  Solving hydrologic inverse problems usually requires repetitive forward
simulations. One approach to mitigate the computational cost is to build a
surrogate model, i.e., an approximate mapping from model parameters (input) to
observable quantities (output), so the forward simulations can be done quickly.
Alternatively, if the surrogate is constructed to approximate the inverse
mapping from model outputs to parameters, the parameter estimates can be
obtained directly by treating measurements as inputs to this inverse surrogate.
Moreover, the uncertainties of parameters can be quantified by propagating the
measurement uncertainties in a straightforward Monte Carlo manner. Based on
this idea, we proposed a novel surrogate-based approach for parameter
estimation and uncertainty assessment, i.e., the Inverse Gaussian Process Monte
Carlo (IGPMC) algorithm. The Gaussian Process (GP) regression is used to
directly approximate the inverse function of the model output-input
relationship. For ill-posed problems, i.e., when there exist non-unique sets of
input parameters that all produce an identical system output, multiple inverse
GP systems are constructed and multiple parameter estimates can be obtained
accordingly. The accuracy and efficiency of this IGPMC algorithm were
demonstrated through four numerical case studies. Results obtained from the
Markov Chain Monte Carlo (MCMC) are used as references to assess our new
proposed method. It was shown that, the IGPMC algorithm can generally obtain
reliable parameter estimates with an affordable computational cost.
