  Climate models are generally calibrated manually by comparing selected
climate statistics, such as the global top-of-atmosphere energy balance, to
observations. The manual tuning only targets a limited subset of observational
data and parameters. Bayesian calibration can estimate climate model parameters
and their uncertainty using a larger fraction of the available data and
automatically exploring the parameter space more broadly. In Bayesian learning,
it is natural to exploit the seasonal cycle, which has large amplitude,
compared with anthropogenic climate change, in many climate statistics. In this
study, we develop methods for the calibration and uncertainty quantification
(UQ) of model parameters exploiting the seasonal cycle, and we demonstrate a
proof-of-concept with an idealized general circulation model (GCM). Uncertainty
quantification is performed using the calibrate-emulate-sample approach, which
combines stochastic optimization and machine learning emulation to speed up
Bayesian learning. The methods are demonstrated in a perfect-model setting
through the calibration and UQ of a convective parameterization in an idealized
GCM with a seasonal cycle. Calibration and UQ based on seasonally averaged
climate statistics, compared to annually averaged, reduces the calibration
error by up to an order of magnitude and narrows the spread of posterior
distributions by factors between two and five, depending on the variables used
for UQ. The reduction in the size of the parameter posterior distributions
leads to a reduction in the uncertainty of climate model predictions.
