# A Method for Extracting Acoustic Water Surface Waves Based on Phase Compensation
## 2 Models
### Phase Discontinuity Model
Therefore, the phase history can be expressed as follows:  
\[\phi(t)=\frac{4\pi y(t)}{\lambda}+\frac{4\pi h(t)}{\lambda}+\frac{4\pi H}{\lambda} \tag{9}\]  
When the sea surface fluctuates, the phase change in the target's range bin may exceed \([-\pi,\pi]\) and phase entanglement occurs. At this time, the phase change will be added or subtracted by an integer multiple of \(2\pi\) to ensure that the phase is within \([-\pi,\pi]\). To obtain the true sea surface changes, phase unwrapping is required. When the sea surface fluctuation is greater than the range resolution of the radar, the echo spans multiple range bins. In this case, there may be a jumping phenomenon in the phase history, as shown in Figure 2. Figure 2a shows the range-time graph after dechirp processing. In Figure 2b, the change in the range bin is shown in green, and the phase history before compensation is in black. It can be observed that when the distance increases or decreases, a phase jump occurs, and a phase compensation is necessary. The specific steps for phase compensation will be provided in the third section. When the sea surface fluctuates greatly, the microwave may be received by the radar-receiving antennas after multiple reflections on the sea surface. In this case, the target's range bin selected according to the peak point is not the real range where the real water surface is located, and the phases of different range bins cannot be uniformly calculated. As is shown at Point b in Figure 3b, the range bin of targets jumps from the 5 th range bin to the 9 th range bin, and there is discontinuity in the phase history. The change in range bins may be due to the generation of abnormal waves caused by the reflection of pool walls [17; 18]. Moreover, when the sea surface dip angle is large, the echoes may exceed the range of the receiving antenna, resulting in data loss. However, the millimeter wave radar board of the TI company in experiments has the phenomenon of data loss, resulting in phase discontinuity, as shown in Box a in Figure 3a. The black curve and the red curve in Figure 3b show the difference in the phase history before and after phase compensation, respectively. It can be seen that there are still phase discontinuous points where the range bin does not change. In summary, the factors that cause phase discontinuity can be divided into two categories: (1) the fluctuation of the sea surface causes the change in range bins, resulting in discontinuous phase history; (2) factors such as frame loss, abnormal waves, a large incident angle leading to missed echoes, or multipath effects can cause a phase jump in the phase history. Based on experimental data, two types of additive phase noise, namely  
Figure 2: Phase jumping example I: **(a)** range–time graph, **(b)** range bin and the phase history. \(\Delta\phi_{1}\) and \(\Delta\phi_{2}\), are added to the radar echo Equation (5) to characterize the phase history discontinuity caused by the two types of factors. \[ \begin{split} s_{rx}\left(\hat{t},\eta\right)=&\ \ \ \text{{{{Arect}}}\left(\frac{{{i-\frac{2R(t)}{T_{p}}}}}{{T_{p}}}\right)}\\ &\ \ \ \text{{{{-}}}exp\left[{{i\left({2\pi{f_{c}}\left({t-\frac{2R(t)}{ c}}\right)+\pi{K_{r}}\left({\hat{t}-\frac{2R(t)}{c}}\right)^{2}-\Delta\phi_{1}- \Delta\phi_{2}}\right)}}\right]}\end{split}  \tag{10}\]  
Therefore, similar to the derivation process of Equation (8), an expression for discontinuous phase history can be derived:  
\[\phi^{\prime}(t)=\frac{4\pi y(t)}{\lambda}+\Delta\phi_{1}+\Delta\phi_{2}+\frac {4\pi H}{\lambda} \tag{11}\]  
Equation (11) indicates that the phase of the radar echo contains vibration information of acoustic water surface waves and the phase history is discontinuous, caused by two factors.