## 1 Introduction
Off-grid methods mainly include two categories [41]: off-grid compressive beamforming that uses Taylor expansion to compensate for grid deviation and off-grid compressive beamforming that uses Newtonized orthogonal matching pursuit to search for the actual sound source location. The first type of method was proposed by [PERSON] [42], which uses grid-based DOA estimation and two-dimensional off-grid DOA compensation as blocks to establish a block-sparse off-grid compressive beamforming model and solve it to obtain source coordinate estimates. [PERSON] [13], [PERSON] [43], [PERSON] [44], and others used the CVX toolbox, group minimax concave penalty function, and weighted block \(L_{1}\) norm regularization to solve this problem, respectively. These methods effectively compensate for off-grid bias. However, the complexity of the model will be increased due to the introduction of additional grid variables. This method is not fully applicable, especially when multiple sound sources exhibit large spatial dispersion, resulting in many discrete points on the source surface. The second type of method [45] takes source coordinates and source strength as parameters to establish a maximum-likelihood estimate. It uses the initial estimate and subsequent Newton optimization to obtain accurate source coordinates and source strength estimates. This type of method is robust and computationally efficient, but its resolution performance is limited. If there are low-frequency sound sources that are closely spaced, the solution of the technique will seriously deviate from the actual sound source location. In summary, among the existing compressive beamforming methods, the fixed on-grid methods suffer from the defect of base mismatch. Grid-free methods require prior information and stringent requirements for array microphone placement. Off-grid methods introduce additional grid position offsets during the solution process, increasing the solution's variable dimensions and model complexity. For source identification problems with significant source surface spans and many potential discrete points, large approximation errors can result. Therefore, a new strategy needs to be proposed. [PERSON] [46] proposed a variable-scale grid sound source localization method. This method first uses a relatively sparse calculation grid to generate a beam map quickly, retains the calculation grids with higher weights, and then uses the deconvolution method to locate the sound source, effectively improving efficiency. Inspired by this, this paper proposes an off-grid compressive beamforming method based on sparse Bayesian learning and grid adaptive split refinement, which can be applied to a large focal region, effectively overcomes the basis mismatch problem, and can be applied to any microphone array. The difference is that [PERSON] mainly uses the reduction in the number of grid points to improve the computational efficiency of the deconvolution method, while this paper mainly uses the reduction in the number of grid points to reduce the column coherence of the sensing matrix, thereby weakening the ill-posed inverse problem solving and improving the robustness. Further, the base mismatch problem is solved by continuously adjusting the positions of the grids, which improves the accuracy. This paper proposes a sound source identification method based on sparse Bayesian learning of grid splitting refinement. The initial source locations are obtained based on a sparse Bayesian learning framework. Then, candidate grids with higher weights are retained, local areas near them are refined, and [PERSON]'s functions and source strength matrices are re-established. The robust principal component analysis model replaces the sound pressure matrix obtained in the previous iteration with a low-rank matrix. The actual source location and source strength estimates are gradually approximated through the dynamically adjusted sound pressure low-order matrix and grid iteration matrix.