# Tomographic Inversion Methods for Retrieving the
## 4 Application of the ERTA Approach to the Reconstruction of WV Fields  
In this section, we analyze the peculiarities of the application of ERTA to the reconstruction of a WV field based on a constellation of LEO satellites and a set of NDSA measurements. As a matter of fact, there are significant differences with respect to the original applications of ERTA shown for instance in [24; 25; 26; 27]. In fact, our setup is characterized by a very narrow annular region (if the width is compared to the inner radius) to be reconstructed and by a very limited number of links on which the IWV can be computed, due to the complexity of the deployment of a highly populated constellation of satellites. Furthermore, the underlying function to be reconstructed (i.e., the 2D WV field) is expected to be quite regular (and in any case, so are the 2D WV fields generated by numerical prediction models that have been used as 'truth' reference in our simulations). This has an impact on the choice of the parameters of the ERTA implementation (for instance, the number of polynomials). To better illustrate the ERTA potential outcomes, we first present and discuss the retrieval results in an ideal case (i.e., in the absence of any impairment source), leaving to Section 5 the analysis in the presence of disturbances. We consider as 'true map' the same atmospheric scenario that we used in [20], i.e., the ECMWF, T511 L91 analysis data of 15 January 2011, 00:00 UTC taken at 20\({}^{\circ}\) longitude East. The vertical and angular sampling intervals are 125 m and 1\({}^{\circ}\), respectively. The WV concentration field is shown in Figure 3a, where the field in polar format is actually shown on a rectangular grid with the horizontal axis and the vertical axis referring to the angular position and the altitude, respectively. For our WV retrieval testing purposes, we did not consider the liquid water content present in such scenario. Figure 3: (**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; (**c**) 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. In our simulation setup, we assumed the following assumptions concerning the acquisition system and satellite constellation geometry:  
* Spherical Earth with radius equal to 6378 km;
* Rectilinear links connecting transmitters and receivers;
* Minimum and maximum tangent altitudes equal to 2 and 10 km, respectively, which implies an opening angle (defined as the angle between the minimum and maximum tangent altitude seen from a transmitting satellite) of 0.2457\({}^{\circ}\);
* 1 Tx satellite and 5, 10 or 15 Rx satellites on the same circular orbit. The directions of the Tx-Rx links are determined by subdividing the opening angle into equal parts. Since the angles are very small, the tangent altitudes of the microwave links in the scanned annular section result almost equally spaced;
* Polar orbit with radius equal to 6651 km (i.e., a satellite altitude equal to 273 km);
* Constant angular speed with a revolution period of 90 min;
* Integration time at the receiver equal to \(\mathrm{T_{s}}=1\) s.  
The simplifying assumptions (i.e., the hypotheses of spherical Earth, circular orbit and rectilinear propagation of the electromagnetic waves) have been made since the focus of this paper is a compared performance analysis of tomographic algorithms. In conclusion, under these assumptions, every \(\mathrm{T_{s}}\) seconds we have a set of 5, 10 or 15 simultaneous measurements of IWV (one from each Rx satellite). In the practical ERTA realization, we truncated the infinite summations involved in the function's decomposition.