## 2 Description of the Proposed Approach
### Forwarding Area of FLRAF  
Figure 2 shows the one-hop transmission range scenario of UASN data transmission, where node S0 is the sink node and node S1 is the current forwarding node. If S4 is selected as the next hop forwarding node, the effective advance distance \(AD\) is defined as  
\[AD=d\times\cos\theta. \tag{11}\]  
From the perspective of the Thorp propagation model, when data are transmitted via multiple hops to the final sink node, maximizing \(AD\) for each hop will minimize the end-to-end delay and the number of hops from the data transmission standpoint. Consequently, this approach provides benefits in both energy efficiency and transmission latency. Specifically, as shown in Figure 2, when node S2 is located on the line connecting nodes S1 and S3, within the one-hop transmission range of S1 with a radius of R, S1 can directly send data to S3. It can also relay data through forwarding node S2. These two forwarding schemes have different energy consumption levels. Assuming the received signal power is \(P_{0}\), according to Equation (9), the node transmission power is \(A(d,f)N(f)P_{0}\). For the two mentioned data transmission schemes, we can use  
\[E_{\text{single}} =A(d_{13},f)N(f)P_{0}T_{tx}+E_{elec}\] \[=N(f)P_{0}T_{tx}d_{13}{}^{k}10^{\frac{a(f)d_{13}}{10}}+E_{elec} \tag{12}\] \[E_{\text{multiple}} =A(d_{12},f)N(f)P_{0}T_{tx}+A(d_{23},f)N(f)P_{0}T_{tx}\] \[\quad+2E_{elec}\] \[=N(f)P_{0}T_{tx}d_{12}{}^{k}10^{\frac{a(f)d_{13}}{10}}\] \[\quad+N(f)P_{0}T_{tx}d_{23}{}^{k}10^{\frac{a(f)d_{23}}{10}}+2E_{ elec} \tag{13}\]  
to represent the energy consumed by single-hop transmission and multihop transmission within the one-hop transmission range, where \(E_{elec}\) denotes the energy required for the electronic equipment of the transmitter and receiver and \(T_{tx}\) represents the transmission time of a data packet. Assuming that multihop transmission is more energy-efficient, then  
Figure 2: Node’s one-hop transmission range diagram. \[d_{1,3}10^{\frac{a(f)d_{1,3}}{30}}-d_{1,2}10^{\frac{a(f)d_{1,2}}{10}}-d_{2,3}10^{ \frac{a(f)d_{2,3}}{30}}>\frac{E_{elc}}{N(f)P_{0}T_{tx}}. \tag{14}\]  
When node S2 lies on the line connecting nodes S1 and S3, there is \(d_{1,3}=d_{1,2}+d_{2,3}\); otherwise, there is \(d_{1,3}<d_{1,2}+d_{2,3}\). In terms of communication hardware, we choose the commonly used underwater acoustic modem model UWM1000 (LinkQuest Inc., San Diego, CA, USA) [40]. After our verification, obviously, Equation (14) is not valid, that is to say, within the one-hop transmission range, \(E_{\text{single}}<E_{\text{multiple}}\), extended to the whole network range, in order to maintain the optimal energy consumption, it is necessary to ensure as few multihop transmission hops as possible. When the current forwarding node is S1, nodes S3 and S4 are within the one-hop transmission range, but \(AD_{53}>AD_{54}\); therefore, to ensure as few multihop transmission hops as possible, nodes with larger \(AD\) within the forwarding node should be selected. To further constrain the forwarding node and make the forwarding area contain as many nodes with larger \(AD\) as possible, we redefine the geometric boundary and selection criteria of the forwarding area, as shown in the blue area in Figure 3.  
The forwarding area of node A can be represented as  
\[\left\{ \begin{aligned} &(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}\leq R^{2} \\ &\cos(\gamma)\leq\frac{\overrightarrow{\text{AB}}\cdot \overrightarrow{\text{AD}}}{\left|\overrightarrow{\text{AB}}\right|\times \left|\overrightarrow{\text{AD}}\right|}\end{aligned} \right. \tag{15}\]  
where the coordinates \((x_{0},y_{0},z_{0})\), \((x_{1},y_{1},z_{1})\), \((x_{2},y_{2},z_{2})\), and \((x,y,z)\) denote the positional information for forwarding node A, sink node B, node C in the previous hop's forwarding region, and location D shown in Figure 3. \(\gamma\) is the semi-vertex angle of the \"cone\" forwarding area. When the node position \((x,y,z)\) satisfies Equation (15), it can be determined that the node is in the forwarding area of the previous forwarding node.