# Tomographic Inversion Methods for Retrieving the
## 6 Conclusions  
The estimation of the atmospheric WV is of extreme interest for the scientific community involved both in meteorological and climatological studies. In particular, the knowledge of the distribution of WV in the troposphere (and specifically, in the low troposphere, where the 95% of the atmospheric WV is concentrated) is of major importance for understanding the dynamics of the meteorological processes. Even though the direct measurements made by radiosondes provide the vertical profiles of WV with good reliability, remote sensing acquisition systems devoted to the estimation of WV on a global scale are of great interest, for instance, for assimilation in numerical weather prediction models. Retrieving the 2D distribution of WV over planes perpendicular to the Earth surface would allow the horizontal structure of the WV fields, which is known in less detail, to be achieved in addition to the vertical profiles. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**24 September 2009 0.m.**} & \multicolumn{4}{c}{**24 July 2020 11.30 0.m.**} & \multicolumn{4}{c}{**27 July 2020 03.00 0.m.**} \\ \hline  & **SEX** & **1 DEX** & **1 DEX** & **3 DEX** & **5 DEX** & **1 DEX** & **1 DEX** & **5 DEX** & **1 DEX** & **1 DEX** & **1 DEX** & **1 DEX** & **1 DEX** \\ \hline
**ERTA** & **LSATR** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** & **ELSTA** \\ \hline
2-10 km & 21.7 & 25.2 & 21.2 & 20.9 & 14.5 & 25.2 & 20.2 & 20.4 & 20.3 & 20.4 & 20.3 & 20.4 & 20.3 & 20.4 & 20.3 & 20.1 & 16.7 & 20.3 & 16.4 \\
2-10 km & 24.7 & 25.3 & 20.5 & 20.9 & 14.6 & 20.2 & 20.3 & 20.3 & 20.4 & 20.5 & 18.0 & 30.2 & 20.4 & 30.3 & 36.7 & 30.5 & 16.4 \\
2-10 km & 27.8 & 29.1 & 27.8 & 18.7 & 27.4 & 18.1 & 37.2 & 34.3 & 38.1 & 21.7 & 37.4 & 17.0 & 37.4 & 25.2 & 37.4 & 20.1 & 27.9 & 17.4 \\ \hline \end{tabular}
\end{table} 
Table 1: Percent NRMSE obtained from the reconstructions of the WV fields for the considered three scenarios, by using ERTA and LSATR. The values refer to the whole field (2–10 km), to a low (2–5 km) and high (5–10 km) tropospheric altitude range. In this paper, we have reported part of the results of the SATCROSS project, funded by ASI, inherent to a novel approach to retrieve 2D fields of WV concentration by means of a constellation of co-rotating LEO satellites and by using the NDSA method for estimating the IWV along the radio links connecting the satellites. The novel approach is based on the External Reconstruction Tomographic Algorithm, which enables one to retrieve a function, defined on an annular region, from a sampling of its Radon transform, defined in the same region. In our case, the function to be retrieved is the WV concentration and its Radon transform samples are represented by the IWV measurements. The ERTA approach has been tested by using a simulator that takes into account realistic impairments as well as synthetic atmospheric scenarios, which allow the reconstruction error to be evaluated. The results have been compared to those obtained with another method developed by the authors, based on a least squares algorithm with Tikhonov regularization (LSATR). In the simulations, we considered constellations with a different number of LEO satellites. The outcomes of the simulations highlight a quite different distribution of the retrieval error between ERTA and LSATR. A quantitative analysis has been made by computing the NRMSE for different altitude ranges corresponding to the low and high troposphere. From both the qualitive and quantitative standpoint, ERTA produces better results at low altitudes and when the number of satellites is smaller; furthermore, its performance seems be poorly affected by the size of the constellation. On the other hand, LSATR exhibits better performance when the number of satellites increases, both in the low and high tropospheric layers. Conceptualization, L.F. and F.C. ; methodology, A.M., F.C., F.A.