Hidden Drought Memory: Fractal Analysis Reveals Brazil’s Cerrado Is Far More Unstable Than Rainfall Trends Suggest

On the surface, the climate of central Brazil appears remarkably stable. Decades of rainfall records across the country’s Midwest region show no statistically significant long-term trends, no dramatic directional shift that conventional statistics can flag as evidence of change. Yet a new study published in Theoretical and Applied Climatology argues that this apparent calm is deceptive. By applying the mathematics of fractals to more than six decades of hydroclimatic data, researchers have uncovered deep, persistent structure in the region’s dry-wet fluctuations — structure that standard drought indices and trend tests simply cannot see. The findings suggest that the Cerrado and Pantanal biomes, the agricultural and ecological heartland of South America, may be experiencing a quiet intensification of hydroclimatic variability that has been hiding in plain sight.

The study, conducted by Amaury de Souza of the Federal University of Mato Grosso do Sul and José Francisco de Oliveira Junior of Fluminense Federal University and the Federal University of Alagoas, focused on the capital cities of Midwest Brazil: Brasília, Goiânia, Cuiabá, and Campo Grande. The team drew on the TerraClimate dataset, a high-resolution global climate product with roughly four-kilometer spatial resolution, to extract monthly precipitation and actual evapotranspiration from 1958 to 2024. Crucially, the dataset contained no missing values, eliminating the need for gap-filling procedures that can introduce artificial structure into a time series. From these two variables, the researchers constructed what they call the Hydroclimatic Balance Index, or HBI — the difference between precipitation and actual evapotranspiration, accumulated over twelve months and standardized as a z-score.

The choice of actual evapotranspiration rather than potential evapotranspiration is what sets the HBI apart from its better-known cousin, the Standardized Precipitation Evapotranspiration Index, or SPEI. Potential evapotranspiration measures how much water the atmosphere would demand if supply were unlimited; actual evapotranspiration measures how much water is genuinely exchanged between the land surface and the atmosphere, constrained by soil moisture, vegetation dynamics, and land-surface feedbacks. In tropical continental environments where evapotranspiration is a major component of the hydrological cycle, that distinction matters. The authors stress that the HBI is not intended to replace established drought indices, but to complement them by capturing realized hydroclimatic conditions rather than atmospheric demand alone. As a benchmark, the team also computed the classic Standardized Precipitation Index at the twelve-month scale, SPI-12, which relies exclusively on rainfall.

Before turning to fractal analysis, the researchers established a rigorous statistical baseline. Linear regression of annual precipitation, evapotranspiration, and HBI series revealed only weak slopes with p-values all exceeding the five percent significance level and coefficients of determination that were vanishingly small. The non-parametric Mann-Kendall test, which operates on data ranks and is robust to the outliers that plague hydroclimatic records, told the same story: Kendall’s tau coefficients hovered near zero, confirming the absence of significant monotonic trends. The Pettitt change-point test, designed to detect abrupt structural shifts without assumptions of normality, found significant breakpoints only in the evapotranspiration series of Goiânia in 2001 and Cuiabá in 2009. By every conventional measure, the region’s climate looked stable — which is precisely what made the next set of results so striking.

That next step was Multifractal Detrended Fluctuation Analysis, or MFDFA, a technique developed by Kantelhardt and colleagues in 2002 that extends earlier methods for detecting long-range correlations in nonstationary signals. The procedure integrates the original series, divides it into segments of varying lengths, removes local polynomial trends from each segment, and then examines how fluctuation magnitudes scale with segment size. For a monofractal process, the generalized Hurst exponent h(q) remains constant regardless of the moment order q. For a multifractal process, h(q) varies systematically with q, revealing that small and large fluctuations obey different scaling rules — a hallmark of complex, intermittently organized systems. Rainfall, temperature, wind, and river flow have all been shown to behave this way, but multifractal assessments of drought indices in tropical continental settings have remained scarce.

The results were unambiguous. Across all four capitals, the generalized Hurst exponent declined systematically with increasing moment order, confirming genuine multifractal scaling in both SPI-12 and HBI-12 series. The Hurst exponent at q equals two ranged from approximately 0.87 to 1.11, values that indicate strong long-range dependence — a form of hydroclimatic memory in which wet and dry conditions cluster together rather than arriving randomly. Persistent systems of this kind are prone to drought clustering and prolonged anomalies, a property with direct implications for agriculture and hydropower in a region that anchors Brazil’s agricultural production and generates a substantial share of its electricity. The width of the singularity spectrum, denoted delta-alpha, served as the study’s measure of multifractal strength: values below 0.5 indicate weak multifractality, values between 0.5 and 1.0 moderate, and values above 1.0 strong.

