  Uncertainty quantification of groundwater (GW) aquifer parameters is critical
for efficient management and sustainable extraction of GW resources. These
uncertainties are introduced by the data, model, and prior information on the
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 a confined GW aquifer. The Bayesian solution of this
inverse problem takes the form of a posterior probability density of the
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 expense of solving the poroelastic
forward problem. However, in many partial differential equation (PDE)-based
Bayesian inversion problems, the data are only informative in a few directions
in parameter space. For the poroelasticity problem, we prove this property
theoretically for a one-dimensional problem and demonstrate it numerically for
a three-dimensional aquifer model. We design a generalized preconditioned
Crank--Nicolson (gpCN) MCMC method that exploits this intrinsic low
dimensionality by using a low-rank based Laplace approximation of the posterior
as a proposal, which we build scalably. The feasibility of our approach is
demonstrated through a real GW aquifer test in Nevada. The inherently two
dimensional nature of InSAR surface deformation data informs a sufficient
number of modes of the permeability field to allow detection of major
structures within the aquifer, significantly reducing the uncertainty in the
pressure and the displacement quantities of interest.
