Time migration is an attractive tool to produce a subsurface image because it is faster and less sensitive to velocities errors than depth migration. However, a highly focused time image is only achievable with well-determined time-migration velocities. Therefore, a refinement of the initial time-migration velocities often is required. We introduced a new technique for prestack time migration, based on the common-migrated-reflector-element stack of common scatterpoint gathers, including an automatic update of time-migration velocities. The common scatterpoint gathers are generated using a new formulation of the double-square-root equation that is parametrized with the common-offset apex time. The common-migrated-reflector-element stack is a multiparameter stacking technique based on the Taylor expansion of traveltimes of time-migrated reflections in the paraxial vicinity of the image ray. Our 2D synthetic and field data examples demonstrated that the proposed method provides updated time-migration velocities that are more robust and have higher resolution compared with the initial time-migration velocities. The prestack time migration method also showed a clear improvement of the focusing of reflections for such geologic features as faults and salt structures.