El Niño and the Southern Oscillation (ENSO) remain one of the main climate puzzles: they influence droughts, floods, and harvests worldwide, but forecasts beyond half a year often lose accuracy. Alongside this work in the archive are studies that seek hidden patterns in complex systems, from predicting El Niño one and a half years ahead to training neural networks on the Ising model. The common motif among neighbors is an attempt to extract more from a time series than linear methods allow. This paper stands in the same field but uses not statistical correlations but the geometry of the attractor, that is, the shape taken by the system's trajectory in state space.
The original SOI series (monthly values from 1866 to 2013) was decomposed into nine components using a discrete wavelet transform with the Morlet mother function. Eight decomposition levels allowed separating intraseasonal, interannual, and decadal oscillations, while the ninth component described the trend. The resulting components formed a nine-dimensional state space E9. For a chosen event (e.g., El Niño 1997–1998), its starting point in this space was taken, and all trajectory points closest to it by Euclidean distance were sought. From each such point, a signal segment of the same length as the event was extracted, shifted to the event's start, and an amplitude was adjusted. Among all candidates, the one with the same number of maxima and minima as the original event and the smallest quadratic deviation was selected. The forecast was averaged over six consecutive months, giving an effective horizon one year shorter than the original.
For the 1997–1998 event, orbits reproducing its shape were found in the past: these are trajectories of the 1911, 1972–1973, and 1982–1983 events. The relative forecast error for two years was below the threshold of 0.5 (that is, less than half the standard deviation of the entire signal) and even when the horizon was extended to four years, it did not exceed 0.37. For the 1982–1983 event, no suitable orbit was found in the past, and the error remained above the threshold. On average across all 20th-century events, the relative error grew with the horizon: 0.177 for two years, 0.298 for three, and 0.389 for four. The histogram of distances to the nearest orbits has a Gaussian shape, meaning the system most often finds a similar trajectory near the event, but there is no deterministic rule for choosing an orbit.
The work shows that the SOI index itself contains nonlinear information sufficient for forecasts years ahead, and that this information is encoded in the topology of the attractor. Unlike neighboring studies that rely on statistical models or machine learning, here the forecast is built directly from the geometry of phase space, without parameter fitting. This is the first test that wavelet embedding of SOI can bypass the spring predictability barrier for the extreme 1997–1998 event, which was usually poorly predicted.
The method can be extended to other climate indices and longer series to test whether predictability persists beyond the 20th century. An open question remains why no suitable orbit was found for the 1982–1983 event: possibly it is a rare state of the attractor that did not recur in the available record. Further study of the attractor's topology and orbit selection rules may lead to more reliable forecasts of extreme events.
The results could influence long-term climate forecasting practice in meteorology and risk assessment for agriculture, energy, and insurance.
It is worth testing the method on independent data, for example on sea surface temperature indices in different regions of the Pacific Ocean, and comparing with forecasts from dynamical models.
For climatology, the key unresolved problem is the predictability of extreme events beyond the seasonal horizon. This work directly addresses that problem by showing that the nonlinear dynamics of the attractor can provide forecasts years ahead, though not equally for all events.
The 1997–1998 El Niño was so strong that it is called the 'climate event of the century': it caused droughts in Indonesia and Australia, floods in Peru and California, and is estimated to have led to the deaths of more than 20,000 people.
In Kim Stanley Robinson's novel 'Forty Signs of Rain', climatologists try to predict abrupt changes in ocean currents, and in the film 'The Day After Tomorrow', a sudden climate shift begins with anomalies in the Pacific Ocean. The theme of long-term forecasting of such events remains fantastically difficult.