Environmental Engineering Reference
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dispersion than on tracer velocity. As expected, increasing the amount of data points
improves the parameters estimation as it is shown in the 40 data case (first column in
Figs.
2
,
3
and
4
). This improvement is directly related to the fact that larger amount
of information is provided. It is important to point out the relevance of the quality of
the information because if the data are not sampled properly, as it may happen even
in synthetic data, the results can be influenced or biased by external factors that are
inconsistent with the estimation methodology.
The number of fitting parameters has important effects on the parameter estima-
tion. This can be seen in the data fromFigs.
2
,
3
and
4
. Clearly, fitting three parameters
is more complicated than fitting only one and therefore the result improves signifi-
cantly. The estimation of
at the same time shows that the last parameter
almost does not affect the transport model and it can be explained in function of
the involved phenomena. As mentioned before
D
ad
and
α
,
D
ad
and
ʸ
are related to the tracer
dispersion, but the first has a strong effect, as it depends on other physical properties
like fluid density, fractal length dimension, etc., whereas the second is an isolated
parameter relevant only at small values of length. If this parameter is fixed and the
others are fitting parameters, the estimation of
ʸ
slightly improves but the estima-
tion of
D
ad
significantly worsens, as shown in Table
4
. This is a consequence of
a rescaling process in the model sensitivity due to the presence of only two fitting
parameters. When only
α
is a fitting parameter, the estimation does not changes
significantly with respect to the previous case. This may suggest that one-parameter
optimizations provide robust results,regardless the noise or the amount of data.
α
7 Conclusions
A methodology for parameter estimation in a given fractal model for tracer transport
is presented. The effect of the data noise level, the amount of data and the amount
of fitting parameters on the parameter estimation for tracer pulse injection has been
analysed.
Even though the parameter estimation is sensitive to the level of data noise, to
the amount of data, and to the amount of fitting parameters, in general the proposed
methodology is robust, particularly in estimating
and
D
ad
. Estimation of each
parameter not only depends on such properties, but mainly on the model sensitivity
to the specific parameter. In this sense the model is more sensitive to
α
α
than to
D
ad
and
has more uncertainty than it has with the other parameters.
Moreover the contribution to the percentage of relative error (PRE) of
ʸ
. The estimation of
ʸ
α
is smaller
than that associated to
D
ad
and
. Finally, care must be taken in the following issues
when working with synthetic data and numerical solutions:
ʸ
•
Estimation depends on the discretization parameters used in the numerical solution.
•
A sensitivity analysis must be performed before the parameter estimation.
•
The results may change if data noise is generated using other distribution functions
different from the uniform one used here.
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