## 1 Introduction  
Sound source identification technology extracts the location and strength information of the sound source from the acoustic signal measured by the microphone array. Accurate identification of sound sources is crucial in sound field characterization and noise control in aerospace [1, 2, 3, 4], power transformers [5, 6], high-speed trains [7, 8], and other scenarios [9, 10, 11]. In these application scenarios, the noise distribution area is wide, and the sound field propagation characteristics are complex. In addition, for safety operational reasons, microphone arrays are often placed in the far field for measurements [12]. The above characteristics bring the following difficulties to the sound source identification process. First, due to the extensive focus area being discrete, it is difficult to ensure that the discrete grid points exactly cover the actual sound source locations, which may cause basis mismatch errors [13]. More importantly, due to the limited array aperture size and number of microphones, it will be much more difficult to obtain high-resolution sound source identification results during far-field measurements when multiple low-frequencysound sources are distributed nearby in the space [14]. In addition, path effects during far-field propagation can amplify the impact of measurement noise [15]. In recent years, compressive beamforming has been widely used to improve sound source identification performance. The method based on the compressed sensing theory first assumes different types of equivalent sources. Then, it applies sparsely constrained prior information [16] to the source distributions in various domains, such as the frequency domain [17] and the spatial-temporal domain [18]. Finally, the optimal sound source distribution from the measured signal is sought. Compared with the traditional beamforming method, the compressive beamforming method can achieve higher resolution based on fewer measurement samples. At the same time, such methods possess a higher dynamic range and allow for dealing with correlated/uncorrelated and sparse/non-sparse distributed sources, as well as obtaining quantitative results on the strength of the sources. However, this is an ill-posed problem in the Hadamard sense. The linear operator matrix in the inverse problem is usually irreversible and ill-conditioned. It has a large number of conditions, which makes the signal reconstruction extremely sensitive to small changes in the initial conditions [19], leading to an unstable solution under small perturbations in the measurement noise. Many methods have been developed to ensure stable solutions for ill-conditioned inverses. Orthogonal matching pursuit (OMP) [20] can effectively approximate \(L_{0}\) norm minimization, but it does not perform well under conditions where the constraint boundaries cannot be guaranteed. Therefore, the original non-convex \(L_{0}\) norm minimization problem is usually reduced to an \(L_{1}\) norm relaxation optimization problem. Methods such as the minimum-angle regression lasso [21] algorithm and the generalized inverse beamforming (GIBF) proposed by [PERSON] [22] can both achieve \(L_{1}\) norm minimization. [PERSON] [23] proposed a weighted matrix method to increase the dynamic range of GIBF, thereby improving the sound source identification resolution. In addition, [PERSON] [24] proposed an iteratively reweighted \(L_{1}\) norm minimization compressive beamforming (IR-CSB) method to replace the relaxed \(L_{1}\) norm penalty, effectively improving the performance of source strength estimation. [PERSON] [25] established an acoustic inverse problem framework based on the alternating direction method of multipliers, using the \(L_{2}\) norm penalty based on the parsimony assumption and the \(L_{\text{p}}\) norm penalty based on the sparsity assumption to solve, respectively, which can be applied to different types of sound sources. However, a common problem with the above methods is difficulty in selecting an appropriate regularization strategy.