## 2 The Basic Principle of Tidal Inversion
### Triple-Frequency, No Geometry, and No Ionosphere Combination  
The triple-frequency combined-phase observation is defined as the linear combination of the carrier-phase deviation of three bands [12]:  
\[\begin{array}{l}M_{1,2,5}(t)=(\lambda_{5}^{2}(\lambda_{1}\widetilde{\phi}_{1 }(t)-\lambda_{2}\widetilde{\phi}_{2}(t))+\lambda_{1}^{2}(\lambda_{2} \widetilde{\phi}_{2}(t)-\lambda_{5}\widetilde{\phi}_{5}(t))\\ \qquad\qquad\qquad\qquad+\lambda_{2}^{2}(\lambda_{5}\widetilde{\phi}_{5}(t)- \lambda_{1}\widetilde{\phi}_{1}(t)))\big{/}(1\text{m}^{2})\end{array} \tag{8}\]  
where \(\lambda_{1}=0.1902937\) m, \(\lambda_{2}=0.2442102\) m, and \(\lambda_{5}=0.2548280\) m represent the wavelengths of L1, L2, and L5 carriers of GPS, respectively. \(\widetilde{\phi}_{i}\) represents the sum of the superimposed value of the carrier-phase value of the direct signal and the carrier-phase offset of the multipath reflected signal, which satisfies the following equation:  
\[\widetilde{\phi}_{i}=\varphi_{i}(t)+\beta_{i}(t) \tag{9}\]  
where, the following table represents specific band signals. In this paper, assuming that the antenna gains of three different frequencies are exactly the same, Equation (8) can be further written as the following equation:  
\[M_{1,2,5}(t)=k_{1}\widetilde{\varphi}_{1}+k_{2}\widetilde{\varphi}_{2}+k_{5} \widetilde{\varphi}_{5} \tag{10}\]  
where  
\[\begin{array}{rcl}\widetilde{\varphi}_{i}(t)&=&2\pi\varphi_{i}(t)+\beta_{i} (t)\\ \mathbf{k}_{1}&=&\lambda_{1}\eta_{1},k_{2}=\lambda_{2}\eta_{2},k_{5}=\lambda_ {5}\eta_{5}\\ \eta_{1}&=&\frac{(\lambda_{1}^{2}-\lambda_{2}^{2})}{12\pi^{2}},\eta_{2}=\frac {(\lambda_{1}^{2}-\lambda_{2}^{2})}{12\pi^{2}},\eta_{1}=\frac{(\lambda_{2}^{2} -\lambda_{1}^{2})}{12\pi^{2}}\\ \mathbf{u}&=&k_{1}N_{1}+k_{2}N_{2}+k_{5}N_{5}\end{array} \tag{11}\]  
After the linear combination of the three carrier-phase observations of GPS, the linear distance between the satellite and the receiver, the ionospheric delay, and the convective layer delay is effectively eliminated. Since the combination amount U of ambiguity has no effect on the periodicity of the carrier-phase offset, the triple-frequency combined-phase observations can be regarded as the linear combination of the carrier-phase offset of the triple-frequency bands, and Equation (10) can be further written as the following equation:  
\[M_{1,2,5}(t)=k_{1}\beta_{1}(t)+k_{2}\beta_{2}(t)+k_{5}\beta(t)_{5} \tag{12}\]  
The effective wavelength of the combined phase is  
\[\lambda_{1,2,5}=\left(\frac{\eta_{1}}{\lambda_{1}}+\frac{\eta_{2}}{\lambda_{2 }}+\frac{\eta_{5}}{\lambda_{5}}\right)^{-1}=465.005\;\mathrm{m} \tag{13}\]  
Like the classical GNSS-MR tide level inversion principle, the frequency of the triple-frequency combined-phase observation value that varjes with the sinusoidal value of the height angle is linear with the effective inversion height. The difference is that the tide level height value needs to be observed in advance for a period of time to establish a linear model between the effective inversion height and the triple-frequency combined-phase observation value.  
The sinusoidal value of the height angle is a variable with unequal spacing, so it is impossible to extract the frequency information from the triple-frequency carrier-phase combination observation value by fast Fourier transform. Therefore, the Lomb-Scargle spectrum method is used to extract the frequency value of the combined signal in this paper. The algorithm is as follows [13; 14]:  
\[P_{x}(f)=\frac{1}{2\delta^{2}}\left\{\frac{\left[\sum\left(\phi_{n}^{j_{n}}- \overline{\phi^{jj}}\right)\cos\omega(t_{n}-\tau)\right]^{2}}{\sum\cos^{2} \omega(t_{n}-\tau)}+\frac{\left[\sum\left(\phi_{n}^{j_{n}}-\overline{\phi^{jj }}\right)\sin\omega(t_{n}-\tau)\right]^{2}}{\sum\sin^{2}\omega(t_{n}-\tau)}\right\} \tag{14}\]  
where \(\overline{\phi^{jj}}\) and \(\delta^{2}\) are the mean and the variance of the triple-frequency combined-phase observations, respectively, and \(\omega\) is the angular frequency.