A quasi-elastic wave equation as a function of the pressure variable is presented which can accurately model PP reflections with elastic amplitude versus offset (AVO) effects under the first-order Born approximation. It uses a virtual source to model the amplitudes of reflections, which is a function of the perturbations of density and Lamé parameters l and µ. The quasi-elastic wave equation is used for true-amplitude elastic inversion of PP reflections, where the perturbations of elastic parameters are iteratively updated by minimizing the L2 norm of the difference between the recorded PP reflections and the predicted pressure data. Numerical tests on synthetic and field data show that true-amplitude waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude waveform inversion provides images with wider bandwidth and fewer artifacts because of the accurate elastic modeling of amplitudes at all pre-critical offsets.