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

Abstract

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 ${{\text{SO}}}_{4}^{2-}$), 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.

Appendix

1.1 Proof of Proposition 1

Given $k>0$ such that $k\ge m$, we set the following quantity:

$$G\left(k\right)={\int }_{0}^{+\infty }\text{max}\left\{X-m,0\right\}\gamma \left(X;k;1\right){\text{d}}X.$$

(52)

Equation (52) is rewritten as follows:

$$\begin{array}{l}G\left(k\right)={\int }_{m}^{+\infty }\left(X-m\right)\gamma \left(X;k,1\right){\text{d}}X\\\;\;\;\;\;\;\;\;\; ={\int }_{1}^{+\infty }\left(mU-m\right)\gamma \left(mU;k,1\right){\text{d}}\left(mU\right)\\ \begin{array}{l}\;\;\;\;\;\;\;\; ={m}^{2}{\int }_{1}^{+\infty }\left(U-1\right)\gamma \left(mU;k,1\right){\text{d}}U\\ \begin{array}{c}\;\;\;\;\;\;\; ={m}^{2}{\int }_{1}^{+\infty }\left(U-1\right)\frac{{\left(mU\right)}^{k-1}}{\Gamma \left(k\right)}{\text{exp}}\left(-mU\right){\text{d}}U\\\;\;\;\;\; ={m}^{k+1}{\int }_{1}^{+\infty }\left(U-1\right){\text{exp}}\left(-mU\right)\frac{{U}^{k-1}}{\Gamma \left(k\right)}{\text{d}}U\end{array}\end{array}\end{array}.$$

(53)

We show that, for each fixed $U>1$, the function ${\vartheta }_{U}\left(k\right)=\frac{{U}^{k-1}}{\Gamma \left(k\right)}$ ($k>0$) is increasing. By fixing $U>1$, we obtain the following equality

$$\begin{array}{l}\frac{{\text{d}}{\vartheta }_{U}\left(k\right)}{{\text{d}}k}=\frac{{\text{d}}\left({U}^{k-1}\right)}{{\text{d}}k}\frac{1}{\Gamma \left(k\right)}-{U}^{k-1}{\left(\frac{1}{\Gamma \left(k\right)}\right)}^{2}\frac{{\text{d}}\Gamma \left(k\right)}{{\text{d}}k}\\\;\;\;\;\;\;\;\;\;=\frac{1}{\Gamma \left(k\right)}{U}^{k-1}{\text{ln}}U-{\left(\frac{1}{\Gamma \left(k\right)}\right)}^{2}{U}^{k-1}\Gamma \left(k\right)\psi \left(k\right)\\\;\;\;\;\;\;\;\;\;=\frac{1}{\Gamma \left(k\right)}{U}^{k-1}\left({\text{ln}}U-\psi \left(k\right)\right)\end{array}$$

(54)

with $\psi \left(k\right)$ in Eq. (23) [Theorem 3.1.2 of Little et al. (2022)]; this $\psi$ is strictly increasing due to the following equality:

$$\frac{{\text{d}}\psi \left(k\right)}{{\text{d}}k}=\sum\limits_{j=0}^{+\infty }\frac{1}{{\left(k+j\right)}^{2}}>0 ,k>0.$$

(55)

Moreover, $\psi \left(k\right)=0$ is solved by a unique $k={k}^{*}\in \left(\mathrm{1,2}\right)$ at which $\Gamma \left(k\right)$ is minimized [Corollary 3.3.2 of Little et al. (2022)]; further, $\psi \left(k\right)<0$ (resp.,$\psi \left(k\right)>0$) if $0<k<{k}^{*}$ (resp.,$k>{k}^{*}$). Consequently, ${\text{ln}}U-\psi \left(k\right)$ is nonnegative if $0<k<{k}^{*}$ because $U>1$. We then obtain the following equality:

$$\frac{\text{d}G\left(k\right)}{\text{d}k}=m^{k+1}\int_1^{+\infty}\left(U-1\right)\exp\left(-mU\right)\frac{\mathrm d\vartheta_U\left(k\right)}{\mathrm dk}\;\mathrm dU\;\mathrm{for}\;0\;<k\mathit\;\mathit<k^{\mathit\ast}$$

(56)

because $\left(U-1\right){\text{exp}}\left(-mU\right)>0$ for $U>1$. The existence of its right-hand side follows from $U>1$ and the following bound:

$$\begin{array}{l}\left|\left(U-1\right){\text{exp}}\left(-mU\right)\frac{{\text{d}}{\vartheta }_{U}\left(k\right)}{{\text{d}}k}\right|=\frac{1}{\Gamma \left(k\right)}{U}^{k}\left|{\text{ln}}U-\psi \left(k\right)\right|exp\left(-mU\right)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \left({C}_{1}{U}^{k+1}+{C}_{2}\right)exp\left(-mU\right)\end{array}.$$

(57)

with sufficiently large constants ${C}_{1},{C}_{2}>0$; the last line of (57) is integrable.

1.2 Model identification results

Tables and figures summarizing the WQI results are summarized in Table 3 (Otsu) and Table  4 (Kisuki). Table 5 shows the Wasserstein distance ${W}_{p}$ between the empirical data and identified model at Otsu. Table 6 shows the results at Kisuki. Table 7 shows the relative errors of the average, variance, and skewness of TN at Otsu for different values of $\omega$. Figure  12 compares the empirical and fitted ACFs of each WQI. Figure 13 compares the empirical and fitted PDFs of each WQI. Note that some of the data at Kisuki (TN and ${{\text{SO}}}_{4}^{2-}$) was analyzed in Yoshioka and Yoshioka (2024).

Table 3 Parameter values of the supCIR processes at Otsu

Table 4 Parameter Values of the supCIR processes at Kisuki

Table 5 Wasserstein distance ${W}_{p}$ between the empirical data and identified model at Otsu

Table 6 Wasserstein distance ${W}_{p}$ between the empirical data and identified model at Kisuki

Table 7 Relative errors of the average, variance, and skewness of TN at Otsu for different values of $\omega$

Fig. 12

Empirical and fitted ACFs of WQIs: a TN, b NO₂-N, c NO₃-N, d NH₄-N, (e) TP, (f) PO₄-P, (g) ${{\text{SO}}}_{4}^{2-}$, (h) COD, (i) TOC. The colors represent empirical (black) and theoretical results (blue). The exponential fit (red) is also presented for comparison

Fig. 13

Empirical and theoretical histograms (Hist in short): (a) TN, (b) NO₂-N, (c) NO₃-N, (d) NH₄-N, (e) TP, (f) PO₄-P, (g) ${{\text{SO}}}_{4}^{2-}$, (h) COD, (i) TOC. Colors are empirical (red) and theoretical results (blue)