###### Abstract  
In this paper, a novel two-dimensional direction of arrival (2D-DOA) estimation method with total variation regularization is proposed to deal with the problem of sparse DOA estimation for spatially extended sources. In a general sparse framework, the sparse 2D-DOA estimation problem is formulated with the regularization of extended source characteristics including spatial position grouping, acoustic signal block sparse, and correlation features. An extended sources acoustic model, two-dimensional array manifold and its complete representation, total variation regularization penalty term, and the regularization equation are built, and are utilized to seek the solutions where the non-zero coefficients are grouped together with optimum sparseness. A total variation sparse 2D-DOA estimation model is constructed by combining total variation regularization with LASSO. The model can be easily solved by the convex optimization algorithm, and the solving process can promote the sparsity of the solution on the  
signal-to-noise ratio, under the condition of coherent signal source algorithm performance degradation or even failure fault [17]. The Orthogonal Matching Pursuit (OMP) [18], as an example of compressive sensing methods, is a type of novel high- resolution DOA estimation algorithm. Despite using the OMP algorithm for DOA estimation which has many advantages in comparison to other algorithms, it cannot distinguish between two adjacent signal sources. The FOMP algorithm [19; 20] is an improved version of the OMP algorithm. It can detect two signal sources exactly when they are very close and the beam former has a single peak corresponding to a direction between right directions. All of the compressive sensing algorithms for DOA estimation should satisfy the incoherence condition [21]. Thus, almost all of these algorithms fail when the maximum pairwise coherence of the measurement matrix exceeds a constant value. However, the DOA estimation problem is usually underdetermined, and a stable solution can be obtained by the regularization technique. The \(\ell_{1}\)-norm regularization has gradually become a hot spot because it can reduce the computational amount of sparse recovery and assume the role of using its sparse promotion characteristics to obtain high resolution [22; 23; 24; 25; 26]. The LASSO proposed by [PERSON] [27] can provide high-resolution DOA maps even in the presence of coherent sources or single snapshot data [28]. However, the LASSO cannot effectively promote the grouping of related positions considering that the extended source has the characteristics of grouping (referring to the grouping of related positions) and the amplitude distribution with the phase linearity and block sparsity [29], therefore it is not suitable for recovering multipole or spatially extended sources. In order to discuss these problems involving the sound features, the elastic net model [30] combines the sparse promotion feature using \(\ell_{1}\)-norm with the component clustering feature based on \(\ell_{2}\)-norm which is proposed to achieve the balance between smoothness and sparsity. However, this model can only achieve a smooth solution and cannot promote a piecewise constant outline. The weighted LASSO [31], namely WL, imposes more general geometric or structural constraints on solutions by constructing weighted structural matrices, and different matrices form total variation models such as focused total variation (FTV) [32], generalized total variation (GTV) [33], and total variation(TV) [25]. The total variation model has been widely used in image processing and one-dimensional (1D) DOA estimation in recent years [29] because it can suppress noise components at smaller scales while retaining larger feature edges in images [34]. Two-dimensional-DOA estimation has been widely used because it can better describe the spatial characteristics of the target. [PERSON] et al.