# Research on Shore-Based River Flow Velocity Inversion Model Using GNSS-R Raw Data
## 2 River Flow Velocity Inversion Model
### Inversion Method  
When GNSS satellite signals are reflected by the river in the target region, the Doppler frequency shift caused by the river flow velocity is the low-frequency component in the spectrum, due to the very low river flow compared with the satellite's velocity. Moreover, the disconnected residual phase caused by random noise \(N_{i}\) and \(N_{q}\) only affects the high-frequency part of the frequency spectrum. In this article, GPS and BDS use the same inversion method. By analyzing the residual phase spectrum of GNSS direct signal and reflected signal of the river to extract the low-frequency component of the spectrum, the inversion of river flow velocity can be realized. The geometric relationship of GNSS-R river flow inversion is shown in Figure 5.  
In this paper, the principle of the reference [30] was expanded to apply to the inversion of river flow velocity under the experimental conditions of shore-based dual antennas. Unlike the experimental conditions of airborne experiments, under the conditions of shore-based experiments, there is no Doppler frequency caused by flight vibration. This paper explains in detail the magnitude and influence of each frequency component. The phase and frequency of the direct signal and reflected signal received by the receiver at time \(t\) are defined as follows:  
\[\varphi_{d}(t)=2\pi f_{d}\cdot t \tag{8}\]  
\[\varphi_{r}(t)=2\pi f_{r}\cdot t\]  
\[f_{d}=f_{GNSS}-f_{shock}+f_{dm}+f_{ds}\]  
\[f_{r}=f_{GNSS}-f_{shock}+f_{m}+f_{ra}+f_{flow} \tag{9}\]  
where \(f_{GNSS}\) is the carrier frequency of satellite signals; the crystal vibration frequency of the receiver is \(f_{shock}\) (the crystal vibration frequency of the GPS receiver used in this paper is 1575 MHz, and that of the BDS receiver is 1561 MHz); \(f_{dm}(t)\) is the carrier Doppler frequency of direct signal due to satellite motion; \(f_{rm}(t)\) is the carrier Doppler frequency of reflected signal carrier due to satellite movement; \(f_{da}(t)\) and \(f_{ra}(t)\) are the Doppler frequencies generated when the satellite signal propagates through the ionosphere and the atmosphere of the direct signal and reflected signal, respectively; the magnitude of \(f_{a}(t)\) is near \(10^{-2}\) Hz [35]. Since the receiver is fixed in the shore-based experiment, the receiver will not generate a Doppler frequency. Finally, \(f_{flow}(t)\) refers to the Doppler frequency generated by the river. Figure 5: Geometric relationship of river flow velocity inversion. Using the open-loop tracking method of Section 2.1, the residual interferometric phase \(\varphi_{out}\) of the unwrapped phase could be obtained:  
\[ \begin{array}{ll}\varphi_{out}(t)&=2\pi(f_{r}-f_{d})\cdot t+\Delta\phi\\ &=2\pi(\Delta f_{m}+\Delta f_{a}+f_{flow})\cdot t+\Delta\phi\end{array}  \tag{10}\]  
where \(\Delta f_{m}(t)=f_{rm}(t)-f_{dm}(t)\). Under shore-based experimental conditions, the delay of reflected signal and direct signal is within 1 ms. The path of direct/reflected signals are almost the same in the shore-based experiment set-up, so the satellite movement during the direct path and that during the reflected path are practically identical, and \(\Delta f_{m}\) will be much lower than \(f_{flow}(t)\). Due to the shore-based experiment setup, the path of direct/reflected signals through the ionosphere and atmosphere is almost the same, \(f_{da}(t)\) approximately equals \(f_{ra}(t)\), and the value is simultaneously very small (\(10^{-2}\) Hz), so \(\Delta f_{a}(t)=f_{ra}(t)-f_{da}(t)\) could also be ignored. Eventually, under shore-based experimental conditions, the parameters that have little influence on the inversion of river flow velocity are removed, and \(\varphi_{out}(t)\) can be written as follows:  
\[\varphi_{out}(t)\approx 2\pi f_{flow}\cdot t+\Delta\phi \tag{11}\]  
By FFT processing, the maximum value of the low-frequency component in the residual phase output spectrum \(f_{flow}\) could be obtained from Equation (11).