Geoscience Reference
In-Depth Information
as the efficiency measure for rainfall-runoff modelling by Nash and Sutcliffe, 1970). This is
defined as
e
o
R
t
=
1
−
(B4.1.18)
where
o
is the variance of the observed output data calculated over all time steps used in
fitting the model, and
e
is the variance of the residual differences between observed and
simulated outputs at each time step assuming the residuals have zero mean. As the model fit
improves, the value of
R
t
approaches one. If the model is no better than fitting the mean of
the observed outputs (when
e
o
), the value of
R
t
will be zero or less. In general the higher
the model order, the better the fit will be, because there are more coefficients or degrees of
freedom in the model than can be adjusted to fit the data. The aim is to find a model structure
that gives a good fit but is
parsimonious
in having a small number of coefficients.
There is a danger in allowing too many coefficients or too high a model order. This might
result in good values of
R
t
, but the model may then be over-fitted or over-parameterised with the
result that the transfer function may not be physically realistic (perhaps with negative ordinates
or oscillations) or, when used in simulation, its predictions may be very sensitive to the input
sequence used. One way of checking for over-parameterisation is by looking at how well the
coefficients of the model are estimated. For example, in the TFM package (see Appendix A),
the transfer function model coefficients are fitted using the Simplified Recursive Instrumental
Variable algorithm (Young, 2011a) which is available in the Matlab CAPTAIN Toolbox (Taylor
et al.
, 2007). This algorithm allows the standard errors associated with each of the
a
and
b
coefficients to be estimated every time a model structure is calibrated to a data set. The higher
the model order (i.e. the greater the number of
a
and
b
coefficients), the larger the standard
errors will tend to be. Very large standard errors on the coefficients are an indication that a
model is over-parameterised. Young has suggested a criterion for model structure evaluation
that combines elements of goodness of fit and standard errors on the coefficients. This “Young
Information Criterion” (YIC) is defined as
=
ln
1
ln
e
o
N
N
P
ii
˛
i
e
YIC
=
+
(B4.1.19)
where
˛
i
,i
1
...N
are the model coefficients and
P
ii
is the i
th
diagonal element of a scaled
parameter covariance matrix. The YIC value should be as negative as possible, indicating on its
logarithmic scale that the variances of the model residuals and coefficient values are as small
as possible. Over-parameterisation leads to a very rapid increase in the value of YIC. However,
before accepting a transfer function model of this type, the user should always check the shape
of the transfer function to ensure that it is physically reasonable for the system being studied.
In the case of rainfall-runoff model or flow routing for example, a transfer function that has
negative ordinates would not normally be considered acceptable.
=
B4.1.4 Factorising a Second-Order Model
If a model is identified as second order (
n
1
,
2), then
it is possible to factorise the model into two first-order components by treating the denominator
as follows. From the equations above, for both a serial and a parallel model:
=
2) and has one or two
b
parameters (
m
=
a
+
a
a
1
=
a
a
a
2
=−
Combining these two equations, having fitted values of the two coefficients
a
1
and
a
2
,gives
a
2
a
1
a
−
−
a
2
=
0
(B4.1.20)
