A system and method involve acoustic source localization using passive sonar and capitalizing on the sparse nature of a source location map (SLM). Two types of sparsity are exploited, namely sparsity in the support of the SLMs and sparsity in the innovations across consecutive SLMs. The first type is motivated by the desire to construct SLMs whose non-zero entries corresponded to locations where sources are present. The second type of sparsity is motivated by the observation that few changes occur in the support of consecutive SLMs. Per time instant, an SLM is obtained as the solution of a regularized least-squares problem, where the regularization terms are chosen to encourage the desired sparse structures in each SLM and the innovations. Each SLM may be obtained via a specifically-tailored, computationally-efficient proximal gradient algorithm.