How to deal with non-stationary conditions in hydrology using neural networks

apricotpigletAI and Robotics

Oct 19, 2013 (3 years and 5 months ago)

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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