## 1 Introduction  
Ozone (O\({}_{3}\)) is the third most important gas that enhances greenhouse effect. In the stratosphere, O\({}_{3}\) prevents ultraviolet radiation from negatively affecting life on Earth's surface. At the same time,\(\text{O}_{3}\) in the troposphere is considered to be a pollutant e.g., [1]. From a modelers' perspective \(\text{O}_{3}\) is one of the most important input parameters for understanding of the fundamental atmospheric processes. It is a prerequisite for many climate and atmospheric studies and is labelled as an Essential Climate Variable (ECV) by the Global Climate Observation System (GCOS). Although satellite-based ozone retrievals from UV-VIS measurements are popular, the nadir-viewing infrared (IR) measurement is particularly advantageous for nighttime retrievals. The first satellite instrument designed to measure the vertical distribution of ozone was the backscatter ultraviolet (BUV) spectrometer instrument on NIMBUS 4 launched in 1970. Since 1970, many publications e.g., [1; 2; 3; 4; 5; 6; 7; 8; 9] have been published on atmospheric \(\text{O}_{3}\) estimation from several satellite-based instruments (e.g., CrIS, MIPAS, OMI, MLS, GOME, TOMS, OMPS, SBUV/2, SCIAMACHY, ACE-FTS, IASI, AIRS and TES, see Appendix A for additional acronyms). Unfortunately most of these retrievals are made using stochastic methods, where errors are treated as information. In this study, we focus on nadir-viewing thermal IR (TIR) instruments for \(\text{O}_{3}\) profile retrieval using deterministic inverse. With nadir-view, retrieval of ozone profiles is relatively easier than those of other trace gases since the mixing ratio of \(\text{O}_{3}\) increases with altitude unlike most other trace gases. This reduces the complexity of the Hilbert function as well as the condition number of the inverted matrix. Also, plenty of \(\text{O}_{3}\) absorption lines with variable line strengths at \(9.7\)\(\upmu\)m increases the information content of the retrieval. To investigate and compare the uncertainty in the satellite retrieval problem, a simulated comparative study with a deterministic method and a prevalent stochastic method for profile retrievals from satellite hyper-spectral IR measurements is undertaken. The definition of deterministic and stochastic methods can be found in [10; 11; 12; 13]. This work is extended from our previous studies [13; 14; 15; 16] by including more case studies to convince the scientific community about the flaws of prevailing strong reliance on Bayesian probability-based methods employed by major space and environmental agencies. The basic inverse method will remain similar to that described in our earlier publication [13; 16], but different fundamental aspects of ill-posed inversion will be discussed including more case studies using both deterministic and stochastic methods. Among the existing deterministic methods e.g., [17; 18; 19; 20; 21] the family of regularized total least squares (RTLS) methods has a well-established heritage in other branches of science, especially medical imaging e.g., [22; 23; 24; 25; 26]. However, it has seldom been exploited in Earth observation science to date. We selected the RTLS that is the only one which has a mathematical form to determine the optimal regularization strength using all noise terms embedded in the residual vector e.g., [13] and monotonically reduces the regularization strength towards a solution point. Other data-driven deterministic methods, i.e., generalized cross-validation (GCV) and L-Curve, are unstable to dynamically calculate the regularization at all iterations. A recent study shows that these methods are unsuitable for satellite inverse problems [18], based on pre-calculated regularization strength using GCV and L-Curve. Our implementation of RTLS employing a Laplacian first derivative operator (LFDO) as a stabilizer injects added information into the inversion.