# Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR
## 2 Methods
### Sea Level Based on Carrier Phase-Combination Measurements  
For carrier phase observables, it is difficult to isolate the multipath from several other effects, such as the satellite-antenna pseudorange, ionospheric and tropospheric delay. However, when considering the multipath, the carrier phase contains two quantities: the phase of the direct signal and the composite excess phase with respect to the direct phase, as shown in Equation (3). Thus, the GNSS phase observation for carriers L1 and L2 can be expressed as  
\[L_{1}=\lambda_{1}\psi_{1}(t)+\lambda_{1}\beta_{1}(t) \tag{9}\]  
\[L_{2}=\lambda_{2}\psi_{2}(t)+\lambda_{2}\beta_{2}(t) \tag{10}\]  
In GNSS positioning, the carrier phase of the direct signal can be expressed as  
\[\lambda_{i}\psi_{i}=\rho+I_{i}+T+\Delta \tag{11}\]  
where \(i=1\) or \(2\), which stands for the two signals of different frequencies. The geometric distance between the GNSS satellite and antenna is represented by \(\rho\), and \(I_{i}\) represents ionospheric delays. \(T\) represents tropospheric delays, and \(\Delta\) accounts for all other carrier-independent effects. To isolate the multipath, the L1-L2 combination of carrier phase measurements, shown in Equation (12), is used. For simplicity, this combination is referred to as L4 in this paper. \[L_{4}=L_{1}-L_{2}=I_{1}-I_{2}+\lambda_{1}\beta_{1}(t)-\lambda_{2}\beta_{2}(t) \tag{12}\]  
Some measurement errors can be eliminated in Equation (12), including clock errors and tropospheric delays. Meanwhile, the geometric distance between the GNSS satellite and antenna is eliminated because of the subtraction of carrier phase measurements with two frequencies. Thus, L4 is also known as the geometry-free linear combination of the carrier phase. However, the ionospheric delays are related to the signal frequency, so they cannot be eliminated using this combination. According to [37], ionospheric delays are first order phenomena that are inversely proportional to the square of the frequency. Ionospheric delays in the L1 or L2 carrier phase signal can be expressed as the first two items [38] in Equation (13),  
\[L_{4}=\frac{1}{\sin\theta}\frac{40.3}{f_{1}^{2}}TVEC-\frac{1}{\sin\theta}\frac{4 0.3}{f_{2}^{2}}TVEC+\lambda_{1}\beta_{1}(t)-\lambda_{2}\beta_{2}(t) \tag{13}\]  
where _TVEC_ is the total electronic density. From Equation (13), it is evident that the L4 data contain two kinds of effects. One is the change from ionospheric delay, and the other is from the multipath. To analyze the multipath, the ionospheric delay should be eliminated. Considering the magnitudes of the items in Equation (13), a high-pass filter can be used to remove the ionospheric error components. A 10-degree polynomial fit was adopted in this paper, such that \(dL4\) can be determined as follows:  
\[dL_{4}=\lambda_{1}\beta_{1}(t)-\lambda_{2}\beta_{2}(t) \tag{14}\]  
In general, \(x=\varepsilon\) when \(\tan x=\varepsilon\) and \(\varepsilon\approx 0\). Making use of this approximation, and in the case of \(A_{m}\ll\)\(A_{d}\), after substituting Equations (4) and (5) into Equation (14), Equation (14) can be further simplified as follows:  
\[dL_{4}\approx\lambda_{1}(\frac{A_{m}}{A_{d}})_{1}\sin(\frac{4\pi h}{\lambda_{ 1}}\sin\theta)-\lambda_{2}(\frac{A_{m}}{A_{d}})_{2}\sin(\frac{4\pi h}{\lambda _{2}}\sin\theta) \tag{15}\]  
According to Equation (14), \(dL4\) is the linear combination of two quasi-sinusoidal functions, oscillating at the sine of the elevation angle with a different frequency of \(f_{i}=2h/\lambda_{i}\). Similar to the SNR method, a spectral analysis, such as LSP, is used to deal with the \(dL4\) data because of the irregular sampling intervals of \(\sin\theta\). As \(dL4\) is the linear combination of L1 and L2, shown as a linear combination of two quasi-sinusoidal functions in Equation (15), there will be two peaks in its spectrogram (Figure 3b,d). The peak at the lower frequency is caused by L2, and the peak at the higher frequency is caused by L1.