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

Abstract Evaluating the performance of long-term water quality monitoring networks against international reference systems requires analytical tools capable of comparing entire empirical distributions — not merely summary statistics — yet distributional benchmarking methods have rarely been applied at the cross-continental scale. This study develops and applies a four-method analytical framework — combining Wasserstein-1 distributional distance (earth mover’s distance), Shannon-entropy-weighted TOPSIS, linear programming (LP) convergence planning, and principal component analysis with hierarchical cluster analysis (PCA/HCA) — to compare ten São Paulo river basins (Brazil) against 23 European countries monitored under the Water Framework Directive (WFD), using a harmonised dataset of 646,356 CETESB records and 998,410 GRQA v1.4 records covering seven common physicochemical parameters over a shared window of 2016–2022. The Wasserstein-1 analysis reveals that the distributional gap is non-uniform: Afluentes do Paraíba do Sul (composite $$W_1 = 0.245$$ W 1 = 0.245 ) and the Cantareira system ( $$W_1 = 0.333$$ W 1 = 0.333 ) are already within the distributional range of lower-ranking European countries, while the Alto Tietê ( $$W_1 = 2.060$$ W 1 = 2.060 ) and Médio Tietê ( $$W_1 = 1.392$$ W 1 = 1.392 ) are distributional outliers. Shannon-entropy-weighted TOPSIS ranking of the 20 entities with complete parameter coverage (all ten Brazilian basins and the ten European countries with complete five-parameter medians) places Afluentes-PSB seventh overall (score = 0.901), above Portugal, Poland, Germany, and Belgium, while the Alto Tietê (0.246) and Médio Tietê (0.194) rank last. Mann–Whitney U tests confirm statistically significant distributional differences for six of seven parameters, with large Cliff’s $$\\delta $$ δ for temperature ( $$\\delta = +0.858$$ δ = + 0.858 ), dissolved oxygen ( $$\\delta = -0.768$$ δ = - 0.768 ), and biochemical oxygen demand ( $$\\delta = +0.571$$ δ = + 0.571 ); nitrate is the sole parameter with no significant difference ( $$p = 0.354$$ p = 0.354 ; $$\\delta = -0.010$$ δ = - 0.010 , negligible). The LP convergence model, constrained by historically observed maximum improvement rates in the CETESB dataset, identifies dissolved oxygen as the binding convergence constraint for seven of ten basins, with critical-path convergence times ranging from 4.4 years (Afluentes-PSB) to 14.8 years (Alto Tietê) under the 90th-percentile feasibility assumption, lengthening to 5.5 and 28.7 years respectively under a more conservative 75th-percentile assumption; total phosphorus carries the highest LP cost weight ( $$w = 0.669$$ w = 0.669 ) but is the rate-limiting parameter for only one basin (Tietê-Médio). Rankings are robust to the weighting scheme (Kendall’s $$\\tau \\ge 0.86$$