## 4 Numerical Results and Discussions
### Sea Trial Data Analysis Results  
To validate the proposed Doppler factor estimation algorithm, we analyzed the data collected from a UWA mobile communication experiment conducted near the Ross Sea in Antarctica on 25 December 2022. The transmitter was deployed on a moving boat drifting with the ocean current and the depth of the transducer was about 12 m. The receipt was deployed on the seabed at (\(163^{\circ}45.42^{\prime}\) E, \(74^{\circ}56.347^{\prime}\) S), and the depth of the transducer was about 130 m. In the sea trial, the communication frame structure, as illustrated in Figure 13, was composed of two up-sweeping LFM signals and one down-sweeping LFM signal. The parameters of LFM signals are consistent with those in the simulations. The proposed Doppler factor estimation algorithm can be indirectly validated by modifying (20) using the second up-sweeping LFM signal and the third down-sweeping LFM signal. As the time duration of the gap between LFM\({}^{+}\) and LFM\({}^{-}\) is \(T_{\rm GAP}\), (20) can be modified as  
\[\Delta t_{2}=t_{2}-t_{0}=-\frac{\alpha(f_{\rm L}+f_{\rm H})T_{\rm P}}{2(f_{\rm L }-f_{\rm H})}+\frac{T_{\rm GAP}}{1+\alpha}=\frac{\alpha(f_{\rm L}+f_{\rm H})T _{\rm P}}{2(f_{\rm H}-f_{\rm L})}+\frac{T_{\rm GAP}}{1+\alpha}. \tag{30}\]  
Solving (19) and (30), the estimation of the Doppler factor can be obtained as  
\[\hat{\alpha}=\frac{(t_{2}-t_{1}-a)+\sqrt{(t_{2}-t_{1}-a)^{2}-4a(T_{\rm GAP}-(t _{2}-t_{1}))}}{2a}, \tag{31}\]  
where \(a=\frac{(f_{\rm L}+f_{\rm H})T_{\rm P}}{f_{\rm H}-f_{\rm L}}\). The moving speed of the boat is calculated using the GPS records, and the sound speed in water is about 1440.5 m/s. The actual Doppler factor can be calculated. Figure 14 shows the estimated Doppler factor, where the proposed Doppler factor estimation algorithm and the block estimation algorithm are compared. In Figure 14, the blue circle denotes the calculated actual Doppler factor, the red triangle denotes the estimated Doppler factor with the proposed Doppler factor estimation algorithm, and the green star denotes the  
Figure 12: The performance of UWA communication system in terms of BER. Figure 13: The communication frame structure in the sea trial conducted near the Ross Sea in Antarctica. estimated Doppler factor with the block estimation algorithm. From Figure 14, we observe that the proposed estimation algorithm can estimate the Doppler factor successfully and outperforms the block estimation algorithm. To validate the proposed channel estimation algorithm, we analyzed the data collected from a UWA communication experiment conducted in the East China Sea in September 2022. Two UWA communication nodes were deployed at (122\({}^{\circ}\)59.025\({}^{\prime}\) E, 30\({}^{\circ}\)38.392\({}^{\prime}\) N) and (122\({}^{\circ}\)58.283\({}^{\prime}\) E, 30\({}^{\circ}\)37.328\({}^{\prime}\) N). The depth of two nodes was about 50 m. The communication frame structure as illustrated in Figure 13 is also adopted. Generally, the channel temporal coherence function is adopted to characterize the channel variation over time. It is indicated in [32] that the channel temporal coherence can be calculated by  
\[p(\tau)=\frac{E[\mathbf{h}^{*}(t)\mathbf{h}(t+\tau)]}{\sqrt{E\Big{[}|\mathbf{h}(t)|^{2} \Big{]}E\Big{[}|\mathbf{h}(t+\tau)|^{2}\Big{]}}}, \tag{32}\]  
where \(\mathbf{h}(t)\) is the CIR at geointe \(t\) and \(\mathbf{h}(t+\tau)\) is the CIR with a lag time \(\tau\). Thus, the larger the channel temporal coherence, the slower the channel variation. By matched-filtering each received LFM signal with its corresponding transmitted LFM signal, an estimate of the CIR is obtained for each LFM signal within a communication frame. To calculate the temporal coherence, the CIRs estimated by the second LFM\({}^{+}\) and the third LFM\({}^{-}\) are used.