# Tomographic Inversion Methods for Retrieving the
## 5 Results  
In this section, we show some reconstruction examples obtained by applying the ERTA and LSATR methods. Unlike the example shown in the previous section, in this case a more realistic scenario is considered by introducing thermal noise at the receiver as well as the phenomenon of scintillation. Such impairments are described in [20]. For the simulations carried out in this work, we used the same values of the parameters as those utilized in that paper, listed below for the sake of convenience:  
* Transmitted power: 3 dBW on each NDSA tone;
* Frequency channels for NDSA measurement: 17, 19, and 21 GHz, with spectral separation \(\Delta\)f = 0.2 GHz. The channels are selected according to the tangent altitude of the link: the altitude intervals for 17, 19, and 21 GHz are 0-3.5, 3.5-8, and 8-10 km, respectively, for the reasons given in [16];
* Transmit and receive antenna gains: 26.4 dB;* Scintillation: the log-amplitude standard deviation was set to \(\sigma\chi=0.3\) and the zero-lag correlation between the two NDSA frequencies to \(\rho=0.85\), whereas for the time correlation, dictated by the scintillation bandwidth, we used B\(\chi\) = 0.1 Hz. * Thermal noise: the variance of the zero-mean additive white Gaussian noise process is kT\({}_{\rm eq}\)/T\({}_{\rm s}\), where k is the Boltzmann constant and T\({}_{\rm eq}\) is the equivalent noise temperature at the receiver, which was set to 25.3 dBK. Before introducing the simulation results, it is opportune to highlight some of the differences between ERTA and LSATR. As shown in [20], LSATR retrieves the WV field as the solution of a linear system, in which the unknowns are the function (i.e., WV concentration) values on the sampling grid chosen for the reconstruction map and the number of equations corresponds to the number of IWV measurements. In general, the Tikhonov regularization term induces a smooth solution, which is driven by a parameter \(\lambda\) to balance smoothness and reconstruction error and whose choice is discussed in [20]. Another advantage of introducing the regularization term is the fact that in the case of an underdetermined problem (i.e., when the number of unknowns is higher than that of the measurements), a unique solution can be achieved. In the case of ERTA, it is always possible to retrieve the WV field, but it is not obtained as the solution of a system. In fact, ERTA is based on the expansion by means of a set of orthonormal functions whose coefficients are derived through projections of the function to be decomposed onto the space generated by the orthonormal functions themselves. As mentioned in the previous section, in the simulations we used three different constellations of satellites (1 Tx-5 Rx; 1 Tx-10 Rx; 1 Tx-15 Rx). This allows us to highlight the behavior of the two retrieval methods as well as the performance of LSATR in both the underdetermined and overdetermined cases. In the following, we present both a qualitative and a quantitative assessment of the retrieval obtained by using the two inversion algorithms. For this purpose, we show first the 'true' and reconstructed maps obtained with ERTA and LSATR, jointly with the local relative error map (i.e., the normalized error field as defined in Equation (17)). After that, we provide a nonlocal quantitative metric of the error by using the percent Normalized Root Mean Square Error (NRMSE) between the 'true' and reconstructed WV maps defined as  
\[NRMSE=\sqrt{\frac{\sum_{i,j}\left[f(i,j)-f(i,j)\right]^{2}}{\sum_{i,j}[f(i,j) ]^{2}}}\cdot 100 \tag{18}\]  
For the sake of conciseness, the figures show the retrievals referring to only one of the tested satellite constellations, whereas the NRMSE are reported for all of them. The first simulation outcome we present refers to the same 'global' atmospheric scenario considered in the previous section. The reconstruction sampling grid coincides with that used in the previous section (1\({}^{\circ}\times\) 250 m).