MODELING OF WATER QUALITY - Water Quality Engineering in Natural Systems - page 395

in the data that is not explained by the model, and the
denominator measures measures the total variation that
could potentially be explained by the model. Values of
E given by Equation (11.26) range from −∞ to 1.0, with
higher values indicating better agreement between the
model-predicted and observed data. If E is greater than
zero, the model is considered to be a better predictor
of system behavior than the mean of the observed data.
Values of E are considered to be better suited to evalu-
ating model goodness-of-fit than R 2 , because R 2 is insen-
sitive to additive and proportional differences between
model simulations and observations (Harmel and
Smith, 2007). However, like R 2 , E is overly sensitive to
extreme values because it squares the values of paired
differences. As a consequence, the NSE can potentially
perform poorly in hydrologic applications when both
low flows and high flows are of concern (Foglia et al.,
2009).
To reduce the effect of high values on the NSE, a
modified coefficient of efficiency, , E ′ , is sometimes used,
where

three distinctive components representing the correla-
tion, bias, and relative variability in the simulated and
observed values. Expressed in the decomposed form the
NSE is given by

NSE = ⋅ 2 α ⋅ − r α β

2

−

2

(11.28)

where

σ
σ

s

α =

(11.29)

o

µ µ
σ

−

s o

β = (11.30)

o

where μ and σ represent the mean and standard devia-
tions, subscripts “o” and “s” represent observed and
simulated data, and r is the correlation coefficient
between the observed and simulated values. The collec-
tive parameter α measures the relative variability in the
simulated and observed values, and β is the bias normal-
ized by the standard deviation in the observed values.
It is clear that two of the three components of NSE (in
Eq. 11.28) relate to the ability of the model to reproduce
the first and second moments of the distribution of the
observations, while the third relates to the ability of the
model to reproduce timing and shape as measured by
the correlation coefficient. The ideal values for the three
components are r = 1, α = 1, and β = 0. From a modeling
perspective, “good” values for each of these three com-
ponents are highly desirable. Therefore, optimizing NSE
is essentially a search for a balanced solution among the
three components, where with optimal values of the
three components, the overall NSE is maximized.
However, it is evident that the bias component ( μ s − μ o )
appears in a normalized form, scaled by the standard
deviation in the observed flows. This means that in
basins with high runoff variability, the bias component
will tend to have a smaller contribution (and therefore
impact) in the computation and optimization of NSE,
possibly leading to model simulations having large
volume balance errors. This is equivalent to using a
weighted objective function with a low weight applied
to the bias component. A second concern made evident
by Equation (11.28) is that α appears twice, exhibiting
a problematic interplay with the linear correlation coef-
icient r . It can be shown that the maximum value of
NSE is obtained when α = r , and since r will always be
smaller than unity, this means that maximizing the NSE
tends to select a value for α that underestimates the
variability in the flows (this will favor models that gen-
erate simulated flows that underestimate the variabil-
ity). It should be noted that when β = 0 and α = r , then
the NSE is equivalent to r 2 . Therefore, r 2 can be inter-

∑
∑

N

ˆ

y − y

j j

j = 1

E ′ = 1.0 − (11.27)

N

y − y

j

j = 1

and y is the mean of the observations (López et al., 2007;
Legates and McCabe, 1999). In some cases, Equation
(11.27) is taken as the definition of the NSE (e.g.,
Moriasi et al., 2007), which is appropriate as long as
there is no model bias (i.e., y = ). For models optimized
using least squares, E is an equivalent measure to R 2 .
The coefficient of efficiency, E , compares the variance
about the 1:1 line ( y j versus y j ) to the variance of the
observed data.
In cases where a model produces biased predictions,
the effectiveness of the Nash-Sutcliffe coefficient of
efficiency is compromised; therefore, it is highly recom-
mended that the bias of a model be considered in
tandem with the model efficiency (McCuen et al., 2006).
In general, it is not advisable to evaluate the perfor-
mance of a model solely on the basis of NSE. Other
statistical tools, such as a scatter plot, which may reveal
important information about the ability of the model to
reproduce the dependent variable in different ranges,
need to be employed to arrive at a definite conclusion
about the model performance. Even a poor model may
yield E values in the vicinity of 0.60, and these cases
definitely warrant a careful look at the model results
before drawing any conclusion about its suitability or
otherwise (Jain and Sudheer, 2008).
gupta et al. (2009) presented an illuminating decom-
position of the NSE, showing that the NSE consists of