Forecast error variance is not a constant quantity. It depends on the presence or absence of instabilities and/or coherent features such as fronts or cyclones. Ensemble forecasting systems attempt to predict forecast error variance as a function of the flow of the day and the position, accuracy and type of observations used to initialize the forecast. Inevitably, such forecast error variance predictions are inaccurate. Is it possible to empirically measure the accuracy of such predictions? How should such inaccuracy be accounted for in data assimilation? To address these questions, we use an implementation of the Ensemble Transform Kalman Filter (ETKF) on a Lorenz-40 variable model to justify a simple model of error variance prediction accuracy.

This simple model has the properties that (i) the “true” forecast error variance is a random draw from an inverse-gamma distribution, and (ii) the ensemble variance is a random draw from a gamma distribution whose mean is a linear function of the “true” forecast error variance. We then show how the parameters defining this simple model may be directly deduced from a large archive of innovation and ensemble variance pairs. Under the assumption that the “true” forecast error variance is equal to the climatologically averaged error variance of the forecasting system, one can measure the accuracy of ensemble predictions of forecast error variance. Further, we show that the accuracy of the ensemble predictions can be quantified in terms of “effective ensemble size”. We also show that optimal prediction of the “true” forecast error variance given an ensemble variance is a weighted linear combination of the climatological variance and the ensemble variance. Hence, the result provides theoretical support for the use of linear combinations of static and ensemble based covariance models in Hybrid 4D-VAR schemes.