In this talk, a reduced-cost ensemble Kalman filter (PC-EnKF) is implemented for the estimation of the model input parameters in the context of a front-tracking problem. The forecast step relies on a probabilistic sampling based on a Polynomial Chaos (PC) surrogate model. The performance of the hybrid PC-EnKF strategy is assessed for synthetic front-tracking test cases as well as in the context of wildfire spread, which features a front-like geometry and where the estimation targets are the unknown biomass fuel properties and the surface wind conditions. Results indicate that the hybrid PC-EnKF strategy features similar performance to the standard EnKF algorithm, without loss of accuracy but at a much reduced computational cost. Reference published in NHESS (2014) ➞ Rochoux, M.C., Ricci, S., Lucor, D., Cuenot, B., and Trouvé, A. (2014) Towards predictive data-driven simulations of wildfire spread. Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation, Natural Hazards and Earth System Sciences, Special Issue: Numerical Wildland Combustion, from the flame to the atmosphere, vol. 14, pp. 2951-2973, doi: 10.5194/nhess-14-2951-2014, published.

1. Reduced-cost ensemble Kalman filter
for parameter estimation!
Application to front-tracking problems!
Mélanie Rochoux!
in collaboration with S.Ricci, D. Lucor, B. Cuenot & A. Trouvé!
*
melanie.rochoux@cerfacs.fr!
MS-10 Reduced-order models for stochastic inverse problems – U626

2. INTRODUCTION ●●●●
Data assimilation: why? how?!
2 !Rochoux et al. – UNCECOMP 2015 – MS-10!
➙ Key idea: “optimal combination of observations and forward model”!
Determine best estimate of a dynamical system
given
Weather forecast!
Atm. chemistry!
Hydrology!
Biomechanics!
- Sparse and imperfect
- Relation between
observations and model
outputs
Observations
Numerical model
Model formulation
Model parameters
Initial condition
Forcing data
Mathematical technique based on estimation theory
• The “true state” is unknown and should be estimated
• Measurements and models are imperfect
• The estimate should be an optimal combination of both
measurements and models ➙ error minimization problem
Ex. applications

3. INTRODUCTION ●●●●
Data assimilation: why? how?!
➙ Key idea: “optimal combination of observations and forward model”!
Ensemble Kalman filter (EnKF)
• Forecast step ➙ uncertainty propagation
- Explicit propagation of the error statistics
- Nonlinear extension of the Kalman filter
• Analysis step ➙ Kalman filter update equation
!
reality
model forecast
Diagnostic!
measurements
analysis
Time!
Sequential approach
=
+
K
[
-­‐
]
Distance to observations!
G( )
Kalman gain matrix!
Stochastic characterization
Estimation of error
covariance matrices
Control variables!
3 !Rochoux et al. – UNCECOMP 2015 – MS-10!

4. INTRODUCTION ●●●●
Uncertainty quantification!
➙ Challenging idea: Use uncertainty quantification to overcome the slow
convergence rate and sampling errors of the Monte Carlo-based EnKF!
!
reality
model forecast
Diagnostic!
measurements
analysis
Time!
Sequential approach
Npc
X
k=1
ˆck k( )
●
Basis functions
4 !Rochoux et al. – UNCECOMP 2015 – MS-10!
=
+
K
[
-­‐
]
G( )
Control variables!
Hybrid Ensemble Kalman filter (PC-EnKF)
• Forecast step ➙ uncertainty propagation
- Use of surrogate model to compute model trajectories
- Polynomial Chaos (PC) expansion
• Analysis step ➙ Kalman filter update equation
!

5. INTRODUCTION ●●●●
Parameter estimation!
➙ Objective: Improvement of the forecast performance
• State estimation limitation ➙ no long persistence of the initial condition for a chaotic system
• Parameter estimation ➙ accounting for the temporal variability in the errors
Difficulties
➙ Possible nonlinear relationship between input parameters and model counterparts of the observations
➙ Existence of an evolution model for parameters?
!
Forward
model
Parameters
Initial condition
Boundary conditions
Comparison
Model outputs
Observations
Ensemble Kalman filter
Parameter estimation
State estimation
5 !Rochoux et al. – UNCECOMP 2015 – MS-10!

6. INTRODUCTION ●●●●
Outline!
!
Reduced-cost ensemble Kalman filter for parameter
estimation (PC-EnKF)!
!
u Algorithm!
u Application to wildfire spread forecasting!
• Front-tracking problem
• Synthetic case
• Controlled fire experiment
6 !Rochoux et al. – UNCECOMP 2015 – MS-10!

