Monotonic Linear Constraints for Deriving Realistic Optimal Unit Hydrographs

Deriving unit hydrographs from rainfall-runoff data using optimization techniques circumvents the cumbersome requirements of the conventional manual method. In one of the two main ways of formulating the optimization problem, the decision variables are the individual ordinates of the unit hydrograph (UH). Because of their high number of degrees of freedom, though, such formulations can easily lead to overfitting the input hydrographs in the training set, resulting in highly irregular shapes for the derived UHs. These may be optimal for back-constructing the hydrographs in the training set, but as they do not resemble a typical hydrograph, they might not be capable of reproducing other unseen events adequately. In this work, we propose a set of linear constraints that can be included in existing optimization formulations to force the UH to feature a monotonically ascending limb, a single peak, and a monotonically receding limb. The ordinate of the peak, where the shift in monotonicity occurs, is not established in advance, but is rather determined within the optimization process. If required, the constraints can also be applied piecewise to obtain multiple peaks and troughs with monotonic limbs between them. Application to two case studies reveals that the reduction in the degrees of freedom does help in preventing overfitting of the input hydrographs, resulting in better performance when reproducing unseen events.