Ensemble smoother with multiple data assimilation for reverse flow routing

The reverse flow routing is an inverse procedure aimed at estimating the inflow to a hydraulic system based on information collected downstream. The hydraulic system can be a river reach or a water reservoir. In this paper, we propose a new approach for the solution of the reverse flow routing problem based on the Ensemble Smoother with Multiple Data Assimilation (ES-MDA). The objective is the estimation of an unknown inflow hydrograph discretized in time by coupling ES-MDA with a given forward routing model that relates inflow hydrograph and downstream observations. Two realistic synthetic examples are presented to show the capabilities of the methodology. The first case is an application of the reverse flow routing problem to a linear reservoir, where the outflow hydrograph and the reservoir characteristics are known; the second one focuses on the estimation of the inflow hydrograph to an open channel from water level information recorded downstream. We also investigate the performance of the inverse algorithm, by looking at different ensemble sizes, and using covariance localization and inflation techniques. Our tests show that the proposed approach provides good results, comparable with those of other optimization methods presented in the recent literature. It accurately reproduces the inflow hydrographs, as well as the observations, with narrow confidence intervals. Although ES-MDA yields better results increasing the ensemble size, significant improvements in the solution are obtained for small ensemble sizes when covariance localization and inflation techniques are applied. The proposed approach can compete in accuracy and speed with other approaches, with the advantage that it is conceptually simple and can be used with almost any forward routing code.