## 1 Introduction  
The flow of water in rivers and streams is of great interest because it represents the easiest access to water, a fundamental natural resource for human beings, animals and cultivations. The ability to quantify the flow of water in terms of river discharge and flow volume depends on the monitoring of water surface height of water bodies [1]. In-situ gauges have been installed along rivers to measure the water height and to describe its variation in the space and time. Unfortunately, the ground network is not uniformly distributed worldwide (many rivers in developing countries are still unmonitored) and since the 1980s, we have also been assisting the decline of gauge stations in the developed countries [2]. For this reason, the advanced capability of satellite sensors to monitor inland water and the direct access to their data motivated scientists to integrate and reinforce the traditional monitoring of surface water with this new source of information. River discharge estimation from satellite remote sensing of river hydraulic variables has been investigated in recent decades [3; 4; 5; 6]. Nadir altimetry has been largely used for measuring the river surface height from space [7; 8]. From the first studies carried out with Geosat [7] to the more recent analyses with Jason-2 and SARAL [8; 9], the improved capability of the altimeters allowed to monitor even narrow rivers, for example, the Po river (\(\sim\)300 m wide) or the Garonne River (\(\sim\)200 m wide). Despite these encouraging results, the use of radar altimetry for narrow rivers is still limited because of uncertainty in the evaluation of the water surface elevation due to the local topography that contaminates the returned radar signal. With the synthetic aperture radar (SAR) technology applied to altimeters, reliable measurements of water level are obtained for rivers of 200 m width [10] and according to the requirements of the next SWOT mission, rivers of 100 m width will be accurately observed by the new Karin sensors [11]. However, the nominal orbit of the satellites often does not guarantee the global coverage of all narrow rivers. The designed inter-track diamond distances (when the ground tracks of the low Earth orbits satellites, traveling on ascending and descending directions, are depicted, geometric figures in the shape of diamonds are obtained) and the revisit time represent obstacles to the monitoring of water courses. To overcome these issues, low cost satellite constellations are investigated in order to provide global coverage and finer temporal resolution. Moreover, the use of high resolution measurements is fundamental to ensure the level information of small water bodies. In the literature, some examples show the use of SAR images to derive information about the water level. For example, Reference [12] showed the use of SAR images from ENVISAT and RADARSAT to indirectly estimate water level of the Severn River (UK) and the Red River (US). Other research has investigated the use of Along Track Interferometry from SAR (ATI-SAR) to obtain water level estimation. However, in order to obtain two interferometric InSAR images with a short time delay from a moving platform, it is necessary to install two antennas separated by the corresponding spatial baseline oriented along the flight direction. Accordingly, the technique is called ATI, which is different if compared to the cross-track interferometry (XTI) used for topographic reconstruction. ATI can be suitably exploited to estimate the surface velocity of water masses and classical dual-sensor ATI geometric configuration was first proposed in Reference [13], in which the authors describe a new method to measure sea surface currents. The experimental results refer to an airborne implementation of the technique, tested over the San Francisco Bay near the time of maximum tidal flow and leading to a map of the east-west component of the water current. This study also underlines that only the line-of-sight (LOS) component of the targets velocities is measured by ATI.