Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion

Abstract

Seismic full waveform inversion (FWI) is a powerful technique 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, and nonlinearity of the forward problem. Probabilistic Bayesian FWI addresses this non-uniqueness by estimating the entire family of possible model solutions described by the posterior probability density function (pdf). The posterior pdf can be estimated using nonlinear inversion methods to quantify full uncertainties. Alternatively, by linearising the physics relating parameters and observations around the maximum a posteriori solution, the posterior pdf is usually approximated by 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 a variational inference algorithm for the nonlinear case, in which a transformed Gaussian is optimised to approximate the posterior pdf. The results can be compared with those from a linearised, locally-Gaussian based method. We also apply Stein variational gradient descent for comparison. The results show that while both the linearised and nonlinear methods recover the posterior mean models accurately, they exhibit significantly different posterior uncertainty structures, especially around layer interfaces, due to the linearisation of wave physics. Linearised uncertainty estimates are shown to be significantly less accurate: they provide far less accurate fits to observed waveform data, and yield biased estimates of inferred meta-properties such as volumes of geological bodies. This work therefore motivates the application of fully nonlinear inversion methods in Bayesian FWI if accurate uncertainty estimates over parameters, or inferred or interpreted meta-properties are important.

Figures (21)

• Figure 1: (a) True P-wave velocity used in the layered model example. Source locations are indicated by red stars and a receiver line is marked by a white line. Dashed black lines display locations of three vertical profiles used to compare the inversion results in the main text. (b) Velocity model used to define the mean of a Gaussian prior distribution, obtained by smoothing the true velocity model in (a).
• Figure 2: Ten common shot gathers with a dominant frequency of 10 Hz, used as the observed waveform data in the first FWI example.
• Figure 3: Inversion results obtained using linearised, PSVI and SVGD methods. (a), (b) and (c) Posterior mean models. The mean model in (a) represents the maximum a posteriori (MAP) solution obtained by a deterministic, linearised inversion. (d), (e) and (f) The corresponding standard deviation maps. Note that velocity values above the receiver line (100 m depth) are fixed at their true values. The corresponding regions in the standard deviation maps are therefore left blank.
• Figure 4: (a) Prior and posterior marginal distributions obtained from (b) linearised, (c) PSVI and (d) SVGD, along three vertical profiles at horizontal locations 1 km, 2 km and 3 km. Their locations are represented by vertical dashed black lines in Figure \ref{['fig:layered_true_initial']}a. From top to bottom, each row represents results along one profile.
• Figure 5: Relative deviation of the linearised uncertainty estimates from the nonlinear estimates ($R_\sigma$) calculated by equation \ref{['eq:relative_deviation']}. $R_\sigma$ values lower than 1 indicate that the linearised uncertainty estimates are roughly as good as the nonlinear ones, given the epistemic uncertainties around the various methods used herein; otherwise, the linearised estimates are different from (worse than) the nonlinear ones.