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The conception of clutter azimuth-elevation (AE) spectrum is brought forward, based on which a novel space-time adaptive processing (STAP) algorithm called AE based STAP (AE-STAP) algorithm is proposed in this paper. Instead of using clutter spatial-temporal spectrum for sparse recovery typed STAP algorithms and maximum likelihood estimation for statistic based STAP algorithms, the proposed algorithm utilizes the clutter AE spectrum, which is obtained by the sparse recovery technique and an elevation filter, to estimate the clutter covariance matrix and design the adaptive filter to suppress clutter. The proposed algorithm could achieve a greatly accurate estimation of clutter covariance matrix with few training samples, as well as avoid the influence of range ambiguous clutter and most false values appearing in the process of sparse recovery, and thus enjoy a great performance of clutter suppression and slowly moving targets detection. Simulation results are provided to illustrate the effectiveness of the proposed algorithm.