  This letter raises the possibility that ergodicity concerns might have some
bearing on the signal-to-noise paradox. This is explored by applying the
ergodic theorem to the theory behind ensemble weather forecasting and the
ensemble mean. Using the ensemble mean as our best forecast of observations
amounts to interpreting it as the most likely phase-space trajectory, which
relies on the ergodic theorem. This can fail for ensemble forecasting systems
if members are not perfectly exchangeable with each other, the averaging window
is too short and/or there are too few members. We argue these failures can
occur in cases such as the winter North Atlantic Oscillation (NAO) forecasts
due to intransitivity or regime behaviour for regions such as the North
Atlantic and Arctic. This behaviour, where different ensemble members may
become stuck in different relatively persistent flow states (intransitivity) or
multi-modality (regime behaviour), can in certain situations break the ergodic
theorem. The problem of non-ergodic systems and models in the case of weather
forecasting is discussed, as are potential mitigation methods and metrics for
ergodicity in ensemble systems.
