  Markov chain Monte Carlo (MCMC) simulation methods are widely used to assess
parametric uncertainties of hydrologic models conditioned on measurements of
observable state variables. However, when the model is CPU-intensive and
high-dimensional, the computational cost of MCMC simulation will be
prohibitive. In this situation, a CPU-efficient while less accurate
low-fidelity model (e.g., a numerical model with a coarser discretization, or a
data-driven surrogate) is usually adopted. Nowadays, multi-fidelity simulation
methods that can take advantage of both the efficiency of the low-fidelity
model and the accuracy of the high-fidelity model are gaining popularity. In
the MCMC simulation, as the posterior distribution of the unknown model
parameters is the region of interest, it is wise to distribute most of the
computational budget (i.e., the high-fidelity model evaluations) therein. Based
on this idea, in this paper we propose an adaptive multi-fidelity MCMC
algorithm for efficient inverse modeling of hydrologic systems. In this method,
we evaluate the high-fidelity model mainly in the posterior region through
iteratively running MCMC based on a Gaussian process (GP) system that is
adaptively constructed with multi-fidelity simulation. The error of the GP
system is rigorously considered in the MCMC simulation and gradually reduced to
a negligible level in the posterior region. Thus, the proposed method can
obtain an accurate estimate of the posterior distribution with a small number
of the high-fidelity model evaluations. The performance of the proposed method
is demonstrated by three numerical case studies in inverse modeling of
hydrologic systems.
