# Underwater Electromagnetic Sensor Networks, Part II: Localization and Network Simulations
## 3 Algorithms for Self-Localization and Navigation-Aid Applications
### Numerical Experiments  
We consider a network of 21 nodes forming a grid, where \(|\mathcal{N}_{a}|=4\) and \(|\mathcal{N}_{u}|=17\). The nodes have a separation of \(15\,\mathrm{m}\) in the horizontal and vertical directions as depicted in Figure 5. We used the channel coefficients derived in Section 2, where the channel model is given by (1). The nodes use \(40\,\mathrm{d}\mathrm{B}\mathrm{m}\) of transmit power, as the commercial product [24], and they have an attenuation of \(-125\,\mathrm{d}\mathrm{B}\) between two adjacent nodes. We set the reach of two distant nodes at \(22\,\mathrm{m}\), which is a reachable distance as explained in Section 4. If the receivers have high sensitivity, about \(-120\,\mathrm{d}\mathrm{B}\mathrm{m}\), the signal at such a distance has a strength of \(-112\,\mathrm{d}\mathrm{B}\mathrm{m}\) and can be detected. This distance allows communication between three neighbors for every node, which makes Problem (2) have a unique solution. Finally, we have considered noise with a standard deviation equal to \(2\,\mathrm{m}\) for the distance estimation procedure. In Figure 5, we represent the qualitative result of Algorithm 1 in the case where the network does not miss any packet. In Figure 6, we represent the absolute mean error of all nodes with unknown position versus the number of iterations (message exchanges) averaged over 100 simulations. We plot multiple curves corresponding with a 0%, 5%, 10% and 20% probability of packet loss and compare our method with the one proposed in [17]. The algorithm presented in [17] is a deterministic state-of-the-art  
Figure 5: Network self-localization qualitative result of Algorithm 1.  
localization algorithm with distributed computation and low communication requirements, such as the one we propose. Probabilistic methods, such as [25], model the nodes' position using particle filters and have higher communication requirements, which make them not appropriate for the application at hand. The packet loss levels are typically caused by different communication impairments, as we show in Section 4. When a packet from some node is lost, Algorithm 1 performs the computation step 4 using the last known estimate of such a node. Figure 6 illustrates that the algorithm is robust against packet losses; in particular, the only effect caused by the packet-loss that can be observed is a slower convergence rate. It also shows a consistent convergence to good estimates of the nodes, whereas the algorithm from [17] converges frequently to local suboptimal solutions, degrading its average performance. For a fair comparison, the same random initialization point was used for the two algorithms. Next, we consider the tracking method proposed in Section 3.2. We plot a qualitative localization result of a target moving in the previous network, which is depicted in Figure 7. The network is the same as in the previous case. We considered that the target node can measure distances from nodes within 22 m. We assumed a random distance deviation of 2 m. The mean absolute error of all points is about 0.96 m. We remark that, although Problem (6) is nonconvex, Algorithm 2 solves Problem (6) in an optimal manner. The statistical significance of these simulations relies mainly in how well the distances are estimated. The algorithms work very well and are robust to missing packets (mainly due to collisions), but the precision that can be achieved depends mainly on the estimated distances. Channel noise is very low and has little influence on the estimates, but the accuracy in the distance-attenuating curve is critical for proper estimation. From our experiments, we conclude that the parametric model fits the  
Figure 6: Network self-localization mean absolute error results versus the number of iterations of Algorithm 1. The curves indicate the probability of packet loss of some node to every neighbor. Figure 7: Tracking performance taking into account 22 m of radio sensitivity.