforward averaging kernel or DFS/DFR analysis to understand the information content in the retrieved profiles is inadequate, as opposed to what has been a prevalent practice in the community. Although the subspace information for two different sensors for same retrieval grids at original Jacobian space is significantly different, the distribution of altitudinal information H\({}_{\text{sb}}\)(reg) under RTLS regularization for both sensors are closer and near to 1. This confirms that the RTLS regularization scheme is optimally preventing the propagation of noise from the measurement space to the state space for almost all altitudinal grids and retrievals are reliable. The most striking result is that the full sinusoidal true profile shapes are closely retrieved (Figure 2) whereas the calculated DFR is only 3. Conventionally, a minimum of five pieces of information is required to resolve a full sinusoidal profile, which raises the question on the validity of the postulate of degree of freedom in retrieval or DFR. However, subspace information content analysis is also not fully conclusive, which we briefly discussed here.