Probabilistic versus deterministic potential seasonal climate predictability under the perfect-model framework

Understanding the relationship between probabilistic and deterministic predictabilities is important for climate predictability studies. Focusing on the actual skill of dynamical seasonal prediction, we previously found that the probabilistic skills of resolution and relative operating characteristic (ROC)/discrimination, but not reliability, have functional relationships with deterministic anomaly correlation (AC). Herein, we further investigate the relationship between probabilistic and deterministic seasonal potential predictabilities. The potential predictabilities are characterized by the potential skills of the AC, resolution, and ROC evaluated using the perfect-model framework, under which reliability is ideal and not considered. A theoretical argument demonstrates that similar theoretical relationships to those for actual skills exist between probabilistic and deterministic potential predictabilities, regardless of how different the potential predictabilities are from the corresponding actual skills. These theoretical relationships are strictly monotonic and characterized by symmetrical probabilistic predictabilities for the below- and above-normal categories, and lower predictability for the near-normal category corresponding to deterministic predictability. A subsequent diagnostic analysis reveals that while the probabilistic and deterministic potential predictabilities in current dynamical climate models differ noticeably from the corresponding actual skills, they exhibit quasi-monotonic relationships as expected theoretically, which effectively and quantitatively validates the theoretical argument. This work, combined with our previous findings, establishes a solid equivalence of the resolution and discrimination aspects of probabilistic predictability to deterministic predictability in seasonal prediction, which can have beneficial implications for further studying probabilistic predictability.