Abstract The acoustic approximation in seismic migration is not allowed when the effects of wave conversion cannot be neglected, as is often the case in data with large offsets. Hence, seismic migration should ideally be founded on the full elastic wave equation, which describes compressional as well as shear waves in solid media (such as rock layers, in which shear stresses may play an important role). In order to cope with conversions between those wave types, the full elastic wave equation should be expressed in terms of the particle velocity and the traction, because these field quantities are continuous across layer boundaries where the main interaction takes place. Therefore, the full elastic wave equation should be expressed as a matrix differential equation, in which a matrix operator acts on a full wave vector which contains both the particle velocity and the traction. The solution of this equation yields another matrix operator. This full elastic two-way wave field extrapolation operator describes the relation between the total (two-way) wave fields (in terms of the particle velocity and the traction) at two different depth levels. Therefore it can be used in prestack migration to perform recursive downward extrapolation of the surface data into the subsurface (at a "traction-free" surface, the total wave field can be described in terms of the detected particle velocity and the source traction). Results from synthetic data for a simplified subsurface configuration show that a multiple-free image of the subsurface can be obtained, from which the angle-dependent P-P and P-SV reflection functions can be recovered independently. For more complicated subsurface configurations, full elastic migration is possible in principle, but it becomes computationally complex. Nevertheless, particularly for the 3-D case, our proposal has improved the feasibility of full elastic migration significantly compared with other proposed full elastic migration or inversion schemes, because our method is carried out per shot record and per frequency component.