Hydrodynamics – exponential smoothing factor

The exponential smoothing factor may be used for unstable links, although it will cause a time lag in the corresponding water levels for the coupled models. The exponential smoothing factor works by applying a weight to the transferred water level from the surface model to the river mode.

\[ H_{r}^{n} = (1 - \alpha) \cdot H_{r}^{n - 1} + \alpha \cdot H_{s}^{n} \]

Equation: Exponential smoothing of the water level transferred to the river model.

Where \(\alpha\) is the smoothing factor, \(H_{r}(n)\) is the water level for the river model at time step \(n\), and \(H_{s}(n)\) is the water level for the surface model at time step \(n\). By applying this expression \(n\) times the water level may be given as

\[ H_{r}^{n} = (1 - \alpha)^{n} \cdot H_{r}^{0} + \alpha \cdot \sum_{i = 0}^{n}{(1 - \alpha)^{i}} \cdot H_{s}^{n - 1} \]

Equation: Exponentially smoothed river water level after n time steps.

Using the above expression, a criterion for selecting the smoothing factor may be obtained. Assuming that the water level in the surface model is constant, then we can expect that after \(n\) time steps the river water level at the link location will be within a fraction \(f\) of the surface water level (i.e., \(H_{r} \approx f \cdot H_{s}\)), as given by

\[ n = \frac{\log f}{\log(1 - \alpha)} \]

Equation: Number of time steps needed to reach a given fraction of the surface water level.

Thus, the smoothing factor \(\alpha\) should be chosen in relation to the time scale at which the dynamics change in the model (\(n\)).

Reversed definition in MIKE+

The definition of smoothing factor used inside MIKE+ GUI is \(\alpha_{\text{MIKE+}} = 1 - \alpha\) and is reversed from a typical definition used in the exponential smoothing equation. So, for \(\alpha_{\text{MIKE+}} = 0\), there is no exponential smoothing.