Uncertainty Quantification in Full Waveform Inversion: Linearised versus Fully Nonlinear Methods

Seismic full waveform inversion (FWI) is a powerful technique that uses seismic waveform data to generate high resolution images of the Earth's interior. However, significant uncertainties exist in FWI solutions due to imperfect acquisition geometries, inherent noise in the data, and the nonlinearity of the forward problem. Probabilistic Bayesian FWI estimates the family of all possible model solutions and quantifies their uncertainties by calculating the so-called posterior probability density function (pdf) of model parameter values of interest. In a linearised framework, the posterior pdf can be represented as a Gaussian distribution centred around the maximum a posteriori (MAP) solution, and the associated uncertainties are described by an a posteriori covariance matrix derived from the inverse Hessian matrix. Recent advancements have introduced nonlinear methods, such as variational inference, to solve Bayesian FWI problems efficiently. Their solutions quantify full uncertainties including those created by the nonlinearity of the problem. In this study, we apply both linearised and fully nonlinear methods to 2D acoustic Bayesian FWI problems. In particular, we use a physically structured variational inference algorithm for the nonlinear case, in which a transformed Gaussian distribution is optimised to approximate the full posterior pdf, such that the results can be compared fairly with those from the linearised, Gaussian-based method. We also employ an independent nonlinear variational algorithm – Stein variational gradient descent – for validation. The results show that while both linearised and nonlinear methods adequately recover the posterior mean models, they exhibit significantly different posterior uncertainty structures, especially at layer interfaces, due to the linearisation of wave physics. In addition, we show that linearised uncertainties are inaccurate since they can not fit observed waveform data and they yield biased estimates of inferred meta-properties such as volumes of geological bodies. This work therefore justifies the application of fully nonlinear inversion methods in Bayesian FWI if accurate uncertainty estimates are needed.