The aim of full waveform inversion is the reconstruction of subsurface material parameters from seismic measurements. The corresponding mathematical task is the solution of an ill-posed nonlinear parameter identification problem for the linear visco-elastic wave equation. To attack this inverse problem in realistic scenarios we apply implicit hierarchical space-time discretizations and efficient parallel schemes for solving the wave equation forward and backward in time. These schemes will finally be combined with a tailor-made regularization method of inexact Newton type for the full nonlinear problem. The convergence is analyzed within a suitable Banach space framework.