SUMMARY Iterative least-squares reverse time migration (LSRTM) is the state-of-the-art linearized waveform inversion method to obtain quantitative subsurface parameters. The main drawback of such an iterative imaging scheme is the significant computational expense of many modelling/adjoint cycles through iterations. In the context of the extended domain, an interesting alternative to LSRTM is the asymptotic linearized direct waveform inversion, providing quantitative results with only a single iteration. This approach was first proposed for constant-density acoustics and recently extended to the variable-density case. The former is based on the application of the asymptotic inverse Born operator, whereas the latter has two more extra steps: building an angle-dependent response of the asymptotic inverse Born operator and then solving a weighted least-squares approach for simultaneous inversion of two acoustic parameters. To examine the importance of accounting for density variations, we compare constant- and variable-density linearized direct waveform inversion techniques applied to a marine real data set from the Eastern Nankai Trough, offshore Japan. The inversion results confirm the efficiency of the asymptotic linearized direct waveform inversion in estimating quantitative parameters within a single iteration. The variable-density direct inversion yields subsurface images that (1) exhibit a superior resolution and (2) better reconstruct the field data than does the constant-density approach, even if the data set does not contain large enough surface offset to fully decompose velocity and density perturbations.