## 5 Results
### Case 2: Optimal Placement Evaluation  
We consider a sensor node \(s_{i}\) located at \((x_{i},y_{i},z_{i})\). Node \(s_{i+1}\) is located at \((x_{i}\pm j.r,y_{i}\pm j.r,z_{i+1}),j\in N\), \(r=4,\)000 m for shallow water, \(r=10,\)000 m for deep water, \(z_{i+1}<z_{i}\). Nodes \(s_{(i\cdot 1)b}\)\(\forall k\in K\) are located at \((x_{i}\pm j.r,\)\(y_{i}\pm j.r,\)\(z_{(i\cdot 1)k}),j\in N\), \(r=4,\)000 m for shallow water, \(r=10,\)000 m for deep water, \(z_{(i\cdot 1)k}>z_{i}\), \(k\in K\). The acoustic links \(l_{i\cdot i+1}\) and \(l_{(i\cdot 1)k\,i}\) are affected by a shadow zone with height \((z_{sz-end}-z_{sz-start})\), where \(z_{sz-start}<z_{i}<z_{sz-end}\), \(z_{(i\cdot 1)k}>z_{sz-end}\), \(z_{i+1}<z_{sz-start}\). Node \(s_{i}\) is uncoupled into sensor nodes \(s_{i\cdot 1}\) and \(s_{i\cdot 2}\). The optimal locations of node \(s_{i\cdot 2}\) and \(s_{i\cdot 1}\) have been found using NLP solving the optimization problems **P1** and **P2**, respectively. Now we evaluate the performance of the proposed scheme for the optimal location of node \(s_{i\cdot 1}\) (Case 2). For shallow water the value of \(z_{sz-end}\) varies between 50 and 85 meters, \(z_{sz-start}\)\(=40\) m and\(z_{(i\cdot 1)k}\in(z_{sz-end},\)100]. The average transmission loss of the acoustic link \(l_{(i\cdot 1)k\,i}\) as a function of the shadow zone height \((z_{sz-end}-z_{sz-start})\) for shallow water and three different frequencies has been computed. The transmission loss of the link \(l_{(i\cdot 1)k\,i}\) between the node \(s_{(i\cdot 1)k},\)\(\forall k\in K\) and \(s_{i\cdot 1}\) with the highest value is minimized for shallow water. The transmission loss values are not affected by the shadow zone height but they are increased with the frequency to 41 dB for  
Figure 16: Average transmission loss of a link as a function of frequency in shallow and deep water. and \(z_{sz-end}\)= 4049 m for deep water. The optimal locations of nodes \(s_{(i-1)k\to 1}\) have been found with NLP solving the optimization problem **P3** with the objective function 1. The optimal locations of nodes \(s_{(i-2)k\to 1}\) have been found with NLP solving the optimization problem **P3** with the objective function 2. In shallow water, sensor nodes are located following a hierarchical structure at the depth levels of 25, 50, 75 and 90 meters. In deep water, sensor nodes are located following a hierarchical structure at the depth levels of 3,900, 3,950, 4,000 and 4,050 meters. We consider four hierarchical levels with one node \(s_{i}\) at the lowest level, three nodes \(s_{(i-1)k}\), \(\forall k\in K\) at the next level, six nodes \(s_{(i-2)k}\), \(\forall k\in K\) at the next level and 12 nodes \(s_{(i-3)k}\), \(\forall k\in K\) at the deepest level. Figure 18 shows the average SNR for the acoustic links \(l_{(i-1)k\,i}\), \(l_{(i-2)k\,(i-1)k}\), \(l_{(i-3)k\,(i-2)k}\) affected by a shadow zone as a function of frequency for shallow and deep water under different propagation phenomena. The average SNR values in the presence of shadow zones are negative, which indicates that the signal is below the noise level and transmission loss, and it can't be properly recovered. They have not been depicted but are decreased when the frequency is increased and are lower for deep water + shadow zone than for shallow water + shadow zone because the transmission loss values are higher for deep water + shadow zone. The average SNR values are increased significantly when the optimal locations are computed and the communication between sensor nodes is again restablished outside the shadow zone. Furthermore, the improved SNR values are maintained for all frequencies. With shallow water the SNR values are better because in shallow water the transmission loss values are lower. The SNR values are higher with deep water + convergence zone than with deep water + deep sound channel. Nevertheless, in all cases the SNR values are positive and they are increased outside the shadow zones.