A pseudo-time formulation for acoustic full waveform inversion

When surface seismic data contain diving or supercritical reflections, full waveform inversion (FWI) of surface seismic data helps improving the macro velocity model under acoustic vertical transversely isotropic assumption. When large discontinuities, that generate pre-critical reflections, are present in the initial model or appear during the iterative optimization process, FWI suffers from the depth/velocity ambiguity because the model parameters are naturally represented in the spatial depth coordinate system. With classic velocity analysis, a time processing in which vertical time replaces depth as vertical axis provides a more robust formulation by reducing the null space of the inverse problem. Classic time processing assumes mild lateral variations and does not apply to complex geological settings. To extend the time processing without the assumption of mild lateral variations, a change of variables based on the vertical velocity is performed. Applied directly to the wave equation, it leads to a modified wave equation in pseudo-time coordinate system where depth is replaced by vertical time.