Reflection Waveform Inversion method: solutions to the reflectivity-background coupling problem and consequences on the convergence

A new formulation of the Full-Waveform Inversion (FWI) method called the Reflection Full Waveform Inversion (RFWI) method has been recently introduced in order to enable background velocity updates from reflection events. However, it has been observed (even with state of the art optimization methods) that numerous iterations in the RFWI workflow are necessary to converge towards a correct background model. This numerous iterations makes the cost of the current approach prohibitive for real scale applications. In this contribution, using an analytic model, we explain how the high number of RFWI iterations is related to the background-reflectivity models coupling. We propose two solutions for solving this issue. One solution considers a joint optimization in an extended space considering independent de-migration and migration background models. Another solution consists in a variable projection technique which enables an optimization in a reduced model parameter space where the background and the reflectivity models are always consistent. In our tests, both solutions provide convergence rates of a least one order of magnitude faster than the conventional RFWI method. Furthermore, we provide a geometrical interpretation of the relation between these different solutions and explain why the methodology based on the variable projection should be preferable.