Multifractal Detrended Cross-Correlation Analysis of Sunspot Numbers and River Flow Fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, $(12-24)\leq s\leq 130$ months, is $\lambda = 1.17\pm0.04$ which is almost similar for all underlying rivers at $1\sigma$confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are $\lambda\in (0.76, 0.85)$ and $\gamma_{\times}\in(0.30, 0.48)$, respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), $ \lambda\approx(h_{\rm sun} + h_{\rm river})/2$ is confirmed.

💡 Research Summary

The paper investigates whether solar activity, as quantified by the monthly sunspot number, exerts a measurable influence on river discharge variability. To this end, the authors apply Detrended Cross‑Correlation Analysis (DCCA) and its multifractal extension (MF‑DXA) to four long‑term river flow records: the Daugava River in Latvia and the Holston, Nolichucky, and French Broad Rivers in the eastern United States. The sunspot series and the river flow series each span more than a century, providing ample data for robust statistical analysis.

Because both data sets contain strong periodic components (seasonal cycles in river flow and the well‑known ~11‑year solar cycle), the authors first remove these trends using a Singular Value Decomposition (SVD) based filtering technique. By reconstructing the series after discarding the dominant singular values associated with sinusoidal patterns, they obtain “cleaned” time series that retain the intrinsic stochastic fluctuations while suppressing deterministic cycles.

Applying DCCA to the filtered series yields the cross‑fluctuation function F×(s) as a function of the time scale s. In a log‑log plot, a clear power‑law regime emerges for intermediate scales, specifically 12–24 months ≤ s ≤ 130 months. Within this window the scaling exponent λ is remarkably consistent across all four rivers, with a mean value of λ = 1.17 ± 0.04 (1σ confidence). The authors interpret this as evidence of a “second universal behavior” of river runoff: despite geographical and climatic differences, the cross‑correlation with solar activity follows the same scaling law at multi‑annual scales.

The multifractal analysis (MF‑DXA) further shows that the q‑order fluctuation exponents h(q) are nearly flat, indicating only weak multifractality in both the sunspot and river flow series. This suggests that the dominant dynamics are governed by a single scaling exponent, while local heterogeneities (topography, land use, dam regulation, etc.) introduce only minor deviations. Moreover, the empirical relation λ ≈ (h_sun + h_river)/2, previously reported for DCCA and DFA, is confirmed, reinforcing the internal consistency of the methodology.

From λ the authors derive the cross‑correlation exponent γ× = 2 − 2λ, obtaining values in the range 0.30–0.48. These numbers imply a persistent, positive long‑range cross‑correlation between solar activity and river discharge. Nevertheless, subtle differences in λ and γ× among the four rivers are observed. The authors attribute these variations to basin‑specific factors such as drainage network morphology, catchment size, human water management, and regional climate regimes. For instance, the French Broad River, with a relatively flat basin and several reservoirs, shows a slightly lower λ than the more mountainous Nolichucky River, which exhibits a higher γ×.

In summary, the study demonstrates that (i) after removing dominant sinusoidal trends, a statistically significant long‑range cross‑correlation exists between sunspot numbers and river flow fluctuations; (ii) this cross‑correlation follows a universal scaling law at intermediate time scales for geographically disparate rivers; (iii) the underlying fluctuations possess only weak multifractal characteristics; and (iv) basin‑specific physical and anthropogenic factors modulate the exact magnitude of the scaling exponents. The combination of SVD detrending, DCCA, and MF‑DXA provides a powerful framework for uncovering hidden couplings in complex climate‑hydrological systems and may be extended to other climate proxies (e.g., precipitation, temperature) and additional river basins to test the robustness of the observed universal behavior.