## Materials and Methods
### Soil Moisture Retrieving Algorithm  
The LUT representation of a complicated forward model has been demonstrated to be an accurate and fast tool for retrieval [47]. This approach is also applied in the SMAP mission as a baseline algorithm for the radar observations [47], and it simplifies the forward model by the soil dielectric constant, the soil surface roughness, and the vegetation water content due to the limited observation channels. By adopting a fine interval of LUT axis, this approach avoids the rather complex retrieval process of the numerical or analytical methods, while achieving similar retrieval accuracy and retaining the same retrieval formula as the forward model. Therefore, we used the LUT approach to estimate the soil moisture of the S1A images, which is similar to the baseline algorithm of the SMAP mission. The approach is divided into the following two steps: Step one: Construct the simulated data cube. In this study, the backscattering coefficients were simulated by the AIEM [48] and the Michigan microwave canopy scattering (MIMICS) [49] model for the soil and vegetation layers, respectively. Only the VV polarization was simulated for the data cube because the HV polarization is not well simulated by the AIEM. The radar configurations in the forward models were a frequency of 5.4 GHz and incident angles ranging from 30 to 46\({}^{\circ}\), with an interval of 2\({}^{\circ}\). Instead of characterizing the soil roughness by just the root-mean-squared height (\(s\)), a discrete combination of \(s\) and correlation length (\(l\)) was set as one axis of the data cube, where \(s\) ranged from 0.5 to 5 cm with an interval of 0.1 cm, and the \(l\) ranged from 5 to 55 cm with an interval of 10 cm. The volumetric soil moisture ranged from 0.03 to 0.45 m\({}^{3}\)/m\({}^{3}\), with an interval of 0.01 m\({}^{3}\)/m\({}^{3}\), and was converted to the soil dielectric constant using the Hallikainen method [50]. The vegetation factor was characterized by the vegetation water content (\(VWC\)), which ranged from 0.02 to 3.6 kg/m\({}^{2}\).  
Step two: Retrieve the soil moisture based on the LUT approach. The cost function is constructed as follows:  
\[C_{\min}(s,l,\epsilon_{r1\dots N})=\left(\sigma^{0}_{vv}(t_{1})-\sigma^{0}_{ vv,datacube}(s,l,\epsilon_{r1},VWC_{1})\right)^{2}+\dots+\left(\sigma^{0}_{vv}(t_{N})- \sigma^{0}_{vv,datacube}(s,l,\epsilon_{rN},VWC_{N})\right)^{2} \tag{1}\]  
where \(\sigma^{0}_{vv}\) and \(\sigma^{0}_{vv,datacube}\) are the backscattering coefficients of the VV polarization from the S1A images and the data cube, respectively; and, \(\epsilon_{r1\dots N}\) are the soil dielectric constants of the S1A time-series. In the function, \(VWC\) is estimated using a set of land-cover-based equations for the NDVI and stem factors. The full details of the \(VWC\) estimation are available in the SMAP Vegetation Water Content Ancillary Data Report [51]. Supposing that soil roughness \(s\) and \(l\) are time-invariant throughout the observation period, then \(\epsilon_{r1\dots N}\) corresponding to the minimal value of the cost function are the desired soil dielectric constants. The obtained dielectric constants were then converted to volumetric soil moisture with the Hallikainen method using the soil texture information from the HWSD.