## 1 Introduction  
Underwater acoustic source localization determines the altitude or depth, range, and bearing angle of the underwater target, that is, the three coordinates of the underwater target in the elliptical coordinate system [1]. The estimation of bearing angle (or direction of arrival (DOA)) of underwater acoustic source is an important and indispensable step in underwater acoustic source localization. In fact, in some underwater acoustic source localization methods, it is the target direction information that is used to estimate the target distance [2]. Using a vector hydrophone (or vector hydrophone array) is perhaps one of the simplest and most straightforward methods of underwater acoustic source DOA estimation. The vector hydrophone is capable of simultaneously measuring sound pressure and particle velocity along one to three orthogonal directions. Therefore, only a single vector hydrophone can generate a directional beam pattern. A Directional Autonomous Seafloor Acoustic Recorder (DASAR) system consisting of several vector hydrophones has been reported [3]. By using a vector hydrophone, directional industrial noise is effectively suppressed, and weak marine mammal sounds can be successfully detected. Conventional beamforming (or delay-and-sum beamforming) is also one of the common DOA estimation methods [4; 5; 6]. Based on the ray-path approximation for the sound channel's impulse response, the frequency-difference beamforming method for the sparse hydrophone array is proposed in [7; 8], which can estimate the signal phase difference by using the conventional delay-and-sum beamforming of the field product at the difference frequency. In contrast, by determining the array weight coefficients in a nonlinear manner, the minimum variance distortionless response (MVDR) method can achieve higher angular resolution than conventional beamforming [4]. In order to cope with the model mismatch problems in the uncertain ocean environment, such as the signal look direction mismatch, the signal spatial signature mismatch due to local scattering or wavefront distortion [9; 10], many robust MVDR-based adaptive beamforming algorithms have been proposed [10; 11; 12]. Similar to the MVDR-based methods, the subspace-based high-resolution DOA estimation techniques also use information carried by the covariance matrix. The most representative subspace-based DOA estimation methods may be the Multiple Signal Classification (MUSIC) algorithm [13], the Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) [14], and the Propagator Method (PM) [15]. The key to the subspace-based DOA estimation methods is the estimation of the signal subspace (or noise subspace). To achieve this purpose, one can first perform eigendecomposition on the sample covariance matrix, then construct the signal subspace with the eigenvectors corresponding to the larger eigenvalues, and form the noise subspace with the eigenvectors corresponding to the smaller eigenvalues. Another more sophisticated approach is to reconstruct the sample covariance matrix according to the Toeplitz structure of the covariance matrix [16], and then obtain the signal subspace in a similar way as above. In contrast to the above eigenvalue-based methods, the eigenvector pruning algorithm implements the estimation of the signal subspace by using the statistical properties of eigenvectors of signal-free sample covariance matrix [17]. In many underwater acoustic source DOA estimation scenarios, the number of acoustic sources distributed in the underwater far field is much smaller than the number of hydrophones in the observation array. This is the intrinsic basis of the popular sparse-based DOA estimation methods. In the least absolute shrinkage and selection operator (LASSO) method [18], the signal amplitude vector is obtained by solving an \(l_{1}\)-norm regularized least-squares problem. The LASSO method contains the \(l_{1}\)-norm constraint on the solution vector, thus making the result of the solution vector sparse [19].