A pseudo-time formulation for acoustic full waveform inversion

Abstract

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. A pseudo-time formulation of FWI is proposed to reduce the depth/velocity ambiguities. This is especially important when the initial model parameter is derived from reflection data and contains large contrasts. The reformulated wave equation in pseudo-time however leads to a rather complicated implementation. A simpler approach consists of applying the change of variables at the level of the misfit functional and computing the gradient with the standard chain rules between the depth and pseudo-time coordinate systems. Synthetic examples illustrate the relevance of this new formulation of FWI. The inversion of a wide-azimuth, long-offset real data set shows that the new formulation can be helpful to retrieve the background velocity in presence of salt bodies. This real example illustrates that the depth position of the salt bodies is changed to keep the vertical time of the reflections on the top of the salt bodies consistent with the velocity changes in the shallow part of the model. The pseudo-time FWI retrieves the shallow velocity variations by also inverting the diving/transmitted waves recorded at the long offsets. This pseudo-time reformulation could also be applied to any migration-based velocity analysis. © The Author 2012. Published by Oxford University Press on behalf of The Royal Astronomical Society.