Geoscience Reference
In-Depth Information
greater, it is implicitly assumed that some average gradient can be used to characterise the flow and that
the effects of
preferential flow
through macropores in the soil (one explanation of the observations of
Figure 1.1) can be neglected. It is worth noting that, in many articles and model user manuals, while the
equations on which the model is based may be given, the underlying simplifying assumptions may not
actually be stated explicitly. Usually, however, it is not difficult to list the assumptions once we know
something of the background to the equations. This should be the starting point for the evaluation of a
particular model relative to the perceptual model in mind. Making a list of all the assumptions of a model
is a useful practice that we follow here in the presentation of different modelling approaches.
The conceptual model may be more or less complex, ranging from the use of simple mass balance
equations for components representing storage in the catchment to coupled nonlinear partial differential
equations. Some equations may be easily translated directly into programming code for use on a digital
computer. However, if the equations cannot be solved analytically given some
boundary conditions
for the
real system (which is usually the case for the partial differential equations found in hydrological models)
then an additional stage of approximation is necessary using the techniques of numerical analysis to define
a
procedural model
in the form of code that will run on the computer. An example is the replacement of
the differentials of the original equations by finite difference or finite volume equivalents. Great care has
to be taken at this point: the transformation from the equations of the conceptual model to the code of
the procedural model has the potential to add significant error relative to the true solution of the original
equations. This is a particular issue for the solution of nonlinear continuum differential equations but
has been the subject of recent discussion with respect to more conceptual catchment models (Clark and
Kavetski, 2010). Because such models are often highly
nonlinear
, assessing the error due only to the
implementation of a numerical solution for the conceptual model may be difficult for all the conditions
in which the model may be used. It might, however, have an important effect on the behaviour of a model
in the calibration process (e.g. Kavetksi and Clark, 2010).
With the procedural model, we have code that will run on the computer. Before we can apply the
code to make quantitative predictions for a particular catchment, however, it is generally necessary to go
through a stage of parameter
calibration
. All the models used in hydrology have equations that involve a
variety of different input and state variables. There are inputs that define the geometry of the catchment
that are normally considered constant during the duration of a particular simulation. There are variables
that define the time-variable boundary conditions during a simulation, such as the rainfall and other
meteorological variables at a given time step. There are the state variables, such as soil water storage or
water table depth, that change during a simulation as a result of the model calculations. There are the
initial values of the state variables that define the state of the catchment at the start of a simulation. Finally,
there are the model
parameters
that define the characteristics of the catchment area or flow domain.
The model parameters may include characteristics such as the porosity and hydraulic conductivity
of different soil horizons in a spatially distributed model, or the mean residence time in the saturated
zone for a model that uses state variables at the catchment scale. They are usually considered constant
during the period of a simulation (although for some parameters, such the capacity of the interception
storage of a developing vegetation canopy, there may be a strong time dependence that may be important
for some applications). In all cases, even if they are considered as constant in time, it is not easy to
specify the values of the parameters for a particular catchment
a priori
. Indeed, the most commonly used
method of parameter calibration is a technique of adjusting the values of the parameters to achieve the
best match between the model predictions and any observations of the actual catchment response that
may be available (see Section 1.8 and Chapter 7).
Once the model parameter values have been specified, a simulation may be made and quantitative
predictions about the response obtained. The next stage is then the
validation
or
evaluation
of those
predictions. This evaluation may also be carried out within a quantitative framework, calculating one or
more indices of the performance of the model relative to the observations available (if any) about the
runoff response. The problem at this point is not usually that it is difficult to find an acceptable model,
