Damiano Pasetto

Laboratory of ecohydrologyEcole polytechnique federalede Lausanne

Data assimilation for distributed models:an overview of applications with CATHY

Damiano Pasetto

Workshop on coupled hydrological modelingPadova, 24 Sept. 2015

Damiano Pasetto DA for distributed models Padova - 24 September 2015

Table of Contents

Table of Contents

1 Introduction

2 Data assimilation methods

3 Hydrological applications


Introduction Motivations

State-space model

{x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0,∞] transient modely∗k

↔ yk = h (x, tk) + vk k = 1, . . . observation model

y∗k observationsx(t) state variables

p (x(t))λ parameters p(λ)q(t) ATM forcingsx(0) initial conditionw(t) model structural errorvk measurement error



State-space model

{x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0,∞] transient modely∗k

↔ yk = h (x, tk) + vk k = 1, . . . observation model


p (x(t))

λ parameters

p(λ)

q(t) ATM forcingsx(0) initial conditionw(t) model structural error

vk measurement error



State-space model

{x(t) = f (x(t), λ, q(t), t) + w(t) t ∈ [0,∞] transient modely∗k ↔ yk = h (x, tk) + vk k = 1, . . . observation model


p (x(t))

λ parameters

p(λ)

q(t) ATM forcingsx(0) initial conditionw(t) model structural errorvk measurement error



State-space model


y∗k observationsx(t) state variables p (x(t))λ parameters

p(λ)

q(t) ATM forcingsx(0) initial conditionw(t) model structural errorvk measurement error



State-space model


y∗k observationsx(t) state variables p (x(t))λ parameters p(λ)q(t) ATM forcingsx(0) initial conditionw(t) model structural errorvk measurement error



MotivationsHydrological forecasting is subject to many sources of uncertainty

Initial conditionForcing termsModel parameters(Model itself?)

Data Assimilation (DA)Correct the model forecast considering the measurements

State . . . xk−1 → x−k xk → 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)






State . . . xk−1 → x−k xk → x−k+1 . . .↓ ↑ ↓


Forecast pdf: π−(x(tk) | y1, . . . , yk−1)

Filtering pdf: π+(x(tk) | y1, . . . , yk−1, yk)






State . . . xk−1 → x−k xk → x−k+1 . . .↓ ↑ ↓


Forecast pdf: π−(x(tk) | y1, . . . , yk−1)Filtering pdf: π+(x(tk) | y1, . . . , yk−1, yk)


Introduction A simple example with CATHY

Example: application to CATHY (CATchment HYdrology)

Coupled surface/subsurface modelRichards 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, electricpotential (ERT).

(Camporese et al. 2010, WRR)Damiano Pasetto DA for distributed models Padova - 24 September 2015


DA: example on the V-catchment

3 m soil depthAssimilation of streamflow

Uncertainty:Initial conditionsATM forcings



Forecast considering model uncertainties (open loop)

0 1800 3600 5400 7200 9000 10800 12600 144000

1

2

3

4

5

6St

ream

flow

(m

3 /s) TRUE

ObservationsOpen Loop

0 1800 3600 5400 7200 9000 10800 12600 14400Time (s)

1.939

1.940

1.941

1.942

1.943

1.944

Wat

er S

tora

ge (

106 m

3 )



Assimilation of measurement of streamflow

0 1800 3600 5400 7200 9000 10800 12600 144000

1

2

3

4

5

6St

ream

flow

(m

3 /s) TRUE

ObservationsSIR

0 1800 3600 5400 7200 9000 10800 12600 14400Time (s)

1.939

1.940

1.941

1.942

1.943

1.944

Wat

er S

tora

ge (

106 m

3 )


Data assimilation methods EnKF and SIR

Forecast step: MC simulation

xi0 ∼ p(x0), i = 1, . . . , N Initial samples

xi,−k = f(xik−1, λi, qik, tk) + wi

k Forecast

Analysis stepEnsemble Kalman filter (EnKF, Evensen 1994): Kalman gain

xik = xi,−k +Kk

(y∗k − h(x

i,−k )

)

Sequential Importance Resampling (SIR):weighted realizations

(xik, ω

ik

)update weights with the likelihood and normalize

ωik = Cωi

k−1L(y∗k |xi,−k )

duplicate particles that have largest weights.



