A mechanism for the skew of ensemble forecasts

It is often not appreciated that forecast ensembles are generally skewed. The skew can arise from the state dependence of the chaotic system dynamics responsible for the ensemble spread. Generation of skew by this mechanism can be demonstrated in even the simplest dynamical system with state-dependent noise, and even when the initial and the asymptotic (i.e., the "climatological") forecast distributions are both symmetric. Indeed, forecast distributions of systems with state-dependent noise in the dynamical tendencies must in general be both skewed and heavy tailed, with implications for forecasting extreme anomaly risks. Ensemble forecast systems that misrepresent such state-dependent noise have state-dependent errors in their forecast probability distributions. Because such errors depend on both the initial condition and forecast lead time, they cannot be removed by simple a posteriori bias corrections of the forecast distributions. The ensemble standard deviation is often used as a simple metric of ensemble spread even when the forecast distribution is not Gaussian. In a similar spirit, the ensemble skew S may be used as a simple metric of the difference D between the ensemble-mean and most likely forecast, as well as the risk ratio R of extreme positive and negative deviations from the ensemble-mean forecast. This is motivated by the facts that (1) the probability distributions of many geophysical quantities are approximately stochastically generated skewed (SGS) distributions, for which simple analytical relationships exist between these quantities, and (2) Gaussian distributions are a sub-class of SGS distributions. However, S may serve as a useful metric of R and D even when the distributions are not strictly SGS distributions.