Stochastic streamflow and dissolved silica dynamics with application to the worst-case long-run evaluation of water environment

Abstract

Management of river environment requires to assess streamflows and water quality dynamics, which are often stochastic as well as not easy to model without errors. A new mathematical framework is proposed in this paper to assess an environment in a long run based on a stochastic optimization theory under model ambiguity. Focusing on the dissolved silica (DSi) load as an environmental indicator of rivers, the coupled discharge and DSi load dynamics as a two-variable continuous-state branching process with immigration is formulated. The ambiguity as a model misspecification is evaluated by a relative entropy measuring deviation between benchmark and distorted models. The model misspecification is expressed as a bound of the relative entropy. Novel stochastic optimization problems are formulated to evaluate the long-run DSi load subject to the misspecification as expectation constraints. Nonlocal degenerate elliptic Hamilton–Jacobi–Bellman (HJB) equations having Lagrangian multipliers are employed to solve these problems and their optimality are verified theoretically. The HJB equations admit closed-form solutions that can be computed efficiently. Our model is finally applied to assessing long-run DSi load and discharge in an upstream Hiikawa River, Japan.

Appendix 1: Model parameter values

Tables

Table 2 The calculated mean discharge $\overline{Q}$ and mean precipitation $\overline{P}$, estimated intake $q$ and the corresponding proportionality coefficient $a$ in each FY

2,

Table 3 The identified parameter values of the streamflow dynamics in each FY

3 summarize model parameter values used in Sect. 4.