On the Relationship Between Reliability Diagrams and the “Signal‐To‐Noise Paradox”

The “signal‐to‐noise paradox” for seasonal forecasts of the winter North Atlantic Oscillation (NAO) is often described as an “underconfident” forecast and measured using the ratio‐of‐predictable components (RPCs) metric. However, comparison of RPC with other measures of forecast confidence, such as spread‐error ratios, can give conflicting impressions, challenging this informal description. We show, using a linear statistical model, that the “paradox” is equivalent to a situation where the reliability diagram of any percentile forecast has a slope exceeding 1. The relationship with spread‐error ratios is shown to be far less direct. We furthermore compute reliability diagrams of winter NAO forecasts using seasonal hindcasts from the European Centre for Medium‐range Weather Forecasts and the UK Meteorological Office. While these broadly exhibit slopes exceeding 1, there is evidence of asymmetry between upper and lower terciles, indicating a potential violation of linearity/Gaussianity. The limitations and benefits of reliability diagrams as a diagnostic tool are discussed. Plain Language Summary The North Atlantic Oscillation (NAO) is an atmospheric phenomenon which can be understood as summarizing large‐scale winter conditions across western Europe. Long‐range forecasts of the NAO have been shown to be skillful, but also to suffer from a so‐called “signal‐to‐noise paradox,” which roughly says that the real world appears to be more predictable than the forecasts think it is. However, interpreting the exact meaning of this “paradox” has proved challenging. We help bring some clarity by showing that one can interpret the “paradox” as a case of a probabilistically underconfident forecast, namely a forecast which tends to underestimate the likelihood of high magnitude NAO events. Key Points The exact relationship between the “signal‐to‐noise paradox” and standard measures of forecast reliability has remained unclear It is shown that the “paradox” is equivalent to the slope of reliability diagrams exceeding 1, assuming linearity/Gaussianity The value of reliability diagrams as a diagnostic tool for understanding the “paradox” better is discussed