# Reconstruction of Rainfall Field Using Earth-Space Links Network: A Compressed Sensing Approach
## 1 Introduction  
High-resolution measurement of rainfall is of great importance for the monitoring and warning of natural disasters such as floods and mudslides [1]. Moreover, high-precision rainfall data play an important role in improving the accuracy of numerical weather prediction [2]. At present, the traditional instruments used to obtain rainfall information mainly include rain gauges, weather radars and meteorological rainfall measuring satellites [3]. The rainfall obtained by a rain gauge only represents the rainfall at the location of the weather station due to the non-uniform spatial and temporal distribution of the rainfall [4, 5]. Weather radar is the primary instrument for ground-based remote sensing of precipitation and plays an important role in the monitoring and warning of extreme rainfall [6]. Additionally, rainfall measuring satellites are capable of global monitoring of widespread precipitation, such as the TRMM (Tropical Rainfall Measuring Mission) and the GPM (Global Precipitation Measurement) mission, but the resolution of rainfall detected by rainfall measuring satellites is very low, and the cost is enormous [7]. The combined use of these traditional instruments has great potential for high-precision detection of rainfall. However, traditional instruments for rainfall detection are unevenly distributed all over the world, especially in remote areas with more sparse density distribution [8]. Therefore, it is of great importance to develop new techniques for detecting rainfall with low cost and high resolution for high-precision detection of rainfall. In recent years, a new method of measuring rainfall using microwave links to retrieve rainfall has been proposed based on the physical phenomenon that a microwave signal is attenuated as it propagates through rainfall [9]. In addition, the use of microwave links to retrieve rainfall data has the advantage of low cost and high resolution compared to the traditional rainfall measuring instruments, as it relies on the widespread distribution of existing communication base stations. Furthermore, the method has been validated for applications such as precipitation classification, radar parameter correction and raindrop size distribution (DSD) inversion [10; 11]. Additionally, the dense distribution of communication base stations has provided the basic conditions for using microwave link networks to retrieve data on large rainfall fields in an area with the development of new-generation communication technologies. [PERSON] et al. [12] and [PERSON] et al. [13] implemented the reconstruction of a rainfall field using a horizontal microwave link network based on inverse distance weighting (IDW) and computer tomography (CT) techniques, respectively, further validating the feasibility of the scheme. However, currently available microwave links, limited by sparse density and irregular topology, are still not capable of supporting IDW and CT algorithms to reconstruct high-precision rainfall fields. In addition to horizontal microwave links between ground-based communication base stations, the method of retrieving rainfall data using the inclined Earth-space links (ESL) between communication satellites in space and ground-based receiving antennas has also attracted widespread interest since its proposed introduction, and has been validated by a number of intensive studies [14; 15; 16]. The research was first initiated with the retrieval of rain rate and accumulated rainfall by a single ESL in the Ku band and Ka band, and the key to that was to accurately extract the rain attenuation from the combined attenuation of the ESL signal caused by multiple factors in the atmosphere [15; 17]. It has been shown that the attenuation caused by light rain is not significant for an ESL in the Ku band, and thus the retrieval results for light rainfall are poor [14; 18].