Forecast step: MC simulation

xi0 ∼ p(x0), i = 1, . . . , N Initial samples

xi,−k = f(xik−1, λi, qik, tk) + wi

k Forecast

Analysis stepEnsemble Kalman filter (EnKF, Evensen 1994): Kalman gain

xik = xi,−k +Kk

(y∗k − h(x

i,−k )

)Sequential Importance Resampling (SIR):

weighted realizations(xik, ω

ik

)update weights with the likelihood and normalize

ωik = Cωi

k−1L(y∗k |xi,−k )

duplicate particles that have largest weights.


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


Iterative particle filter

(Manoli et al. 2015, JCP)Damiano Pasetto DA for distributed models Padova - 24 September 2015

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 controlledgreenhouse facility

Landscape instrumentation

rainfall simulator(3-45 mm/h)

10 load cells

6 flow meters forseepage faceoutflow

1,835 sensorsembedded in thesoil



First experiment at LEO (18 February 2013)

Experiment setup:

Unsaturated initialconditions

Imposed rainfall:≈12 mm/h

With homogeneous soil,steady state expectedafter 36 h

After the experiment: the rainfall wasstopped after 22 h due to the occurrenceof overland flow.



Synthetic scenario reproducing Experiment 1 at LEO

Assumption: Y = log(KS) is a Gaussian random field with exponentialcovariance function. E[KS ] = 10−4 m/s with coefficient of variation100% (µY = −9.56, σY = 0.83)

Test case 1 (TC1): λx = λy = 8 m; λz= 0.5 mTest case 2 (TC2): λx = λy = 4 m; λz= 0.25 m

Number of grid cells: 60×22×20= 26400

Sensor failure analysisThe assimilation is repeated decreasing the number of measurements,from m=496 to m= 21 active sensors.

(Pasetto et al. 2015, AWR)



−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

K S (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

K S (m

/s)

m= 496

True and estimated spatial distributions of KS in TC2.Damiano Pasetto DA for distributed models Padova - 24 September 2015


0

0.5

1

1.5

Ove

rland

Flo

w (m

3 /h)

EnsembleEnsemble MeanTrue90% C.I.

TC1 (long correlation length)

0

0.5

1

1.5

Seep

age

Face

Flo

w (m

3 /h)

0 4 8 12 16 20 24 28 32 36Time t (h)

40

60

80

100

120

Wat

er S

tora

ge (m

3 )

TC2 (short correlation length)

0 4 8 12 16 20 24 28 32 36Time t (h)

Open loop: model response with 200 random realizations of the priordistribution of Y = log(KS) without data assimilation.



0

0.5O

verla

nd (m

3 /h)

Truem= 496m= 196m= 46m= 21

TC1 (long correlation length)

0

0.5

1

1.5

Seep

age

(m3 /h

)

406080

100120

Stor

age

(m3 )

0 4 8 12 16 20 24 28Time t (h)

0.001

0.01

RM

SE o

n vw

c

TC2 (short correlation length)

0 4 8 12 16 20 24 28 32 36Time t (h)

Model response with the calibrated saturated hydraulic conductivity


Conclusions

Conclusions

Data assimilation methods help improve the forecast and reduce theuncertainty of high dimensional hydrological models.

Data assimilation methods allow the online estimation of both the statevariables and the model parameters.

Work in progress

Covariance localization and ensemble inflation to minimizeill-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.


Conclusions

Thank you for your attention

ReferencesD Pasetto, M Camporese, and M Putti. Ensemble Kalman filter versus particle filter for aphysically-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 iterativeparticle filter approach for coupled hydro-geophysical inversion of a controlled infiltrationexperiment, 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 multiplehydro-geophysical data, Adv Water Resources, 2015.D Pasetto, G-Y Niu, L Pangle, C Paniconi, M Putti, PA Troch. Impact of sensor failure onthe observability of flow dynamics at the Biosphere 2 LEO hillslopes, Adv WaterResources, 2015.


Conclusions


Damiano Pasetto

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