# Tomographic Inversion Methods for Retrieving the
## 1 Introduction  
It is widely recognized that tropospheric water vapor (WV), which represents about 99% of the total WV, is of paramount importance in meteorology [1, 2, 3, 4]. As a matter of fact, the amount of water available for precipitation in any form depends on its global and local dynamics. Furthermore, the transformation of the water from the gas phase to the liquid phase--and vice versa--produces the release and absorption of energy in the form of latent heat and, thus, it affects the thermic profiles of the atmosphere, which are at the basis of the atmospheric transport processes. Therefore, the measure of the WV concentration and of its variation as a function of the altitude is essential to guarantee a correct forecast of the rainfall and of the related atmospheric phenomena. However, achieving appropriate resolution and sensitivity to measure the time and spatial (both vertical and horizontal) variations of WV still represents a challenging goal [5, 6, 7]. Furthermore, it is recognized that to precisely represent the global water cycle, a large number of observations is needed as input to atmospheric models [8]. There are several methods to estimate the atmospheric WV content. Roughly speaking, they can be classified into methods that measure the local content of humidity--e.g., in situ methods that provide vertical profiles of WV through radio soundings (possibly incombination with Raman lidar measurements or even through standalone lidars, see for instance [9])--and methods, both active and passive, which estimate the WV remotely through the sensitivity of an electromagnetic signal to the WV content at different frequencies. The former methods are the most precise techniques to measure the WV concentration, but they are an expensive solution and they do not allow for continuous time-spatial measures, as only spaceborne systems may guarantee them. The latter methods are real remote sensing techniques, which can be implemented on satellites, thus providing global coverage: they are based either on ground radiometers working in the infrared spectrum [10] or on the phase difference of signals sent by global navigation satellite systems (GNSS)--such as GPS, Galileo, GLONASS, BeiDou--and measured by fixed stations [2]. This second class of methods is able to provide the average WV content along the electromagnetic path of the sounding signals or, using a specific tomographic approach, also to provide estimates of water vapor above a network of receivers [11; 12]. Average vertical profiles of the WV concentration over an area are provided by specific inversion techniques of such signals by Radio Occultation (RO) methods, with the aid of independent auxiliary information [13]. Other kind of methods have been proposed to observe WV even from geostationary satellites by exploiting infrared measurements [14], or to utilize spaceborne differential absorption radars to provide the columnar WV content, which is a key parameter in numerical prediction models (NWP) [15]. All the global monitoring systems available at present, however, suffer from an accuracy issue of the WV measurements in the lower troposphere where, on the other hand, the WV concentration is maximum [5]. A measurement approach that has the potential to overcome such an issue has been proposed some years ago. It is based on the so-called Normalized Differential Spectral Attenuation (NDSA) method applied to microwave links established between two Low Earth Orbiting (LEO) satellites [16]. The NDSA method enables one to get estimates of the integrated water vapor (IWV, intended as absolute mass of water vapor) along such links; by means of specific inversion methods, WV concentration (intended as absolute humidity) profiles or fields can then, in turn, be estimated. Being based on limb soundings of the troposphere, such an approach gains more sensitivity to the presence of WV in the lower layers with respect to other kinds of measurements from space. The NDSA approach is characterized by a certain simplicity _per se_.