Bayesian inverse problems for quasi-static poroelasticity with application to ground water aquifer characterization from geodetic data

Effective and sustainable management of groundwater (GW) resources under the present unprecedented global-scale GW demand is a complex task that requires high fidelity models of the subsurface. The goal of our work is to characterize the lateral heterogeneity in GW aquifer permeability to potentially improve the accuracy of subsurface models for GW management applications. To achieve this goal, our methodology integrates surface deformation data into a high-dimensional Bayesian inversion framework to infer the laterally heterogeneous permeability field in a three-dimensional quasi-static fully-coupled poroelastic model of the subsurface. Using poroelastic models, in particular, has the advantage of simulating the three-dimensional displacement components, in addition to the pressure field, which can be compared directly with the deformation measurements. We use centimeter-level Interferometric Synthetic Aperture Radar (InSAR) surface deformation maps and GPS deformation time series as input data sets to our inversion framework.

Exploring the resulting Bayesian posterior distribution is a computationally challenging task due to the large dimension of the discretized permeability field. We overcome this challenge by deriving adjoint-based expressions for the gradient and Hessian of the negative log posterior, employing a Matérn class sparse prior precision operator, and taking advantage of posterior properties such as smoothness, intrinsic low dimensionality, and geometry. The latter is achieved using inexact Newton methods to determine the maximum a posteriori (MAP) point and a low-rank based Laplace approximation of the posterior as a proposal for a discretization-invariant MCMC sampling technique---generalized preconditioned Crank--Nicolson (gpCN) [Villa et al., 2018]. Together, these guarantee that the cost of finding the MAP point, measured in the number of forward/adjoint poroelasticity solves, together with the convergence of the MCMC chains, are independent of the parameter dimension. To provide theoretical evidence for the plausibility of the low-rank based approximation, we provide analytical results demonstrating a decay rate of order O(1/i⁴) for the eigenvalues λ [subscript i] the Hessian of the negative log likelihood (the data misfit Hessian) for a one-dimensional steady-state poroelastic inverse problem.

We demonstrate the feasibility of our approach through application of our framework to a test case for a municipal well in Mesquite, Nevada, in which InSAR and GPS surface deformation data are available. We devise an approach based on the data misfit Hessian eigendecomposition to quantify the information content of each data set about the aquifer permeability and hence its effectiveness in improving the accuracy of the GW aquifer model. Our results show that the use of InSAR data significantly improves the characterization of lateral aquifer heterogeneity when compared to, for example, GPS time series. For validation, we show that the InSAR-based aquifer characterization additionally recovers complex lateral displacement trends observed by the independent GPS measurements. Due to the consistent treatment of the InSAR data noise level that we employ, the aquifer characterization result is independent of the InSAR data pixel spacing. 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. We solve problems with up to 320,824 state variable degrees of freedom (DOFs) and 16,896 parameter DOFs.

Our research illustrates the potential of combining the widely available and continuously growing global InSAR data set along with scalable poroelasticity-based Bayesian inversion frameworks to provide a valuable tool for characterizing GW aquifer properties in GW management applications. Our framework additionally provides a systematic way for quantifying the information content of different data sets about the inferred model properties, a useful tool for evaluating data acquisition decisions.