Correlogram, predictability error growth, and bounds of mean square error of solar irradiance forecasts

Solar forecast verification must incorporate a notion of “relativity.” Particularly for deterministic forecasts, one wishes to know the upper and lower bounds of the mean square error (MSE) for the forecasting situation of concern, such that the performance of some forecasts of interest can be quantified relative to these bounds. This work proposes a method for estimating these bounds. For the upper bound of MSE, the theory builds upon the premise that no forecast worse than that provided by the standard of reference should be used. Under some mild assumptions, the MSE of the standard of reference, which is herein defined to be the optimal convex combination of climatology and persistence, varies solely with the lag-h autocorrelation of the clear-sky index. Then, by fitting a theoretical isotropic correlogram with a nugget effect to the empirical autocorrelations, the upper bound of MSE, as a function of forecast horizon, can be derived. For the lower bound of MSE, it can be approximated by the predictability error growth, which is the difference between the controlled and perturbed forecasts from a perfect dynamical weather model. The empirical part of this work considers the irradiance forecasts from the European Centre for Medium-Range Weather Forecasts, at seven locations in United States, over a period of two years (2019–2020). Various MSEs reported in the literature are put in perspective of the estimated bounds of MSE. The proposed bounds have a profound impact on predictability quantification and skill score computation, which are essential for comparative forecast verification.