# Research on Localization Algorithms Based on Acoustic Communication for Underwater Sensor Networks
## 3 Localization Algorithm
### Spatial Coverage
#### 3.2.2 3D-Networks  
Similarly, 3D-networks are classified into two subcategories, namely, anchor-free and anchor-based schemes. Firstly, we introduce the anchor-free scheme. An Anchor-Free Localization Algorithm (AFLA) is presented in [33], which is suitable for the active-restricted UWSNs. This algorithm takes advantage of the relationship of adjacent nodes and does not require an anchor node. In this scheme, the active-restricted nodes mean, when they anchor at the bottom of the sea, they can float in the sea and move within a hemisphere area (shown in Figure 5). The anchor and the cable length (denoted as L) is the center and radius of the hemisphere, respectively. Note that the pressure sensor can obtain the depth (H) of the node. The node with the unknown location broadcasts its spherical center's coordinates, depth, and cable length. When two messages are received by the node, it calculates the position. In addition, the authors consider the various characteristics of the mobile network and describe the mobile node localization process in detail. As for the anchor-based scheme in 3D-UWSNs, a relatively larger amount papers will be introduced. In [34], the authors analyze the problem existing in the nonlinear least-square-based node self-localization scheme, point out the biased distribution of multi-hop distance estimation error, and employ the orthogonal regression method to solve the problem of normal nodes localization in the case of error anchor position. The multi-hop localization framework is shown in Figure 6. The normal node and anchor node can be denoted as \(N_{u}\) and \(N_{a}\), and their coordinates are\(X_{a}=[x_{u},y_{u},z_{u}]^{T}\in R^{3}\) and \(X_{a}=[x_{a},y_{a},z_{a}]^{T}\in R^{3}\),respectively, where \(a=(1,2,...,K)\), and \(K\) is the anchor node number. \(N_{u}\) can obtain all multihop distance estimations to the anchor nodes by multihop information exchange. The problem of locating the node \(N_{u}\) can be shown as the following nonlinear regression model:  
\[d_{a}+e_{a}=||X_{u}-X_{a}||_{2}, \tag{2}\]  
where \(X_{u}\) and \(X_{a}\) are the regression parameters and the independent variables, respectively. Observations of \(X_{a}\) are anchors' declared coordinates \(\{X^{\prime}_{1},X^{\prime}_{2},...,X^{\prime}_{K}\}\). \(d_{a}\) is the dependent variable, and its observations are the estimated distances between \(N_{u}\) and \(N_{a}\). \(e_{a}\) is the distance estimation error. Then, the Nonlinear Least Squares Estimator (NLSE) can be used to solve the unknown parameter \(X_{u}\):  
\[ \begin{split}\hat{X}_{u}&=\arg\min_{X_{u}}\sum \limits_{a=1}^{K}{(e_{a})^{2}}\\ &=\arg\min_{X_{u}}\sum\limits_{a=1}^{K}{(||X_{u}-X_{a}||_{2}-d^{ \prime}{}_{a})^{2}},\end{split}  \tag{3}\]  
where \(\hat{X}_{u}\) are the estimated coordinates of \(N_{u}\). Two sources of distance estimation errors of the multi-hop scenarios are discussed and empirical analysis of multi-hop distance estimation bias for 3D-UWSNs is given in this paper. To address these issues, an anchor position error-tolerant multi-hop localization method based on the orthogonal regression for UWSNs is proposed. Figure 5: Model of an active-restricted underwater sensor node [33]. In [35], a localization scheme is proposed that only requires two references to approximate locations that are suitable for use in low-density deployment. It is a Range-based Low-overhead Localization Technique (RLLT). The anchor node broadcasts its position, and the nodes within the transmission range of the anchor estimate the distance from the anchor. However, there still some nodes that are out of the transmission range of an anchor. Hence, in this scheme, once a node is located, it becomes a new reference node and broadcasts its position. When a node can obtain some distances from anchor nodes, it uses those distances to calculate its position. In [36], a Top-down Positioning Scheme (TPS) that is suitable for UWSNs increases the localization coverage while keeping the localization error small.