Potentials and limitations of detrended fluctuation analysis for complex geoscientific time series: The case of global mean sea-level

Detrended fluctuation analysis (DFA), introduced by Peng et al. in 1994, along with its numerous algorithmic variants and multifractal extensions have become standard tools in nonlinear time series analysis and have found a vast body of applications across a wide range of scientific disciplines. However, many successful applications have in common that the time series under study are of sufficient length and exhibit unique scaling characteristics that are not overprinted by the action of any external dynamical factors. However, the latter two conditions can be violated in the context of complex geoscientific time series. In this work, we are interested in how such situations affect the general behavior of the resulting detrended fluctuation functions and attempt to provide a mechanistic explanation of the observed features, expanding on previous works that have largely been based on the thorough analysis of different kinds of stochastic model systems.

As a paradigmatic example for geoscientific data with particularly complex variability properties, we focus on a time series of satellite altimetry based global mean absolute sea-level variations (GMSL), which is available for the period from 1993 to present day at multi-day temporal resolution. This time series exhibits a variety of non-trivial fluctuation properties, including seasonality and a long-term nonlinear trend, but also reflections of nonlinear inter-annual climate variability modes like the El Nino-Southern Oscillation (ENSO) and Pacific Decadal Oscillation (PDO). While the first two components can be largely removed from the time series by standard approaches like phase averaging and detrending/smoothing, the latter two manifest in GMSL in more subtle manners. We show that the resulting fluctuation functions of GMSL obtained with different orders of local detrending indeed do not exhibit simple and unique scaling characteristics across the full range of accessible scales, even when being subject to de-seasoning and de-trending prior to analysis. Instead, we find that depending on the scale considered, the scale-local fluctuation exponents exhibit a marked pattern, with consistent values across different DFA orders exclusively being observed below multi-annual to sub-decadal time scales. We present a simplistic explanation of those findings by studying the fluctuation functions for different types of stochastic signals with superposed oscillatory components with periodicities resembling those of the different variability modes in GMSL. Additionally, we discuss the potentials and limitations of different statistical approaches (including regression on potential external drivers as well as different time-scale decomposition techniques) to remove the effects of complex externally driven oscillatory modes from the original time series to obtain a more realistic picture of the intrinsic stochastic fluctuation properties of GMSL.

This work was partially supported by INESC TEC via the International Visiting Researcher Programme 2024 and by CMUP - Centro de Matemática da Universidade do Porto, member of LASI, which is financed by national funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UID/00144/2025. Doi: https://doi.org/10.54499/UID/00144/2025.