The spatial patterns proved revealing. Brasília exhibited the weakest multifractality, with SPI-12 producing a narrow spectrum of just 0.194, though even there the HBI-12 spectrum widened to 0.517 when evapotranspiration entered the calculation. Goiânia showed strong multifractality for both indices, at 1.333 and 1.382 respectively, suggesting that rainfall variability alone already accounts for much of its complexity. The most dramatic contrasts emerged in Cuiabá and Campo Grande. Cuiabá jumped from weak multifractality under SPI-12 at 0.254 to strong multifractality under HBI-12 at 1.008, while Campo Grande surged from a moderate 0.755 to 1.488 — the strongest multifractal signature of any capital. In these locations, evapotranspiration processes, vegetation dynamics, soil moisture availability, and land-surface feedbacks amplify nonlinear fluctuations far beyond what precipitation alone can explain.

Perhaps the most consequential finding came from splitting the record into two sub-periods, 1958 to 1990 and 1991 to 2024. The recent period showed broader singularity spectra in several capitals, with Campo Grande reaching a delta-alpha of 1.938 and Cuiabá 1.341, indicating intensified intermittency and hydroclimatic heterogeneity in recent decades. Brasília, by contrast, showed a modest reduction, underscoring the spatially uneven character of the change. Spectral asymmetry values predominantly exceeded one, meaning that large-magnitude fluctuations — the extreme wet and dry events — contribute disproportionately to the multifractal organization. Yet throughout, generalized Hurst exponents remained above unity, confirming persistent long-range dependence across all cities and both periods. The intensification of variability, in other words, is happening within a persistently memory-laden system, a combination that favors clustering of extremes.

The contrast between the classical and fractal diagnoses carries a pointed message for drought monitoring. Linear regression, Mann-Kendall tests, and Pettitt analysis detected almost nothing, yet the multifractal framework revealed substantial structural variability, growing intermittency, and shifts in scaling behavior that no trend statistic could capture. The authors argue that hydroclimatic change in Central Brazil should not be judged solely by the presence or absence of monotonic trends; increasing persistence, enhanced intermittency, and greater scaling heterogeneity may be equally important indicators of environmental change. This resonates with a broader body of climate-complexity research showing that scaling diagnostics can expose structural variability invisible to conventional methods, particularly in regions governed by monsoon dynamics and land-atmosphere interactions.

The study’s implications extend beyond Brazil. The measured spectrum widths, ranging from roughly 0.19 to 1.94, are comparable to or exceed those reported for Mediterranean rainfall systems, Indian monsoon basins, African semi-arid regions, and East Asian monsoon domains — but the gap between precipitation-only and water-balance multifractality appears unusually large in Central Brazil, underscoring the outsized role of evapotranspiration in tropical continental hydroclimates. The authors acknowledge limitations: the analysis rests entirely on gridded TerraClimate data, which may smooth local extremes; only the twelve-month accumulation scale was evaluated; and causal attribution to land-use change, though plausible given the Cerrado’s rapid agricultural transformation, would require integration with remote sensing and land-cover datasets. Still, the core conclusion stands. Drought monitoring frameworks built on precipitation alone may systematically underestimate hydroclimatic complexity in monsoon-influenced tropical regions, and as warming and land-use change continue, complexity-based diagnostics may become essential tools for anticipating when quiet variability turns into visible crisis.

Subject of Research: Multifractal assessment of hydroclimatic variability and drought indices in Midwest Brazil

Article Title: Multifractal assessment of hydroclimatic variability in Midwest Brazil using the hydroclimatic balance index (HBI): a comparison with the standardized precipitation index (SPI)

Article References: de Souza, A., & de Oliveira Junior, J. F. (2026). Multifractal assessment of hydroclimatic variability in Midwest Brazil using the hydroclimatic balance index (HBI): a comparison with the standardized precipitation index (SPI). Theoretical and Applied Climatology, 157(10), Article 682. https://doi.org/10.1007/s00704-026-06467-6

Image Credits: AI Generated

DOI: 10.1007/s00704-026-06467-6

Keywords: multifractal analysis, MFDFA, drought, hydroclimatic variability, Brazil, Cerrado, evapotranspiration, Standardized Precipitation Index, Hurst exponent, TerraClimate, South American Monsoon, water balance