  We propose a methodology for improving the accuracy of surrogate models of
the observable response of physical systems as a function of the systems'
spatially heterogeneous parameter fields with applications to uncertainty
quantification and parameter estimation in high-dimensional problems.
Practitioners often formulate finite-dimensional representations of spatially
heterogeneous parameter fields using truncated unconditional Karhunen-Lo\'{e}ve
expansions (KLEs) for a certain choice of unconditional covariance kernel and
construct surrogate models of the observable response with respect to the
random variables in the KLE. When direct measurements of the parameter fields
are available, we propose improving the accuracy of these surrogate models by
representing the parameter fields via conditional Karhunen-Lo\'{e}ve expansions
(CKLEs). CKLEs are constructed by conditioning the covariance kernel of the
unconditional expansion on the direct measurements via Gaussian process
regression and then truncating the corresponding KLE. We apply the proposed
methodology to constructing surrogate models via the Basis Adaptation (BA)
method of the stationary hydraulic head response, measured at spatially
discrete observation locations, of a groundwater flow model of the Hanford
Site, as a function of the 1,000-dimensional representation of the model's
log-transmissivity field. We find that BA surrogate models of the hydraulic
head based on CKLEs are more accurate than BA surrogate models based on
unconditional expansions for forward uncertainty quantification tasks.
Furthermore, we find that inverse estimates of the hydraulic transmissivity
field computed using CKLE-based BA surrogate models are more accurate than
those computed using unconditional BA surrogate models.
