Bayesian Inference of Groundwater Aquifer Parameters via InSAR Data, High-Dimensional Poroelastic Models, and Dimension-Invariant MCMC Methods

Abstract:

Uncertainty quantification (UQ) of groundwater (GW) aquifer parameters is critical for efficient GW resources management and, hence, for mitigating the global-scale problem of unsustainable GW extraction. These uncertainties stem from uncertainties in the data, model, and prior information on the parameters, along with the insensitivity of observables to parameters. Here we develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of GW aquifers. The Bayesian solution of this inverse problem takes the form of a posterior probability density quantifying uncertainties in the aquifer permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expensive poroelasticity forward solves. However, in many PDE-based Bayesian inversion problems, the data are informative in only relatively few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for an idealized 1D problem and demonstrate it numerically for the 3D aquifer model. We apply an MCMC method that exploits this intrinsic low dimensionality using a low-rank based Laplace approximation of the posterior as a proposal for a generalized preconditioned Crank--Nicolson (gpCN) method (Villa et al., 2018). We build this proposal scalably by deriving adjoint-based expressions for the Hessian of the negative log posterior, and using randomized algorithms to obtain its low-rank approximation. We carry out the implementation using the FEniCS library for finite element discretization in space and the hIPPYlib library for state-of-the-art Bayesian and deterministic PDE-constrained inversion algorithms. The feasibility of our approach is demonstrated by applying this Bayesian inversion framework to a real GW aquifer test site in Nevada. We find that the use of InSAR data significantly reduces the uncertainty in the permeability field and chosen posterior predictive quantities of interest. Moreover, the Laplace approximation provides a good approximation of the far more expensive MCMC-based posterior.