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| Titel |
Weak-formed based objective functions in hydrology. Application to a 1D transport/degradation model |
| VerfasserIn |
Carole Delenne, Vincent Guinot, Bernard Cappelaere, Blaise Delmotte |
| 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 |
250055821
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| Zusammenfassung |
| The process of model calibration consists in bringing the simulated variable as close as
possible to the measured one. The intuitive term "as close as possible" can be translated
into a mathematical formulation of the error between simulation and observation:
the objective function provides a measure of how close to zero the modeling error
is.
The most widely used objective function in any scientific application is probably the Root
Mean Squared Difference (RMSD), which is the euclidian distance between the observed and
simulated variable. In hydrology, the well-known Nash-Sutcliffe efficiency, may also be
considered as a measure of the distance between observations and simulation. Using
distance-based objective functions, turns the model calibration process in an optimization
problem: minimization if a cost function is used or maximization for an efficiency
function. A general formulation of a distance-based cost function can be written
as:
-
Jp = α |F (Ui,Ï) - Vi|p
i
where α depends on the function definition, F(Ui,Ï) is the model output (for different
locations or times), with Ï the parameter to be calibrated and V i are the measurements. In
case of an efficiency function, the form 1 -Jp can be used. In this formulation, the model is
considered perfect if Jp = 0, i.e. if it reproduces exactly the measurements, but
uncertainty on the latter can also be taken into account in the definition of the objective
function.
Another category of objective functions is based on a weak formulation of the evaluation
problem. This can be for example, the bias-indicator or cumulative error. The weak
formulation can be written in the general form:
- p-1
Jp = α (F (Ui,Ï)- Vi)|F(Ui,Ï)- Vi|
i
In contrast with the distance-based formulation, the weak-form based objective functions are
monotone (and thus, can be negative). This has several practical consequences. The first one,
is that the model calibration process is transformed into a root-finding problem, that can be
solved using a simple, efficient gradient-based method (such as Newton’s algorithm) rather
than generally more complex global optimization algorithms. Moreover, the function
monotony implies the unicity of the solution, thus avoiding multiple local optima. A model
with N parameters can be calibrated by defining N weak form-based objective functions and
finding their intersection in the parameter space. These functions can be defined by using
different observation variables (or functions of these variables) or different values of p.
Parameter redundancy can be evidenced when no intersection can be found between
objective functions defined with the same observe variable but different values of
p.
The weak-form based objective function is applied on the multi-criteria calibration of a
hydrodynamic transport-degradation model. The parameter with respect to which the model
is calibrated, is the degradation coefficient k. However, since it is time dependent, a piecewise
linear function is defined with n values of k, leading to a n-criteria calibration. It appeared in
this study that the classical distance RMSD failed to converge to a global optimum of
the kn vector. The use of n weak-form based objective functions, defined with
n different values of p enables the model calibration with the classical Newton’s
algorithm. |
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