Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion

Summary 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 uncertainty exists in all FWI solutions due to imperfect acquisition geometries, inherent noise in the data, nonlinearity of the forward problem, and the under-determined nature of real-world tomographic problems in which the target is heterogeneous over all length scales. Probabilistic Bayesian FWI addresses this non-uniqueness by estimating the entire family of possible model solutions and thus the solution uncertainty, described by the so-called posterior probability density function (pdf) over model parameter values. The posterior pdf can be estimated using nonlinear inversion methods to quantify full uncertainties, including those created by nonlinearity in the physics. Alternatively, by linearising (approximating) the physics relating parameters and observations around a chosen reference model solution, the posterior pdf is usually approximated by a compact distribution centred around the maximum a posteriori solution, typically a Gaussian pdf. This is referred to as the linearised method. In this work, we apply both nonlinear and linearised methods to 2D acoustic Bayesian FWI problems. We use one variational inference algorithm for the nonlinear case, in which a transformed Gaussian distribution is optimised to approximate the unknown, full posterior pdf, and a second, independent nonlinear variational algorithm – Stein variational gradient descent – for comparison. The results of both are then compared with those from a linearised, locally-Gaussian based method. The results show that while both the linearised and nonlinear methods recover the posterior mean models accurately, they exhibit different posterior uncertainty structures, especially around layer interfaces, due to the linearisation of wave physics. The differences become most obvious in partially constrained regions of the model, where posterior solutions are constrained jointly by data, prior information, and the nonlinearity of wave physics rather than being dominated by any single factor. We also demonstrate that linearised uncertainty estimates are significantly less accurate: they provide far less accurate fits to observed waveform data, and yield biased estimates of inferred or interpreted meta-properties such as volumes of geological bodies. This work therefore motivates the application of fully nonlinear inversion methods in Bayesian FWI if either accurate uncertainty estimates over parameters, or inferred or interpreted meta-properties are important.