The seismic imaging problem entails recovering an image of the earth's subsurface from data that is recorded on the surface of the earth, determined by the propagation of seismic (vibrational) waves through the body of the earth. The 2D or 3D acoustic wave equation is commonly used as a simplified mathematical model for this seismic wave propagation. The full waveform inverse problem aims to deduce the physical parameters of the (acoustic) medium of propagation from recorded data of impulsive waves that are transmitted or reflected through the medium, and thus form an image of the subsurface.

In this study, we demonstrate a numerical algorithm that uses factorization in the PDE solver of the 2D acoustic wave model, a multi-scale approach to the inverse solution, and a projection-based linearization search for the solution to the inverse problem. The multi-scale approach is used to decrease the rank of the inverse problem, thus decreasing the ill-posedness and under-determinedness of the solution. With a few examples, we show the robust properties of the inversion algorithm, a fast numerical convergence rate, and the advantages of multi-scaling.