## 1 Introduction  
Underwater geodetic networks (UGNs) are pivotal for ocean resources exploration, environmental monitoring, and underwater navigation [1]. Due to the attenuation of the electromagnetic field in water, the global navigation satellite system (GNSS) is unsuitable for underwater scenarios [2, 3]. Based on the combination of GNSS and acoustic ranging technology (GNSS-A), acoustic underwater geodetic networks can accurately obtain seafloor displacement information [4, 5, 6, 7, 8, 9]. Moreover, geodetic networks can provide location information for underwater targets, including autonomous underwater vehicles (AUV), remotely operated underwater vehicles (ROV), underwater gliders, underwater buoys, etc., which can be deployed in seafloor surveying and ocean precision engineering [2, 3, 10, 11]. Therefore, the construction of UGNs is currently an important research topic in marine geodesy. Acoustic positioning systems, such as Long Baseline (LBL) and Ultra-Short Baseline (USBL), are commonly used for underwater positioning [12]. The seafloor beacon array of LBL can be considered a simple underwater geodetic network. With the appropriate sonar and acoustic ranging technology, the positioning range of UGN can reach 10 km with the same ranging precision at a certain depth [13; 14]. However, the topography of the seabed could obstruct the bending acoustic signal ray and limit the positioning performance of the geodetic network in seafloor surveying and ocean precision engineering. In general, the longer the required acoustic ranging distance, the higher the seafloor base station's erection height should be. For the same erection height, the larger the seabed slope, the shorter the acoustic ranging distance becomes [15]. Similar to UGN, [PERSON] et al. [16] established a GPS intelligent buoy (GIB) using a sea surface GPS buoy, which can also provide positioning for underwater targets. [PERSON] et al. [10] developed an LBL system with float devices on the water's surface, which was utilized for underwater glider positioning. In addition, researchers have studied the combined configuration of acoustic positioning systems. The Scripps Institution of Oceanography (SIO) kept the ship in the horizontal center of three seafloor acoustic transponder arrays and performed GPS-A observations [6; 7]. To compensate for acoustic ray bending, the SIO also proposed a configuration in that one end of an acoustic interrogator floated tens of meters above the seafloor [17]. [PERSON] et al. [18] added a seafloor-mooring acoustic beacon to the underwater positioning network, whose position was determined by responding acoustic signals to the four seafloor base stations. In addition to acoustic ray bending, the acoustic transmission characteristics, including the multipath effect caused by the acoustic signal reflected from the seabed or the sea surface [19], the Doppler effect caused by the motion of the acoustic source [20; 21], the transmission loss of acoustic energy, etc., also affect the acoustic signal propagation. The construction of an UGN also comprises the maintenance of underwater geodetic stations. To obtain the position of the underwater stations, GNSS-A measurements are primarily performed using sea surface vessels, which restricts the revisit observation frequency due to the high costs and low mobility [22]. To address these shortcomings, alternative observation platforms have been investigated. [PERSON] et al. [23], [PERSON] et al. [21], and [PERSON] et al. [22] conducted GNSS-A measurements using a moored buoy and an autonomous surface vehicle (ASV). Some researchers also investigated the capability of wave gliders and unmanned aerial vehicles (UAVs) as platforms for GNSS-A measurements [11; 24; 25]. However, the geometric structures formed by the platforms and the station network affect the positioning accuracy. [PERSON] et al. [26] searched for the lowest geometric dilution of precision (GDOP) and discovered that the circular track can obtain the highest positioning accuracy. [PERSON] et al.