7. ALGORITHM ●●●
Standard EnKF!
Cxy = Pt
f
Gt
T
xt
a,(k)
= xt
f,(k)
+Cxy (Cyy + R)−1
(yt
o
+ξo,(k)
− yt
f,(k)
)
Prior
parameters
Prior
fire fronts
Posterior
parameters
xt
f,(1)
xt
f,(2)
xt
f,(Ne )
yt
f,(1)
yt
f,(2)
yt
f,(Ne )
Covariance matrices
EnKF
update
Cyy = GtPt
f
Gt
T
xt
a,(1)
xt
a,(2)
xt
a,(Ne )
yt
o
+ξo,(1)
Ke
t
Posterior
fire fronts
yt
a,(1)
yt
a,(2)
yt
a,(Ne )
EnKF
prediction
FORECAST ANALYSIS
EnKF
prediction
yt
o
+ξo,(Ne )
yt
o
+ξo,(2)
Gt Gt
➙ Key idea: 3D-Var approach with stochastically-based estimation of the error
covariance matrices over the assimilation cycle [t-1, t]
Specificities!!
• Random walk model for
parameter evolution
• Data randomization
➙ Burgers et al. 1998
• Limitations
➙ Slow convergence
rate (large number of
members)
➙ Sampling errors (Li
2008) – local & global
7 !Rochoux et al. – UNCECOMP 2015 – MS-10!

8. ALGORITHM ●●●
Hybrid PC-EnKF!
➙ Objective: Reduce computational cost of forward model integration
• Integrating Polynomial Chaos (PC) into forecast
• Control parameters projected onto a stochastic space spanned by orthogonal PC functions of
independent Gaussian random variables
Surrogate model!
Model inputs!
Model outputs!
Random event!
• Easy access to statistics (mean, covariance, ensemble sampling)
Ensemble sampling!
• Integrating Polynomial Chaos (PC) into observation
• Use the same basis for the model and for the data space (not obvious since observations and model
counterparts should remain uncorrelated, Evensen 2009)
8 !Rochoux et al. – UNCECOMP 2015 – MS-10!

9. ALGORITHM ●●●
Coupling PC and EnKF approaches!
EnKF prediction
Surrogate model
Surrogate model
Forward modelHermite quadrature Simulated fire fronts
Hermite polynomials Surrogate model
Forecast !
distribution!
➀!
Monte-Carlo sampling Predicted fire front
positions
Posterior estimate of
parameters
Updated fire front
positions
➁
➂
EnKF update
('q)q = 1, · · · , Npc
k = 1, · · · , Ne
k = 1, · · · , Ne
EnKF prediction
k = 1, · · · , Ne
k = 1, · · · , Ne
FIREFLY
j = 1, · · · , (Nquad)n
j = 1, · · · , (Nquad)n
⇣
x
f,(j)
t , !j
⌘ ⇣
y
f,(j)
t
⌘
pf
(xt)
yf
t = Gpc,t(xf
t)
⇣
x
f,(k)
t
⌘
⇣
x
a,(k)
t
⌘ ⇣
y
a,(k)
t
⌘
⇣
y
f,(k)
t
⌘
➙ Non-intrusive approach: PC used to build a surrogate model
of the observation operator
9 !Rochoux et al. – UNCECOMP 2015 – MS-10!

10. WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
Experimental grassland fire
(100m x 100m), N.S. Cheney,
Annaburroo site (Australia)!
➙ Wildfires feature a front-like
geometry at regional scales!
FRONT!
• Scales ranging from meters up to
several kilometers
• Thin flame zone propagating normal
to itself towards unburnt vegetation
• Local propagation speed of the front
called “rate of spread” (ROS)
10 !Rochoux et al. – UNCECOMP 2015 – MS-10!

11. WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
• 2-D state variable: reaction progress variable c
• Front marker: contour line c = 0.5
• Submodel for the local ROS along the normal
direction to the front
• Semi-empirical formulation (Rothermel)
• Function of the local environmental conditions
➙ Level-set-based front propagation simulator
ROS = f(uw, ↵sl, Mv, v, ⌃v, ...)
Simulated front c= 0.5 !
(x1
, y1
)
(x2
, y2
)
(x3
, y3
)
(x4
, y4
)
@c
@t
= ROS |rc|
11 !Rochoux et al. – UNCECOMP 2015 – MS-10!

12. WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
• 2-D state variable: reaction progress variable c
• Front marker: contour line c = 0.5
• Submodel for the local ROS along the normal
direction to the front
• Semi-empirical formulation (Rothermel)
• Function of the local environmental conditions
➙ Level-set-based front propagation simulator
ROS = f(uw, ↵sl, Mv, v, ⌃v, ...)
Simulated front c= 0.5 !
(x1
, y1
)
(x2
, y2
)
(x3
, y3
)
(x4
, y4
)
@c
@t
= ROS |rc|➙ Observation represented as a
discretized fire front!!
raw data: infrared imagery
12 !Rochoux et al. – UNCECOMP 2015 – MS-10!

13. WILDFIRE SPREAD APPLICATION ●●●
Synthetic experiment!
• Estimation of a uniform proportionality coefficient P in the ROS formulation
• True parameter at the tail of the Gaussian distribution associated with the forecast estimates
• Reduced-cost approach:
• 5 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 2 m
!
75 85 95 105 115 125
75
85
95
105
115
125
x [m]
y[m]
m
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]x−coordinate[m]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]
y−coordinate[m]
forecast true
trueforecast
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]
x−coordinate[m]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
y−coordinate[m]
analysis
analysis true
true
- forecast
- analysis
+ observations
quadrature points
▾ Response surface for the x-coordinate front marker m
◀ Fire front positions at time 50 s (analysis time)
13 !Rochoux et al. – UNCECOMP 2015 – MS-10!

14. WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
!
1min32s
50s
1min46s
1min04s
1min18s
Wind 1m/s
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
x [m]
y [m]
• Mid-Infrared imaging
• Quasi-homogeneous short
grass (22% moisture content)
• Mean wind speed: 1m/s in
northwestern direction
• Mean ROS = 2 cm/s
• Max. ROS = 5 cm/s
ANALYSIS
TIME
FORECAST
TIME
14 !Rochoux et al. – UNCECOMP 2015 – MS-10!

15. WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
• Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv)
• Reduced-cost approach:
• 25 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 5 cm
!
15 11500
Σv [1/m]Mv [%]
Σv [1/m]Mv [%]
15 11500
x-coordinate[m]y-coordinate[m]
13.8 2234513.8 22345
13.8 22345
Σv [1/m]Mv [%]
x-coordinate[m]y-coordinate[m]
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
m
▾ Response surface for the x-coordinate front marker m
◀ Fire front positions at time 1min18 s
quadrature
points
forecast!
analysis!
- Forecast (PC-EnKF)
- Analysis (PC-EnKF)
□ Analysis (standard EnKF)
+ observations
ANALYSIS TIME
15 !Rochoux et al. – UNCECOMP 2015 – MS-10!

16. WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
• Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv)
• Reduced-cost approach:
• 25 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 5 cm
!
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
m
ANALYSIS TIME
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
- Forecast (PC-EnKF)
- Analysis (PC-EnKF)
□ Analysis (standard EnKF)
+ observations
FORECAST TIME
Good behavior
of the PC
surrogate model!
16 !Rochoux et al. – UNCECOMP 2015 – MS-10!

17. CONCLUSION
Key ideas!
Rochoux et al (2014), NHESS!
Rochoux et al (2012), CTR brief!
➙ Reduced-cost ensemble Kalman filter (PC-EnKF)
for parameter estimation in front-tracking problems!
• Stand-alone parameter estimation ➙ forecast improvement!
• Prototype able to address multi-parameter sequential estimation at a reduced cost
• Spatially-uniform and constant parameters over the time window
• Application: Reduced-scale wildfire spread problem
➙ Need to extend the strategy at regional scales
➙ Need to combine parameter estimation and state estimation approaches to
treat anisotropic uncertainties
!
17 !Rochoux et al. – UNCECOMP 2015 – MS-10!

18. CONCLUSION
Key ideas!
Rochoux et al (2014), NHESS!
Rochoux et al (2012), CTR brief!
➙ Reduced-cost ensemble Kalman filter (PC-EnKF)
for parameter estimation in front-tracking problems!
• Stand-alone parameter estimation ➙ forecast improvement!
• Prototype able to address multi-parameter sequential estimation at a reduced cost
• Spatially-uniform and constant parameters over the time window
• Application: Reduced-scale wildfire spread problem
➙ Need to extend the strategy at regional scales
➙ Need to combine parameter estimation and state estimation approaches to
treat anisotropic uncertainties
• Front-tracking problem ➙ dynamically-evolving observation operator over time!
• Prototype able to track coherent features
• Unusual application of the EnKF algorithm
➙ Need to test the sensitivity of the hybrid data assimilation algorithm to different
representations of the front
!
18 !Rochoux et al. – UNCECOMP 2015 – MS-10!

19. *
Melanie.Rochoux@cerfacs.fr
Thank you for your attention!
Any question?

20. Cxy = Pt
f
Gt
T
xt
a,(k)
= xt
f,(k)
+Cxy (Cyy + R)−1
(yt
o
+ξo,(k)
− yt
f,(k)
)
Prior
parameters
Prior
fire fronts
Posterior
parameters
xt
f,(1)
xt
f,(2)
xt
f,(Ne )
yt
f,(1)
yt
f,(2)
yt
f,(Ne )
Covariance matrices
EnKF
update
Cyy = GtPt
f
Gt
T
xt
a,(1)
xt
a,(2)
xt
a,(Ne )
yt
o
+ξo,(1)
Ke
t
Posterior
fire fronts
yt
a,(1)
yt
a,(2)
yt
a,(Ne )
EnKF
prediction
FORECAST ANALYSIS
EnKF
prediction
yt
o
+ξo,(Ne )
yt
o
+ξo,(2)
Gt Gt
ALGORITHM ●●●
Standard EnKF!
➙ Key idea: 3D-Var approach with stochastically-based estimation of the error
covariance matrices over the assimilation cycle [t-1, t]
!
• Local error (over one assimilation cycle)
!
• Global error (over all assimilation cycles)
Specificities!!
• Random walk model for
parameter evolution
• Data randomization
➙ Burgers et al. 1998
• Limitations
➙ Slow convergence
rate (large number of
members)
➙ Sampling errors (Li
2008) – local & global
20 !Rochoux et al. – UNCECOMP 2015 – MS-10!