# Analysis of the Influence of Flood on the L4 Combination Observation of GPS and GLONASS Satellites
## 2 Relationship between Multipath Error, Flood, and L4  
The multipath error means that both direct and indirect (such as reflection and refraction) signals can be received during the process of GNSS signal receiving. The amplitude of the indirect signal is affected by the size, shape, and reflection coefficient of the object surface, as well as the incident angle. The reflection coefficient is also dependent on the characteristics of the reflective surface. For example, when the incident angle is \(90^{\circ}\), the reflection coefficient of dry soil is \(0.27\) and the attenuation of the signal is approximately \(-11.4\) dB on GPS L1 frequency. However, the reflection coefficient of fresh water is \(0.80\) and the attenuation is \(-1.95\) dB [10]. Thus, the multipath error during the flood is different from that of under dry soil.  
To demonstrate the relationship between multipath error and L4 observation, a schematic multipath error model is given first. In addition, the reflection signal is taken as an example to analyze the multipath error for convenience. The simple multipath error model on the GNSS receiver is presented in Figure 1.  
For a given incident angle \(\theta\) and the height h of the antenna from the reflection surface, the delay \(\delta\) of the reflection signal relative to the direct signal can be calculated by the trigonometric function, which is expressed as follows:  
\[\delta=2\mathrm{h}\sin\theta \tag{1}\]  
In theory, the angle in Equation (1) is not a fixed value, but a value that changes over time. Moreover, the difference in phase between the reflection signal and the direct signal can be calculated as follows [26]:  
\[\varphi(\mathrm{t})=\frac{2\pi\delta(\mathrm{t})}{\lambda}=\frac{4\pi\mathrm{ h}}{\lambda}\mathrm{sin}\theta(\mathrm{t}) \tag{2}\]  
where h is the height of the antenna; \(\lambda\) is the wavelength of the signal; \(\theta(\mathrm{t})\) denotes the satellite elevation angle. The phase of the direct signal is defined as follows [27]:  
\[\psi(\mathrm{t})=2\pi(\varphi(\mathrm{t})+\mathrm{N}) \tag{3}\]  
where N represents the integer ambiguity. The relationship between the direct signal, reflection signal, and the received signal is given by [23]:  
\[\mathrm{S}(\mathrm{t})=A_{\mathrm{d}}\sin\psi(\mathrm{t})+\alpha A_{\mathrm{m} }\sin(\psi(\mathrm{t})+\varphi(\mathrm{t})) \tag{4}\]  
Figure 1: Simple multipath error model on the GNSS receiver.  
where \(A_{d}\) and \(A_{m}\) are the amplitude of the direct and reflected signals; \(\alpha\) denotes the amplitude of attenuation factor (AAF), which is related to the elevation angle, antenna surrounding environment, and the antenna gain [21]. Equation (4) can be written as:  
\[S(t)=(A_{d}+\alpha A_{m}\cos\psi(t))\sin\psi(t)+\alpha A_{m}\sin\psi(t)\cos\psi(t) \tag{5}\]  
It should be noted that the above analysis mainly discusses the relationship between multipath error and GNSS received signal in theory. However, in the practical application, L4 can be directly obtained by making a difference between L1 and L2 frequency, which is expressed as follows [19]:  
\[L4=L1-L2 \tag{6}\]  
where L1 and L2 denote the GNSS signal on different frequencies. The L4 consists of the constant bias (integer ambiguity), ionospheric error, and multipath error. Thus, to analyze the multipath error, the average moving model and low-pass filter method are used to remove the constant bias and the ionospheric error component in Equation (6). Based on the above analysis, the relationship between multipath error and L4 can be denoted by the residuals of L4. Thus, the residuals of L4 are used to analyze the influence of flood on L4 in this experiment.