Geoscience Reference
In-Depth Information
Box 2.1 Linearity, Nonlinearity and Nonstationarity
Unit hydrograph and linear transfer function models in general are based on an assumption of
linearity and stationarity in time. In this context this may be simply understood in terms of the
relationship between inputs and outputs. A response that is stationary in time means that a unit
of input will always produce the same output response (remember that for the unit hydrograph,
the inputs are
effective rainfalls
not total rainfalls). A response that is linear means that if two
units of input fall in the same time step, we would expect twice the output response. If two units
of input fall in successive time steps, the two associated output responses, suitably delayed,
can be simply added together to produce a total output. More complex input sequences can be
dealt with as the simple addition of unit responses. This is called the principle of
superposition
.
It follows directly from the assumption of a linear model.
In a
nonlinear
model, the principle of superposition breaks down since it cannot be assumed
that a unit of input will always produce the same output. In rainfall-runoff modelling, nonlinear
responses are primarily due to two causes. The most important is the effect of antecedent
conditions. Thus the relationship between total rainfall and runoff is generally considered to
be nonlinear because the wetter the catchment prior to a unit input of rainfall, the greater the
volume of runoff that will be generated.
Figure B2.1.1
Nonlinearity of catchment responses revealed as a changing unit hydrograph for storms
with different volumes of rainfall inputs (after Minshall, 1960).
A secondary cause of nonlinearity is due to the change of flow velocities and celerities with
discharge. In Chapter 1, it was explained that inputs to the system will cause a response in
the output faster than the pore water velocity (see also Section 5.5.3) due to the way in which
pressure changes associated with disturbances to the system propagate with a wave celerity. In
