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):