## 4 Problem Formulations and Optimization Models
_Case 1: Only the uplink \(l_{i\ i+1}\) between \(s_{i}\) and \(s_{i+1}\) is located in the shadow zone._  
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 Figure 3. These sensor nodes will be connected to each other inside the shadow zone by a wire to maintain robust communication. The node \(s_{i\to 1}\) remains in the same location as the node \(s_{i}\) was and continues sensing data at the required depth level. We should now determine the optimal location for the node \(s_{i\to 2}\) We formulate this problem as a Nonlinear Program (NLP). We introduce the following notation:  
* \(u_{i+1}\) = (\(x_{i+1}\), \(y_{i+1}\), \(z_{i+1}\)) is the location of node \(s_{i+1}\). * \(u_{i\to 2}\) = (\(x_{i}\), \(y_{i}\), \(z_{i\to 2}\)) is the new location of node \(s_{i\to 2}\). * \(H\) is the sea depth. * \(R_{MAX}\) is the maximum transmission range. * \(TL_{Th}\) is the detection threshold. * \(r_{i\to 2\ i+1}\) refers to the transmission range of the acoustic link between nodes \(s_{i\to 2}\) and \(s_{i+1}\) expressed in meters. * \(\alpha\) represents the absorption coefficient and has the units dB/Km. * \(\chi\) is a variable used to compute the transmission loss with the value \(\chi=10\) for shallow water (cylindrical spreading) and \(\chi=20\) for deep water (spherical spreading) [3]. * \(A\) is called the transmission loss anomaly and summarizes the combined effects of several complex factors such as multiple path propagation due to the variations of the speed with temperature, depth and salinity, refraction effects, diffraction and scattering of sound by particles, bubbles and plankton within the water column. Detailed formulas of the multipath propagation loss according to the different propagation phenomena (surface reflections, bottom reflections, convergence zones, deep sound channel, etc.) can be found in [3]. It is given in dB. * \(w_{sz-start}=(x_{i},\,y_{i},\,z_{sz-start})\) and \(w_{sz-end}=(x_{i},\,y_{i},\,z_{sz-end})\) are the coordinates of the points delimiting the shadow zone at \((x_{i},\,y_{i})\) with \(z_{sz-start}<z_{sz-end}\) and height \(z_{sz-end}-z_{sz-start}\). * \(E_{Bw}\) is the energy consumed moving the sensor \(s_{i\to 2}\). * \(E_{Th}\) is the energy consumption threshold. The problem can be formulated as follows:  
**P1: Location Optimization Problem**  
\[ \begin{array}{ll}\mbox{\it Given:}&x_{i},x_{i+1},y_{i},y_{i+1},z_{i+1},H \,,R_{MAX},TL_{TII},\alpha,\chi,\\ &A,z_{sz-start},z_{sz-end},E_{bw},E_{th}\\ \mbox{\it Find:}&z_{i\to 2}\in[0,H],\,\,r_{i\to 2,\,i\,i\, 1}\in R^{+}\\ \end{array} \]  
Figure 3: Scenario in the 3D UWSN with a shadow zone (Case 1). \[Min:\quad TL_{i\to 2~{}i+1}=\chi\log r_{i\to 2~{}i+1}+\alpha r_{i\to 2~{}i+1}.10^{-3}+A\]  
_Subject to_ :  
\[r_{i\to 2~{}i+1}=\left\|u_{i+1}-u_{i\to 2}\right\| \tag{1}\] \[r_{i\to 2~{}i+1}<R_{MAX}\] (2) \[TL_{i\to 2~{}i+1}<TL_{n}\] (3) \[z_{i\to 2}<z_{zz-start}\] (4) \[E_{Bw}<E_{T_{h}} \tag{5}\]  
The objective function of problem **P1** aims at finding the best location for \(s_{i\to 2}\) that minimizes the transmission loss and preserves the connectivity with node \(s_{i+1}\) preventing from network partition. The absorption coefficient \(\alpha\) is computed as derived in [17,18] as a function of the frequency \(f\), the salinity \(S\), the temperature \(T\), the water ph _pH_ and the depth \(z\). We consider a two-path Rayleigh fading channel model, where each path is Rayleigh distributed. More specifically, the envelope of the signal from each path in the two-path model is modeled as an independent Rayleigh distributed random variable, \(\alpha_{i}\), \(i\in\){1,2} [19]. Consequently, for each path, the received energy per bit per noise power spectral density is given by, \(\psi=\alpha^{2}\). (\(E_{b}\)/\(N_{0}\)) which has a distribution, \(f_{\rm{T}}\)(\(\psi\))=(\(1/\psi_{0}\)).exp(\(-\psi/\psi_{0}\)) where \(\psi_{0}=\mathrm{E}[\alpha^{2}\)).