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.
Highlights
As the water sector moves into the digital era it is inevitable that detailed hydraulic models will be used for real-time monitoring, forecasting and real-time control purposes
This shows that the updating works well, and thereby that the Kalman gain used for the point
This shows that the updating works well, and thereby that the Kalman gain used for the updating is appropriate
Summary
As the water sector moves into the digital era it is inevitable that detailed hydraulic models will be used for real-time monitoring, forecasting and real-time control purposes. The EnKF is a Monte Carlo implementation of the extended Kalman filter in which the error statistics required for computing the Kalman gain (a matrix that determines how much each state in a model is corrected in response to a given deviation between the model and observations) are derived from an ensemble of models that are propagated forward in time in parallel. The constant Kalman gain can be used in the online situation to correct the model via the standard Kalman filter state update equation, which implies to increment the state vector with the product of the Kalman gain and the measurement residual (the difference between observed and modeled values) whenever observations are available This eliminates the need for an online ensemble, but it is possible only if the Kalman gain can be assumed to be approximately constant in time, which, as we show in the following, is not the case for urban drainage systems exposed to surcharge. We document through a synthetic experiment for a single pipe stretch with a downstream CSO structure, that the Kalman gain varies dynamically in both time and space during backwater periods, and this critical finding should be taken into account when developing future DA schemes for DUDMs
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