Parameter Uncertainty Quantification in an Idealized GCM With a Seasonal Cycle

Abstract

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). UQ 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 the non‐Gaussian posterior distributions by factors between two and five, depending on the variables used for UQ. The reduction in the spread of the parameter posterior distribution leads to a reduction in the uncertainty of climate model predictions.

Calibration and Uncertainty Quantification of a Gravity Wave Parameterization: A Case Study of the Quasi‐Biennial Oscillation in an Intermediate Complexity Climate Model

“…From this, we can see that that one can learn the optimal $$p(\boldsymbol{\theta }\vert \boldsymbol{y})$$ by optimizing $$p(\boldsymbol{y}\vert \boldsymbol{\theta })$$, that is, maximum likelihood. It is standard to choose a Gaussian likelihood (e.g., Cleary et al., 2021; Dunbar et al., 2021; Howland et al., 2022): $$$p(\boldsymbol{y}\vert \boldsymbol{\theta })=\frac{1}{\sqrt{2{\Gamma}}}\mathrm{exp}\left(-\frac{1}{2}{(\boldsymbol{y}-\mathcal{G}(\boldsymbol{\theta }))}^{T}{{\Gamma}}^{-1}(\boldsymbol{y}-\mathcal{G}(\boldsymbol{\theta }))\right).$$$ Where superscript $$T$$ denotes the transpose. This equation is the probability that the data $$\boldsymbol{y}$$ originates from $$\mathcal{G}(\boldsymbol{\theta })$$, allowing for the Gaussian noise with variance $${\Gamma}$$ as described by Equation .…”

“…To do this, we need samples from the optimal distribution of model parameters that produce model outputs in agreement with an observed dataset. We employ the Calibrate, Emulate and Sample (CES) method (Cleary et al., 2021; Dunbar et al., 2021; Howland et al., 2022). This involves (a) calibration of model parameters so that the model output agrees with the observed dataset, (b) emulation of the expensive model given model parameters to allow for quick evaluations and (c) sampling from the calibrated distribution of model parameters with the emulator.…”

“…We take a probabilistic approach to calibration, where the goal is to learn probability distributions of From this, we can see that that one can learn the optimal 𝐴𝐴 𝐴𝐴(𝜽𝜽|𝒚𝒚) by optimizing 𝐴𝐴 𝐴𝐴(𝒚𝒚|𝜽𝜽) , that is, maximum likelihood. It is standard to choose a Gaussian likelihood (e.g., Cleary et al, 2021;Dunbar et al, 2021;Howland et al, 2022):…”

Ensemble‐Based Experimental Design for Targeting Data Acquisition to Inform Climate Models

“…The year length in the GCM is 360 days. We stack four 90-day seasons of data together (Howland et al, 2022) and define the forward map…”

“…With fixed insolation at the top of the atmosphere, the statistics are also statistically stationary. Prescribing seasonally (but not diurnally) varying insolation generates seasonally varying (cyclostationary) statistics, with symmetry between the northern and southern hemisphere (i.e., winter in the northern hemisphere winter is statistically identical to winter in the southern hemisphere) (Bordoni & Schneider, 2008;Howland et al, 2022). Dunbar et al (2021) and Howland et al (2022) have shown that the parameters 𝐴𝐴 RH and 𝐴𝐴 𝐴𝐴 of the convection parameterization in the GCM can be calibrated in the stationary and cyclostationary regimes.…”