Environmental Engineering Reference
In-Depth Information
that can be inferred using procedures that are indepen-
dent of observable catchment responses, for example,
local hydraulic conductivities obtained using core
samples or slug tests. Conceptual parameters, such as
discharge coefficients, have no formal physical interpre-
tation and can only be inferred by matching model
output to observed data. The calibration of models by
adjusting conceptual parameters is required since, by
definition, these parameters cannot be independently
measured.
Conventional calibration techniques use measured
values of the forcing variables, X , to estimate the model
response f ( X , θ ), and the error matrix, E , is estimated
as the difference between the model response f ( X , θ )
and the observed response Y , hence
ters (Kavetski et al., 2003). The most serious consequence
of this is that the model will produce biased results
when used for prediction, particularly when errors in
the forcing variables deviate from those found under
calibration conditions. Additional limitations relate to
limitations on the regionalization of the model and
obscuring inadequacies in the model itself.
Statistics commonly used in error analysis include the
bias, mean symmetry error (MSyE), mean absolute
error (MAE), residual sum of squares (RSS), mean
square error (MSE), root mean square error (RMSE),
standard error (SE), and relative error (RE).
Bias. The bias of model predictions is given by
N
1
ˆ
Bias =
(
y
y
)
(11.10)
E
=
f (
X
,
q
)
Y
(11.8)
j
j
N
j
=
1
where ( X , Y ) denote measured values of ( X , Y ), that
differ due to observational errors that are endemic to
field measurements. Using Equation (11.8), the esti-
mated error matrix E and associated error statistics are
expressed as a function of the parameter vector, θ .
Parameter estimation then consists of varying θ to maxi-
mize or minimize selected error statistics.
In cases where θ is varied to minimize the sum of
squares of the errors given by Equation (11.8), the
parameter estimation approach is called the standard
least squares approach . Although the standard least
squares approach is quite common, it has a significant
shortcoming in that it neglects possible (and likely)
errors in the forcing variables. In reality, the actual error
matrix, E , is given by
The bias is usually more easily interpreted when it is
normalized (i.e., divided by) the mean of the measure
values; this is called the relative bias or percent bias
(Moriasi et al., 2007), which can be expressed as
N
(
)
ˆ
y
y
j
j
j
=
1
Relative Bias =
(11.11)
N
y
j
j
=
1
Relative bias has the ability to clearly indicate poor
model performance, and relative biases greater than 5%
in absolute value are considered significant (McCuen et
al., 2006). In cases where streamflow is the variable of
interest, the relative bias measures the relative stream-
flow volume error that is commonly used in calibration
hydrologic models (e.g., Fernandez et al., 2005).
(11.9)
E
=
f (
X
,
q
)
Y
which recognizes that model prediction should be
derived from the actual value of the forcing variable, X ,
rather than the measured value of the forcing variable,
X , which might be in error. Precipitation estimates are
typically the most uncertain forcing variables in hydro-
logic models. Derivation of the error statistics from
Equation (11.9) is problematic in that the actual values
of the forcing variable, X , are not observed and must be
estimated. The parameter estimation approach that
minimizes the sum of squares of the errors and takes
into account the errors in the forcing variables is called
the total least squares approach , which is sometimes
referred to as the error in variables method .
A significant concern in using the standard least
squares approach is that the least squares approach
does not identify the actual model parameters, since it
neglects uncertainty in the forcing variables, and typi-
cally produces biased estimates of the model parame-
Mean Symmetry Error. The mean symmetry error (in
percent) is given by
N
ˆ
100
y
y
j
j
MSYE =
(11.12)
N
y
j
j
=
1
where y j is the observed value, y j is the predicted value
of the j th observation, and N is the number of observa-
tions. MSyE measures the prediction symmetry with
respect to the line of perfect agreement.
Mean Absolute Error. The mean absolute error is given
by
N
1
ˆ
MAE =
y
y
(11.13)
j
j
N
j
=
1
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