1. Laboratory of ecohydrology
´Ecole polytechnique f´ed´erale
de Lausanne
Data assimilation for distributed models:
an overview of applications with CATHY
Damiano Pasetto
Workshop on coupled hydrological modeling
Padova, 24 Sept. 2015
Damiano Pasetto DA for distributed models Padova - 24 September 2015
2. Table of Contents
Table of Contents
1 Introduction
2 Data assimilation methods
3 Hydrological applications
Damiano Pasetto DA for distributed models Padova - 24 September 2015
3. Introduction Motivations
State-space model
˙x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0, ∞] transient model
y∗
k
y∗
k observations
x(t) state variables
Damiano Pasetto DA for distributed models Padova - 24 September 2015
4. Introduction Motivations
State-space model
˙x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0, ∞] transient model
y∗
k
y∗
k observations
x(t) state variables
λ parameters
q(t) ATM forcings
x(0) initial condition
w(t) model structural error
Damiano Pasetto DA for distributed models Padova - 24 September 2015
5. Introduction Motivations
State-space model
˙x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0, ∞] transient model
y∗
k ↔ yk = h (x, tk) + vk k = 1, . . . observation model
y∗
k observations
x(t) state variables
λ parameters
q(t) ATM forcings
x(0) initial condition
w(t) model structural error
vk measurement error
Damiano Pasetto DA for distributed models Padova - 24 September 2015
6. Introduction Motivations
State-space model
˙x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0, ∞] transient model
y∗
k ↔ yk = h (x, tk) + vk k = 1, . . . observation model
y∗
k observations
x(t) state variables p (x(t))
λ parameters
q(t) ATM forcings
x(0) initial condition
w(t) model structural error
vk measurement error
Damiano Pasetto DA for distributed models Padova - 24 September 2015
7. Introduction Motivations
State-space model
˙x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0, ∞] transient model
y∗
k ↔ yk = h (x, tk) + vk k = 1, . . . observation model
y∗
k observations
x(t) state variables p (x(t))
λ parameters p(λ)
q(t) ATM forcings
x(0) initial condition
w(t) model structural error
vk measurement error
Damiano Pasetto DA for distributed models Padova - 24 September 2015
8. Introduction Motivations
Motivations
Hydrological forecasting is subject to many sources of uncertainty
Initial condition
Forcing terms
Model parameters
(Model itself?)
Data Assimilation (DA)
Correct the model forecast considering the measurements
State . . . ˆxk−1 → x−
k ˆxk → x−
k+1 . . .
↓ ↑ ↓
Observations . . . y−
k ↔ y∗
k . . .
Damiano Pasetto DA for distributed models Padova - 24 September 2015
9. Introduction Motivations
Motivations
Hydrological forecasting is subject to many sources of uncertainty
Initial condition
Forcing terms
Model parameters
(Model itself?)
Data Assimilation (DA)
Correct the model forecast considering the measurements
State . . . ˆxk−1 → x−
k ˆxk → x−
k+1 . . .
↓ ↑ ↓
Observations . . . y−
k ↔ y∗
k . . .
Forecast pdf: π−(x(tk) | y1, . . . , yk−1)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
10. Introduction Motivations
Motivations
Hydrological forecasting is subject to many sources of uncertainty
Initial condition
Forcing terms
Model parameters
(Model itself?)
Data Assimilation (DA)
Correct the model forecast considering the measurements
State . . . ˆxk−1 → x−
k ˆxk → x−
k+1 . . .
↓ ↑ ↓
Observations . . . y−
k ↔ y∗
k . . .
Forecast pdf: π−(x(tk) | y1, . . . , yk−1)
Filtering pdf: π+(x(tk) | y1, . . . , yk−1, yk)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
11. Introduction A simple example with CATHY
Example: application to CATHY (CATchment HYdrology)
Coupled surface/subsurface model
Richards equation:
Sw(ψ)Ss
∂ψ
∂t
+ φ
∂Sw(ψ)
∂t
= · [KsKrw(Sw(ψ)) ( ψ + ηz)] + qss(h)
1-D path-based surface routing:
∂Q
∂t
+ ck
∂Q
∂s
= Dh
∂2Q
∂s2
+ ckqs(h, ψ)
BC-switching/forcing algorithm
(Camporese et al. 2010, WRR)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
12. Introduction A simple example with CATHY
Example: application to CATHY (CATchment HYdrology)
Coupled surface/subsurface model
Richards equation:
Sw(ψ)Ss
∂ψ
∂t
+ φ
∂Sw(ψ)
∂t
= · [KsKrw(Sw(ψ)) ( ψ + ηz)] + qss(h)
1-D path-based surface routing:
∂Q
∂t
+ ck
∂Q
∂s
= Dh
∂2Q
∂s2
+ ckqs(h, ψ)
BC-switching/forcing algorithm
State variables: x = {ψ, Q}.
