Environmental Engineering Reference
In-Depth Information
In performing sensitivity analyses, specifying a per-
centage change in parameter value and reporting a
corresponding change in output variable is not always
appropriate, since the expected variation of some
parameters is much more than others. For example,
curve numbers typically vary within a fairly narrow
range compared with the saturated hydraulic conductiv-
ity of the vadose zone, which typically varies over
several orders of magnitude. Therefore, the variation
of values for a particular parameter for sensitivity analy-
sis has to be in accordance with the range appropriate
for that parameter. Some modelers identify a high,
medium, and low value for each parameter, and then
calculate the value of the output variable for each of
these values. The model output is deemed sensitive to
a parameter if the output changes significantly, as the
parameter value varies from low to high (Kannan et
al., 2007b). Calibration efforts then focus primarily on
sensitive parameters.
The aforementioned sensitivity analyses calculate the
sensitivity of model output to individual parameters and
require fixing the values of the other model parameters;
this type of sensitivity analysis is sometimes called
local
sensitivity analysis
(Foglia et al., 2009). In contrast,
global sensitivity analysis
considers the sensitivity of
model output to various parameter sets, thereby provid-
ing a more complete perspective of the relationship
between model output and parameter values. Com-
monly used methods used in global sensitivity analysis
are the generalized likelihood uncertainty estimation
(gLUE) method and Markov chain Monte Carlo
(MCMC) simulation. These methods are discussed in
greater detail in subsequent sections.
1
( |
, computed from
Equation (11.3), depends on the values chosen for the
other elements of the parameter vector
a
. A distribution
of
S y a
j
The elementary sensitivity,
S y a
j
1
( |
is obtained by sampling at different points
of the parameter space, that is, different choices of
parameter set
a
. The mean of
S y a
j
1
( |
over the param-
eter space indicates the overall influence of the param-
eter
a
j
on the model output,
y
, while the variance of
S y a
j
1
( |
demonstrates interactions with other parame-
ters
a
i
≠
j
and nonlinear effects (Arabi et al., 2007).
The sensitivity coefficient given by Equation (11.1)
can be normalized by the parameter value so that the
sensitivity coefficient with respect to any parameter has
the same units as that of the dependent variable, in
which case the sensitivity coefficient is taken as
S
i
2
,
where
ˆ
(
ˆ
(
y
a a
∂
ˆ
y a
+
∆
∆
a
)
−
y a
)
i
j
j
i
j
i
S
2
=
≈
(11.4)
ij
∂
/
a a
/
j
j
j
j
Sensitivity analysis is extremely useful in determin-
ing the level of accuracy that input variables need to be
measured and that adjustable parameters need to be
calibrated.
Some modelers assess sensitivity using the
relative
sensitivity
,
S
i
3
, defined by the relation
a
y
ˆ
+
10
ˆ
−
10
=
y
−
−
y
j
i
i
S
3
(11.5)
ij
ˆ
a
+
10
a
−
10
i
j
j
where
a
j
is the parameter corresponding to the model
output
y
i
,
a
j
+10
,
a
j
−10
and
y
i
+10
,
y
i
−10
correspond to ±10% of
the parameter and corresponding output values, respec-
tively (James and Burges, 1982; White and Chaubey,
2005). In cases where the relative sensitivity is used to
measure the sensitivity of a single model output,
y
(
a
),
to a single parameter,
a
j
, of a parameter vector
a
, the
elementary effect of a small perturbation Δ
a
j
of
a
j
is
11.3.2 Performance Analysis
The performance of a model is measured by the level
of agreement between model predictions and observa-
tions. Quantitative measures used for evaluating the
level of agreement between observations and model
predictions include statistical criteria, hypothesis testing,
linear regression, goodness-of-fit criteria with residual
error analysis, nonparametric comparisons, and graphi-
cal comparisons. If the model meets the specified per-
formance criteria associated with these approaches,
then the model is considered to be validated, otherwise
the model is not validated and the modeler should con-
sider an alternative model that might perform better.
Measures used to assess model performance gener-
ally depend on which components of the model output
are most relevant to the particular application. The com-
monly used approaches for evaluating agreement
between experimental measurements and model pre-
dictions are summarized below.
ˆ
ˆ
∆
∆
y y
a a
/
/
(
)
=
3
ˆ
S y a
|
(11.6)
j
j
j
where
y
is the base value of the model output variable,
and
∆
y
is the change in the output variable. In general,
the greater the value of
S
i
3
, the more sensitive the model
output variable
y
i
is to the parameter
a
j
. Limitations
to using
S
i
3
to assess the sensitivity of output variables
to model parameters are related to the assumption of
linearity, the lack of consideration of correlations
between parameters, and the lack of consideration to
the different degrees of uncertainty associated with
each parameter.
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