## 2 The Basic Principle of Tidal Inversion  
### Multipath Signal and Carrier-Phase Offset  
Affected by the sea-surface multipath effect, the sea-surface reflected signal causes the measured carrier signal-to-noise ratio of the receiver to produce periodic oscillation in the low-altitude angle region of the satellite. It also makes the measured carrier-phase value produce a small (millimeter level) periodic variation of carrier measurement offset; the frequency of signal-to-noise ratio periodic term and the periodic variation frequency of carrier measurement offset are related to the height variation between the antenna phase center and the sea surface. The former is the theoretical basis of classical GNSS-MR tide level inversion, and the latter is the theoretical basis of this method.  
Figure 1 shows the basic model of tide level inversion of the shore-based receiver. In the figure, H represents the height from the antenna phase center to the reflecting surface, also known as the effective inversion height; \(h\) is the vertical height from the celestial center phase center to the datum of the tide gauge station; and \(\theta\) represents the altitude angle of the satellite.  
Figure 1: The tidal-level inversion model.  
The amount of phase delay generated by the reflected signal received by the receiver compared with the direct signal satisfies the following relationship:  
\[\delta\varphi=\frac{4\pi h}{\lambda}\sin\theta \tag{1}\]  
where \(h\) is the height from the antenna phase center to the sea surface, \(\lambda\) is the carrier wavelength, and \(\theta\) is the satellite altitude angle.  
The direct signal of satellite \(S_{d}\) can be simplified as the following formula:  
\[S_{d}=Ap(t)\sin(2\pi ft) \tag{2}\]  
where \(A\) is the signal amplitude, of which \(p\)(t) is the exclusive or sum of \(\pm 1\) data code and pseudo code, \(f\) is the carrier frequency, and the reflected signal of the direct satellite direct is  
\[S_{m}=\alpha Ap(t-\tau)\sin(2\pi f(t-\tau)+\delta\varphi) \tag{3}\]  
where \(\alpha\) is the reflection coefficient, \(t\) is time variable, \(\tau\) is the propagation time delay, and \(\delta\varphi\) is the phase delay represented by Equation (1). The satellite signal actually received by the receiver can be expressed as  
\[S(t)=Ap(t)\sin(2\pi ft)+\alpha Ap(t-\tau)\sin(2\pi f(t-\tau)+\delta\varphi) \tag{4}\]  
When \(S(t)\) performs a correlation operation with the copied \(C/A\) code inside the receiver, the phase deviation \(\theta_{e}\) of the detected synthetic signal compared with the direct signal satisfies the following equation [13]:  
\[\theta_{e}=\arctan\biggl{(}\frac{\alpha R(t-\tau)\sin\delta\varphi}{R(t)+ \alpha R(t-\tau)\cos\delta\varphi}\biggr{)} \tag{5}\]  
\(R(t)\) is the autocorrelation function of \(C/A\). If the difference between \(R(t-\tau)\) and \(R(t)\) is ignored, the above formula can be simplified as  
\[\theta_{e}=\arctan\biggl{(}\frac{\sin\delta\varphi}{\alpha^{-1}+\cos\delta \varphi}\biggr{)} \tag{6}\]  
Affected by the fact that the receiver antenna can only receive right-handed circular polarity signals, \(\alpha<<1\). Therefore, the above formula can be further simplified to the following equation:  
\[\theta_{e}=\alpha\sin\delta\varphi=\alpha\sin(\frac{4\pi h}{\lambda}\sin\theta) \tag{7}\]  
It can be seen from Equation (7) that the carrier-phase offset changes periodically with the sinusoidal value of the satellite altitude angle, and the frequency is related to the wavelength of the satellite signal and the height from the antenna phase center to the sea surface. Because the carrier-phase offset is difficult to directly separate from the carrier-phase offset, the triple-frequency combined-phase observation resolution carrier-phase offset is used in this paper.