  For quantitative seismic imaging, iterative least-squares reverse time
migration is the recommended approach. The existence of an inverse of the
forward modelling operator would considerably reduce the number of required
iterations. In the context of the extended model, such a pseudo-inverse exists,
built as a weighted version of the adjoint and accounts for the deconvolution,
geometrical spreading and uneven illumination. The application of the
pseudo-inverse Born modelling is based on constant density acoustic media,
which is a limiting factor for practical applications. To consider density
perturbation, we propose and investigate two approaches. The first one is a
generalization of a recent study proposing to recover acoustic perturbations
from angle-dependent response of the pseudo-inverse Born modelling operator.
The new version is based on weighted least-squares objective function. The
method not only provides more robust results, but also offers the flexibility
to include constrains in the objective function in order to reduce the
parameters cross-talk. We also propose an alternative approach based on Taylor
expansion that does not require any Radon transform. Numerical examples based
on simple and the Marmousi2 models using correct and incorrect background
models for the variable density Born modelling, verify the effectiveness of the
weighted least-squares method when compared with the other two approaches. The
Taylor expansion approach appears to contain too many artifacts for a
successful applicability.
