## 2 Methods
### Kinematic Positioning Performance Assessment Methods  
To evaluate the kinematic positioning performance of the UGN, an underwater space with a depth of 3000 m and a horizontal range of 25 km \(\times\) 25 km was simulated in grids, and the grid spatial resolution was 500 m on the horizontal and 200 m on the vertical. For ocean near-bottom surveying and precision engineering applications, the user can distribute at any vertex of the grids 100 m above the seafloor, and the total number of grids was 39,015. The underwater user's single-epoch kinematic positioning can be realized by receiving acoustic signals from UGN. The acoustic sonar properties and the underwater environment influence the kinematic positioning performance. Considering the current acoustic ranging technology [13; 32], the omni-direction acoustic sonar's operating frequency is 8-16 kHz, and the maximum positioning range of one network station can reach 10 km with the same precision at a depth of 3000 m. Additionally, the time delay measurement error was set to 0.1 ms, which was equivalent to the ranging error of approximately 0.15 m. To simplify the impact of the underwater environment, only the constant sound speed error of 0.1 m/s was considered in the simulation [35]. In terms of the positioning range and precision, the Available Station Number (ASN), Effective Positioning Grid (EPG), and Position Dilution of Precision (PDOP) indicators are considered. * Available Station Number (ASN) and Effective Positioning Grid (EPG)  
For the user's random position, the Euclidean distance between the user and the station (seafloor base station, subsurface buoy, or sea surface buoy) is set to \(L\). If \(0<L\leq 10\) km, then the user is within the effective positioning range of a station, and we consider this station as an available station for this user grid point; otherwise, it is unavailable. For different user grid points, the available station number (ASN) is variable. Similar to GNSS, if the sonar signal delay is ignored and the clock of the synchronization-type sonar is precisely synchronized, the user single-epoch kinematic positioning can be achieved by receiving acoustic signals from at least 3 stations (seafloor base stations and/or buoys). We defined the Effective Positioning Grid (EPG) as the user grid point with at least 3 available station numbers (ASNs). Therefore, the EPG index is intended to evaluate the effective positioning range of the network configuration in the underwater space. * Position Dilution of Precision (PDOP)  
To realize user single-epoch kinematic positioning, the least-squares adjustment can be applied, and the basic positioning model is  
\[l_{i}^{j}=L_{i}^{j}(\mathbf{x}_{i})+v_{i}^{j} \tag{1}\] \[L_{i}^{j}(\mathbf{x})=\left\|\mathbf{c}^{j}-\mathbf{x}_{i}\right\| _{2}=\sqrt{\sum\limits_{k=1}^{3}\left(x_{i,k}-c^{j,k}\right)^{2}}\]  
where \(l_{i}^{j}\) is the acoustic measurement in distance between user \(i\) and station \(j\), \(L_{i}^{j}(\mathbf{x}_{i})\) is the Euclidean distance, \(v_{i}^{j}\) is the distance error, and \(\mathbf{x}_{i}\) and \(\mathbf{c}^{j}\) are the positions of user \(i\) and station \(j\), respectively. The linearization of the Equation (1) is  
\[ \begin{split} v_{i}^{j}&=\mathbf{e}_{i}^{j}\delta \mathbf{x}_{i}-\left(l_{i}^{j}-L_{i}^{j}(\mathbf{x}_{i}^{0})\right)\\ \mathbf{e}_{i}^{j}&=\left[c^{j,1}-x_{i,1}^{0},\ c^{ j,2}-x_{i,2}^{0},\ c^{j,3}-x_{i,3}^{0}\right]/\left\|\mathbf{c}^{j}-\mathbf{x}_{i}^{0 }\right\|_{2}\end{split}  \tag{2}\]  
where \(\delta\mathbf{x}=\mathbf{x}-\mathbf{x}^{0}\), and \(\mathbf{x}^{0}\) is the initial value of \(\mathbf{x}\). Accumulating the acoustic measurements from all of the available stations, we obtain the error equations  
\[