## 5 Simulation Results and Analysis  
In computer simulations, it is assumed that the linear array consists of 10 omnidirectional hydrophones, and the spacing of adjacent hydrophones is set to half the wavelength of the narrowband acoustic signal. In addition, assuming that there are two underwater acoustic sources in the far field, their directions of arrival are set to 30 degrees and 60 degrees, respectively. In the simulations below,the intensity of the acoustic source at 30 degrees is set to be variable for testing the DOA estimation performance of the SSC-MVDR algorithm, while the intensity of the acoustic source at 60 degrees is set to always be 30 dB (relative to noise) for testing the algorithm performance in a strong interference environment. Meanwhile, the received noise of the linear array is assumed to be spatially white Gaussian noise. It should be noted that various unfavorable factors that may be encountered in the complex uncertain ocean environment are also considered in the simulations, including direction error, random wavefront fluctuations, hydrophone position errors, and hydrophone amplitude phase errors. Therefore, the steering vector model in Equation (7) is used while assuming that the amplitude deviation coefficients of the steering vector and the phase deviation coefficients of that are known. In Figures 2 and 3, the pseudo spatial power spectrums of the SSC-MVDR algorithm are given under different acoustic source intensities and different snapshot numbers (i.e., the number of snapshots used to calculate the sample covariance matrix in Equation (5)). Specifically, Figure 2 corresponds to a friendly simulation environment where the signal-to-noise ratio (SNR) of the acoustic source in the direction of 30 degrees is set to 20 dB, the number of snapshots is set to 100, while Figure 3 corresponds to a poor simulation environment where the SNR of the same acoustic source is only \(-10\) dB, and the number of snapshots is only 30. For performance comparison, the spatial power spectrums of the MVDR method [4], the MUSIC method [13] and the PM method [15] under the same simulation conditions are also given in Figures 2 and 3. Compared to other algorithms, the SSC-MVDR algorithm exhibits sharper peaks near the true DOA of the acoustic signal in Figures 2 and 3. Figures 4 and 5 show the DOA estimation accuracy results of the SSC-MVDR algorithm. In Figure 4, the number of snapshots is fixed at 100, and the intensity of the acoustic source is gradually increased from \(-20\) dB to 20 dB. In Figure 5, the acoustic source intensity is fixed at \(-10\) dB, and the number of snapshots is gradually increased from 10 to 100. The DOA estimation accuracy is evaluated by the root mean square error (RMSE) of DOA estimation, which is defined as follows,  
\[\text{RMSE}=20\log\sqrt{\frac{1}{S}\sum_{i=1}^{S}(\hat{\theta}_{i}-\theta_{a} )^{2}} \tag{17}\]  
Figure 2: Normalized spatial power spectrums obtained by the MVDR method, the MUSIC method, the PM method and the SSC-MVDR method when the acoustic source intensity is 20 dB and the snapshot number is 100. In Equation (17), RMSE is actually the logarithm of the root mean square error, so the unit of RMSE is dB. The number of independent computer simulations is represented by \(S\). In the following simulations (i.e., from Figures 4-7), \(S=100\) is set, which means that each simulation result requires 100 independent runs. \(\hat{\theta}_{i}\) represents the DOA estimate obtained by the \(i\)th computer simulation, and \(\theta_{a}\) is the corresponding true DOA value. Meanwhile, the results of the DOA estimation accuracy of the MVDR method [4], the MUSIC method [13], the Root-MUSIC method [34], the TLS-ESPRIT method [35] and the PM method [15] are also given in Figures 4 and 5. It can be seen from Figure 4 that the RMSE results of the SSC-MVDR algorithm is significantly lower than that of other methods when the acoustic source intensity is in the range of \(-20\) dB to \(-10\) dB.