Why does the signal-to-noise paradox exist in seasonal climate predictability?

Abstract. Estimates of the potential predictability limit (PPL) for seasonal climate, typically based on a perfect model framework, sometimes encounter challenges of being paradoxical, as actual skill surpasses the PPL. The signal-to-noise paradox (SNP) gets its name from the use of model-based signal-to-noise ratios to estimate the PPL. Here, we study seasonal climate predictability in the tropical and subtropical regions during the boreal summer (June to September), with a focus on the SNP. We estimate PPL within the perfect model framework, only considering error growth from initial conditions. Signal and noise components display temporal non-orthogonality and a weak association between estimates of PPL and actual prediction skill, contradicting its intended purpose. Moreover, paradoxical regions do not align with significant correlations between signal and noise, indicating that the accurate separation of seasonal forecasts into signal and noise components alone is not sufficient to avoid paradoxes. We have also demonstrated that sub-seasonal components, which are building blocks of seasonal mean, substantially contribute to seasonal anomalies in association with major global predictors. The co-variability between sub-seasonal components and seasonal anomalies is wide-ranging and often skewed compared to observations, thereby influencing seasonal prediction skills and PPL. Therefore, a robust PPL estimation should consider errors from initial conditions and model-related factors such as physics, dynamics, and numerical methods. In this context, we propose a novel method to estimate the PPL of seasonal climate, which can be free from paradoxical situations.