A note on Marchenko-linearised full waveform inversion for imaging

Abstract

Full waveform inversion and least-squares reverse time migration are the leading technologies for imaging with seismic waves. Both of them usually rely (in one way or another) on a single-scattering approximation, i.e. the Born approximation, to compute gradients and obtain an updated model. This approximation linearises the relation between modelled data and model by ignoring multiple scattering. We propose to use the Marchenko integral, an equation originating from inverse scattering theory, to obtain an alternative linear equation. Using the Marchenko method we can retrieve Green's functions, including all orders of scattering, for virtual sources anywhere within the volume of interest - without prior knowledge of the high-wavelength model variations that induce scattering. Plugging these estimated Green's functions into the Lippmann-Schwinger integral delivers a Marchenko-linearised relation between the full waveform data and the model. We present this new linearisation strategy and illustrate its advantages and disadvantages by comparing numerical results for different inversion kernels. Our new linearisation is exact, i.e. it does not exclude any orders of scattering, however, it relies on the quality of the Marchenko-derived Green's functions. These Marchenko-based Green's functions require an estimate of the first arrivals of the Green's functions - commonly obtained by modelling in a background medium. Although these first arrival estimates strongly bias our results for inaccurate background models, we find the Marchenko-linearisation to deliver overall slightly better inverted models than the single-scattering approximation.