where the weighting function L accounts for the decreasing grid-cell area towards the poles and is defined as\(L\left(i\right)=cos\left(\text{lat}\left(i\right)\right)\).
This commonly used metric provides a single number summarising the mismatch between the predictions (\(x\)) and the target variables (\(x^{t}\)). By squaring the difference, the RMSE also weighs large discrepancies more heavily, more heavily penalising larger errors. We average the target variables over the three available ensemble members (\(n\)) and the RMSE over a long period of the target scenario (2050 – 2100) in order to minimise the contribution of internal variability.
Estimates of this internal variability can be very valuable for climate projections however and since ClimateBench includes three ensemble members for each training dataset emulators are encouraged to include estimates of it if they are able. A natural extension of the RMSE for probabilistic estimates commonly used in weather forecasting is the Continuous Ranked Probability Score (CRPS):