## 3 Simulated Theoretical Smooth Profile Retrieval
### Information Content Analysis Using Subspace  
Information content analysis of retrieval is a relative measure of the quality of retrieval for an applied method in some degrees. There is no absolute meaning of information content for parameter estimation from a remote sensing measurement and the estimated values are dependent on both forward and inverse modeling. For example, the number of degrees of freedom for a profile is effectively infinite (approximately the number of molecules along the path in the atmosphere) for RT modelling. It is not feasible to model this in a realistic situation due to the constraints in numerical modelling. Thus, in practice, a forward model is conventionally made with a finite number of grids (e.g., in this study, less than 1 km up to tropopause and then more than 1 km is considered). Such approaches restrict the degrees of freedom of the entire modelling scheme and the maximum number of pieces of information is restricted by the number of the state space parameters or the number of the measurements, whichever is lesser. One of the approaches for measuring the information content of stochastic-based retrieval is the degree of freedom from signal (DFS) [41], which is the trace of the averaging kernel (**A**) matrix of retrieval method. Analogous to **A** in the stochastic approach, model resolution matrix (\(\mathbf{M}_{\mathrm{rm}}\)) is used in the deterministic approach [42]. The trace of the \(\mathbf{M}_{\mathrm{rm}}\) matrix is the degree of freedom in retrieval (DFR) (in the deterministic approach [43]). The total information content in terms of DFR can be calculated using the parameters of RTLS methodology at the final iteration as:  
\[\mathrm{H}_{\mathrm{dfr}}=\mathrm{trace}\big{(}\big{(}\mathbf{K}^{T}\mathbf{K}-\mathrm{ g}(\mathbf{x})\mathbf{L}+\mathbf{\alpha}\mathbf{L}^{T}\mathbf{L}\big{)}^{-1}\mathbf{K}^{T}\mathbf{K}\big{)} \tag{7}\]  
Although \(\mathrm{H}_{\mathrm{dfr}}\) is a measure for understanding of the quality of retrieval, it is unable to give a complete picture of how information is distributed along the altitude grid at every iteration. An assumption was made that the diagonal elements of \(\mathbf{M}_{\mathrm{rm}}\) is the information distribution along the latitudinal grids, which is plotted in Figure 3. The values of \(\mathrm{H}_{\mathrm{dfr}}\) are always less than 0.2 and well distributed. By definition the \(\mathrm{H}_{\mathrm{dfr}}\) of single altitudinal grid is 1 for LS solution when measurement is more than or equal to the state space parameters. However, this reduces to \(\sim\)0.2 (e.g., Figure 3) under regularization using RTLS. One may argue that the solution contains 80% of noise if \(\mathrm{H}_{\mathrm{dfr}}\) is 0.2 but this is not true in reality. Put differently, the value of \(\mathrm{H}_{\mathrm{dfr}}\) is not representative to understand how much error is propagated from the measurement space to the state space at the time of inversion. Figure 2: Similar to Figure 1 but for TES retrievals. Thus, we extend our study using subspace [44] analysis here. This idea is extended for vector space in such a way that the angle between two measurement vectors can represent the information added into the system by the second measurement with respect to the first measurement. One degree of freedom will be added if two measurement vectors are mutually perpendicular and an angle of 0 implies no information improvement by second measurement. It can be applied to state-space instead of measurement space, e.g., two state space parameters can be uniquely solved if two column vectors (a set of measurement) of the Jacobian, which are the derivatives of measurement/model with respect to the state space parameter, are perpendicular. [Please note that the variation of O\({}_{3}\) number density can be viewed as a two-dimensional model, one within a given altitude and the other along the altitudinal path].