Environmental Engineering Reference
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preted as the maximum (potential) value for NSE if the
other two components are able to achieve their optimal
values.
Statistical significance (
P
-value) for
E
can be esti-
mated with the bootstrap percentile-
t
method (Zoubir
and Boashash, 1998) or by the sampling distribution
given by McCuen et al. (2006). Using the method pro-
posed by McCuen et al. (2006), the sampling distribu-
tion of
E
can be approximated by the following
cumulative distribution function
Characterization of model performance (e.g., “good”)
based on values of the NSE generally depend on the
type of model. For example, a “good” hydrologic model
would have a higher NSE threshold than and “good”
model for predicting microbial concentrations in surface
runoff.
11.3.2.5 Index of Agreement.
The
index of agreement
,
d
, was originally developed by Willmott (1981), and is
given by
2
N
x
e
e
−
+
1
1
∑
(
)
(11.31)
2
P E E N
(
|
,
)
=
ˆ
y
−
y
0
j
j
x
j
=
1
d
= −
1
(11.37)
N
(
)
∑
2
ˆ
ˆ
ˆ
y
−
y
+
y
−
y
where
j
j
j
=
1
1
1
+
−
E
E
2
z
+
x
=
ln
(11.32)
The index of agreement varies between 0 and 1,
where a value of 1 indicates perfect agreement between
the measured and predicted values, and 0 indicates no
agreement at all. The index of agreement was not
designed to be a measure of correlation, but of the
degree to which a model's predictions are error free. It
has been suggested that
d
is better suited to model
evaluation than
R
2
, but is too overly sensitive to extreme
values (Legates and McCabe, 1999).
(
N
−
3
0 5
)
.
ε
m
S
ε
z
=
(11.33)
ε
1
1
+
−
E
E
0 5
.
ε=
0 5
.
ln
(11.34)
0 5
.
1
1
+
−
E
E
0 5
.
0
(11.35)
m
ε
=
0 5
.
ln
0 5
.
0
S
ε
=
(
N
−
3
)
−
0 5
.
(11.36)
11.3.2.6 Hydrologic Measures.
Hydrologic measures
typically used in in model calibration include: (1) flow
volumes as it relates to the water budget; (2) agreement
in shape between observed and predicted hydrographs;
(3) agreement of the peak flow characteristics, such as
timing and peak flow rate; and (4) agreement of reces-
sion limbs and low-flow periods.
It is generally recommended that a sufficient length
of continuous rainfall runoff events be used in calibrat-
ing hydrologic models to provide a warming-up period
so that the effect of initial conditions become negligible
at the forecast time. However, it has been shown that
for small watersheds having moderate impact of initial
moisture memory, rainfall runoff models can still be
calibrated reliably over a set of representative events
provided that the events cover a wide range of peak
flow, total runoff volume, and initial soil moisture condi-
tions (Tan et al., 2008). The event-based parameter set
can provide more accurate results than for continuous
calibration in terms of overall hydrograph shape, time
to peak, and peak flow rate, while the continuous event
parameter set tends to perform better in runoff volume
estimation.
where
E
0
is the actual (population) model efficiency.
Equation (11.31) can be used to test various hypotheses
as to whether a proposed model efficiency (
E
0
) is sup-
ported by a calculated model efficiency (
E
) based on
N
measurements.
There have been several reported studies where the
NSE (=
E
) was used to evaluate model performance.
White and Chaubey (2005) used the
E
statistic to assess
the performance of the SWAT model in simulating the
flow, sediment, and nutrient yields from the Beaver Res-
ervoir watershed (Arkansas). Values of
E
ranged from
0.50 to 0.89, which were characterized as typical of mul-
tisite, multivariable model calibrations. Muñoz-Carpena
et al. (2005) used the
E
statistic to assess models of
groundwater-quality in south Florida and classified
model performance as satisfactory when
E
≥ 0.5. Mishra
et al. (2003) classified model performance as very good
if
E
≥ 0.95, good if 0.90 ≤
E
< 0.95, satisfactory if
0.75 ≤
E
< 0.90, and poor if
E
< 0.75. Henriksen et al.
(2003) used the
E
statistic to assess models of discharge
hydrographs at a particular river gauging station. Model
performance was considered excellent if
E
≥ 0.85, very
good if 0.65 ≤
E
≤ 0.85, good if 0.50 ≤
E
< 0.65, poor if
0.20 ≤
E
< 0.50, and very poor if
E
< 0.20.
11.3.2.7 Selection of Performance Measures.
A com-
bination of graphical techniques and error statistics
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