Augmented formulation for a Bayesian approach for frequency-domain full-waveform inversion to estimate the material properties of a layered half-space

Seismic full-waveform inversion (FWI) facilitates the generation of high-resolution subsurface images using wavefield measurements. Seismic FWI in the frequency domain is preferable because it allows consideration of the multiscale nature of FWI, controls the numerical dispersion of the media, and represents the hysteretic damping of the material. The Bayesian approach can be considered for FWI problems to alleviate the ill-posedness of inverse problems and quantify the uncertainty of the estimated parameters. This study rigorously formulates a Bayesian approach for seismic FWI in the frequency domain, assuming Gaussian probability distributions for the prior information of parameters to be estimated and the likelihood functions of observations. Conventional and augmented formulations are provided. In the augmented formulation, complex dynamic responses in the frequency domain are augmented by their complex conjugates. Rigorous expressions are derived for the posterior covariance matrix of estimated parameters to assess the uncertainty in these parameters. The proposed augmented formulation is demonstrated using various elastic inverse problems to estimate the shear-wave velocities of layered half-spaces. Excellent inverted profiles for the shear-wave velocities are obtained, and their posterior probability distributions are estimated using the Bayesian approach.