We have developed an efficient convex optimization strategy enabling the simultaneous attenuation of random and erratic noise with interpolation in prestack seismic data. For a particular analysis window, frequency slice spatial data were reorganized into a block Toeplitz matrix with Toeplitz blocks as in Cadzow/singular spectrum analysis methods. The signal and erratic noise were, respectively, modeled as low-rank and sparse components of this matrix, and then a joint low-rank and sparse inversion (JLRSI) enabled us to recover the low-rank signal component from noisy and incomplete data thanks to joint minimization of a nuclear norm term and an -norm term. The convex optimization framework, related to recent developments in the field of compressed sensing, enabled the formulation of a well-posed problem as well as the use of state-of-the-art algorithms. We proposed an alternating directions method of multipliers scheme associated with an efficient singular value thresholding kernel. Numerical results on field data illustrated the effectiveness of the JLRSI approach at recovering missing data and increasing the signal-to-noise ratio.