## 4 Methods
### Preprocessing
#### 4.1.5 Vegetation Correction  
In terms of the first-order radiative transfer model, the total backscattering signal can be written as the sum of the bare soil component and the vegetation affected scattering component:  
\[\sigma_{0}^{F}=\ \left(f_{b}\ \times\ \sigma_{soil}^{F}\right)+\left(1-f_{b}\ \right) \times\left(\sigma_{\text{reg}}^{F}+\gamma^{2}\sigma_{soil}^{F}+\sigma_{int}^ {F}\right) \tag{7}\]  
with \(\sigma_{soil}^{F}\), \(\sigma_{\text{reg}}^{F}\) and \(\sigma_{int}^{F}\) being the soil surface, vegetation and vegetation--surface interaction backscattering component, respectively, \(\gamma^{2}\) being the vegetation attenuation factor and \(f_{b}\) being the bare soil fraction [76]. Applying a short-term change detection approach, the ratio of two consecutive backscatter signals are as follows:  
\[\frac{\sigma_{0}^{F2}}{\sigma_{0}^{F1}}=\frac{\left(f_{b}\ \times\ \sigma_{soil}^{F2} \right)+\left(1-f_{b}\ \right)\times\left(\sigma_{\text{reg}}^{F2}+\gamma^{2}\sigma_{soil}^{F2}+ \sigma_{int}^{F2}\right)}{\left(f_{b}\ \times\ \sigma_{soil}^{F1} \right)+\left(1-f_{b}\ \right)\times\left(\sigma_{\text{reg}}^{F1}+\gamma^{2}\sigma_{soil}^{F1}+ \sigma_{int}^{F1}\right)} \tag{8}\]  
For areas dominated by agricultural crop production, the surface changes from bare soil conditions to fully vegetated conditions during one growing cycle. For agricultural areas, this leads to different equations during bare soil and vegetation period, assuming that the bare soil fraction changes from 0 to 1:  
\[\frac{\sigma_{\text{baresoll}}^{F2}}{\sigma_{\text{baresoll}}^{F1}}=\frac{ \sigma_{\text{soil}}^{F2}}{\sigma_{\text{soil}}^{F1}} \tag{9}\]  
and  
\[\frac{\sigma_{\text{vegetation}}^{F2}}{\sigma_{\text{vegetation}}^{F1}}=\frac{ \sigma_{\text{reg}}^{F}+\gamma^{2}\sigma_{\text{soil}}^{F2}+\sigma_{int}^{F}} {\sigma_{\text{reg}}^{F}+\gamma^{2}\sigma_{soil}^{F1}+\sigma_{int}^{F}} \tag{10}\]  
In this regard, a change in soil moisture leads to different changes in the total backscatter signal when comparing bare soil and vegetation periods:  
\[\frac{\sigma_{soil}^{F2}}{\sigma_{soil}^{F1}}\ \
eq\ \frac{\sigma_{\text{reg}}^{F}+ \gamma^{2}\sigma_{soil}^{F2}+\sigma_{int}^{F}}{\sigma_{\text{reg}}^{F}+ \gamma^{2}\sigma_{soil}^{F1}+\sigma_{int}^{F}} \tag{11}\]  
Here, the related parameters of bare soil fraction \(f_{b}\), vegetation attenuation \(\gamma^{2}\), as well as vegetation and vegetation-surface backscatter component \(\sigma_{\text{reg}}^{Fourier}\) and \(\sigma_{int}^{Fourier}\), causing the difference in backscatter sensitivity of the total backscatter signal, are all related to vegetation. For simplification, we therefore assume that a systematic change in the backscatter ratios between bare soil and vegetation conditions can be observed and described by a linear relationship using an appropriate estimator for the transition between bare soil and vegetation conditions. Comparing the backscatter signals from both agricultural areas as well as meadow areas for the Rur catchment, having similar temporal soil moisture dynamics and soil moisture ranges the intensity of the co-polarized backscatter signal was chosen as an estimator for differentiating the influence of vegetation on the related backscatter ratios. As shown in Figure 5, the backscatter signal from agricultural areas ranges between 0.025 and 0.200, with significantly higher values within the bare soil period, compared to a range of 0.025 and 0.120 for the meadow stations. Especially during the shooting-phase in spring, the increasing vegetational effect on the backscatter signal leads to sharply declining total backscattering values, becoming partly uncorrelated to the temporal behaviour of soil moisture. Looking at the agricultural areas, this results in overall minima in backscattering values within the period between March and May, where the influence of changing vegetation is high on the total backscattering signal, while the overall minimum of in-situ measured soil moisture is occurring during August and September.