Uncertainty-Aware Source Apportionment of Soil Heavy Metals in a Coal–Steel Industrial Zone Using Inter-Metal Graph-Informed Dirichlet Neural Posterior Estimation

Industrialization and mining raise the heavy metal burden of soil ecosystems and create risks for both the environment and human health. Apportionment becomes difficult when neighboring facilities emit overlapping metal signatures, and conventional receptor models compound the difficulty by returning contribution shares without an attached uncertainty statement. This study develops a probabilistic apportionment framework for the coal–steel industrial zone of Zonguldak, Turkey, using 93 georeferenced topsoil stations and eight priority metals. Graphical Lasso estimates a sparse conditional dependency graph whose nodes are the measured metals; fixed propagation over that graph expands each observation into a 24-dimensional relational representation; and a Dirichlet neural posterior estimator maps the representation to source contribution distributions. The Dirichlet parameterization enforces non-negativity and unit sum by construction and delivers 95% credible intervals at every station; empirical coverage on held-out simulations was 0.957 against a nominal 0.95. Averaged across the province, posterior mean contributions were 0.25 for the iron-and-steel sector, 0.25 for thermal power generation, and 0.50 for a composite residual class that combines geogenic background with unmodelled anthropogenic inputs. The two industrial signals concentrate around Ereğli and Çatalağzı, while the residual class dominates elsewhere. The graph employed here encodes chemical relationships among metals and not geographical relationships among stations; maps of the posterior estimates are reported as post hoc visualizations. The framework is offered as an uncertainty-aware methodological contribution, and comparison against established receptor models remains to be carried out.