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How to deal with non-stationary conditions in hydrology using neural networksVirgile 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 predictiono For forecasting floodso 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 possibleo Regularization methods must be used
Neural Networks Methods Results Conclusion 2
Neural Networks
• Neuron definition:o Weighted sumo 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 (Pi, i=1,5)
o One (sub)-set for stopping (early stopping with records ≠(Pi, 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 (ui)
• 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-directedNN model
StaticNN model
DirectedNN model
• Postulated model 1
• Postulated model 2: noise on the state
• Postulated model 3 : noise on the measurement
φ
q-1
u (k)
yp (k)
b (k+1)
yp(k+1)φRNu (k)
yp (k)g(k+1)
yp: observed output of the physical process
u (k): observed input of the physical process (rain)
b (k): noise
φ
q-1
u (k)xp (k)
b (k+1)
yp(k+1)
xp (k+1)
φu (k) yp(k+1)φRNu (k) g(k+1)
φRN
q-1
u (k)g (k)
g (k+1)
Neural Networks Methods Results Conclusion 7
1 NN model for each postulated model
Un-directedNN model
StaticNN model
DirectedNN model
• Postulated model 1
• Postulated model 2: noise on the state
• Postulated model 3 : noise on the measurement
φ
q-1
u (k)
yp (k)
b (k+1)
yp(k+1)φRNu (k)
yp (k)g(k+1)
yp: observed output of the physical process
u (k): observed input of the physical process (rain)
b (k): noise
φ
q-1
u (k)xp (k)
b (k+1)
yp(k+1)
xp (k+1)
φu (k) yp(k+1)φRNu (k) g(k+1)
φRN
q-1
u (k)g (k)
g (k+1)
Neural Networks Methods Results Conclusion 8
1 NN model for each postulated model
Non-directedNN model
StaticNN model
DirectedNN model
• Postulated model 1
• Postulated model 2: noise on the state
• Postulated model 3 : noise on the measurement
φ
q-1
u (k)
yp (k)
b (k+1)
yp(k+1)φRNu (k)
yp (k)g(k+1)
yp: observed output of the physical process
u (k): observed input of the physical process (rain)
b (k): noise
φ
q-1
u (k)xp (k)
b (k+1)
yp(k+1)
xp (k+1)
φu (k) yp(k+1)φRNu (k) g(k+1)
φRN
q-1
u (k)g (k)
g (k+1)
Neural Networks Methods Results Conclusion 9
1 NN model for each postulated model
Non-directedNN model
StaticNN model
DirectedNN model
• Postulated model 1
• Postulated model 2: noise on the state
• Postulated model 3 : noise on the measurement
φ
q-1
u (k)
yp (k)
b (k+1)
yp(k+1)φRNu (k)
yp (k)g(k+1)
yp: observed output of the physical process
u (k): observed input of the physical process (rain)
b (k): noise
φ
q-1
u (k)xp (k)
b (k+1)
yp(k+1)
xp (k+1)
φu (k) yp(k+1)φRNu (k) g(k+1)
φRN
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 AssimilationA 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-directedNN model
DirectedNN modelφRNu (k)
yp (k)g(k+1)
φRN
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 km2)
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
Durance watershed, France ( 2170 km2)
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.
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, Temperature, 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 catchmentPoor 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, AssimilationFernow 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 criterionBad simulations on low flows
Non-Directed, Assimilation, T°
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