# Tomographic Inversion Methods for Retrieving the
## 5 Results
Such sampling implies that, in the LSATR case, all three satellite constellations produce an overdetermined inversion problem. Figure 4 presents the reconstructions and the relative errors obtained by applying the ERTA and LSATR approaches for the 1 Tx-5 Rx configuration. The following Figures 5-7, instead, refer to the ERTA and LSATR results obtained using as 'true' maps some high resolution WV fields obtained from atmospheric scenarios generated over Tuscany, Italy, through the WRF numerical weather prediction model, initialized with the IFS model of the ECMWF (only the 1 Tx-10 Rx constellation case is presented in the figures). The sampling grid of the original maps has an angular and radial spacing of 0.1\({}^{\circ}\) and 125 m, respectively, whereas the sampling grid of the reconstructed maps has an angular and radial spacing of 0.25\({}^{\circ}\) and 250 m, respectively. Such reconstruction map sampling implies that, in the LSATR case, the 1 Tx-5 Rx configuration produces an underdetermined inversion problem (in the absence of the Tikhonov regularizer), while the other two satellite configurations generate overdetermined inversion problems. The horizontal axis of the figures reports the latitude, as it was assumed that the a meridian. The WV fields refer to three different atmospheric scenarios corresponding to three different dates of 2020. As in the ideal case shown in Section 4, the liquid water content component of such scenarios was not accounted for. Figure 4: (**a**) True WV field (ECMWF analysis T511 L91 of 15 January 2011 taken at 20\({}^{\circ}\) longitude East) as a function of angle and altitude; (**b**) retrieved WV field with ERTA and (**c**) the corresponding normalized error field; (**d**) retrieved WV field with LSATR and (**e**) the corresponding normalized error field defined as in Equation (17). The angular coordinate has the origin in the North Pole and descends along the 20\({}^{\circ}\) East longitude meridian. Satellite configuration: 1 Tx-5 Rx. Observing the retrieval results, it is apparent that the two methods produce error maps with very different spatial behavior. In fact, ERTA exhibits a spatially regular trend while LSATR is characterized by 'bumps' in correspondence of the tangent altitudes of the satellite links. The reason for this behavior lies in the quadratic minimization at the base  
Figure 5: (**a**) True WV field (analysis of 24 September 2020 9 p.m.) as a function of angle and altitude; (**b**) retrieved WV field with ERTA and (**c**) the corresponding normalized error field; (**d**) retrieved WV field with LSATR and (**e**) the corresponding normalized error field defined as in Equation (17). Satellite configuration: 1 Tx-10 Rx. of LSATR, which tends to force to zero the error in the measurement points (the tangent altitudes). As a matter of fact, such bumps could be attenuated by increasing the amount of the regularization term with respect to the 'fidelity' term in the Tikhonov unconstrained minimization, but this would produce poorer reconstructions. The balance between the regularization and fidelity terms is driven by the aforementioned parameter \(\lambda\) (chosen as suggested in [20]). On the contrary, ERTA inherently produces a quite smooth result in the overall reconstruction area. Figure 6: (**a**) True WV field (analysis of 24 July 2020 11:30 a.m.) as a function of angle and altitude; (**b**) retrieved WV field with ERTA and (**c**) the corresponding normalized error field; (**d**) retrieved WV field with LSATR and (**e**) the corresponding normalized error field defined as in Equation (17). Satellite configuration: 1 Tx-10 Rx. Notice also that the behavior of the error vs. the altitude is different for the two methods. In fact, at the lowest altitudes the reconstruction error produced by ERTA is lower than that produced by LSATR, while above a certain altitude, depending on the WV field, ERTA constantly overestimates. A reasonable explanation of this behavior of the error vs. the altitude is that the WV field is more likely to be located in the WV field.