The El Niño–Southern Oscillation (ENSO) remains the primary source of climate variability on the planet. Its phases trigger droughts, floods, malaria outbreaks, and coral bleaching far beyond the tropical Pacific. Satellite data and ground-based observations show that ENSO affects phytoplankton, ocean salinity, and vegetation. However, models diverge from reality: the forecast for the strong 2015–2016 El Niño was overestimated. The discrepancy likely stems from models failing to capture features of ENSO's temporal structure that are visible in observations. This work stands apart: its nearest neighbors investigate El Niño prediction a year and a half ahead, ocean conveyor belts, and gravitational waves from neutron stars, but they do not analyze ENSO index statistics using a multifractal approach.
The authors used two time series: the Southern Oscillation Index (SOI) from January 1951 to June 2017 (798 points) and the Best ENSO Index (BEI) from January 1870 to June 2017 (1770 points). SOI is computed as the normalized difference in atmospheric pressure between Tahiti and Darwin, while BEI combines SOI with sea surface temperature in the Niño 3.4 region. Seasonality was removed using the classical Wiener method. For analysis, they applied detrended fluctuation analysis (DFA), which reveals a power-law relationship between root-mean-square fluctuations and time scale. The multifractal variant (MF-DFA2) allowed estimation of the singularity spectrum and the generalized Hurst exponent. The distribution of extreme increments was tested against the normal law using Kolmogorov–Smirnov, chi-square, and Anderson–Darling tests, and then the empirical exceedance probability was constructed.
The distributions of SOI and BEI increments are not Gaussian: goodness-of-fit tests rejected normality at the 95% significance level. The tails follow power laws with exponents μ = 4.52 ± 0.17 for SOI (amplitudes from 1.3 to 2.6) and μ = 5.75 ± 0.14 for BEI (amplitudes from 0.99 to 2.09). This means extreme fluctuations occur more often than the Gaussian model predicts. DFA revealed a crossover at about 2.5 years for SOI and about 2 years for BEI. Before the crossover, the slope α ≈ 1.12 (SOI) and 1.41 (BEI) indicates persistent 1/f-type behavior; after the crossover, the slope drops to 0.65, suggesting weakened memory. After removing seasonality, a single slope of 0.89 ± 0.02 for SOI and 0.84 ± 0.02 for BEI confirmed long-range correlations. Multifractal analysis showed that the generalized Hurst exponent h(q) depends on the moment q, and the singularity spectrum is broad, indicating multifractal structure in the series.
This work differs from its archive neighbors: they investigate physical mechanisms or forecasts, whereas here multifractal analysis is applied for the first time to long ENSO index records. The discovery of power-law tails and long memory implies that models assuming Gaussian fluctuations and short memory will systematically underestimate the probability of strong events. This explains why the 2015 forecast was too high. The results highlight the need to incorporate nonlinear dynamics and long-range correlations into models. For science, this is a step toward a more realistic description of the coupled ocean–atmosphere system.
The topic could develop by applying similar analysis to other climate indices, such as ozone data, as the authors suggest. It may be possible to build predictive models that account for power-law distributions and multifractality. Comparing with CMIP model outputs and identifying systematic errors is also promising. Another avenue is exploring the link between multifractal parameters and physical mechanisms such as Kelvin and Rossby waves.
The results will influence climate modeling, forecasting of extreme weather events, and risk assessment for agriculture, health, and marine ecosystems.
The next step is to apply the same analysis to climate model data and compare their statistical properties with observations. It is also important to check whether the found patterns hold in shorter satellite records.
This work directly addresses an unresolved problem in climatology: why models underestimate extreme events and fail to reproduce the long-term memory of the climate system. The detection of multifractality points to the need to revise stochastic parameterizations in models.
The power-law tail exponent μ = 4.52 for SOI means that an event with amplitude twice the threshold occurs roughly 23 times less often than an event at the threshold amplitude. That is much slower than the Gaussian distribution predicts.