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non-uniqueness of the identified parameters. The instability and non-uniqueness is as-
sociated with the numerical or observational errors, possibility of the multiple solu-
tions, as well as the limited field data [3, 11].
In groundwater modeling, usually two type parameterization approaches are used
to identify the parameter structures: interpolation and zonation [25]. In the
interpolation approach, the parameter structure of the flow domain is generated using
some interpolation functions. However, this approach requires known parameter
values at some locations to determine the whole parameter structure [26]. If there is
no known parameter values available in the flow domain, this approach may not be
used. On the other hand, in the zonation approach, the flow domain is subdivided into
the zones where associated parameter values are assumed to be constant. This may be
the more realistic approach if distribution of the aquifer parameters in the field is
suitable for using the zonation approach.
Note that, identification of the zone structure of an aquifer system is more difficult
than the identifying the associated parameter values within zones [27]. Hence, the
number of the published studies dealing with the use of this approach is limited [3,
10-11, 20, 28-32].
In zonation approach, each zone includes a basis point location which is the key
parameter of determining the shape of zone structure. Several zonation approaches
have been proposed to characterize the parameter structures. One of the most widely
used approaches is the Voronoi Tesselation (VT) [33]. In VT, zonation process is per-
formed based on the metric distances between data points and basis points. VT has
very simple computational procedure and always produces convex shaped zone struc-
tures. Another zonation approach is the Fuzzy c-Means Clustering (FCM) approach
[34] in which each zone is considered as a fuzzy set and data points are assigned to
the zones by a fuzzy membership grade. The key difference between VT and FCM is
that VT uses the metric distance for assigning a data point into a zone whereas FCM
uses fuzzy memberships [11]. Another version of the FCM algorithm is the kernel
based FCM (KFCM). The main idea of the KFCM algorithm is to transform the origi-
nal low-dimensional input space into a higher dimensional feature space. Through this
transformation, more general zone shapes may be generated [3]. After partitioning the
flow domain through zonation approach, homogeneous parameter values are assigned
into each zone.
Note that inverse parameter structure identification problem can be solved through
the combined S/O models. In these models, the locations of the basis points and the
associated parameter values are treated as the decision variables of the optimization
model. The objective of the optimization model is to identify the associated zone
structure by minimizing the Residual Error (RE) between observed and simulated
hydraulic heads at available observation wells. RE is used to identify the aquifer's
response for each parameter zone structure as follows [11]:
⎡
⎤
N
N
T
d
∑∑
(
)
(
)
( )
2
(1)
RE
=
Min
h
Ω
,
t
−
h
t
⎢
⎥
c
μ
c
μ
⎢
⎥
⎣
⎦
t
=
1
μ
=
1
where RE
c
is the residual error value in the parameter zone structure Ω
c
,
N
T
is the total
simulation time,
N
d
is the number of observation wells,
h
μ
and
h
μ
are the simulated
and observed hydraulic heads in observation well
μ
.
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