Physics Tool Reveals Brazilian Rivers Split Between European-Level Clean and Global Worst

A team of Brazilian researchers has borrowed a piece of mathematics from physics and economics — the Wasserstein distance, better known as the earth mover’s distance from optimal transport theory — and used it to answer a question that has long resisted conventional statistics: how far, really, are the rivers of São Paulo from the rivers of Europe? The answer, published in Environmental Monitoring and Assessment, is startlingly uneven. Some São Paulo basins sit comfortably inside the European quality range, ranking above Germany and Belgium. Others, draining the metropolitan sprawl of a city of 22 million people, are so distributionally distant that no European country in the reference set comes close.

The study, conducted by Hugo Pimentel Tavares and Nilo Antônio de Souza Sampaio of the State University of Rio de Janeiro, is billed as the first application of Wasserstein distance to cross-continental river water quality benchmarking. The metric, formalised in its modern form by mathematician Cédric Villani, measures the minimum cost of transforming one probability distribution into another — literally, how much ‘earth’ must be moved to reshape one histogram into a second. Unlike a median or a compliance rate, it captures differences in location, spread, skewness and tail behaviour simultaneously. Two monitoring networks with identical medians can differ profoundly in their distributional architecture, and only a distributional metric can expose that divergence.

The evidence base is enormous. The researchers harmonised 646,356 physicochemical records from 141 stations operated by CETESB, the São Paulo State environmental agency, with 998,410 records from the Global River Water Quality Archive version 1.4, which aggregates monitoring from 23 European countries under the Water Framework Directive. Seven parameters common to both networks — pH, dissolved oxygen, biochemical oxygen demand, total phosphorus, turbidity, temperature and nitrate — were compared over a shared analytical window of 2016 to 2022. For each of ten São Paulo basins, the Wasserstein-1 distance between its empirical distribution and the pooled European distribution was computed for every parameter, then normalised by the 95th-to-5th percentile range of the combined data to allow cross-parameter comparison.

The results reveal a bifurcation that aggregate reporting has systematically hidden. The Afluentes do Paraíba do Sul basin records a composite normalised distance of just 0.245, and the Cantareira system 0.333 — both within the distributional range of lower-ranking European countries. At the opposite extreme, the Alto Tietê, which receives the cumulative discharge of the São Paulo Metropolitan Region, scores 2.060, a full 8.4 times the cleanest basin, with a dissolved oxygen distance of 7.661 milligrams per litre reflecting a substantial mass of observations near anoxia. The Médio Tietê follows at 1.392. Mann–Whitney U tests confirm statistically significant distributional differences for six of the seven parameters, with large Cliff’s delta effect sizes for temperature, dissolved oxygen and biochemical oxygen demand.

To convert these distances into rankings, the team built a decision matrix covering twenty entities — all ten Brazilian basins and the ten European countries with complete five-parameter median coverage — and applied TOPSIS, a multi-criteria method that scores each entity by its closeness to an ideal solution. Crucially, the criteria weights were not chosen by the analysts but derived objectively through Shannon entropy weighting, which assigns influence in proportion to each parameter’s cross-entity variability. Total phosphorus, spanning more than a thirty-fold range from Austria’s 0.030 milligrams per litre to the Tietê-Médio’s 1.000, dominated with a weight of 0.472. The resulting ranking places Afluentes do Paraíba do Sul seventh of twenty, ahead of Portugal, Poland, Germany and Belgium, while the Alto and Médio Tietê occupy the last two positions, below every complete-coverage European country.

Perhaps the most actionable component is a linear programming convergence model that translates distributional gaps into timelines. The model’s key innovation is its feasibility constraint: rather than assuming engineering targets, it anchors the maximum achievable annual improvement for each parameter to the 90th percentile of historically observed year-on-year improvements in the CETESB record itself, which extends back to 1978. Solved for each basin, the programme identifies which parameter is the binding constraint — the one whose gap-to-feasibility ratio is largest — and computes a critical-path convergence time. Dissolved oxygen emerges as the bottleneck for seven of the ten basins, with convergence times ranging from 4.4 years for the Afluentes do Paraíba do Sul to 14.8 years for the Alto Tietê under the optimistic feasibility assumption, stretching to 5.5 and 28.7 years respectively under a more conservative 75th-percentile assumption.

