## 4 Problem Formulations and Optimization Models
(\(E_{b}\)/\(N_{0}\)) and \(E_{b}\)/\(N_{0}\) can be found from the SNR of the channel. Since the 2-path Rayleigh model does not have a closed-form expression for SNR distribution, it is found through simulations [19]. The value of \(\alpha\) is used to find the transmission loss and the SNR of the channel. Constraint (2) imposes that the transmission range should be lower than a threshold to ensure connectivity. Constraint (3) expresses that the transmission loss should be lower than a transmission loss threshold, that is, the maximum propagation loss for properly receiving the transmitted signal. The SNR of an emitted underwater signal at the receiver can be expressed in dB by the passive sonar equation [3]:  
\[\psi_{0,dB}=SL-TL-NL+DI\geq\psi_{T_{h}} \tag{6}\]  
where \(\psi_{T_{h}}\) has been defined as the detection threshold, _SL_ is the source level, _TL_ is the transmission loss, _NL_ is the noise level and _DI_ is the directivity index. The signal level _SL_ is related to the intensity \(I_{t}\) and hence to the transmission power of the transceiver for shallow water as follows:  
\[I_{t}=\frac{P_{t}}{2\pi~{}1m~{}z},~{}SL=10\log\biggl{(}\frac{I_{t}}{0.67\times 1 0^{-18}}\biggr{)} \tag{7}\]  
where \(P_{t}\) is the transmission power in watts and \(z\) is the depth in meters. In deep water the equation (7) becomes:  
\[I_{t}=\frac{P_{t}}{4\pi~{}1m},~{}SL=10\log\biggl{(}\frac{I_{t}}{0.67\times 1 0^{-18}}\biggr{)} \tag{8}\]  
In our analysis we assume a fixed detection threshold for the received SNR. Based on this assumption we define the transmission loss threshold. It is given by [20]:\[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]0  
where \(SL\) is fixed and \(NL\) is also fixed for a particular operating frequency. We consider five different channel models:  
* Shallow water. * Deep water with convergence zones. * Deep water with deep sound channel. * Shallow water with shadow zones. * Deep water with shadow zones. Details of these models can be found in [3]. Constraint (4) imposes that the sensor node \(s_{i\to 2}\) should be located outside the shadow zone. Constraint (5) states that the energy consumed by the electronically controlled engine to move the sensor to the optimal position should be lower than a threshold. It is defined as [21]:  
\[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]1  
where \(P_{Bw}\) is the consumed power to move the sensor upwards or downwards and depends on the power required to drive the electronics, \(T_{Bw}\) is the time required for the movement, \(v_{s}\) is the moving speed and \(\lambda\) is a constant depending on factors such as the drag force, sensor size and shape, friction of the motor, etc. With the proposed mechanism some extra energy will be consumed to locate the shadow zone and to relocate the sensor to the required position, but communication in the shadow zone will be possible. _Case 2: The uplink \(\mathsf{l}_{i\ i+1}\) between \(s_{i}\) and \(s_{i+1}\) and the uplinks \(l_{(i\to 1)k\ i}\) between \(s_{(i\to 1)k},\ \forall k\in K\) and \(s_{i}\) are located in the shadow zone. \(K\) represents the set of neighbours connected to \(s_{i}\) through uplinks using hierarchical routing._  
In this case under the presence of a shadow zone the sensor node \(s_{i}\) is uncoupled into two sensor nodes \(s_{i\to 1}\) and \(s_{i\to 2}\), as shown in \[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]2 These sensor nodes will be connected to each other inside the shadow zone by a wire to maintain robust communication. Now we should determine the optimal location for the nodes \(s_{i\to 1}\) and \(s_{i\to 2}\) using NLP. Since the topology is tree-like, we assume several nodes \(s_{(i\to 1)k},\ \forall k\in K\) forward their data messages towards node \(s_{i}\). Our objective consists of minimizing the transmission loss while preventing from network partition.