# Tomographic Inversion Methods for Retrieving the
## 2 The NDSA Measurement Concept  
The NDSA technique, described in [16] and [17] (which the reader is referred to for details), is based on the conversion of the spectral sensitivity \(S\) to the total content of water vapor (IWV) along the propagation path between two LEO satellites, one with a transmitter onboard and the other with a receiver. Defining \(A(f)\) as the total spectral attenuation undergone by a signal tone at frequency \(f\), the spectral sensitivity function is the normalized derivative of \(A(f)\), that is  
\[S(f)=\frac{1}{A(f)}\frac{dA(f)}{df} \tag{1}\]  
The spectral sensitivity \(S\) at a given frequency \(f_{0}=(f_{1}+f_{2})/2\) is obtained by transmitting two sinusoidal signals at two close frequencies \(f_{1}\) and \(f_{2}\) (\(f_{1}>f_{2}\)), such that their distance \(\Delta f=f_{1}-f_{2}\), referred to as \"spectral separation\", does not exceed 2.2% of \(f_{0}\). Under this hypothesis, and assuming also that the two transmit signals have the same power, the spectral sensitivity \(S\) referred to \(f_{0}\) (pointed out as a subscript whenever needed) is simply the finite difference approximation of the spectral sensitivity function defined in Equation (1)  
\[S_{f_{o}}=\frac{P_{2}-P_{1}}{\Delta fP_{2}} \tag{2}\]  
where \(P_{1}\) and \(P_{2}\) are the powers of the two received tones measured at the receiver.  
In [16] and [19], the authors have shown that the IWV along a given radio path and the spectral sensitivity \(S\) measured along that link are tightly related, so that one can estimate the IWV from the knowledge of \(S\) by using specific IWV-S relationships. Such relationships have been computed in ideal conditions (i.e., no signal impairments) by means of a regression analysis, for different values of \(f_{o}\) and tangent altitudes, showing that for different intervals of tangent altitudes an optimal \(f_{o}\) exists for which the IWV-S relationships are tight. The analysis was carried out by using a large set of atmospheric profiles (temperature, pressure, and WV concentration) coming from available radiosonde observations (in [16]) and from ECMWF (European Centre for Medium range Weather Forecasts) analysis data (in [19]). Each of such profiles was in turn used to generate a spherically symmetric atmosphere. In brief, the procedure to derive IWV-S relationships accounting for the seasonal, daily and geographical variations of the atmospheric conditions was the following. For a given a profile (from radiosondes or ECMWF analysis data) and for a link at a given tangent altitude, we calculated the IWV by integrating the WV concentration along the path, while we computed \(S_{f_{o}}\) assuming that \(P_{1}\) and \(P_{2}\) were measured in ideal conditions. After having repeated this procedure for all available profiles, we obtained the optimal value of \(f_{o}\) and the corresponding coefficients of a linear IWV-S relationship for any tangent altitude below 10 km. The conclusion of such analysis was that the optimal frequencies to provide IWV measurements between two LEO satellites up to 11-km tangent altitude are 17, 19, and 21 GHz.  
The Signal to Noise Ratio (SNR) at the receiver is one of the most important parameters that impact the measure of \(S\). It increases with the integration time used to estimate \(P_{1}\) and \(P_{2}\). Increasing the integration time, however, means a lower number of available measurements in a certain area and then a worse resolution in the IWV measurements. In particular, in the case of counter-rotating satellites, increasing the integration time limits the vertical resolution of the IWV. In [17], it has been demonstrated that accurate measure of \(S\) can not be achieved with a SNR lower than 20 dB, which implies that the integration time has to be at least 10 ms. In the case of co-rotating satellites, which is the case analyzed in this paper, much higher integration times can be used, and this guarantees a greater accuracy in the estimate of the IWV.