How to deal with non

stationary conditions in
hydrology using neural networks
Virgile TAVER
1, 2
Anne JOHANNET
2
Valérie BORRELL ESTUPINA
1
Séverin PISTRE
1
(1) HSM, HydroSciences Montpellier, UMR5569,
Université de Montpellier
2,
France
(2) Ecole des Mines d’Alès
, France
Made possible by a collaboration
IAHS General Assembly in Göteborg

July 2013
Session : Testing simulation and forecasting models in non

stationary conditions
1
Neural Networks (NN)
•
The Neural Networks are increasingly used in hydrology:
o
For prediction
o
For forecasting floods
o
For modelling unknown relations
•
The Neural Networks learn their behavior by training, nevertheless:
o
They are sensible to overfitting (bias

variance dilemma)
o
Model complexity must be chosen as simple as possible
o
Regularization methods must be used
Neural Networks
Methods Results Conclusion
2
Neural Networks
•
Neuron definition:
o
Weighted sum
o
Non

linear function (
f
)
•
Neural Network architecture:
o
Multilayer perceptron
➥
Universal approximation
➥
Parsimony (for statistical models)
Neural Networks
Methods Results Conclusion
3
Neural Network: design methodology
•
Minimization of the quadratic error during training by Levenberg

Marquardt rule
•
Data base utilization:
o
One (sub)

set for training (
P
i
,
i
=1,5)
o
One (sub)

set for stopping (early stopping with records ≠(
P
i
,
i
=1,5))
o
One (sub)

set for test (in level 3), different from training and stop sets
•
Complexity selection
o
Definition of architecture using cross

validation (included inside the training period) :
•
Input variables (
u
i
)
•
Number of hidden layers and hidden neurons
o
Selection using Nash criterion
Neural Networks
Methods Results Conclusion
4
3 ways of modelling
•
3 models can be investigated regarding the
postulated model
•
For example let us consider an
analogy
: calculate the price of a
“
baguette”
, 3 methods can used to estimate such a price :
1.
Take into account the
price of primary ingredients
(flour, water …), energy, and
compute the price for a specific recipe
2.
If the state measurement is good: take into account the
measured price
yesterday
, and anticipate a one

day evolution
3.
If the state measurement isn't good: take into account the
estimated price
yesterday
, and anticipate evolution
Neural Networks
Methods
Results Conclusion
5
3 ways of modelling
•
3 models can be investigated regarding the
postulated model:
1.
Computing discharge from
rainfall and physics
2.
Computing discharge from
the state measurement
3.
Computing discharge from
the state estimation
Neural Networks
Methods
Results Conclusion
6
Static
system
modelling
Dynamic
system
modelling
=> Non

directed NN model
=> Static NN model
=> Directed NN model
1 NN model for each postulated model
Un

directed
NN model
Static
NN model
Directed
NN model
•
Postulated model 1
•
Postulated model 2: noise on the state
•
Postulated model 3 : noise on the measurement
φ
q

1
u
(
k
)
y
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
φ
RN
u
(
k
)
y
p
(
k
)
g
(
k
+1)
y
p
:
observed
output
of
the
physical
process
u
(
k
)
:
observed
input
of
the
physical
process
(
rain
)
b
(
k
)
:
noise
φ
q

1
u
(
k
)
x
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
x
p
(
k
+1)
φ
u
(
k
)
y
p
(
k
+1)
φ
RN
u
(
k
)
g
(
k
+1)
φ
RN
q

1
u
(
k
)
g
(
k
)
g
(
k
+1)
Neural Networks
Methods
Results Conclusion
7
1 NN model for each postulated model
Un

directed
NN model
Static
NN model
Directed
NN model
•
Postulated model 1
•
Postulated model 2: noise on the state
•
Postulated model 3 : noise on the measurement
φ
q

1
u
(
k
)
y
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
φ
RN
u
(
k
)
y
p
(
k
)
g
(
k
+1)
y
p
:
observed
output
of
the
physical
process
u
(
k
)
:
observed
input
of
the
physical
process
(
rain
)
b
(
k
)
:
noise
φ
q

