On Monitoring Changes in the Earth’s Subsurface using 4D Bayesian Variational Full Waveform Inversion

This work discusses the use of full waveform inversion (FWI) with fully nonlinear estimates of uncertainty, to monitor changes in the Earth’s subsurface due to dynamic processes. Typically, FWI is used to produce high resolution 2D and 3D static subsurface images by exploiting information in full acoustic, seismic or electromagnetic waveforms, and has been applied at global, regional and industrial spatial scales. To avoid the over-interpretation of poorly constrained parts of resulting subsurface images or models, it is necessary to know their uncertainty – the range of possible subsurface models that are consistent with recorded data and other pertinent constraints. Almost all estimates of uncertainty on the results of FWI approximate the model-data relationships by linearisation to make the calculation computationally efficient; unfortunately this throws those uncertainty estimates into question, since their raison d’etre is to account for possible model and data variations which are themselves related nonlinearly.

In a related abstract and associated manuscript we use variational inference to achieve the first Bayesian uncertainty analysis for 3D FWI that is fully nonlinear (i.e., involves no linearisation of model-data relationships: https://arxiv.org/abs/2210.03613 ). Variational inference refers to a class of methods that optimize an approximation to the probability distribution that describes post-inversion parameter uncertainties.

Here we extend those methods to perform nonlinear uncertainty analysis for 4D (time-varying 3D) FWI monitoring of the subsurface. Specifically we apply stochastic Stein variational gradient descent (sSVGD) to seismic data generated synthetically for two 3D seismic surveys acquired over a changing 3D subsurface structure based on the 3D overthrust model (Aminzadeh et al., 1997: SEG/EAGE 3-D Modeling Series No. 1). Iterated linearised inversion of each data set fails to image changes (~1%) in the wave speed of the medium, both when each inversion begins independently from the same (good) reference model, or when the best-fit model from inversion of the first survey’s data was used as reference model for the second inversion. Nonlinear inversion of each data set from the same prior distribution also fails to detect these ~1% changes. However, the changes can be imaged and their uncertainty estimated if variational methods applied to invert data from the second survey are initiated from their final state in the inversion of the first survey data. In addition, the methods then converge far more rapidly, compared to running each inversion independently.

We conclude that the probability distributions describing 3D seismic velocity uncertainty are sufficiently complex that the computations of 3D parameter uncertainty for each survey independently have not converged sufficiently to detect small 4D changes. However, the change in these probability distributions between surveys must be sufficiently small that the final solution found from the first survey could evolve robustly into the second survey solution, such that changes are resolved above the uncertainty using variational methods. Nevertheless, this change must be sufficiently complex that linearised methods can not evolve smoothly from one solution to the next, explaining why linearised methods fail, and highlighting why the estimation of nonlinear uncertainties is so important for imaging and monitoring applications.