[Paper Review] Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies

Eva Koscielny–Bunde, Jan W. Kantelhardt+3 more|ArXiv.org|May 19, 2003

Complex Systems and Time Series Analysis19 references4 citations

TL;DR

This study applies detrended fluctuation analysis (DFA) and wavelet techniques to 41 global river runoff records, revealing long-term persistence (H = 0.55–0.95) and weak multifractality. The multifractal structure is best described by a two-parameter model τ(q) = −ln(a^q + b^q)/ln 2, indicating time-correlated clustering rather than broad marginal distributions.

ABSTRACT

We study temporal correlations and multifractal properties of long river discharge records from 41 hydrological stations around the globe. To detect long-term correlations and multifractal behaviour in the presence of trends, we apply several recently developed methods [detrended fluctuation analysis (DFA), wavelet analysis, and multifractal DFA] that can systematically detect and overcome nonstationarities in the data at all time scales. We find that above some crossover time that usually is several weeks, the daily runoffs are long-term correlated, being characterized by a correlation function C(s) that decays as C(s) ~ s^(gamma). The exponent gamma varies from river to river in a wide range between 0.1 and 0.9. The power-law decay of C(s) corresponds to a power-law increase of the related fluctuation function F_2(s) ~ s^H where H = 1-gamma/2. We also find that in most records, for large times, weak multifractality occurs. The Renyi exponent tau(q) for q between -10 and +10 can be fitted to the remarkably simple form tau(q) = -ln(a^q+b^q) /ln 2, with solely two parameters a and b between 0 and 1 with a+b >= 1. This type of multifractality is obtained from a generalization of the multiplicative cascade model.

Motivation & Objective

• To detect long-term correlations in river runoff records despite the presence of trends.
• To investigate multifractal properties of runoff time series using advanced detrending techniques.
• To determine whether multifractality arises from fat-tailed marginal distributions or temporal correlations.
• To assess whether scaling exponents H and multifractal width Δα are universal or basin-size dependent.
• To provide a physical interpretation of multifractal scaling in hydrological systems.

Proposed method

• Detrended fluctuation analysis (DFA) is used to quantify long-term correlations by analyzing the scaling of fluctuation functions F2(s) with time scale s.
• Wavelet-based methods (WTMM) are applied to cross-validate multifractal spectrum estimation and detect multifractal scaling.
• A generalized multifractal DFA is used to compute the generalized Hurst exponent h(q) and the singularity spectrum f(α).
• The multifractal spectrum τ(q) is fitted to the analytical form τ(q) = −ln(a^q + b^q)/ln 2, with parameters a and b between 0 and 1.
• Shuffled data are generated to test whether multifractality is due to temporal correlations or marginal distribution shape.
• The crossover time scale is identified as the transition point between short-term and long-term scaling behavior.

Experimental results

Research questions

• RQ1 Do long-term correlations persist in global river runoff records after removing trends?
• RQ2 Is the observed multifractality in runoff time series due to non-Gaussian marginal distributions or temporal correlations?
• RQ3 Can the multifractal spectrum of runoff be described by a universal two-parameter model?
• RQ4 Does the fluctuation exponent H vary systematically with basin size or geographic region?
• RQ5 What is the origin of multifractality—temporal clustering or distributional properties?

Key findings

• The fluctuation function F2(s) scales as s^H with H ranging from 0.55 to 0.95 across 41 rivers, indicating strong long-term persistence.
• The multifractal spectrum τ(q) for all records is well described by τ(q) = −ln(a^q + b^q)/ln 2 with two parameters a and b, suggesting a universal multifractal model.
• Shuffling the data reduces h(q) to 0.5 for all q, proving that multifractality arises from time correlations, not fat-tailed marginal distributions.
• The multifractal width Δα decreases slightly with increasing basin area, indicating less multifractality in larger catchments.
• No significant difference is found in H or Δα between southern German rivers and international rivers, suggesting universal scaling behavior.
• The crossover time scale, above which long-term correlations dominate, is typically several weeks, marking the onset of persistent behavior.