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![Map of the Maumee river hydrological network in the State of Ohio (USA) with water quality gauging stations represented as points. In yellow the St. Marys tributary taken into consideration for the present work (geographic coordinates 41∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}04’58” N 85∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}07’56” W)](publication/371162138/figure/fig2/AS:11431281190228399@1695261417780/Map-of-the-Maumee-river-hydrological-network-in-the-State-of-Ohio-USA-with-water.png)
Map of the Maumee river hydrological network in the State of Ohio (USA) with water quality gauging stations represented as points. In yellow the St. Marys tributary taken into consideration for the present work (geographic coordinates 41∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}04’58” N 85∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}07’56” W)
Source publication
Normal and anomalous diffusion are ubiquitous in many physical complex systems. Here we define a system of diffusion equations generalized in time and space, using the conservation principles of mass and momentum at channel scale by a master equation. A numerical model for describing the steady one-dimensional advection-dispersion equation for solu...
Citations
... In the previous work (Part I) a numerical technique, computationally less expensive than the existing diffusive methods, was presented (Rizzello et al. 2023). The numerical model was applied at the channel scale with the aim of observing its applicability from a computational point of view, monitoring the first results with the purpose of extending the obtained results to a larger scale (De Bartolo et al. 2006Bartolo et al. , 2009aBartolo et al. , 2022. ...
... The transition probability, P ij , plays a noteworthy role in transport scales, particularly concerning river networks. Specifically, from a physical point of view, P ij represents the conservation of the momentum characterising the process in its evolution, while from a stochastic point of view, it takes into account uncertainties arising e.g. from limited spatial and temporal resolution, accuracy and availability of input data (Rodriguez-Iturbe et al. 2009;Rinaldo et al. 2018;Rizzello et al. 2023). ...
... The results presented here are a continuation of the research conducted in Part I (Rizzello et al. 2023). ...
At basin scale the physical phenomenon of diffusion involves the intricate spreading and dispersion of substances within complex systems as networks of interconnected channels, streams, and land surfaces. Understanding this process is crucial for many purposes as management and conservation of water resources. We extend the model application of our previous work (Part I, Rizzello et al. in Stoch Environ Res Risk Assess 37:3807–3817, 2023) from channel to basin scale. We use conservation of mass and momentum to formulate and apply the Master Equation system at basin scale. The results on simulated events highlight the transition of the model from channel scale to basin scale.
River water quality often follows a long-memory stochastic process with power-type autocorrelation decay, which can only be reproduced using appropriate mathematical models. The selection of a stochastic process model, particularly its memory structure, is often subject to misspecifications owing to low data quality and quantity. Therefore, environmental risk assessment should account for model misspecification through mathematically rigorous and efficiently implementable approaches; however, such approaches have been still rare. We address this issue by first modeling water quality dynamics through the superposition of an affine diffusion process that is stationary and has a long memory. Second, the worst-case upper deviation of the water quality value from a prescribed threshold value under model misspecifications is evaluated using either the divergence risk or Wasserstein risk measure. The divergence risk measure can consistently deal with the misspecification of the memory structure to the worst-case upper deviation. The Wasserstein risk measure is more flexible but fails in this regard, as it does not directly consider the memory structure information. We theoretically compare both approaches to demonstrate that their assumed uncertainties differed substantially. From the application to the 30-year water quality data of a river in Japan, we categorized the water quality indices to be those with truly long memory (Total nitrogen, NO3-N, NH4-N, and SO42-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{SO}}}_{4}^{2-}$$\end{document}), those with moderate power-type memory (NO2-N, PO4-P, and Total Organic Carbon), and those with almost exponential memory (Total phosphorus and Chemical Oxygen demand). The risk measures are successfully computed numerically considering the seasonal variations of the water quality indices.
