Risk assessment of river water quality using long-memory processes subject to divergence or Wasserstein uncertainty

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.