## 2 Background
### Problem Formulation  
With the purpose of developing an accurate and high-performance UABL scheme used in the complex underwater environments, the assumptions are made as below:  
Both the anchors and the mobile nodes have the same communication range \(R\) and are equipped with arrays of wireless antennas in order to communicate with each other by acoustic signals which are used to transmit the depth information, the anchors' coordinates and the measured AOA values. The mobile node transmits acoustic signals regularly. Once the acoustic signal is received, each anchor replies a message including its own three dimensional coordinates and its depth information. When the information exchange is completed, the mobile node has sufficient information to localize itself. The information refers to locations and the depth information of all task anchors and the AOA measurement from the task anchors estimated by the mobile node. The depth information is measured by the pressure sensors with which all the nodes are equipped. In UASNs, it is difficult to achieve time synchronization precisely as a result of the characteristics of acoustic signal propagation. While by employing the AOA values of the acoustic signals from the anchors, synchronous request between the nodes is not so necessary, so the PCFL and ACFL techniques advanced here have the advantage in synchronization. Let's consider a 3D UASNs _UN_ with \(p\) anchors and \(q\) mobile nodes and express the location of each node as:  
\[n_{i}=(x_{i},y_{i},z_{i}),n_{i}\in UN,i=1,2,...p+q \tag{1}\]  
We assume that the locations of \(p\) anchors in \(\mathrm{A=\{}n_{1},n_{2},...n_{p}\mathrm{\}}\) are known but the locations of the other \(q\) mobile nodes in \(\mathrm{M=\{}n_{p+1},n_{p+2},...n_{p+q}\mathrm{\}}\) are undetermined and to be localized, _UN_\(=A\cup M\). In our localization methods, the only obtainable information is the smaller angle of acoustic signal \(\alpha_{y}\) from the anchors \(n_{j}(j=1,2,...p)\) received by the mobile node \(n_{i}(i=p+1,...p+q)\) based on the AOA measurement and the depth information obtained by the corresponding pressure sensors. The Euclidean distance between the node \(n_{i}\) and the node \(n_{j}\) can be calculated as:  
\[d(n_{i},n_{j})=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2} +\left(z_{i}-z_{j}\right)^{2}},(n_{i}\in UN,n_{j}\in UN) \tag{2}\]  
At time instant \(t\), the set of task anchors corresponding to the mobile node \(n_{i}\) can be expressed as  
\[A_{i}^{t}=\{n_{j}\big{|}d(n_{i},n_{j})\leq R,i=p+1...p+q,j=1,2,...p\}\subset A \tag{3}\]  
Let \(p^{i}\) denote the number of task anchors in \(A_{i}^{t}\) and it can be seen that:  
\[\left|A_{i}^{t}\right|=p^{t}\leq p \tag{4}\]  
As all the nodes have the same communication range \(R\), at time instant \(t\), the mobile node \(n_{i}(i=p+1,...p+q)\) can exchange information with anchor \(n_{j}\) directly if and only if \(n_{j}\in A_{i}^{t}\). The set of task projections \(B_{i}^{t}\) for \(n_{j}\in A_{i}^{t}\) is defined as:  
\[B_{i}^{t}=\{n_{j}^{t}\big{|}d(n_{i},n_{j})\leq R,x_{j}^{t}=x_{j},y_{j}^{t}=y_{ j},z_{j}^{t}=z_{j}\}\quad\left|B_{i}^{t}\right|=p^{t}\leq p \tag{5}\]where \(n^{\prime}_{j}=(x^{\prime}_{j},y^{\prime}_{j},z^{\prime}_{j})\).