## 2 Description of the Proposed Approach
### Network Model of the FLRAF  
The transmission attenuation of the underwater acoustic signal depends on center frequency and transmission distance. In this paper, we use the Urick model to characterize the underwater acoustic signal's propagation loss [36], which can be expressed as  
\[A(d,f)=d^{k}\alpha(f)^{\frac{d}{1000}} \tag{1}\]  
where \(d\) represents the Euclidean distance between nodes (in m), \(k\) is the spreading factor describing the propagation geometry with a value ranging from 1 to 2, typically set to 1.5, \(f\) denotes the acoustic carrier frequency (in kHz), and \(\alpha(f)\) represents the absorption coefficient. For the Euclidean distance \(d\) between nodes, \(N_{i}(x_{i},y_{i},z_{i})\) is used to represent the 3D coordinates of any node and \(d_{i,j}\) can be used to represent the Euclidean distance between any two nodes. The calculation formula is as follows:  
\[d_{i,j}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left( z_{i}-z_{j}\right)^{2}}. \tag{2}\]  
Figure 1: Network architecture for underwater acoustic sensor network (UASN) data collection. We note that a variety of underwater localization technologies have been extensively studied and applied, which can effectively mitigate the impact of signal distortion on localization accuracy. These technologies provide a solid foundation to ensure the feasibility of position-based forwarding strategies in practical scenarios [37; 38]. The absorption coefficient \(\alpha(f)\) can be determined according to [PERSON]'s empirical formula [36], which is calculated as  
\[10\text{log}_{10}\alpha(f)=0.003+\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100 +f^{2}}+\frac{2.75f^{2}}{10^{4}}. \tag{3}\]  
Besides path loss, underwater acoustic signals face interference from ocean noise during transmission. Ocean noise \(N(f)\) comprises various components, which can be broadly categorized into the following environmental noise sources: shipping noise \(N_{s}(f)\), turbulence noise \(N_{t}(f)\), thermal noise \(N_{th}(f)\), and wave noise \(N_{w}(f)\). The calculation can be determined using  
\[N(f) =N_{t}(f)+N_{s}(f)+N_{w}(f)+N_{th}(f) \tag{4}\] \[10\text{log}_{10}N_{t}(f) =17-30\text{log}_{10}f\] (5) \[10\text{log}_{10}N_{s}(f) =40+20(s-0.5)+26\text{log}_{10}f\] \[\quad-60\text{log}_{10}(f+0.03)\] (6) \[10\text{log}_{10}N_{w}(f) =50+7.52\sqrt{w}+20\text{log}_{10}f\] \[\quad-40\text{log}_{10}(f+0.4)\] (7) \[d_{i,j}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left( z_{i}-z_{j}\right)^{2}}. \tag{2}\]0  
where \(s\) represents the shipping activity factor, which is a constant ranging from 0 to 1, typically set to \(s\) = 0.8, and \(w\) is the speed of wind (in m/s) set according to specific conditions. After obtaining the transmission loss of the underwater acoustic signal, if only path transmission loss and noise interference are considered, the \(SNR\) between nodes can be estimated using  
\[d_{i,j}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left( z_{i}-z_{j}\right)^{2}}. \tag{2}\]1  
where \(P\) represents the transmitted acoustic source power (in dB/\(\upmu\)Pa) and \(N(f)\) denotes the noise power spectral density. It is important to note that nodes convert electrical signals into acoustic signals to transmit data via underwater acoustic transducers. The conversion between the transmitted acoustic source power and the electrical transmission power is determined by  
\[d_{i,j}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left( z_{i}-z_{j}\right)^{2}}. \tag{2}\]2  
where \(\eta\) is the efficiency of the underwater acoustic transducer, represented as a constant between 0 and 1, typically set to \(\eta\) = 0.8, \(\hat{P}\) is the electrical transmitting power (in W). In this study, the acoustic characteristics of the transmission channel are primarily considered through the conversion between electrical power and acoustic source power, along with modeling the attenuation properties of sound propagation in underwater environments.