Analysis of the Primal and Dual Problem for Long-Term Groundwater Monitoring Spatial Optimization

Abstract

Long term monitoring (LTM) can be costly given the large number of sampling locations monitored at given site. Redundant monitoring wells in the existing LTM network make it possible to remove some of them while maintaining data reliability from the remaining wells. We can optimize a monitoring network design by maximizing cost-effectiveness without compromising data quality. Alternatively, decision makers may define their LTM goal depending on the budget; they may set a low number of monitoring wells and identify the best combination of sampling points among all remaining monitoring wells. The problem of LTM spatial optimization is a non-linear one. We formulate the LTM optimization problem in two ways: one is to minimize the number of remaining wells given constraint rules on data quality and estimation errors; the other is given the number of remaining wells, the objective is to determine the optimal combination of a reduced set of wells from among the original ground water monitoring network. Here we reverse the role of the number of remaining wells from objective function to constraint, and we call the former optimization formulation the primal problem and the latter the dual problem. An ant colony optimization (ACO) method is developed to solve these problems based on field data. The ACO method is inspired by the fact that ants are able to find the shortest route between their nest and a food source. Individual ants can contribute their own information by pheromones; and the shorter the path, the higher the density of pheromones. Increased pheromones will attract the ant colony to choose the shortest route. Two ACO algorithms for LTM optimization are developed in this work — one based on a binary combinatorial formulation and the other is analogous to the traveling salesman problem (TSP).