Phase-diverse speckle imaging is a novel imaging modality that makes use of both speckle-imaging and phase-diversity concepts. A phase- diverse speckle data set consists of one conventional image and at least one additional image with known phase diversity for each of multiple atmospheric phase realizations. We demonstrate the use of a phase-diverse speckle data set collected at the Swedish Vacuum Solar Telescope on La Palma to overcome the effects of atmospheric turbulence and to restore a fine-resolution image of solar granulation. We present preliminary results of simultaneously reconstructing an object and a sequence of atmospheric phase aberrations from these data using a maximum-likelihood parameter- estimation framework. The consistency of the reconstructions is demonstrated using subsets of the sequence of images pairs. The use of different phase-aberration parameterization schemes and their affect on parameter estimates are discussed. Insight into the desired number of atmospheric realizations is provided.