A statistical perspective on the signal-to-noise paradox

An anomalous signal-to-noise ratio (also called the signal-to-noise paradox) present in climate models has been widely reported, affecting predictions and projections from seasonal to centennial timescales and encompassing prediction skill from internal processes and external climate forcing. An anomalous signal-to-noise ratio describes a situation where the mean of a forecast ensemble correlates better with the corresponding verification than with its individual ensemble members. This situation has severe implications for climate science, meaning that large ensembles might be required to extract prediction signals. Although a number of possible physical mechanisms for this paradox have been proposed, none has been universally accepted. From a statistical point of view, an anomalous signal-to-noise ratio indicates that forecast ensemble members are not statistically interchangeable with the verification, and an apparent paradox arises only if such an interchangeability is assumed. It will be demonstrated in this study that an anomalous signal-to-noise ratio is a consequence of the relative magnitudes of the variance of the observations, the ensemble mean, and the error of the ensemble mean. By analysing the geometric triangle formed by these three quantities, and given that for typical seasonal forecasting systems both the correlation and the forecast signal are relatively small, it is concluded that an anomalous signal-to-noise ratio should, in fact, be expected in such circumstances.