Technical Note on the Dynamic Changes in Kalman Gain when Updating Hydrodynamic Urban Drainage Models

Abstract

To prevent online models diverging from reality they need to be updated to current conditions using observations and data assimilation techniques. A way of doing this for distributed hydrodynamic urban drainage models is to use the Ensemble Kalman Filter (EnKF), but this requires running an ensemble of models online, which is computationally demanding. This can be circumvented by calculating the Kalman gain, which is the governing matrix of the updating, offline if the gain is approximately constant in time. Here, we show in a synthetic experiment that the Kalman gain can vary by several orders of magnitude in a non-uniform and time-dynamic manner during surcharge conditions caused by backwater when updating a hydrodynamic model of a simple sewer system with the EnKF. This implies that constant gain updating is not suitable for distributed hydrodynamic urban drainage models and that the full EnKF is in fact required.

Assimilating flow and level data into an urban drainage surrogate model for forecasting flows and overflows

“…Branisavljevic et al (2014) applied the Extended Kalman filter to update the states in a simplified model based on water level and flow observations, and further found that subsequent bias reduction can improve the results. Borup et al (2014Borup et al ( , 2018 showed that the EnKF can be used to update the water levels throughout a HiFi model based on flow as well as water level observations. Even though the EnKF is much cheaper to run for large models than a traditional KF, the propagation of the ensemble forward in time is still computationally expensive for large HiFi models.…”

Improving flood forecasting using real-time data to update urban models in poorly gauged areas

“…The updating process is carried out in a deterministic manner and demonstrated improvements in the simulations, despite the fact that the uncertainties of the model structure and observational data are not considered. Borup et al (2018) tested the use of Ensemble Kalman filter for the MIKE URBAN model in an experiment to evaluate the possibility of using constant Kalman gain updating to address the problem of high computational demand when the ensemble is calculated in real-time. The results show that the gain is nonlinear and varies greatly in time, requiring the use of the complete Ensemble Kalman Filter scheme.…”

“…In contrast to large basins or medium-scale basins, small urban catchments may be affected by intense local precipitation (WMO, 2011), which combined with increased runoff by urbanisation, require faster responses from hydrological models to ensure their utility for flood forecasting (Yang et al, 2011;Chen et al, 2013;Yin et al, 2016). Furthermore, urban models are highly non-linear with many physical state variables and, consequently, computationally costly, posing a further challenge for their real-time updating (Hansen et al, 2014;Borup et al, 2018).…”