## 2 Data and Methods
### Methods  
#### 2.3.1 The SM2 RAIN Algorithm  
The SM2 RAIN algorithm is based on the soil water balance equation (Equation (1)) to derive accumulated precipitation:  
\[p(t)=Z^{*}\frac{dS(t)}{dt}+r(t)+e(t)+g(t), \tag{1}\]  
where \(Z^{*}\) (mm) is the saturated soil water capacity (soil depth times soil porosity), \(S(t)\) (-) is the surface relative soil saturation (0-5 cm), \(t\) (d) denotes time, \(p(t)\) (mm/d), \(r(t)\) (mm/d), \(e(t)\) (mm/d), and \(g(t)\) (mm/d) are precipitation, surface runoff, actual evapotranspiration, and drainage (deep percolation plus subsurface runoff) rate, respectively. In this study, a simplified version of the SM2 RAIN algorithm is adopted by neglecting the contributions of runoff (i.e., \(r(t)\) = 0) over large areas (>20 km) following [PERSON] et al. [15]. The evapotranspiration rate (\(e(t)\)) and drainage rate (\(g(t)\)) in Equation (1) are calculated following [PERSON] et al. [15] and [PERSON] et al. [14]:  
\[e(t)=K_{c}ET_{PML}(t), \tag{2}\]  
\[g(t)=\frac{aS(t-1)^{b}+aS(t)^{b}}{2}, \tag{3}\]  
where \(ET_{PML}(t)\) (mm/d) is the actual evapotranspiration rate obtained from the PML-V2 (China), \(K_{c}\) (-) is a correction factor for taking into account potential bias in the PML-V2 (China) estimates, \(a\) (mm/d) is the saturated hydraulic conductivity, and \(b\) (-) is the exponent related to the pore size distribution index [62]. Finally, by combining Equations (1)-(3), the precipitation amount can be estimated as:  
\[p(t)=Z^{*}\frac{dS(t)}{dt}+K_{c}ET_{PML}(t)+\frac{aS(t-1)^{b}+aS(t)^{b}}{2}, \tag{4}\]  
where \(Z^{*}\), \(K_{c}\), \(a\), and \(b\) are parameters that need to be calibrated. Additionally, prior to the SM2 RAIN implementation, a nonlinear exponential filter proposed by [PERSON] et al. [63] was applied to remove noise from the input soil moisture. This introduced two additional parameters, namely, \(T_{base}\) and \(T_{pot}\). They represent the minimum value of \(T\) (the characteristic time length) corresponding to saturated soil conditions and the parameter allowing to increase \(T\) by decreasing the Soil Water Index [63], respectively. The upper and lower boundaries of these parameters were set as listed in Table 1, mostly following [PERSON] et al. [15] and the associated SM2 RAIN source code (available at [[http://hydrology.irpi.cnr.it/download-area/](http://hydrology.irpi.cnr.it/download-area/)]([http://hydrology.irpi.cnr.it/download-area/](http://hydrology.irpi.cnr.it/download-area/)), accessed on 18 June 2022). [PERSON] et al. [16]also illustrated the general range of each parameter over the globe. However, parameter \(a\) was determined based on the saturated conductivity measured by [PERSON] et al. [43]. These six parameters (\(Z^{*}\), K\({}_{c}\), \(a\), \(b\), \(T_{base}\), and \(T_{pot}\)) were calibrated by minimizing the root mean square error between estimated and observed precipitation (i.e., rain gauge observations from the CMA station) and then fed back to SM2 RAIN to estimate precipitation over the entire study period. The IMERG precipitation with high spatiotemporal resolution (half hourly and 0.1\({}^{\circ}\)) was used to investigate the spatial representativeness of the CMA rain gauge station. Specifically, cumulative precipitation curves of the IMERG grid where the CMA station is located and the regional average IMERG precipitation across the 1\({}^{\circ}\) network were compared during the rainy season (JJAS) of 2015-2017. The results (as shown in Figure 4) demonstrate close accumulated precipitation and high correlation (0.80, 0.73, and 0.71 for 2015, 2016, 2017, respectively) between them, thereby indicating the feasibility of using centroid CMA rain gauge measurements to represent precipitation over the entire 1\({}^{\circ}\) grid. Figure 5a displays the number of effective retrievals for SMAP ascending and descending soil moisture products during the JJAS 2015-2018. Obviously, SMAP had fewer retrievals in 2015 and 2016, both for ascending and descending data.