Objective Function Definition
in Conceptual Hydrological Modelling
V. Guinot, B. Cappelaere, C. Delenne, D. Ruelland
HydroSciences Montpellier UMR 5569 (CNRS, IRD, UM1, UM2)
SimHydro 2010
1
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Reservoir‐based models
• discretized versions of balance equations written in differential form
dU
= R (U , ϕ , t ) − Q (U , ϕ , t )
dt
R
U n +1 = f (U n , Rn , ϕ ,...)
Qn = g (U n , ϕ )
• Predefined flux functions
U
Q
• Parameters to be calibrated
R (U , ϕ , t )
SimHydro 2010
Q (U , ϕ , t )
2
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
The calibration process
R
• The primary variable can often
not be used for calibration
U
• A « secondary variable » is used
instead (e.g. the discharge Q) because it is easier to observe
Q
Q
Observed
Modelled
t
SimHydro 2010
3
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
The calibration process
V
• Try to make the modelled signal as close as possible to the observed one BUT…
• What does « as close as possible » mean ? F(U, ϕ)
Observed
Modelled
t
Q
t
SimHydro 2010
4
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Classical measures of « closeness »
• Nash‐Sutcliffe Efficiency (NSE)
• Mean Square Error (MSE)
• Root Mean Square Error (RMSE)
J=
∑ [F (U
N
J=
n , ϕ ) − Vn
]
2
=
∑
V
en2
e : model error
e
Ω
t
N
2
2
[
]
F
(
U
(
t
),
ϕ
)
−
V
(
y
)
d
t
=
e
∫
∫ (t ) dt
Ω
SimHydro 2010
Ω
5
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Classical measures of « closeness »
V
• Generalized Nash‐Sutcliffe
Efficiency (NSE)
e : model error
e
Jp =
∑
en
∑
F (U (t ), ϕ ) − V ( y ) dt =
∫
F (U n , ϕ ) − Vn
p
=
∫
Ω
SimHydro 2010
Ω
N
N
Jp =
p
p
t
p
e (t ) dt
Ω
6
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Classical measures of « closeness »
V
• Generalized Nash‐Sutcliffe
Efficiency (NSE)
e : model error
e
Jp =
∑
en
∑
F (U (t ), ϕ ) − V ( y ) dt =
∫
F (U n , ϕ ) − Vn
p
=
∫
p
Ω
Ω
N
N
Jp =
p
t
p
e (t ) dt
Ω
Perfect model
SimHydro 2010
Jp = 0
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Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Classical measures of « closeness » verify
• Positivity
Jp =
∫
e(t )
p
dt ≥ 0
∀( p > 0, Ω)
Ω
• Triangle rule
• Zero property
J p = 0 ⇔ e(t ) = 0∀t
∫
p
⎤
dt ⎥
⎥
⎦
1/ p
⎡
⎢ e1 + e2
⎢Ω
⎣
⎡
≤ ⎢ e1
⎢Ω
⎣
∫
p
⎤
dt ⎥
⎥
⎦
1/ p
⎡
+ ⎢ e2
⎢Ω
⎣
∫
p
⎤
dt ⎥
⎥
⎦
1/ p
These are the properties of a norm (distance)
SimHydro 2010
8
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Another possible approach: the weak form
• Desired property: • Weak form:
e(t ) = 0 ∀t ∈ Ω
∫ e(t ) w(t ) dt = 0
∀w(t )
Ω
• Particular case: SimHydro 2010
w(t ) = e(t )
p −1
,
p ≥1
9
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Another possible approach: the weak form
• Desired property: • Weak form:
e(t ) = 0 ∀t ∈ Ω
∫ e(t ) w(t ) dt = 0
∀w(t )
Ω
• Particular case: w(t ) = e(t )
p −1
,
p ≥1
∫
J p = a + b e(t )
Proposed Objective Function
p −1
e(t ) dt
Ω
Jp = a+b
∑
en
p −1
en
N
SimHydro 2010
10
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Very small differences…
Distance-based
p=1
J1 =
∫Q
sim
Weak form-based
− Qobs dt
VE =
Ω
∫ (Q
sim
− Qobs ) dt
Ω
p=2
MSE =
2
(
)
Q
−
Q
∫ sim obs dt
Ω
SimHydro 2010
J2 =
∫Q
sim
− Qobs (Qsim − Qobs ) dt
Ω
11
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
… yield very different objective function behaviours
0.5
cP
dU
= cP(t ) − kU
dt
k
U
kU
0
0
c
0.2
Distance‐based, p = 1
Observed variable: discharge
SimHydro 2010
12
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
… yield very different objective function behaviours
0.5
0.5
k
k
0
0
0
c
0.2
Distance‐based, p = 1
Observed variable: Q
SimHydro 2010
0
c
0.2
Weak form‐based, p = 1
Observed variable: Q
N.B.: This is the Volume Error VE
13
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
… yield very different objective function behaviours
0.5
0.5
k
k
0
0
0
c
0.2
Distance‐based, p = 2
Observed variable: Q
SimHydro 2010
0
c
0.2
Weak form‐based, p = 2
Observed variable: Q
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Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Response surface monotony
If the following assuptions hold
• Q > 0, R > 0
• R – Q is an increasing function of U
• There exists as steady state solution
• Initial condition: U(0) ≥ 0
Then (Guinot & al., submitted)
• Weak form‐based objective functions are monotone in ϕ (no local minimum)
• Distance‐based objective functions
are not (local minima may appear)
SimHydro 2010
dU
= R (U , ϕ , t ) − Q(U , ϕ , t )
dt
Jp
D
W
ϕ
15
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
The calibration process
Distance‐based
• Optimization process
• Local minima
• Equifinality
• Gradient‐based methods
may fail
• Global search procedures
• Optimum depends on p
Weak form‐based
• Root finding problem
• Monotone function
• Gradient‐based methods
can be used
• Search the intersection between hypersurfaces
defined for different
values of p
Both types of objective functions lead to similar optimal parameter sets
SimHydro 2010
16
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Moreover…
(from Guinot & al, submitted to Journal of Hydrology)
• Using different p in Jp allows redundant parameters to be
identified
• In reservoir‐based models, applying the same objective function to several model variables is more efficient for calibration than applying different objective functions to the same model variable
• Correspondence (Parameters ÍÎ Variables) for optimal calibration efficiency
SimHydro 2010
17
Objective Function Definition in Conceptual Hydrological Modelling
Guinot, Cappelaere, Delenne & Ruelland
Conclusions
• Weak form‐based objective functions may prove
easier to use than distance‐based objective functions in practice
• It is well‐admitted that calibration and validation should be carried out using different data sets
• We may also admit that they should be carried out using different types of objective functions !
SimHydro 2010
18
Objective Function Definition
in Conceptual Hydrological Modelling
V. Guinot, B. Cappelaere, C. Delenne, D. Ruelland
HydroSciences Montpellier UMR 5569 (CNRS, IRD, UM1, UM2)
SimHydro 2010
19