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Another important issue for the solution of the inverse parameter structure identifi-
cation problem is the determination of the optimum zone structures. Although the
combined S/O models are the efficient tools of identifying the parameter zone struc-
tures, in practical cases, the number of zones is also an unknown parameter to be
determined. In order to identify the optimum zone structure, the inverse solution pro-
cedure starts with a zone and increases the zone number until the best zone structure
is identified [10].
In this solution procedure, an increase in the zone numbers results with the
decrease in the calculated RE values. However, this situation may result with the in-
crease in the uncertainty of the identified parameters. In such cases, parameter struc-
ture identification process is stopped since the reliability of the identified parameters
usually decreases. Therefore, it is necessary to use an additional measure, Parameter
Uncertainty (PU), to determine the optimum zone structure. Note that PU is calcu-
lated based on the covariance matrix of the estimated parameters as follows [35]:
⎛
⎞
⎡⎤
N
T
1
∑∑
d
(
)
(
)
( )
−
1
2
Τ
(2)
Τ
Cov
=
G
⎜
h
Ω
,
t
−
h
t
J
J
⎟
G
⎣⎦
c
c
μ
c
μ
NTc
×−
⎝
⎠
d
t
=
1
μ
=
1
where the superscript T is the transpose operation,
J
is the Jacobian matrix of hydrau-
lic heads with respect to transmissivities,
G
is the structure matrix whose elements
represent the weight of the parameter value form the
i
th
basis point to the
j
th
data point.
After obtaining the covariance matrix, the trace or the norm of Eq. 2 is used to com-
pute the PU values.
3.1 Model Application
Tsai et al. [10] first proposed the following hypothetical example, which is a two-
dimensional confined aquifer system, to solve the parameter zone structure identifica-
tion problem. Figure 2, which follows, shows the geometry and boundary conditions
of the aquifer model under consideration.
It can be seen form Figure 2 that the boundary conditions of the aquifer is 100 m
specified head at east, and no-flow at the other boundaries. Aquifer has 5 pumping
wells (PWs) and it is assumed that all the wells are continuously operated for 10 days.
The associated pumping rates of the wells are: 4,000 m
3
/day for PW
1
to PW
4
, and
2,000 m
3
/day for PW
5
. The storage coefficient is 0.0002. There are seven observation
points available where head observations are measured at each day. In order to simu-
late the observational errors in the field, all the head observations are corrupted with a
Gaussian noise of zero mean and 0.1 m standard deviation.
In this example, the aquifer's transmissivity is selected as a parameter to be identi-
fied. Therefore, the main objective is to identify the transmissivity zone structure
through inverse modeling approach described in the previous section. The true trans-
missivity field of the aquifer system is shown in Figure 3. Note that in this field, the
largest and smallest transmissivity values are about 595 m
2
/day and 33 m
2
/day.
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