The sparse Radon transform (RT) represents seismic data by the superposition of a few constant amplitude events, and thus it has trouble dealing with amplitude-versus-offset (AVO) variations. We integrated the gradient and curvature parameters of AVO into the RT. With these additional properties, the lateral continuity of the events’ amplitude was modeled in the transformation and it could be fitted with orthogonal polynomials. This resulted in a higher order RT, which included AVO terms. The high-order RT is a highly underdetermined problem, which was solved by extracting the major model parameters from energy distribution in a high-order Radon domain and by decreasing the number of inversion parameters. Thus, a high-order sparse RT was achieved. The proposed method can be used for data interpolation as well as extrapolation. The AVO-preservation performance of the proposed algorithm in data reconstruction was illustrated using both synthetic and field data examples, and the results showed the feasibility of the method.