  Parameters in climate models are usually calibrated manually, exploiting only
small subsets of the available data. This precludes an optimal calibration and
quantification of uncertainties. Traditional Bayesian calibration methods that
allow uncertainty quantification are too expensive for climate models; they are
also not robust in the presence of internal climate variability. For example,
Markov chain Monte Carlo (MCMC) methods typically require $O(10^5)$ model runs,
rendering them infeasible for climate models. Here we demonstrate an approach
to model calibration and uncertainty quantification that requires only
$O(10^2)$ model runs and can accommodate internal climate variability. The
approach consists of three stages: (i) a calibration stage uses variants of
ensemble Kalman inversion to calibrate a model by minimizing mismatches between
model and data statistics; (ii) an emulation stage emulates the
parameter-to-data map with Gaussian processes (GP), using the model runs in the
calibration stage for training; (iii) a sampling stage approximates the
Bayesian posterior distributions by using the GP emulator and then samples
using MCMC. We demonstrate the feasibility and computational efficiency of this
calibrate-emulate-sample (CES) approach in a perfect-model setting. Using an
idealized general circulation model, we estimate parameters in a simple
convection scheme from data surrogates generated with the model. The CES
approach generates probability distributions of the parameters that are good
approximations of the Bayesian posteriors, at a fraction of the computational
cost usually required to obtain them. Sampling from this approximate posterior
allows the generation of climate predictions with quantified parametric
uncertainties.
