  Bayesian inverse modeling is important for a better understanding of
hydrological processes. However, this approach can be computationally
demanding, as it usually requires a large number of model evaluations. To
address this issue, one can take advantage of surrogate modeling techniques.
Nevertheless, when approximation error of the surrogate model is neglected, the
inversion result will be biased. In this paper, we develop a surrogate-based
Bayesian inversion framework that explicitly quantifies and gradually reduces
the approximation error of the surrogate. Specifically, two strategies are
proposed to quantify the surrogate error. The first strategy works by
quantifying the surrogate prediction uncertainty with a Bayesian method, while
the second strategy uses another surrogate to simulate and correct the
approximation error of the primary surrogate. By adaptively refining the
surrogate over the posterior distribution, we can gradually reduce the
surrogate approximation error to a small level. Demonstrated with three case
studies involving high dimensionality, multimodality, and a real-world
application, it is found that both strategies can reduce the bias introduced by
surrogate approximation error, while the second strategy that integrates two
methods (i.e., polynomial chaos expansion and Gaussian process in this work)
that complement each other shows the best performance.