The contrast between absolute gap size and binding constraint is instructive. The Alto Tietê’s biochemical oxygen demand gap is the largest in the entire matrix at 11.4 milligrams per litre, yet it resolves in just 4.6 years at the historically sustained improvement rate of 2.5 milligrams per litre per year. Dissolved oxygen, with a smaller absolute gap but a far slower feasible recovery rate of 0.59 milligrams per litre per year, dictates nearly fifteen years of sustained effort. In other words, even if every other parameter reached European levels tomorrow, oxygen recovery alone would remain a generational project for the metropolitan corridor — a finding the authors argue supports differentiated governance strategies rather than basin-uniform targets.

One result stands out for its counterintuitive character: nitrate is the only parameter showing no significant difference between Brazilian and European distributions, with a p-value of 0.354 and a negligible effect size. The authors interpret this parity cautiously, as a plausible transient coincidence of trajectories rather than evidence of equivalent governance. Europe’s nitrate levels reflect decades of agricultural intensification followed by regulatory pressure from the Nitrates Directive; Brazil has no equivalent instrument, and the basins with the highest nitrate medians — Piracicaba, Capivari and Jundiaí, at 1.2 to 2.0 milligrams per litre — drain the state’s most intensive sugarcane, citrus and soybean country. Nitrate in the Jundiaí basin has risen more than 800 percent since the early 1990s. The aggregate parity conceals a steep internal gradient, from near-pristine headwaters at 0.200 milligrams per litre to sub-basins already exceeding European concentrations, signalling an emerging eutrophication risk.

The framework’s typological analysis, using principal component analysis and Ward hierarchical clustering, positions the twenty entities along a pollution axis and a nitrate axis that together explain 77.5 percent of the variance. Four clusters emerge: a predominantly European mainstream, an intermediate-impaired group containing the agricultural Piracicaba–Capivari–Jundiaí catchments, the metropolitan Tietê arc as a statistical outlier cluster with no European equivalent, and — the study’s most encouraging finding — a clean-reference group in which four São Paulo basins cluster alongside Portugal. The conclusions proved robust to the analytical choices: rankings held under equal-weight and CRITIC alternative weighting schemes, under wet-season and dry-season stratification, and under three levels of European benchmark stringency.

To test whether the method transfers beyond the network it was built on, the researchers applied the benchmarking and ranking components to 33 Polish and 26 Italian sub-basins under a leave-one-country-out design, recomputing the European reference each time with the target country excluded. Basin scores correlated negatively with an independent measure of anthropogenic pressure — population density from the European Commission’s Global Human Settlement Layer — in both countries, with Spearman coefficients of −0.44 and −0.42. That the association replicates across two hydrologically dissimilar settings that played no role in the method’s development is the substantive validation. The authors are careful to note what remains untested: the convergence-planning component requires multi-decadal basin-level records that the European archive does not provide, and the analysis covers only seven physicochemical parameters, excluding biological quality elements, heavy metals and emerging contaminants. Even so, the framework offers monitoring agencies something conventional indices cannot: a distribution-sensitive key performance indicator, a parameter-by-parameter decomposition of divergence, and a convergence horizon grounded in what the monitoring record itself has shown to be achievable.

Subject of Research: Cross-continental benchmarking of surface water quality monitoring networks using Wasserstein distributional distance and linear programming convergence planning

Article Title: Wasserstein distributional distance and linear programming convergence planning as monitoring network benchmarking tools: cross-continental surface water quality assessment across Brazilian and European river basins

Article References: Pimentel Tavares, H., & de Souza Sampaio, N. A. (2026). Wasserstein distributional distance and linear programming convergence planning as monitoring network benchmarking tools: cross-continental surface water quality assessment across Brazilian and European river basins. Environmental Monitoring and Assessment, 198(10), Article 1081. https://doi.org/10.1007/s10661-026-15914-w

Image Credits: AI Generated

DOI: 10.1007/s10661-026-15914-w

Keywords: Wasserstein distance, optimal transport, water quality monitoring, São Paulo river basins, Water Framework Directive, TOPSIS, linear programming, CETESB, GRQA, dissolved oxygen, total phosphorus, nitrate