# Tomographic Inversion Methods for Retrieving the
## 1 Introduction
It requires the measurement of a parameter called spectral sensitivity, which is the normalized incremental ratio of the spectral attenuation at a given frequency. This turns out to be a quite simple function of the received powers of two simultaneously transmitted sinusoidal signals which travel from the transmitter (Tx) onboard a first LEO satellite to the receiver (Rx) onboard a second one. The frequencies of the signals are very close to each other, as their difference is about two orders of magnitude smaller than their average frequency, which is conventionally taken as the \"channel\" of the spectral sensitivity measurement. As shown in [16] and [17], which the reader is referred to for details, the spectral sensitivity is highly correlated to the IWV if frequency channels in the Ku-K bands are employed, so that the IWV pertaining to the Tx-Rx link can be estimated through simple pre-set linear relationships. Furthermore, being the spectral sensitivity based on normalized incremental measures, the NDSA approach limits the negative effects of tropospheric scintillation [18]. In a multichannel (i.e., multifrequency) approach, higher frequency bands can be used both to estimate WV in the upper tropospheric layers and to detect the presence of liquid water along the radio path, giving the possibility to correct the bias it generates on WV estimates [16; 19]. As pointed out in [17], where the authors aimed at estimating the accuracy with which the spectral sensitivity is measured at different tangent altitudes of the Tx-Rx link, the NDSA approach can be exploited both in a \"counter-rotating\" configuration and in a \"co-rotating\" one. In the former case, a couple of LEO satellites orbit in the same plane and in opposite directions, thus having relative sunrises or sunsets. A vertical IWV profile is obtained over a certain position of the Earth, and the WV profile can be obtained by proper inversion methods. In the latter configuration, which is the case focused in this paper, we have a set of LEO satellites orbiting in the same plane and along the same direction: in this manner, one or more Tx LEO satellites chase a group of Rx LEO satellites so that they are always in view of each other, as illustrated in Figure 1. The result is that there is a number of constantly active radio links that cross the troposphere at fixed tangent altitudes with respect to the Earth, covering and scanning an annular region of the orbital plane. The minimum and maximum altitude of such annular region are defined by the two microwave links with the minimum and maximum tangent altitude, respectively (For the sake of simplicity, we assume a spherical Earth so that the tangent altitude of each link does not vary.). Such a configuration of LEO satellites, whose realization is nowadays made easier with affordable costs thanks to the Cubesat technology, allows integration intervals longer than in the counter-rotating configuration (where the integration time is limited by the requested vertical resolution), with a clear benefit to reduce the noise and scintillation impacts at the receiver. In the co-rotating scenario, the set of IWV measurements collected during the observation time can be inverted to obtain the two-dimensional (2D) concentration of WV in the scanned annular region (i.e., on planes perpendicular to the Earth surface). In [20], an inversion approach based on least squares minimization and Tikhonov regularization (in the following referred to as LSATR, i.e., Least Squares Algorithm with Tikhonov Regularization) was proposed. In this paper, we focus on an alternative approach of tomographic inversion that is perfectly coherent with the geometry of the inversion problem at hand and with the path-integrated nature of NDSA measurements; to the best of our knowledge, this technique has never been applied before in the field of remote sensing.