Measures: piezometric head, soil moisture, streamflow, electric
potential (ERT).
(Camporese et al. 2010, WRR)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
13. Introduction A simple example with CATHY
DA: example on the V-catchment
3 m soil depth
Assimilation of streamflow
Uncertainty:
Initial conditions
ATM forcings
Damiano Pasetto DA for distributed models Padova - 24 September 2015
14. Introduction A simple example with CATHY
Forecast considering model uncertainties (open loop)
0 1800 3600 5400 7200 9000 10800 12600 14400
0
1
2
3
4
5
6Streamflow(m
3
/s)
TRUE
Observations
Open Loop
0 1800 3600 5400 7200 9000 10800 12600 14400
Time (s)
1.939
1.940
1.941
1.942
1.943
1.944
WaterStorage(10
6
m
3
)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
15. Introduction A simple example with CATHY
Assimilation of measurement of streamflow
0 1800 3600 5400 7200 9000 10800 12600 14400
0
1
2
3
4
5
6Streamflow(m
3
/s)
TRUE
Observations
SIR
0 1800 3600 5400 7200 9000 10800 12600 14400
Time (s)
1.939
1.940
1.941
1.942
1.943
1.944
WaterStorage(10
6
m
3
)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
16. Data assimilation methods EnKF and SIR
Forecast step: MC simulation
xi
0 ∼ p(x0), i = 1, . . . , N Initial samples
xi,−
k = f(xi
k−1, λi
, qi
k, tk) + wi
k Forecast
Damiano Pasetto DA for distributed models Padova - 24 September 2015
17. Data assimilation methods EnKF and SIR
Forecast step: MC simulation
xi
0 ∼ p(x0), i = 1, . . . , N Initial samples
xi,−
k = f(xi
k−1, λi
, qi
k, tk) + wi
k Forecast
Analysis step
Ensemble Kalman filter (EnKF, Evensen 1994): Kalman gain
ˆxi
k = xi,−
k + Kk y∗
k − h(xi,−
k )
Damiano Pasetto DA for distributed models Padova - 24 September 2015
18. Data assimilation methods EnKF and SIR
Forecast step: MC simulation
xi
0 ∼ p(x0), i = 1, . . . , N Initial samples
xi,−
k = f(xi
k−1, λi
, qi
k, tk) + wi
k Forecast
Analysis step
Ensemble Kalman filter (EnKF, Evensen 1994): Kalman gain
ˆxi
k = xi,−
k + Kk y∗
k − h(xi,−
k )
Sequential Importance Resampling (SIR):
weighted realizations xi
k, ωi
k
update weights with the likelihood and normalize
ωi
k = Cωi
k−1L(y∗
k | xi,−
k )
duplicate particles that have largest weights.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
19. Data assimilation methods EnKF and SIR
Damiano Pasetto DA for distributed models Padova - 24 September 2015
−x ,N−1
{ }π
−
k 1:k−1
(x |y ) k
20. Hydrological applications 1. Geophysical coupled inversion
1. Geophysical coupled inversion: Electrical Resistivity Tomography
(Rossi et al. 2015, AWR)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
21. Hydrological applications 1. Geophysical coupled inversion
Iterative particle filter
(Manoli et al. 2015, JCP)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
22. Hydrological applications 1. Geophysical coupled inversion
Damiano Pasetto DA for distributed models Padova - 24 September 2015
23. Hydrological applications 2. Landscape Evolution Observatory (LEO)
2. Landscape Evolution Observatory (LEO)
Three convergent landscapes
30 m long, 11 m wide, 1 m soil
10 degrees average slope
Environmentally controlled
greenhouse facility
Landscape instrumentation
rainfall simulator
(3-45 mm/h)
10 load cells
6 flow meters for
seepage face
outflow
1,835 sensors
embedded in the
soil
Damiano Pasetto DA for distributed models Padova - 24 September 2015
24. Hydrological applications 2. Landscape Evolution Observatory (LEO)
First experiment at LEO (18 February 2013)
Experiment setup:
Unsaturated initial
conditions
Imposed rainfall:
≈12 mm/h
With homogeneous soil,
steady state expected
after 36 h
After the experiment: the rainfall was
stopped after 22 h due to the occurrence
of overland flow.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
25. Hydrological applications 2. Landscape Evolution Observatory (LEO)
Synthetic scenario reproducing Experiment 1 at LEO
Assumption: Y = log(KS) is a Gaussian random field with exponential
covariance function. E[KS] = 10−4 m/s with coefficient of variation
100% (µY = −9.56, σY = 0.83)
Test case 1 (TC1): λx = λy = 8 m; λz= 0.5 m
Test case 2 (TC2): λx = λy = 4 m; λz= 0.25 m
Number of grid cells: 60×22×20= 26400
Sensor failure analysis
The assimilation is repeated decreasing the number of measurements,
from m=496 to m= 21 active sensors.
