# Review on the Role of GNSS Meteorology in Monitoring Water Vapor for Atmospheric Physics
## 2 GNSS Methodologies
### Tomography  
Although there have been some experiments with single-station retrievals of the vertical structure of water vapor [33; 34], profiles are generally obtained from network solutions. Otherwise, a distribution has to be decided a priori (typically a decreasing exponential with height). One of the first efforts to water vapor tomography was made by [PERSON] et al. [35], who developed a method for obtaining slant water vapor (water vapor integrated along the slant path) and compared it to water vapor radiometer (WVR), with RMSE less than 4 mm. Voxel approaches, however, discretize the space in cubes (voxels) and try to obtain the concentration of water vapor in each of these voxels. This way, they provide a 3D (or 4D, if time is also considered) representation of water vapor. There are, however, many different approaches to accomplish this and none of them are trivial. One way to optimize the process was provided by [PERSON] and [PERSON] [36]. They proposed the use of a long-term mean radiosonde water vapor profile in the location of the GNSS network. The horizontal voxel definition was optimized using a non-uniform symmetrical division, based on the geometry of the network. Figure 2 shows a representation of the voxel tomography, with rays crossing the voxels. At a specific time, one voxel can be crossed by many rays, while other by very few rays. This fact and other complexities of the tomography problem make it need regularization and optimization. [PERSON] and [PERSON] [37] showed a method to solve some of these issues, obtaining tomography values of water vapor density with high spatio-temporal resolution, resulting in RMSE of 0.39 g m\({}^{-3}\). The method was to use a ray-tracing technique in which rays are allowed to move in the 3D space with a hybrid regularization method, which mixed direct and iterative methods. The main idea was to project the large-scale problem into a small-scale one that can be regularized by a direct method, and then transform back to large-scale. The technique, apart from being more efficient computationally, improved the results in some cases with respect to other methods. A different method by [PERSON] et al. [38] tried to maximize the number of rays by considering water vapor density in the upper region outside the boundaries as four (one per direction) new parameters to obtain. This addition to the traditional method allowed the use of rays that do not come from the top boundary of the study region, but from the sides. An experiment in Hong Kong was carried out to test this method, showing better results than traditional tomography. [PERSON] et al. [38] also showed that, to improve tomography, it is more relevant to increase the number of voxels crossed by rays than than increasing the number of rays. However, the number of possible approaches is large and, therefore, [PERSON] et al. [39] compared several variations of approaches to tomographic techniques with radiosonde and ECMWF reanalysis. This study recommended using stacked data over a 30 min window (increasing the number of available rays) and pseudo-slant observations in case studies and nowcasting. Regarding parametrization, another study compared some methods using the European Cooperation in Science and Technology (COST) data-set (central Europe) [40]. Among them, ray-tracing was found to be more adequate than straight-line methods. In an effort to increase the number of observations with respect to the number of unknowns, [PERSON] et al. [41] proposed another method, which consisted on modeling the height factor for both the isotropic and anisotropic part of slant wet delay (SWD). This would allow signals that do not cross the tomography top boundary to be included in the calculations, increasing the number of valid observations.