\(\mu\) 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.