A Residual Ordering of SST Koopman Spectra for the Identification of Fundamental Modes

El Niño-Southern Oscillation (ENSO) is a prominent driver of global climate variability, with significant impacts on ecosystems and societies. While existing empirical-dynamical forecasting methods, such as Linear Inverse Models (LIMs), are limited in capturing ENSO's inherent nonlinearity, Koopman operator theory offers a framework for analyzing such complex dynamics. Recent advancements in Koopman-based methods, such as DMD-based methods, have enabled exploration of nonlinear ENSO-related modes. However, they often suffer from challenges in robustness and interpretability. Specifically, k-EDMD algorithms tend to produce a large number of modes, complicating their physical relevance and reliability. In this study, we address these limitations by employing Colbrook’s Residual EDMD (Res-EDMD) framework as a tool to classify and prioritize modes based on their residuals. This approach enables us to systematically identify robust and physically meaningful modes, distinguishing them from less reliable counterparts. Furthermore, leveraging the property that eigenfunctions of Koopman operators can generate higher-order harmonics through powers and multiplications, we introduce a methodology to detect fundamental modes and their associated harmonics. Applying this framework to tropical Pacific SST data, we demonstrate that k-EDMD, together with Res-EDMD, are capable of isolating fundamental modes of tropical SST dynamics. These fundamental modes provide insights into the system's physical evolution and facilitate the retrieval of meaningful dynamical information. By systematically identifying and interpreting the modes, we establish a pathway to overcome the limitations of conventional Koopman-based methods, thereby enhancing their applicability for studying and forecasting complex climatic systems like ENSO. This study underscores the potential of Res-EDMD to refine mode selection in Koopman spectral analysis, paving the way for robust, physically interpretable insights into tropical SST variability.