## 3 Methods  
### Linear Mixed Effects Model  
For bare soil surfaces with given soil condition (roughness or texture), radar backscattering is found to be linearly dependent upon the volumetric soil moisture in the upper few centimeters [51]. Over vegetated surfaces, the return signal depends both on the volume scattering of vegetation and on the attenuated backscattering from the underlying soil. As in [52], we assumed a linear SM-\(\sigma^{0}\) relationship for both bare and vegetated soil surfaces  
\[\sigma^{0}(dB)=A\times M_{v}+B, \tag{1}\]  
where, \(\sigma^{0}(dB)\) is the backscattering coefficient of bare or vegetated surfaces in logarithmic scale and \(M_{v}\) is the surface soil moisture. The parameter \(A\) is the sensitivity of the backscattering coefficient to changes in soil moisture content, which is primarily related to the vegetation cover and varies seasonally with the vegetation cycle. The parameter \(B\) is a function of soil roughness and vegetation cover effects on radar signal, which varies strongly in both space and time. Unlike some other empirical models that considered the regression parameters as constant, we added random effects to the parameters, allowing for them to vary temporally and spatially. This could be realized by introducing the linear mixed effects model with random slopes and intercepts. The underlying premise of linear mixed effects models is that some subset of the regression parameters varies randomly from one individual to another, thereby accounting for sources of natural heterogeneity in the population. The distinctive feature of linear mixed effects models is that the mean response is modelled as a combination of population characteristics that are assumed to be shared by all the individuals, and subject-specific effects that are unique to a particular individual [53]. The former are referred to as fixed effects, while the latter are referred to as random effects. To represent the varying SM-\(\sigma^{0}\) relationship, a LME model was developed using the soil moisture ground measurements and the collocated Sentinel-1 backscattering coefficients. The LME model regressed \(\sigma^{0}\) as a predictor of SM by including day-specific random intercepts and slopes for \(\sigma^{0}\), and site-specific random intercepts for spatial adjustment. The basic form of the model can be expressed as follows:  
\[ \begin{array}{c}SM_{ij}=\left(\alpha+u_{j}\right)+\left(\beta+v_{j}\right) \times\sigma^{0}_{ij}+s_{i}+\varepsilon_{ij}\\ \left(u_{j}v_{j}\right)\,\sim N\big{[}(00),\Sigma_{\beta}\big{]},\end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**ID** & **Site Name** & **Height (m)** & **Land Cover** & **Mean (\%)** & **Min (\%)** & **Max (\%)** & **Std. (\%)** \\ \hline
1 & Dongxiang & 36.6 & Nursery & 33.08 & 23.20 & 39.10 & 3.93 \\
2 & Yongxiu & 14 & Paddy field & 47.30 & 32.90 & 56.90 & 7.17 \\
3 & Duchang & 37.7 & Dryland & 20.83 & 4.50 & 31.00 & 5.90 \\
4 & Xingzi & 62 & Dryland & 30.63 & 16.50 & 39.90 & 4.93 \\
5 & Fengcheng & 26 & Paddy field & 39.70 & 30.90 & 45.50 & 3.87 \\
6 & Wannian & 51.7 & Grassland & 30.03 & 19.30 & 40.80 & 4.20 \\
7 & Fenggin & 75 & Orchard & 30.66 & 24.00 & 34.90 & 2.50 \\
8 & Hukou & 39.6 & Grassland & 26.81 & 11.30 & 35.80 & 5.95 \\
9 & Nanchang & 29.9 & Dryland & 27.35 & 20.80 & 30.80 & 2.96 \\
10 & Ruichang & 23.6 & Dryland & 35.15 & 26.40 & 38.70 & 3.04 \\
11 & Leping & 26 & Dryland & 30.16 & 22.20 & 35.70 & 3.15 \\
12 & Yuigan & 19 & Dryland & 26.94 & 12.00 & 33.90 & 5.20 \\
13 & Zhangshu & 29 & Dryland & 32.80 & 27.50 & 36.70 & 2.40 \\
14 & Gaoan & 34 & Dryland & 27.17 & 19.90 & 30.30 & 2.54 \\
15 & Xinjian & 7 & Dryland & 20.75 & 13.30 & 24.90 & 2.19 \\ \hline \hline \end{tabular}  
* Mean, min, max and std. represent mean, minimum, maximum values and standard deviation of soil moisture measurements (0–10 cm) respectively.  
\end{table} 
Table 2: Descriptive statistics of 15 soil moisture observation sites.