1
u
(
k
)
x
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
x
p
(
k
+1)
φ
u
(
k
)
y
p
(
k
+1)
φ
RN
u
(
k
)
g
(
k
+1)
φ
RN
q

1
u
(
k
)
g
(
k
)
g
(
k
+1)
Neural Networks
Methods
Results Conclusion
8
1 NN model for each postulated model
Non

directed
NN model
Static
NN model
Directed
NN model
•
Postulated model 1
•
Postulated model 2: noise on the state
•
Postulated model 3 : noise on the measurement
φ
q

1
u
(
k
)
y
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
φ
RN
u
(
k
)
y
p
(
k
)
g
(
k
+1)
y
p
:
observed
output
of
the
physical
process
u
(
k
)
:
observed
input
of
the
physical
process
(
rain
)
b
(
k
)
:
noise
φ
q

1
u
(
k
)
x
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
x
p
(
k
+1)
φ
u
(
k
)
y
p
(
k
+1)
φ
RN
u
(
k
)
g
(
k
+1)
φ
RN
q

1
u
(
k
)
g
(
k
)
g
(
k
+1)
Neural Networks
Methods
Results Conclusion
9
1 NN model for each postulated model
Non

directed
NN model
Static
NN model
Directed
NN model
•
Postulated model 1
•
Postulated model 2: noise on the state
•
Postulated model 3 : noise on the measurement
φ
q

1
u
(
k
)
y
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
φ
RN
u
(
k
)
y
p
(
k
)
g
(
k
+1)
y
p
:
observed
output
of
the
physical
process
u
(
k
)
:
observed
input
of
the
physical
process
(
rain
)
b
(
k
)
:
noise
φ
q

1
u
(
k
)
x
p
(
k
)
b
(
k
+1)
y
p
(
k
+1)
x
p
(
k
+1)
φ
u
(
k
)
y
p
(
k
+1)
φ
RN
u
(
k
)
g
(
k
+1)
φ
RN
q

1
u
(
k
)
g
(
k
)
g
(
k
+1)
Neural Networks
Methods
Results Conclusion
10
3 ways to deal with non stationary
•
How to adapt the model to the changing environment and process?
o
Changing process or environment
•
The observed data are used to adapt parameter values
at
different time steps
Adaptativity
o
The observed data are used as input data
at different time step
Directed Model
•
The observed data are used to modify inaccurate inputs
at
different time steps
Data Assimilation
A variationnal approach is used in this work to modify rainfalls,
temperature and snow at each time step
Neural Networks
Methods
Results Conclusion
11
Possible on
the 3 models
Only for
Directed
model
Possible on
the 3 models
Application:

Fernow watershed,

Durance watershed
Only models able to represent dynamic systems were developed :
Directed (non

recurrent model)
Non

Directed (recurrent model)
Neural Networks Methods
Results
Conclusion
12
Non

directed
NN model
Directed
NN model
φ
RN
u
(
k
)
y
p
(
k
)
g
(
k
+1)
φ
RN
q

1
u
(
k
)
g
(
k
)
g
(
k
+1)
2 ways of dealing with non

stationary :
No option
Adaptativity
Assimilation
Fernow watershed, USA (0,2 km
2
)
Neural Networks
Methods
Results
Conclusion
13
•
Complete period: 01/01/1959

31/12/2009
•
Snowmelt and sampling too distant (day for a very small basin)
•
Calibration periods:
–
P1: 01/01/1959

31/12/1968: forest cut of the lower part of the basin (Mar

Oct 1964); forest cut of the upper part of the basin (Oct 1967

Feb 1968)
–
P2: 01/01/1969

31/12/1978: plantation of firtrees (Mar

Apr 1973)
–
P3: 01/01/1979

31/12/1988
–
P4: 01/01/1989

31/12/1998
–
P5: 01/01/1999

31/12/2008
Fernow model
Neural Networks
Methods
Results
Conclusion
14
Fernow model : illustration
Neural Networks
Methods
Results
Conclusion
15
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
Directed
Directed
Adaptativity
Directed
Assimilation
Non Directed
Non Directed
Adapatativity
Non Directed
Assimilation
Fernow basin
Train P
1
test P1
test P2
test P3
test P4
test P5
Nash criterion
Durance watershed, France ( 2170 km
2
)
Neural Networks
Methods
Results
Conclusion
16
•
Observed non

stationary: Temperature higher implying decrease of glaciers
•
Discharge during spring due to snowmelt
•
Complete period: 01/01/1904

