Environmental Engineering Reference
In-Depth Information
10
5
4
8
ʱ
= 0.752
D
ʱ
= 0.721
D
ʱ
= 0.786
D
4
= 0.0119
FO = 11.7
3
= 0.0217
FO = 2.62
= 0.0099
FO = 0.289
ad
ad
ad
6
3
2
4
2
1
2
1
0
0
0
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
t
D
t
D
t
D
6
3.5
8
3
5
ʱ
= 0.744
D
ad
= 0.0151
FO = 2.65
ʱ
= 0.754
D
ad
= 0.0125
FO = 0.354
ʱ
= 0.751
D
ad
= 0.0104
FO = 0.251
2.5
6
4
2
3
4
1.5
2
1
2
1
0.5
0
0
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
0.5
1
1.5
2
t
D
t
D
t
D
Fig. 3
Qualitative comparison between the synthetic data and the model predictions for the case
where
α
and
D
ad
are the fitting parameters. Description is the same as in Fig.
2
. The original
parameters are
α
=
.
75,
D
ad
=
.
ʸ
=
.
0
0
0133 and
0
05
Ta b l e 3
Comparison between original and estimated fractal parameters for different amount of
data and noise levels
40 Data
20 Data
10 Data
Parameter
Original
5%
10%
5%
10%
5%
10%
α
0.75
0.744
0.752
0.754
0.721
0.751
0.786
D
ad
0.0133
0.0151
0.0119
0.0125
0.0217
0.0104
0.0099
OF
2.65
11.7
0.334
2.62
0.25
0.29
Here
α
and
D
ad
are the fitting parameters
5.3 Sensitivity to the Amount of Fitting Parameters
To analyse the effect of the amount of fitting parameters on the results, two additional
cases are included: one where the fitting parameters are
α
and
D
ad
, (see Fig.
3
),
and the other where only
is considered as fitting parameter, (see Fig.
4
). The fixed
parameters are set according to the original values. Regarding to the case where three
parameters were estimated, in the first case a general improvement in the estimation
of
α
is observedwhereas the estimation of
D
ad
worsened. This can be seen in Tables
3
and
4
. When only one fitting parameter is considered the PRE does not exceed the 5%
regardless the amount of data or the amount of noise. It is very similar to the values
obtained where two or three fitting parameters are considered. Therefore, estimation
strongly depends on the amount of fitting parameters, as expected. It improves as the
number of parameters is reduced, especially if the model is sensitive to the selected
fitting parameters.
α
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