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Characterizing the properties of groundwater aquifers is essential for predicting aquifer response and managing groundwater resources. In this work, we develop a high-dimensional scalable Bayesian inversion framework governed by a three-dimensional quasi-static linear poroelastic model to characterize lateral permeability variations in groundwater aquifers. We determine the maximum a posteriori (MAP) point of the posterior permeability distribution from centimeter-level surface deformation measurements obtained from Interferometric Synthetic Aperture Radar (InSAR). The scalability of our method to high parameter dimension is achieved through the use of adjoint-based derivatives, inexact Newton methods to determine the MAP point, and a Matérn class sparse prior precision operator. Together, these guarantee that the MAP point is found at a cost, measured in number of forward/adjoint poroelasticity solves, that is independent of the parameter dimension. We apply our methodology to a test case for a municipal well in Mesquite, Nevada, in which InSAR and GPS surface deformation data are available. We solve problems with up to 320,824 state variable degrees of freedom (DOFs) and 16,896 parameter DOFs. A consistent treatment of noise level is employed so that the aquifer characterization result does not depend on the pixel spacing of surface deformation data. Our results show that the use of InSAR data significantly improves characterization of lateral aquifer heterogeneity, and the InSAR-based aquifer characterization recovers complex lateral displacement trends observed by independent daily GPS measurements.