  Parameters in climate models are usually calibrated manually, exploiting only
small subsets of the available data. This precludes both 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 and are sensitive to internal variability noise, 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 sampling the GP emulator with 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 synthetic data
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.
