# Sea Level Estimation Based on GNSS Dual-Frequency Carrier Phase Linear Combinations and SNR
## 2 Methods
### Theoretical Model for Calculating Reflector Height  
The values of peak frequency from the spectral analysis based on SNR and geometry-free linear combinations of the carrier phase can be determined using the methods described in Sections 2.2 and 2.3. The peak frequencies for L4 are determined from the \(d\)L4 sequence using LSP, and they consist of two components with different frequencies. There is an interaction between the two different frequencies; thus, peak frequencies will contain some errors. Therefore, the reflector height usually cannot be calculated accurately using Equation (8).  
To address this problem, the relationship between the multipath signature peak frequency and the reflector height can be established using a theoretical model [25; 27]. First, the dL4 for different reflector heights are computed using Equation (14) using simulated carrier phase multipath errors [39]. Then, a series of spectral peak frequencies, which vary with reflector heights, can be determined using LSP. A linear relationship between them can be established as follows:  
\[h=a\times f+b \tag{16}\]  
where \(a\) and \(b\) represent constants, \(f\) is the peak frequency determined from the spectral analysis and \(h\) is the reflector height.  
The L1, L2 and L4 carrier phase multipath errors are simulated using forward modeling of the GPS multipath, as proposed by [PERSON] and [PERSON] [39]. This simulator is mainly used in multipath research to simulate the multipath error in GNSS SNR, carrier phase and code pseudorange observables. Different surfaces, including soil, snow and seawater, and different antenna types and orientations, together with polarization and coherence, are considered. The surface height standard deviation also serves as a parameter when considering the loss of coherent power in the simulator, which may be of importance for multipath research into rough surfaces, such as the ocean surface.  
The simulated results are shown in Figure 3c,d, in which two peaks in the LSP signature are clearly evident. These peaks were derived from the modeled L4 multipath pattern, and can also be  
Figure 3: (**a**) Multipath error of double-frequency phase combination; (**b**) Lomb–Scargle Periodogram from satellite PRN30 recorded on 2 and 3 January 2015; (**c**) Simulated double-frequency phase multipath using the software developed by [39]; (**d**) Lomb–Scargle Periodogram of the simulated L4 multipath.  
seen from the observed data shown in Figure 3a,b. The modeled data are almost identical to the simulated results shown in Figure 3c,d.  
Simulation results for the relationship between antenna height and peak frequency are shown in Figure 4. It is evident that the antenna height can be well-modeled as a linear function of the spectral peak frequency within specific elevation ranges. For (5-12\({}^{\circ}\)) and (5-15\({}^{\circ}\)), they are as follows:  
\[\begin{split} h&=0.127*f-0.3937\\ h&=0.1282*f-0.4324\end{split} \tag{17}\]  
Equation (17) is obtained from simulations of the particular antenna used in the following experiment and the common surface permittivity, and does not consider the actual temperature, salinity or surface roughness. The fitting error in terms of root-mean-square error (RMSE) of the linear model is 1 cm. After obtaining the spectral peak frequency of L2 from the combined raw phase measurements, the reflector height can be calculated using Equation (17).