## 6 Path Loss Channel Model  
Rotational Orbit-Based Inter-Node Adjustment (ROBINA) and its Co-variant Path-Adjustment (PA-ROBINA) work for Path-Loss (PL-ROBINA) respectively. Underwater, the acoustic channels are determined by the operating frequency as well as the distance. Here, the operating frequency is only considered between receiver and transmitter as mentioned in [31; 32; 33]. Underwater path loss dependability is quite different in practice in Terrestrial Wireless Sensor Networks (TWSNs). Path Loss (PL) is determined only by the distance between communication nodes in Radio Frequency (RF) [34] between two communication nodes; particularly, the acoustic signal undergoes a frequency 'offshore for a distance '\(I\)'. From the Urick's Model enlisted in [34], the overall PL is denoted by '\(A(l_{J})\)'; finally, we have to modify it according to the scenario. \[A(l,f)=l^{j}(l(f))^{\frac{d}{1000}} \tag{4}\]  
The '\(j\)' represents the circulating factor and describes acoustic signal propagation geometry. Typically, the practical value for '\(K\)' in the literature is 1.5. The acoustic pressure into heat for the absorption loss factor \(\alpha(f)\) is modeled in Equation (4). According to [PERSON]'s formula [35], usually '\(f\)' measured is \(K\)Hz as following in Equation (5):  
\[ \begin{array}{c}A(f)=10\log(\alpha(f))\ \mathrm{dB/km}\\ =\frac{0.11/^{2}}{1+f^{2}}+\frac{40/^{2}}{4100+f^{2}}+2.75\times 10^{-4}f^{2}+ 0.003\end{array}  \tag{5}\]  
According to [36], there are some main backers defined to the ambient noise like thermal noise \(N_{th}\) (\(f\)), turbulence \(N_{th}\) (\(t\)), shipping and human activities \(N_{s}\) (\(f\)), and wind-driven wave \(N_{w}\) (\(f\)), respectively. The continuous Power Spectral Density (PSD) and Colored Gaussian Statistics (CGS) are the major supportive parameters for modeling these equations. Furthermore, the PSD formulae of all these four types of noises with taking frequency '\(f\)' (measured in \(kHz\)) are given empirically below in Equation (6):  
\[ \begin{array}{c}10\log(N_{t}(f))=19-40\log(f)\\ 10\log(N_{s}(f))=50+30(s-0.6)+28\log(f)-80\log(f+0.04)\\ 10\log(N_{w}(f))=60+7.5w^{\frac{1}{2}}+30\log(f)-50\log(f+0.5)\\ 10\log(N_{th}(f))=-18+30\log(f)\end{array}  \tag{6}\]  
Wheres \(\in\) [0, 1] is defined as 0 in legal shipping activity, '\(f\)' in high-level shipment movement, and '\(w\)' is the storm velocity measure in m/s; the '\(w\)' is also utilized by wind-driven waves that cause surface motion to be captured. Furthermore, the total ambient noise' \(N\) (\(f\))' is calculated as in Equation (7):  
\[N(f)=N_{t}(f)+N_{s}(f)+N_{w}(f)+N_{th}(f) \tag{7}\]  
The SNR of an inbound acoustic sound of frequency 'fi' transmitted over a distance' \(d_{j}\), to node '\(n\)' is represented as [36]. \[SNR\big{(}d_{j,n},f_{i}\big{)}=\frac{\left|a_{j,n}^{i}\right|^{2}p_{j,n}^{i}} {A\big{(}d_{j,n},f_{i}\big{)}N(f_{i})B_{S}} \tag{8}\]  
where:  
(1) \(\mathrm{P}_{j,n}^{i}\) shows the power transmission between two nodes. (2) \(N(f_{i})\) is the noise power spectral density. (3) '\(B_{s}\)' denotes bandwidth on the receiver side. In multiple scenarios in Underwater, we only assume the fading ratio of a particular channel between two nodes on a limited scale owing to multipath \(\mid\!d_{j,n}^{i}\!\mid\). Here, we assume \(\mid\!a_{j,n}^{i}\!\mid\) follows the same work as described in [37; 38]. Optimum Acoustic Carrier for Data Transmits Model  
The suggested protocol is based on mathematical models that show the smallest amount of transmission that is normally compulsory for data rate per connection. Assume \(P_{j,n}^{i}\) is used for transmitting packets from node '\(j\)' to node '\(n\)' node subcarrier '\(i\)'. The goal is to determine the lowest feasible transmit power over a set of '\(x\)' subcarriers while maintaining the required transmission rate, indicated by '\(R_{0}\)' between the nodes '\(j\)' and '\(n\)'.