## 6 Simulation Results and Analysis  
In this section, computer simulation results are used to compare the performance of the proposed methods with the SDP method (RSS-based approach) in [12] in underwater acoustic environments. The propagation model, Equation (1), is used to generate the range measurements. The anchor nodes and target node are assumed to be randomly located at a square region 100 m \(\times\) 100 m in each Monte carlo (_Mc_) run. Unless otherwise stated, the reference distance is set to be \(d_{0}=1\) m, the reference power is \(P_{0}=-10\) dBm, the maximum number of steps in the bisection procedure is \(K_{max}=30\), and the path-loss exponent is fixed as \(\gamma=2\), which corresponds to an UWA spherical spreading case. Different values of \(\alpha_{f}\) are given in each figure. The noise correlation coefficient \(\rho=0.8\) for \(i\
eq j\). Three thousand Monte carlo runs are used to compute the root mean square error (RMSE) defined as RMSE\(=\sqrt{\sum_{i=1}^{\text{Mc}}\frac{\|x-\sqrt{x}\|^{2}}{\text{Mc}}}\), where \(\widehat{x}\) is the estimated location in the \(i\)-th Monte Carlo run. The SDP is solved using the MATLAB package CVX (Version 2.1, Stanford University, Stanford, CA, USA, 2014) [24], where the solver is SeDuMi (Version 1.02, Tilburg University, Tilburg, Netherlands, 1999) [25]. The corresponding ML problems are solved by the MATLAB function \"Isqnonlin,\" which adopts the Levenberg-Marquardt algorithm. Figure 4 compares the RMSE versus the standard derivation \(\sigma\). It is observed that the RMSE grows in nature as the standard derivation increases for all the methods. The proposed localization method WLS-K in the case of known transmit power provides superior performance over the SDP method, and approximates the CRLB-K even when the standard derivation increases. Furthermore, to the best of the authors' knowledge, few people has discussed localization methods based on a UWA RSS propagation model in unknown transmit power cases in UWSN. Therefore, we compared the performance of the proposed WLS-U with CRLB-U in this case. The results show that the proposed method in this case also maintains an optimal performance. Figure 5 compares the RMSE versus the number N of the anchor nodes. As predicted, the RMSE decreases in nature as \(N\) increases for both the WLS-K and WLS-U methods. The figure also shows that adding more anchor nodes in the network can enhance the performance of the proposed methods and outperform the SDP method for all chosen \(N\). Moreover, the performance of WLS-K is slightly better than WLS-U. Figure 4: Root mean square error (RMSE) versus the \(\sigma\) when \(N=10\), \(\alpha_{f}=0.001\) dB/m (\(f=9\) kHz), and \(\rho=0.8\). Figure 6 depicts the cumulative density function (CDF) of localization comparison versus mean error (ME) among the discussed methods. The figure shows that the proposed methods show an improvement in performance compared with the SDP method. Figure 7 compares the RMSE versus the absorption coefficient \(\alpha_{f}\). The figure shows that the RMSE crease in with the increase in absorption coefficient for the discussed methods, and WLS-K method shows a performance that is superior to the SDP method. Furthermore, the gap between the WLS-K method and the WLS-U method is small. This is due to the fact that, in the proposed WLS-U method, with some approximation, the unknown transmit power \(P_{0}\) can be estimated from the GTRS optimization problem, Equation (22). In this case, the problem is transformed into the same problem in the known transmitter power case. The results show that the proposed methods have robust performance for \(\alpha_{f}\) from 0.001 to 0.1 dB/m. Figure 5: RMSE versus the anchor node number \(N\) when \(\sigma=4\) dB, \(\alpha_{f}=0.01\) dB/m (\(f=34\) kHz), and \(\rho=0.8\). Figure 6: Cumulative density function (CDF) versus mean error (ME) when \(\sigma=4\) dB, \(\alpha_{f}=0.1\) dB/m (\(f=454\) kHz), \(N=10\), and \(\rho=0.8\). Figure 8 illustrates the RMSE versus transmit power \(P_{0}\).