Geodetic surveys now provide detailed time series maps of anthropogenic land subsidence and uplift due to injection and withdrawal of pore fluids from the subsurface. A coupled poroelastic model allows the integration of geodetic and hydraulic data in a joint inversion and has therefore the potential to improve the characterization of the subsurface and our ability to monitor pore pressure evolution. We formulate a Bayesian inverse problem to infer the lateral permeability variation in an aquifer from geodetic and hydraulic data and from prior information. We compute the maximum a posteriori (MAP) estimate of the posterior permeability distribution and a Gaussian approximation of the posterior. Computing the MAP estimate requires the solution of a large-scale minimization problem subject to the poroelastic equations, for which we propose an efficient Newton-conjugate gradient optimization algorithm. The covariance matrix of the Gaussian approximation of the posterior is given by the inverse Hessian of the log posterior, which we construct by exploiting low-rank properties of the data misfit Hessian. First and second derivatives are computed using adjoints of the time-dependent poroelastic equations, allowing us to fully exploit transient data. Using three increasingly complex model problems, we find the following general properties of poroelastic inversions: Augmenting standard hydraulic well data by surface deformation data improves the aquifer characterization. Surface deformation contributes the most in shallow aquifers but provides useful information even for the characterization of aquifers down to 1 km. In general, it is more difficult to infer high-permeability regions, and their characterization requires frequent measurement to resolve the associated short-response timescales. In horizontal aquifers, the vertical component of the surface deformation provides a smoothed image of the pressure distribution in the aquifer. Provided that the mechanical properties are known, coupled poroelastic inversion is therefore a promising approach to detect flow barriers and to monitor pore pressure evolution.