Excessive nitrate threatens a wide range of water resources, aquatic habitats, and sensitive infrastructure. Despite this problem, tracing a nutrient from its eventual fate back to its origin remains an elusive challenge due to heterogeneity in how nutrient sources and hydrologic pathways are connected. Typically, this problem is underdetermined (i.e., too many unknowns, not enough equations) and cannot be solved with existing methodologies. The theory of optimal transport allows for the solution of underdetermined systems, and here we construct a novel formulation for its use in water quality modeling. Our objective was to develop an optimal transport modeling framework—coupled to Bayesian source unmixing, loadograph pathway separation, and geospatial connectivity analysis—to apportion nitrate loading from three sources (soil, fertilizer, and manure) across three pathways (quick, intermediate, and slow), resulting in nine possible source‐pathway couplings (soil‐quick, soil‐intermediate, …, manure‐slow). We apply this model to a 30 month elemental (NO₃⁻) and isotopic (δ¹⁵N and δ¹⁸O) nitrate data set from a karst watershed in Kentucky, USA. Modeling results indicate that—of the nine possible source‐pathway couplings—nearly 60% of nitrate export is facilitated by just three: fertilizer‐quick (16.4%), manure‐intermediate (15.4%), and soil‐slow (27.2%). Further, we reinforce the need to explicitly consider heterogeneity in source‐pathway connectivity as homogeneous assumptions lead to erroneous inferences. The applicability of the model, its input requirements, and transferability to other sites is discussed. Lastly, we simulated two land management scenarios (field buffers and septic repair) and demonstrate how optimal transport can be used to test nutrient reduction strategies.

中文翻译：

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评估硝酸盐源-途径连通性的最佳运输

过量的硝酸盐威胁着广泛的水资源，水生生境和敏感的基础设施。尽管存在这个问题，由于养分来源和水文路径之间的异质性，从最终的命运追寻养分回到其来源仍然是一项艰巨的挑战。通常，这个问题是不确定的（即，太多的未知数，没有足够的方程式），并且不能用现有的方法来解决。最优输运理论可以解决不确定系统的问题，在此我们为水质模型的使用构建了一个新颖的公式。我们的目标是开发一个最佳的运输模型框架，并与贝叶斯源分解，载荷仪路径分离和地理空间连通性分析相结合，以分摊三种来源（土壤，肥料，和肥料）跨越三个路径（快速，中间和缓慢），从而导致九种可能的源－路径耦合（土壤－快速，土壤－中间，…，肥料－缓慢）。我们将此模型应用于30个月的元素（否₃ ^- ）和同位素（δ ¹⁵ N和δ ¹⁸从美国肯塔基州流域喀斯特O）硝酸盐数据集。建模结果表明，在九种可能的源－路径耦合中，仅三项就促进了近60％的硝酸盐出口：化肥速效（16.4％），肥料中度（15.4％）和土壤缓慢（27.2％） 。此外，由于同质假设会导致错误的推论，因此，我们强调需要明确考虑源-路径连通性中的异构性。讨论了模型的适用性，其输入要求以及到其他站点的可移植性。最后，我们模拟了两种土地管理方案（田间缓冲区和化粪池修复），并演示了如何使用最佳运输来测试养分减少策略。