An orchestra plays in a dark hall, and the listener hears only a hum: hundreds of instruments blend into a dense noise in which no melody can be discerned. That is roughly what spectra extracted from climate data by methods like the Koopman operator look like: hundreds of components, each with its own frequency, and it is unclear which ones are true musicians and which are the rattling of chairs. The Koopman operator is a mathematical conductor that describes nonlinear dynamics through a linear action on functions of the state, but its orchestra is too large to conduct.
The authors set out to find stable notes in this noise. They took monthly sea surface temperature anomalies from the tropical Pacific from two independent sources: the ERA5 reanalysis and the HadISST reconstruction. Data from 1940 to 2010 served as the training set, and the following fifteen years as the validation set. For each computed eigenpair, they estimated the residual using the ResDMD method, which measures how well that pair satisfies the Koopman equation. They then constructed a pseudospectral residual function along the unit circle. Local minima of this function indicate frequencies that are not knocked off by a slight perturbation, like a tuning fork that keeps ringing even if you clap your hands nearby.
These nineteen stable frequencies form a Koopman framework. Among them is a component with a period of about 3.6 years, whose spatial structure mirrors the canonical El Niño with anomalies in the central and eastern equatorial Pacific. Longer modes around 12.6 years have a broad basin-wide structure with noticeable meridional asymmetry, while shorter modes around 2.1–2.4 years are concentrated closer to the central equator. Reconstructing the Niño-3.4 index from these nineteen modes explains 92% of the variance in the training period. For comparison, the largest-amplitude modes explain only 18% with standard eigenfunctions and 81% with pseudo-eigenfunctions.
The remaining hundreds of eigenpairs are not independent musicians. Many of them coincide with integer powers of the fundamental modes, with the residual growing with the power order. It is as if, in addition to the nineteen main instruments, the orchestra also produced their overtones and combinations: a flute and an oboe together generate a third tone that is not a separate instrument but merely their joint sound. Forecasts based on the Koopman framework retain skill at lead times of 8–18 months, outperforming a linear inverse model and models that use the full spectrum or only the amplitude modes.
Now a melody can be heard in the dark hall. The nineteen notes that make up El Niño are not just a mathematical abstraction but real oscillations that can be tracked and used for prediction. The next time the Pacific Ocean decides to reorganize atmospheric circulation across the globe, we will have the score.
Out of about 800 eigenpairs that k-EDMD produces for each dataset, just 19 fundamental modes suffice to explain 92% of the variability of the main El Niño index. Meanwhile, the largest-amplitude modes, which are typically chosen in reduced models, explain only 18% of the same variability when using standard eigenfunctions.
The idea of a hidden low-dimensional scaffold behind a dense spectrum echoes the plot of Ted Chiang's 'Story of Your Life', where a complex nonlinear signal turns out to be generated by a small set of underlying patterns, and understanding them changes the perception of time and the prediction of the future.