Seismic waveform inversion with Fourier-based gradient shaping

Abstract We present a new gradient shaping scheme designed to favor ripple-style footprint suppression by Fourier-based model reconstruction with a truncated number of Fourier coefficients. It is motivated by improving the computational efficiency of full waveform inversion (FWI) with shot subsampling. Indeed, shot subsampling takes a small number of shots into each iteration, effectively turning all the shots inversion into a multiple-shot subsampling inversion, and finally reducing the high computational demands. A practical requirement of FWI with small-numbered shot subsampling is the removal of footprints without damaging the gradient. Unfortunately, conventional Gaussian smoothing does not provide such a removal. Thus, we introduce a Fourier-based shaping scheme that shares the benefits of removing the footprints and preserving the gradient amplitudes. The contributions of the forward and inverse Fourier transforms in a fast manner are key factors in the implementation of estimating Fourier coefficients and reconstructing the model, which are the main procedures for a Fourier-based shaping scheme. After the behavior of the proposed shaping scheme is analyzed, its performance is studied for the velocity model recovery of the Marmousi and overthrust models. The findings on synthetic data examples indicate improvements in the inverted results using the Fourier-based shaping scheme compared to those inverted by the Gaussian smoothing scheme.