\(\mu\) is a shape coefficient, \(L\) represents the weir width (assumed as the width of the downstream river channel), \(g\) is the acceleration of gravity and \(h_{i}\) the height over the weir (equal to the water level increase). The subscript i refers to the time step: the outflow discharge is calculated at each time step as a function of the varying height over the weir, derived at the previous time step using equation (2). The shape coefficient was initially assumed as equal to 0.5; this value was identified as the value producing the smallest errors by some calibration tests. The parabolic weir equation was also tested: it proved to be less effective whilst also requiring more detailed information about the spillway.
The ideal test case to validate this model with observed data would be a lake with three gauges providing time series (instant values) of the water level and the upstream and downstream discharge. This way the inflow hydrograph of an event can be used as input for the model and the computed outputs can be compared with observed records. The lake should be small enough to be influenced by the inflow hydrograph in terms of fluctuations in the water level – a very large lake’s water elevation won’t vary during a short single event – and without a dam or any regulation device that could influence drastically the water level and the discharge downstream. Unfortunately, we were unable to identify such an ideal test case in either Canada or the United States, where most medium size lakes are dammed and/or do not have a gauging station upstream. Despite this, three test cases were used to compare the physically-based model output with water level observations: lake Maskinongé, lake Brulé in Quebec and Waterbury reservoir in Vermont. For lake Maskinongé and Waterbury reservoir three gauging stations were available, although both water bodies are dammed. There is no gauging station downstream Lake Brulé and the main inflow is influenced by a dam (barrage Ludger). Different peak hydrographs isolated from gauging stations upstream the lakes were used as inputs for the model, after being appropriately scaled to the watershed area at the lake’s outflow. This procedure allows to consider the inflow from ungauged tributaries.