## 4 Simulated Profile Retrievals Using Radiosonde Data  
The OEM is one of the stochastic inverse methods gaining popularity in many remote sensing applications, since more than three decades ago [41] and maintaining it until recently [45]. However, some studies focused on the \"deficiency\" of OEM and employ additional constraints to the OEM method, e.g., by using Tikhonov regularization e.g., [46; 47] or others. These enhancements are technically not parts of the original OEM approach but have been implemented to improve the OEM results. As such, there is nothing wrong in including additional constraints, but the issue is the lack of clarity in the retrieval, whether it is coming from adopted deterministic regularizations or from stochastic approximation processes inherent in OEM. Further to this, any occasional good result in operations may or may not be related to the core inverse method itself but is often attributed to OEM's success. In contrast, a dynamic data-driven regularization is intrinsic to the RTLS method that also can be applied to a single measurement instance, unlike any stochastic approach, which by definition rely on assuming distributions of a prior, a priori error and measurement error. The iterative form of OEM [41] is:  
\[\mathbf{x}_{\text{i+1}}=\mathbf{x}_{\text{a}}+\left(\mathbf{K}^{\text{T}}\mathbf{ S}_{\text{e}}^{-1}\mathbf{K}+\mathbf{S}_{\text{a}}^{-1}\right)^{-1}\mathbf{K}^{\text{T}} \mathbf{S}_{\text{e}}^{-1}((\mathbf{y}_{\text{s}}-\mathbf{y}_{\text{i}})+\mathbf{K }(\mathbf{x}_{\text{i}}-\mathbf{x}_{\text{a}})) \tag{10}\]  
where, \(\mathbf{x}_{\text{a}}\) is the a priori profile, \(\mathbf{S}_{\text{a}}\) and \(\mathbf{S}_{\text{e}}\) are the a priori and measurement error covariance matrices. These are additional parameters required only for OEM and a set of occurrences is required to develop a priori and measurement error covariance matrices. Thus, we conducted retrievals using a set of ozonesonde data collected from the Global Monitoring Division, Earth System Research Laboratory (ESRL) ([[ftp://aftp.cmdl.noaa.gov/data/ozwv/Ozonesonde/](ftp://aftp.cmdl.noaa.gov/data/ozwv/Ozonesonde/)]([ftp://aftp.cmdl.noaa.gov/data/ozwv/Ozonesonde/](ftp://aftp.cmdl.noaa.gov/data/ozwv/Ozonesonde/))), representative of the earth's atmosphere in conjunction with other collocated in situ parameters. We have collected 277 different in situ O\({}_{3}\) profiles from this database (locations: Boulder, Colorado; Hilo, Hawaii; Huntsville, Alabama; Narragansett, Rhode Island; Pago Pago, American Samoa; South Pole, Antarctica and Suva, Fiji) to perform this simulation. The surface temperature (ST) was not available in this data base, thus adding2 K to the near surface temperature data is set as true ST values. For simplicity, emissivity of the surface is assumed to be 1 for this simulation study, which is close to that of the ocean surface. The simulation has been made on the grid of the individual radiosonde profiles, thus, the different atmospheric grids are considered for different profile retrievals. The dataset has a mixture of different altitudinal coverage, e.g., some are up to 5 km while others are up to 30 km and the rest are in between. We have considered all the profiles to study extreme cases and construct a sound stochastic distribution. Although the major signal of measurement on O\({}_{3}\) band is coming from the stratospheric level, the experiment has been done using only tropospheric O\({}_{3}\) profiles up to 30 km because no in situ profiles data up to 50 km are readily available. This experiment can be viewed as a comparative lower-troposphere and upper-troposphere lower-stratosphere (UTLS) O\({}_{3}\) retrieval using different inverse methods assuming that stratospheric O\({}_{3}\) signal from satellite measurement can be successfully deducted. The plot of all profiles is shown in Figure 4 and an approximated middle profile (red) is considered to be a priori and IG for OEM and only IG for RTLS.