(Pasetto et al. 2015, AWR)
Damiano Pasetto DA for distributed models Padova - 24 September 2015
26. Hydrological applications 2. Landscape Evolution Observatory (LEO)
−5 0 5
5
10
15
20
25
d= 0.00÷0.05 m
x (m)
y(m)
−5 0 5
5
10
15
20
25
d= 0.15÷0.20 m
x (m)
−5 0 5
5
10
15
20
25
d= 0.30÷0.35 m
x (m)
−5 0 5
5
10
15
20
25
d= 0.50÷0.55 m
x (m)
−5 0 5
5
10
15
20
25
d= 0.80÷0.85 m
x (m)
−5 0 5
5
10
15
20
25
d= 0.95÷1.00 m
x (m)
10
−5
10
−4
10
−3
KS
(m/s)
True
−5 −2 0 2 5
5
10
15
20
25
d= 0.00÷0.05 m
x (m)
y(m)
−5 −2 0 2 5
5
10
15
20
25
d= 0.15÷0.20 m
x (m)
−5 −2 0 2 5
5
10
15
20
25
d= 0.30÷0.35 m
x (m)
−5 −2 0 2 5
5
10
15
20
25
d= 0.50÷0.55 m
x (m)
−5 −2 0 2 5
5
10
15
20
25
d= 0.80÷0.85 m
x (m)
−5 −2 0 2 5
5
10
15
20
25
d= 0.95÷1.00 m
x (m)
10
−5
10
−4
10
−3
KS
(m/s)
m=
496
True and estimated spatial distributions of KS in TC2.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
27. Hydrological applications 2. Landscape Evolution Observatory (LEO)
0
0.5
1
1.5
OverlandFlow(m
3
/h)
Ensemble
Ensemble Mean
True
90% C.I.
TC1 (long correlation length)
0
0.5
1
1.5
SeepageFaceFlow(m
3
/h)
0 4 8 12 16 20 24 28 32 36
Time t (h)
40
60
80
100
120
WaterStorage(m
3
)
TC2 (short correlation length)
0 4 8 12 16 20 24 28 32 36
Time t (h)
Open loop: model response with 200 random realizations of the prior
distribution of Y = log(KS) without data assimilation.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
28. Hydrological applications 2. Landscape Evolution Observatory (LEO)
0
0.5
Overland(m
3
/h)
True
m= 496
m= 196
m= 46
m= 21
TC1 (long correlation length)
0
0.5
1
1.5
Seepage(m
3
/h)
40
60
80
100
120
Storage(m
3
)
0 4 8 12 16 20 24 28
Time t (h)
0.001
0.01
RMSEonvwc
TC2 (short correlation length)
0 4 8 12 16 20 24 28 32 36
Time t (h)
Model response with the calibrated saturated hydraulic conductivity
Damiano Pasetto DA for distributed models Padova - 24 September 2015
29. Conclusions
Conclusions
Data assimilation methods help improve the forecast and reduce the
uncertainty of high dimensional hydrological models.
Data assimilation methods allow the online estimation of both the state
variables and the model parameters.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
30. Conclusions
Conclusions
Data assimilation methods help improve the forecast and reduce the
uncertainty of high dimensional hydrological models.
Data assimilation methods allow the online estimation of both the state
variables and the model parameters.
Work in progress
Covariance localization and ensemble inflation to minimize
ill-conditioning and filter inbreeding in the EnKF update.
Update step performed with a combination of EnKF and SIR
(Gaussian Mixture Filters)
Surrogate models to accelerate the Monte Carlo simulation.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
31. Conclusions
Thank you for your attention
References
D Pasetto, M Camporese, and M Putti. Ensemble Kalman filter versus particle filter for a
physically-based coupled surface-subsurface model, Adv Water Resources, 2012.
G Manoli, M Rossi, D Pasetto, R Deiana, S Ferraris, G Cassiani, and M Putti. An iterative
particle filter approach for coupled hydro-geophysical inversion of a controlled infiltration
experiment, J Comp Phys, 2015.
M Rossi, G Manoli, D Pasetto, R Deiana, S Ferraris, C Strobbia, M Putti, G Cassiani.
Coupled inverse modeling of a controlled irrigation experiment using multiple
hydro-geophysical data, Adv Water Resources, 2015.
D Pasetto, G-Y Niu, L Pangle, C Paniconi, M Putti, PA Troch. Impact of sensor failure on
the observability of flow dynamics at the Biosphere 2 LEO hillslopes, Adv Water
Resources, 2015.
Damiano Pasetto DA for distributed models Padova - 24 September 2015
