Elastic full-waveform inversion (FWI) provides a better description of the subsurface information than those given by the acoustic assumption. However, it suffers from a more serious cycle-skipping problem compared with the latter. Reflection waveform inversion (RWI) is able to build a good background model, which can serve as an initial model for elastic FWI. Because, in RWI, we use the model perturbation to explicitly fit reflections, such perturbations should include density, which mainly affects the dynamics. We applied Born modeling to generate synthetic reflection data using optimized perturbations of the P- and S-wave velocities and density. The inversion for the perturbations of the P- and S-wave velocities and density is similar to elastic least-squares reverse time migration. An incorrect background model will lead to misfits mainly at the far offsets, which can be used to update the background P- and S-wave velocities along the reflection wavepath. We optimize the perturbations and background models in an alternate way. We use two synthetic examples and a field-data case to demonstrate our proposed elastic RWI algorithm. The results indicate that our elastic RWI with variable density is able to build reasonably good background models for elastic FWI with the absence of low frequencies, and it can deal with the variable density, which is required in real cases.