  We present a new approach for constructing a data-driven surrogate model and
using it for Bayesian parameter estimation in partial differential equation
(PDE) models. We first use parameter observations and Gaussian Process
regression to condition the Karhunen-Lo\'{e}ve (KL) expansion of the unknown
space-dependent parameters and then build the conditional generalized
Polynomial Chaos (gPC) surrogate model of the PDE states. Next, we estimate the
unknown parameters by computing coefficients in the KL expansion minimizing the
square difference between the gPC predictions and measurements of the states
using the Markov Chain Monte Carlo method. Our approach addresses two major
challenges in the Bayesian parameter estimation. First, it reduces
dimensionality of the parameter space and replaces expensive direct solutions
of PDEs with the conditional gPC surrogates. Second, the estimated parameter
field exactly matches the parameter measurements. In addition, we show that the
conditional gPC surrogate can be used to estimate the states variance, which,
in turn, can be used to guide data acquisition. We demonstrate that our
approach improves its accuracy with application to one- and two-dimensional
Darcy equation with (unknown) space-dependent hydraulic conductivity. We also
discuss the effect of hydraulic conductivity and head locations on the accuracy
of the hydraulic conductivity estimations.
