## 2 The System Model  
Consider an UWSN with \(N\) anchor nodes and one target node in a two-dimensional (\(2D\)) localization scenario, where the locations of anchor nodes, noted as \(s_{1},s_{2},\ldots,s_{N}\), are known, while the location of the target node, noted as \(x\), is unknown. For simplicity and without loss of generality, we assume that the anchor nodes are equipped with omnidirectional antennas. Under a centralized processing mode, all sensors convey their RSS measurements with respect to the target node to the central processor. We assume that the locations of all the nodes are supposed to be unchanged during the computation period [11; 15]. Figure 1 shows the \(i\)-th sensoring link in the UWSN model.  
For \(i=1,2,\ldots,N\), the RSS of the UWA propagation between the \(i\)-th anchor node and the target node is measured in dBm and has the log-normal shadowing model as follows [12]:  
\[P_{i}=P_{0}-10\gamma\log_{10}\frac{d_{i}}{d_{0}}-\alpha_{f}(d_{i}-d_{0})+n_{i} \tag{1}\]  
where \(d_{i}=\|x-s_{i}\|\) is the Euclidean distance, \(n_{i}\) is a zero-mean Gaussian random variable with covariance matrix \(Q\) that represents the log-normal shadowing effect, \(P_{0}\) is the reference power at the reference distance \(d_{0}(d_{0}\leq d_{i})\), \(\gamma\) is the path-loss exponent between 2 and 4 according to different  
Figure 1: The \(i\)-th sensoring link in the underwater acoustic wireless sensor network (UWSN) model.  
propagation environments, and \(\alpha_{f}\) is the absorption coefficient that can be obtained from [PERSON]'s formula as a function of signal frequency \(f\) in dB/km, which is as follows [16]:  
\[\alpha_{f}=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+2.75\times 10^{- 4}f^{2}+0.003. \tag{2}\]  
Let \(\mathbf{n}=(n_{1},n_{2},\ldots,n_{N})^{T}\in\mathbb{R}^{N}\) be the collection vector of Gaussian random variables with a symmetric and positive-definite covariance matrix \(\mathbf{Q}\) as follows:  
\[[\mathbf{Q}]_{ij}=\begin{cases}\sigma^{2},&i=j\\ \rho\sigma^{2},&i\
eq j\end{cases} \tag{3}\]  
where \(0\leq\rho<1\) is the correlation coefficient of any two different links between \(i\) and \(j\) to the target node for \(i,j=1,2,\ldots,N\). It is known from Equation (3) that \(\mathbf{Q}\) can be transformed into white noise using the Cholesky decomposition as \(\mathbf{Q}=\mathbf{B}\mathbf{B}^{T}\) with lower-triangular matrix \(\mathbf{B}\). This implies a linear transformation \(\mathbf{n}=\mathbf{B}\mathbf{x}\), where \(\mathbf{w}\sim\mathcal{N}(\mathbf{0},\sigma^{2}\mathbf{I}_{N})\), and \(\mathbf{I}_{N}\) is the order-\(N\) unity matrix [17].  
Let \(\mathbf{P}=(P_{1},P_{2},\cdots,P_{N})^{T}\in\mathbb{R}^{N}\) be the vector of observations given by Equation (1). The joint probability density function (PDF) of \(\mathbf{P}\), conditioned on a given \(\mathbf{x}\), has the following form  
\[f(\mathbf{P}|\mathbf{x})=\frac{1}{(2\pi)^{\frac{N}{2}}|\mathbf{Q}|^{\frac{1}{2}}}e^{-\frac {1}{2}\left(\mathbf{P}-\bar{\mathbf{P}}\right)^{T}\mathbf{Q}^{-1}\left(\mathbf{P}-\bar{\mathbf{P} }\right)} \tag{4}\]  
where \(\bar{\mathbf{P}}=\left(\widehat{P}_{1},\widehat{P}_{2},\ldots,\widehat{P}_{N} \right)^{T}\), with each component  
\[\widehat{P}_{i}=P_{0}-10\gamma\log_{10}\frac{\|\mathbf{x}-\mathbf{s}_{i}\|}{d_{0}}- \alpha_{f}(\|\mathbf{x}-\mathbf{s}_{i}\|-d_{0}).\]  
Therefore, the joint ML estimation of the target location \(\mathbf{x}\) can be formulated as  
\[\operatorname*{arg\,min}_{\mathbf{x}}\,\,(\mathbf{P}-\bar{\mathbf{P}})^{T}\mathbf{Q}^{-1}(\bm {P}-\bar{\mathbf{P}}). \tag{5}\]  
In the special case where \(\rho=0\), which corresponds to added white Gaussian noise, the estimation can be reduced as  
\[\operatorname*{arg\,min}_{\mathbf{x}}\sum_{i=1}^{N}\left(P_{i}+10\gamma\log_{10} \|\mathbf{x}-\mathbf{s}_{i}\|+\alpha_{f}\|\mathbf{x}-\mathbf{s}_{i}\|-P_{0}-10\gamma\log_{10}d _{0}-\alpha_{f}d_{0}\right)^{2}. \tag{6}\]  
Clearly, the ML estimator based on Equation (5) or Equation (6) is non-convex, hence the difficulty in finding a global optimal solution.