Designing inspection frequency to efficiently track stochastic dynamics is a fundamental engineering problem. Especially, tracking environmental variables like water quantity and quality is of vital importance for sustainable and sound development. However, theoretical understanding of methodologies to design cost-efficient inspection schemes of environmental dynamics is still limited. To tackle this issue, a new Poison inspection problem focusing on coupled streamflow and water quality dynamics in rivers was proposed as a model problem of designing inspection schemes in industries. The coupled dynamics are represented by Levy-driven stochastic differential equations. A long-run performance index, an effective Hamiltonian, containing inspection cost and penalization of the information loss between successive inspections is then formulated. The design variable of the proposed model is the inspection frequency. Our optimization problem has two levels where the effective Hamiltonian is firstly obtained in a closed-form from a Hamilton − Jacobi − Bellman equation. The effective Hamiltonian is then minimized concerning the inspection frequency. An application of the proposed model to designing cost-efficient inspection schemes of dissolved silica concentration, a key index in inland fishery industries, of a river in Japan, is also discussed with unique real data.

设计检查频率以有效跟踪随机动力学是一个基本的工程问题。特别是，跟踪水量和水质等环境变量对于可持续和健康发展至关重要。然而，对设计具有成本效益的环境动力学检查方案的方法的理论理解仍然有限。为了解决这个问题，提出了一个新的毒物检查问题，该问题侧重于河流中耦合的水流和水质动力学，作为设计工业检查方案的模型问题。耦合动力学由 Levy 驱动的随机微分方程表示。然后制定一个长期性能指标，一个有效的哈密顿量，包含检查成本和连续检查之间信息丢失的惩罚。所提出模型的设计变量是检查频率。我们的优化问题有两个层次，其中有效哈密顿量首先从哈密顿 - 雅可比 - 贝尔曼方程的封闭形式中获得。然后关于检查频率最小化有效哈密顿量。还讨论了所提出的模型在设计具有成本效益的溶解二氧化硅浓度检测方案中的应用，溶解二氧化硅浓度是日本一条河流的内陆渔业的关键指标，还使用独特的真实数据进行了讨论。