###### Abstract  
Precipitation is closely related to the production and daily life of human beings, so accurate precipitation measurement is of great significance. Spaceborne synthetic aperture radar (SAR) is a microwave remote sensing technology with high resolution, which provides an opportunity to improve the accuracy of precipitation inversion. In this paper, the radar attenuation expression is analyzed according to the scattering characteristics of rain, snow and ground. Combined with the Volterra integral equation of the second kind, the solution to the expression, the precipitation horizontal variation of the double-layer model, can be obtained. The simulated result of this method is in good agreement with the given horizontal variation of precipitation. Compared with the original VIE method, which only considers the effect of rainfall, the method in this paper considers both rainfall and snowfall; compared with the Model Oriented Statistical (MOS) method, the method in this paper not only reduces the number of empirical coefficients used and thus reduces the workload in the early stage and retrieval process and its application limits, but it will also increase the accuracy of the inversion of the horizontal variation. precipitation measurement; synthetic aperture radar; Volterra integral equation; attenuation expression 202214357.14.357.14. The frequency of radio waves working at X band is only 4 GHz lower than that working at Ku band, the working frequency band of precipitation radar (PR). Furthermore, studies have shown that X-SARs are more sensitive to precipitation than SARs operating under lower frequencies [4; 5]. Many countries have successfully launched SAR systems carrying high-frequency X-band synthetic aperture radar (X-SAR) sensors, such as SRTM launched by America in 2000, Cosmo-Skymed launched by Italy in 2006, TerraSAR-X launched by German in 2007, etc. The benefits of using X-SAR to retrieve precipitation are as follows. First, the scale of the spaceborne radar antennas is limited, leading to a serious non-uniform beam filling effect and bringing great deviation to precipitation inversion [6]. Fortunately, this problem can be solved with a high-resolution X-SAR and microwave radiation measurement data. Second, precipitation measurement in mountainous areas is also an important topic. The terrain of such areas brings challenges to PRs while X-SARs provides a method for precipitation measurement in mountainous areas. Third, X-SARs can observe mid-latitude storms which Tropical Rainfall Measuring Mission (TRMM) might miss [7]. Thus, the observation results of X-SARs will supplement and enhance those of Global Precipitation Mission (GPM). Fourth, X-SARs used in the precipitation retrieval increases practical use of X-SAR satellite images and can improve the investment benefits of X-SARs. [PERSON] et al. established a single-layer, the rainfall layer, model for radar observation in 1991 [8]. They obtained the analytical solution to the horizontal rainfall variation of a single-layer model by solving the rainfall attenuation expression combining the Volterra integral equation (VIE) of the second kind. However, because the attenuation coefficients of different condensates (such as rain and snow) are different, this method cannot be directly applied to the model containing two or more condensates. [PERSON] et al. proposed a Model Oriented Statistical (MOS) method. They considered a more complex double-layer precipitation model with a snowfall layer and a rainfall layer completely separated in 2008 [9; 10]. They analyzed the characteristics of radar backscattering cross section and obtained the statistical solution to the horizontal precipitation variation of a double-layer rainfall model. However, this method relies on many empirical values, thus having a large workload in the early stage. What makes the result less accurate is that it makes an approximation in retrieval of the shape of the precipitation area. [PERSON] et al. proposed a MOSVI method, combining the MOS method and the VIE method [11].