Spectral implementation of full waveform inversion based on reflections

Using the reflection imaging process as a source to model reflections for full waveform inversion (FWI), referred to as reflection FWI (RFWI), allows us to update the background component of the model, and avoid using the relatively costly migration velocity analysis (MVA), which usually relies on extended images. However, RFWI requires a good image to represent the current reflectivity, as well as, some effort to obtain good smooth gradients. We develop a spectral implementation of RFWI where the wavefield extrapolations and gradient evaluation are performed in the wavenumber domain, obtaining clean dispersion free and fast extrapolations. The gradient, in this case, yields three terms, two of which provide us with each side of the rabbit ear kernel, and the third, often ignored, provides a normalization of the reflectivity within the kernel, which can be used to obtain a reflectivity free background update. Since the image is imperfect (it is an adjoint, not an inverse), an optimization process for the third term scaling is implemented to achieve the smoothest gradient update. A rare application of RFWI on the reflectivity infested Marmousi model shows some of the potential of the approach.