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| Titel |
Calibration and sensitivity analysis of a one-dimensional model of pollutant transport and degradation in maturation ponds. |
| VerfasserIn |
Blaise Delmotte, Carole Delenne, Vincent Guinot, Elena Gomez |
| Konferenz |
EGU General Assembly 2011
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 13 (2011) |
| Datensatznummer |
250055846
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| Zusammenfassung |
| The following hyperbolic system of conservation laws is used as a model of pollution
dynamic in maturation ponds:
-AC-(x,t)- -QC-(x,t)-
-t + -x = - k(t)AC (x,t)
(1)
-A(x,t)+ -Q-(x,t) = 0
-t -x
(2)
where A is the flow cross-section, C the contaminant concentration, Q the flow discharge, k
the contaminant degradation coefficient. In maturation ponds, the water elevation is generally
maintained to a given level so that the section A can be considered independent of
time, leading to a discharge Q independent of space, according to the continuity
equation (2). This model is discretized using an explicit finite volume approach and a
Godunov scheme, that enables the flux QC to be computed by solving a Riemann
problem.
In this study, the model is applied on a waste water system composed of three consecutive
waste stabilization ponds, with a residence time of approximately five weeks. The system is
considered uni-dimensional but with three different widths, so that A(x) is piecewise
constant with two discontinuities. Several measurements of the input and output discharge
and concentration of chemical oxygen demand (COD) are available, approximatively each 15
days during one year. The model gives an estimation of the concentration in space and time,
but will be calibrated according to the concentration at the system output as a function of
time.
The only parameter to calibrate in this model is the degradation coefficient. However,
since it is time dependent, a piecewise linear function is defined with N values of k, leading
to a N-criteria calibration. The model calibration is classically performed by minimizing an
error function. However, in this study, it appeared that the classical Root Mean Squared
Difference (RMSD) failed in finding a global optimum of the kn, since the convergence was
rarely reached. Another kind of objective functions can be used for model calibration,
that are based on a weak formulation of the distance between model outputs and
measurements:
-
Jp = (Ci(k)- Cmi )|Ci(k)- Cmi |p-1
i
(3)
were Ci(k) and Cim are the output concentrations respectively given by the model and
observed and where p is a real value, that will give more weight to the high differences
between outputs and observations when it is high whereas a small value of p will lead the
model to fit the observations in average. Because these functions are monotone, the model
calibration is transformed from a global optimization problem to a local root finding problem,
that can be solved using a simple gradient-based method such as Newton’s algorithm.
Dealing with multi-criteria calibration, the N optimum parameters can be found by the
intersection of N objective functions.
The gradient-based algorithms require the derivative of the function Jp, which depends
on the derivative of C with respect to k i.e. the sensitivity of C with respect to k. An
equation-based approach is used to compute the sensitivity. It consists in differentiating the
model equation (1) with respect to the parameter of interest (here k). The sensitivity equation
is then solved using the same framework as the model. The sensitivity analysis is also
performed with respect to boundary conditions (here the input concentration and discharge)
or to geometrical data such as the cross-section. However, the sensitivity equation
can be solved only when the model equation is differentiable. In this study, the
discontinuity in cross-sections leads to locally infinite values of the sensitivity. It has been
shown in previous studies that this problem can be eliminated by introducing an
additional source term in the sensitivity equation, that is applied only in case of
discontinuity.
This calibration process, involving weak form-based objective functions and sensitivity
solution, has shown good performance in this first study, despite the lack of accurate
observation data. Its application to a more instrumented system (with more data and less
uncertainties) is thus planed. |
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