## 3 The Proposed Methods
### Target Localization in the Unknown Transmit Power Case  
In most practical applications, the target transmit power is unknown. Therefore, the related \(P_{0}\) or \(\eta\) in Equation (11) is unknown. Therefore, the method proposed in Section 3.1 cannot be used directly in this case. To cope with the difficulties caused by this additional unknown parameter, we developed the following method. By using the notations of \(\lambda_{i}\), \(\beta_{i}\) and \(\eta\), from Equation (9), we have  
\[\|\mathbf{x}-\mathbf{s}_{i}\|\approx\frac{\eta}{\beta_{i}+\frac{\alpha_{f}\ln 10}{10 \gamma}\eta} \tag{14}\]  
and thus  
\[\|\mathbf{x}-\mathbf{s}_{i}\|\approx\frac{\frac{10\gamma}{\alpha_{f}\ln 10}(\beta_{i}+ \frac{\alpha_{f}\ln 10}{10\gamma}\eta)-\frac{10\gamma}{\alpha_{f}\ln 10}\beta_{i}}{ \beta_{i}+\frac{\alpha_{f}\ln 10}{10\gamma}\eta}. \tag{15}\]  
Furthermore, Equation (15) can be rewritten as  
\[\|\mathbf{x}-\mathbf{s}_{i}\|\approx\frac{10\gamma}{\alpha_{f}\ln 10}-\frac{\frac{10 \gamma}{\alpha_{f}\ln 10}\beta_{i}}{\beta_{i}+\frac{\alpha_{f}\ln 10}{10 \gamma}\eta}. \tag{16}\]  
We divide \(\beta_{i}\) from both the numerator and denominator of the second term on the right hand side of Equation (16), yielding  
\[\|\mathbf{x}-\mathbf{s}_{i}\|\approx\frac{10\gamma}{\alpha_{f}\ln 10}-\frac{10\gamma}{ \alpha_{f}\ln 10}\cdot\frac{1}{1+\frac{\alpha_{f}\ln 10}{10\gamma}\cdot\frac{ \eta}{\beta_{i}}}. \tag{17}\]  
Let \(\
u=\frac{\alpha_{f}\ln 10}{10\gamma_{i}\beta_{i}}\eta\), in which \(\alpha_{f}\) is related to the frequency, \(\beta_{i}\) is related to the signal power, \(\gamma\) is a constant, and \(\eta\) is an independent unknown parameter. We will verify that \(|\
u|\ll 1\) in most UWSN environments. Figure 2 is the \(\alpha_{f}\) versus the frequency band 0-30 kHz, within which most acoustic systems operate [20; 21; 22]. The figure shows that \(\alpha_{f}\ll 1\). Figure 3 shows \(\
u\) versus the transmit signal power when \(\alpha_{f}\) takes three different values (in dB/m). The figure shows that \(|\
u|\ll 1\) in this case. Therefore, when the absorption coefficient \(\alpha_{f}\) is sufficiently small, it is reasonable to approximate Equation (17) by using the first-order Taylor expansion about \(\
u=0\) as follows:  
\[\frac{1}{1+\frac{\alpha_{f}\ln 10}{10\gamma}\cdot\frac{\eta}{\beta_{i}}} \approx 1-\frac{\alpha_{f}\ln 10}{10\gamma}\cdot\frac{\eta}{\beta_{i}}. \tag{18}\]  
Taking Equation (18) into Equation (17), we have the following simplified form:  
\[\|\mathbf{x}-\mathbf{s}_{i}\|\approx\frac{\eta}{\beta_{i}}. \tag{19}\]  
As in the previous section, squaring both sides of Equation (19), we have  
\[\|\mathbf{x}-\mathbf{s}_{i}\|^{2}\approx\frac{\eta^{2}}{\beta_{i}^{2}}. \tag{20}\]  
Thus, we can obtain the following WLS problem:  
\[\min_{\mathbf{x},\eta}\ \sum_{i=1}^{N}(\|\mathbf{x}-\mathbf{s}_{i}\|^{2}-\frac{\eta^{2}}{ \beta_{i}^{2}})^{T}\mathbf{Q}^{-1}(\|\mathbf{x}-\mathbf{s}_{i}\|^{2}-\frac{\eta^{2}}{\beta _{i}^{2}}). \tag{21}\]Let \(\widetilde{\mathbf{y}}=\left[\mathbf{x}^{T},\|\mathbf{x}\|^{2},\eta,\eta^{2}\right]^{T}\), and Equation (21) can be written in equivalent GTRS form as  
\[ \begin{split}\min_{\widetilde{\mathbf{y}}}&\|\mathbf{Q}^{- \frac{1}{2}}(\widetilde{\mathbf{A}}\widetilde{\mathbf{y}}-\widetilde{\mathbf{b}})\|^{2}\\ \text{s.t.}&\widetilde{\mathbf{y}}^{T}\widetilde{\mathbf{D}} \widetilde{\mathbf{y}}+2\widetilde{\mathbf{v}}^{T}\widetilde{\mathbf{y}}=0\end{split}  \tag{22}\]  
where \(\widetilde{\mathbf{D}}=diag([1,1,0,1,0]),\widetilde{\mathbf{v}}=[0,0,-\frac{1}{2},0,- \frac{1}{2}]^{T}\), and  
\[\widetilde{\mathbf{A}}= \begin{bmatrix}-2s_{1}^{T}&1&0&-\frac{1}{\beta_{1}^{T}} \\ \vdots&\vdots&\vdots&\vdots\\ -2s_{N}^{T}&1&0&-\frac{1}{\beta_{N}^{T}}\end{bmatrix} ,\widetilde{\mathbf{b}}= \begin{bmatrix}-\|\mathbf{s}_{1}\|^{2}\\ \vdots\\ -\|\mathbf{s}_{N}\|^{2}\end{bmatrix} . \tag{23}\]  
Equation (22) is still non-convex, but both the objective function and the constraint are quadratic and can therefore be solved by using the bisection method [19], as discussed in Section 3.1.