30/12/2010
•
Calibration periods:
–
P1: 01/01/1904

31/12/1924
–
P2: 01/01/1925

31/12/1945
–
P3: 01/01/1946

31/12/1966
–
P4: 01/01/1967

31/12/1987
–
P5: 01/01/1988

31/12/2008
Durance model
Neural Networks
Methods
Results
Conclusion
17
Rain
7j
Temp
10j
PET
4j
Qcalc
2j
Hidden
Layers
3
Architecture defined on P1
Durance model : illustration
Neural Networks
Methods
Results
Conclusion
18
During the spring period, discharge of the Durance is due to snowmelt.
To take into account this process, positive temperatures of winter and spring are
preserved (from 1st of January to 30th of June). All the other temperature are set
to zero.
0
100
200
300
400
500
600
30
20
10
0
10
20
30
19/11/1905
27/02/1906
07/06/1906
15/09/1906
24/12/1906
Discharge (m3/s)
Temperature (
°
C)
Temperature modification for snowmelt process
Temperature
Snowmelt
Discharge
Durance model : illustration
Neural Networks
Methods
Results
Conclusion
19
Model
Input
Temperature
Assimilation
on
Directed
The supplied
ones
Rainfall
Non

Directed
The supplied
ones
Rainfall,
Temperatu
re, PET
Non
directed
Snowmelt
Rainfall
Fernow
Durance
Neural Networks
Methods
Results
Conclusion
20
Directed, no option
With the Directed model with Adaptativity
or Assimilation on the Fernow catchment :
•
Improvement of the Nash
•
But decrease of the performance on the
low flows on some periods
Not a Gain, nor a deterrioration on the
Durance catchment
Fernow
Durance
Neural Networks
Methods
Results
Conclusion
21
Non

Directed, no option
Best results on the Durance catchment
Poor Nash on Fernow
Very bad low flows simulations
Neural Networks
Methods
Results
Conclusion
22
Adaptation and Assimilation options can
strongly improve the Nash criterion (in
particular for the Durance catchment)
But have no effect on low flows
Non

Directed, Adaptation
Non

Directed, Assimilation
Fernow
Durance
Neural Networks
Methods
Results
Conclusion
23
Fernow
Non

Directed, Assimilation
The data assimilation :

improves low flows while
deterioring the Nash on the
Ferrow catchment on some
periods
It was the oppositive result for
the Durance catchment :
improvement of the Nash while
deterioring low flows on
(previous slide)
Neural Networks
Methods
Results
Conclusion
24
Durance
Non

Directed, no Option, T
°
The treatment of temprature (Snowmelt) improves the Nash criterion
Bad simulations on low flows
Non

Directed, Assimilation, T
°
0
1000
100
0
100
19/11/1905
07/06/1906
24/12/1906
Dischar
ge
…
Temper
ature
…
Temperature
…
Non

Directed,
no Option
Conclusions
Neural Networks Methods Results
Conclusion
25
•
The best way (reliable , simple, easy) to adapt the model to the changing
environment consists in using the Directed Model (feedforward model)
•
When using Directed Model, there has been no appreciable progress when using
adaptativity or assimilation
•
When using Non

Directed model, the improvement provided by adapatativity and
data assimilation can be high
•
Neural Network Modelling is more efficient for the largest studied catchment
•
Work on progres : data assimilation must be studied more deeply (some parameters
to adjust), the criteria used for otpimization have to be complixified (to avoid that
the improvement on high flows appears when decreasing performance on low flows
and vice versa)
Thank you for your time
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