The inversion of complete seismic waveforms offers new perspectives to better constrain the elastic properties of Earth’s interior. However, models of density and seismic velocities obtained from full waveform inversions are generally characterized by very different and uneven spatial resolutions. Because the 3-D structure of the Earth represents small deviations from average reference Earth models, the absolute values of density, VP and VS in the Earth are strongly correlated. Here, we exploit this strong correlation between model parameters as a priori information introduced into a new full waveform inversion algorithm, by considering a non-diagonal 3-D model covariance matrix in which the spatial correlations of elastic properties are described with an exponential covariance function. The inverse of such a model covariance matrix is easy to compute, and we thus have all the ingredients to construct a consistent Bayesian full waveform inversion scheme. We show that taking into account the correlations between density and seismic velocities can lead to dramatic improvements on the reconstructed models of density, seismic velocities and VP/VS ratio. This new imaging approach opens new perspectives for refining tomographic images of density and seismic velocities in the lithosphere and upper mantle on a regional scale by full waveform inversion of teleseismic body waves.