Geoscience Reference
In-Depth Information
V
E
C
max
n
y
i
¼
y
i
;
y
j
¼
y
j
Þ
V
i ;j
¼
ð
V
i
V
j
is the second-order conditional variance.
V
E
C
max
n
y
i
¼
y
i
;
y
j
¼
y
j
;
y
k
¼
y
k
Þ
V
i ;j;k
¼
ð
V
i
V
j
V
k
V
i;j
V
i;k
V
j;k
is the
third-order conditional variance.
In GSA, the sensitivity of a model output is estimated by sensitivity indices.
The first-order sensitivity index
S
i
quantifies the sensitivity of a model output
C
max
to parameter
y
i
and is obtained by normalizing a first-order conditional by the
output unconditional variance:
V
i
S
i
¼
C
max
Þ
A high value of the first-order sensitivity index
S
i
indicates that the parameter
V
ð
y
i
plays an important role in explaining the variance of a model output singularly,
independently of interactions. Hence, identification of this parameter should be
prioritized to reduce the output uncertainty.
The total sensitivity index
S
Ti
represents sensitivity of a model output
C
max
to the
parameter
y
i
and its interactions with the other parameters, and is defined as:
C
max
n
y
i
¼
y
i
Þ
V
ð
C
max
Þ
VE
ð
S
Ti
¼
V
ð
C
max
Þ
where
y
i
are all parameters except of
y
i
.
A high value of the total sensitivity index
S
Ti
corresponds to parameter
y
i
affecting a model output both singularly and with interactions, which means that
the parameter is relevant. A low value of the total effect
S
Ti
indicates that the
parameter is unessential and its value could be fixed to simplify a model, and
consequently, to reduce the uncertainty.
In the study, the Sobol' method, which uses Monte Carlo simulations, was
applied to estimate sensitivity indices. In order to prepare Sobol' samples, Monte
Carlo simulations of 100,000 parameter sets were performed. Parameters were
drawn from the same distributions as described in Sect.
4.3
.
Results of the SA are presented in Tables
1
and
2
. According to the main effect
S
i
, parameters
A
and
K
x
are significant in explaining the output variation. Conse-
quently, the uncertainty of results depends mostly on these two parameters. In other
words, during model calibration the main effort should be put on evaluating
A
and
K
x
to reduce the uncertainty.
On the other hand, parameters characterizing transient storage,
and
A
s
, have no
impact on the results, as both sensitivity indices are near to zero. It means that
transient storage process is irrelevant in the studied case.
a
4.5 Simplification of the Dead Zone Model
to the Fickian Model
As was shown in the sensitivity analysis, two model parameters,
A
and
K
x
,are
sufficient to model contaminants transport in the studied reach, and
and
A
s
have
a
