## 4 Problem Formulations and Optimization Models  
We consider the example of the tree-like topology shown in Figure 2, where sensor nodes sense data and sent it upwards to the surface station using multi-hop routing. We assume that the distributed sensor nodes are double units. The cost of implementing double sensors will be limited because software modems will be coupled with generic microphones and speakers that are built-in to create off-the-shelf sensor modules [14-16]. In this way, the need for specialized communication hardware will be eliminated and the system cost will be reduced. The cost per node (\(\$\)) for hardware modems that use especialized hardware is around \(10^{4}\)\(\$\), whereas the cost of software modems that use generic hardware (off-the-shelf configuration) is around \(10^{2}\)\(\$\); this cost decrease facilitates the deployment of sensor nodes to form underwater acoustic sensor networks. The double sensor nodes operate as a single sensor when the underwater communication is reliable. This fact changes in the presence of shadow zones. Each intermediate node \(s_{i}\) should be able to detect if the quality of the uplink \(l_{i\,i+1}\) towards one-hop neighbor \(s_{i+1}\) is affected by the presence of a shadow zone computing the transmission loss \(TL_{i\,i+1}\)[3]. If \(TL_{i\,i+1}(t+\Delta t)=TL_{i\,i+1}(t)\,+\,TL_{shadow\,zone}\) and this value persists during a period longer than \(T\) to eliminate insignificant variations, the sensor node assumes that the uplink is located in a shadow zone  
Figure 2: Architecture for a 3D underwater sensor network. because \(TL_{i\ i+1}(t+\Delta t)\) has been increased by the transmission loss value \(TL_{shadow\ zone}\) inside the shadow zone. We distinguish between three different cases according to the number of links in the underwater sensor network affected by the shadow zone:  
* Case 1: Only the uplink \(l_{i\ i+1}\) between \(s_{i}\) and \(s_{i+1}\) is located in the shadow zone. * Case 2: The uplink \(l_{i\ i+1}\) between \(s_{i}\) and \(s_{i+1}\) and the uplinks \(l_{(i-1)k\ i}\) between \(s_{(i-1)k}\), \(\forall k\in K\) and \(s_{i}\) are located in the shadow zone. \(K\) represents the set of neighbours connected to \(s_{i}\) through uplinks using hierarchical routing. * Case 3: The uplink \(l_{i\ i+1}\) between \(s_{i}\) and \(s_{i+1}\) and the uplinks \(l_{(i-1)k\ i}\) between \(s_{(i-1)k}\), \(\forall k\in K\) and \(s_{i}\), \(l_{(i\geq 2)k\ i-1}\) between \(s_{(i-2)k}\), \(\forall k\in K\) and a particular \(s_{(i-1)k}\),..., \(l_{(i\cdot x)k\ i\ i+1}\) between \(s_{(i\cdot x)k}\), \(\forall k\in K\), \(\forall x\in N\) and a particular \(s_{(i\cdot x+1)k}\) are located in the shadow zone. Each case will be solved differently. In all cases we assume that the sensor node \(s_{i}\) knows the location and dimensions of the shadow zone. The sound speed profile is affected by the time of the day, weather, depth, variations in the water temperature and salinity. With the aid of a forecast model [8] and the estimation of the sound speed profile, the different underwater propagation phenomena can be predicted, including the presence of shadow zones. The location of shadow zones can be performed studying the trajectory of the acoustic rays under the different underwater propagation phenomena [3]. Autonomous Underwater Vehicles (AUVs) can analyze and inform underwater sensors periodically about the existence and location of shadow zones. We assume that sensor nodes are located in a grid (see Figure 6). The axes of the coordinate system are set at the water surface. Sensors know their fixed two coordinates (\(x\),\(y\)). They are also aware of their depth (\(z\) coordinate), which varies with time depending on the networks tasks. The \(z\) coordinate is zero at surface and increases with depth. Sensor nodes also know the location of their neighbours because they exchange their depth information.