Abstract: ‍ ‍The Sparse Linear Array （SLA） exhibits superior anti-coupling effects and a larger array aperture than the Uniform Linear Array （ULA） with the same number of elements. Direction of Arrival （DOA） estimation， which offers higher Degrees Of Freedom （DOFs）， has been extensively researched in recent years. Among various DOA estimation algorithms， the Multiple Signal Classification （MUSIC） algorithm provides high-resolution estimations but requires the covariance matrix of the original array. To achieve accurate DOA estimation with high DOFs， scholars have begun investigating the covariance matrix corresponding to the differential virtual elements of the SLA. This augmented covariance matrix has larger dimensions than the covariance matrix of the primitive element， resulting in increased estimated DOFs. When the differential virtual elements of the SLA are continuously valued， we can obtain its covariance matrix by utilizing information from existing elements and construct a covariance matrix of differential virtual elements based on this for DOA estimation. However， if the SLA differential virtual element values have holes where continuity is disrupted， direct usage of the reconstructed covariance matrix becomes unfeasible for DOA estimation. In such cases， information should be recovered from a fully augmented covariance matrix before performing DOA estimation. To address this problem， considering that errors between vectorized SLA and covariances of virtual difference matrices follow an asymptotic normal distribution， we use normalization and further transformation to formulate an inequality constraint problem that can be easily solved. Consequently， we establish a relationship between the fully augmented covariance matrix， SLA covariance matrix， and the covariances of virtual difference matrices. Finally， leveraging the low rank and semi-definite properties of the fully augmented covariance matrix， we construct a convex optimization problem that can be solved using the MATLAB CVX toolkit. Through this approach， we can recover a higher-dimensional fully augmented covariance matrix. Subsequently， we decompose this completely augmented covariance matrix using singular value decomposition （SVD） and employ the MUSIC algorithm to estimate the spatial spectrum of our target signal source. Numerical simulation results demonstrate that our proposed algorithm enables underdetermined DOA estimation （i.e.， when more sources than elements exist） while achieving high DOFs with respect to DOA estimation through the reconstruction of the fully augmented covariance matrix. Compared with existing algorithms， our method exhibits superior resolution towards targets and performs closer to Cramér-Rao bound （CRB） under varying SNR levels and snapshots scenarios. For multi-source DOA estimates， spectral peaks become sharper while bias is reduced. In conclusion， our paper presents an effective approach for achieving highly precise DOA estimation with ample DOFs for sparse linear arrays with holes.