Preliminary Results of Data Assimilationwithin the Modeling Platform Polyphemus

L.Wu, V.Mallet, M.Bocquet, and B.Sportisse

CLIME Project (INRIA / ENPC)Research and Teaching Center in Atmospheric Environment

École Nationale des Ponts et Chaussées / EDF R&D

Les premières Journées Nationales des ARC, 17-18octobre 2006

Outline

1 Quick Introduction to PolyphemusPurpose and Overall StructureQuick Review of its Content

2 Data AssimilationIntroductionData Assimilation System Within PolyphemusAssimilation Algorithms and Preliminary Results

Polyphemus: Purpose

Purpose

Comprehensive and perennial platform for air quality modeling,with advanced forecasting methods such as data assimilationand ensemble forecast.

Highlights

Designed to share developments and to host other models

Wide range of applications

How?

Modern programming (priority on C++ and Python)

Open source (GNU GPL)

Developed by CEREA and CLIME, supported by IRSN andINERIS

Polyphemus: Overall Structure

Quick Review of Polyphemus Content

Libraries: AtmoData, AtmoPy, etc.

Data post- and preprocessing, physical parameterizations

Models: Castor, Polair3D, etc.

3D chemistry-transport models, Gaussian models

Modules

Photochemistry, aerosols, radionuclides, transport

Drivers

Data assimilation (OI, EnKF, RRSRQT, 4D-Var), local-scalesimulation

Introduction to Data Assimilation

Objective and content

Universal model-experimentproblem

State estimation from diverseinformation

Components

Model (physics)

Data (observation)

Assimilationalgorithms

State-of-the-artComponents In general ADOQA in action

Model Meteorology, oceanography Photochemistry (Polair3D),hydrology, agronomy, ... aerosol, ...

Data In situ, radar, Ground stations,Satellite, ... Ozone column info (Berroir)

Image sequences (Huot et al.)

Algorithms Sequential and variational Maximum entropy (Bocquet)

Difficulties

Nonlinearity + high dimensionConstraint on data assimilation window; reduction on modelor error covariance matricesHighly nonlinear reaction item of chemistry; but stablebecause of the eigenvalues of the Jacobian are negative

Error modelingNeeded because of the high uncertainties of the modelMulti-species, multivariate error covariance

Data Assimilation: System Design

Idea: Object-orient; data, model and algorithmcomponents are independent of one another.

Model facts

∂ci∂t

= −div(Vci) + div(K∇ci) + χi(c) + Si − PiGround BC: K∇ci · n = Ei − DiDimension of state variables 105 − 107

Uncertainties due to physics parameterization andnumerical approximations (Mallet and Sportisse 2006).

Ozone daily profiles from 48members

Relative standard deviationover 15% over Europe

Observation managements

EMEP observation network supported, to be extended toPioneer, BDQA; real and synthetic observations.

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Algorithms: Sequential and Variational

Data assimilation: determining xa given background xb,observation y, and statistics information R, B.

Notations

x : model state vector.

xt : true state.

xb : background estimates.

xa : analysis.

y : observation vector.

�b : xb − xt backgrounderrors.

�a : xa − xt analysis errors.

H: Observation operator

B: (�b − �̄b)(�b − �̄b)T

background errorcovariance matrix.

A: (�a − �̄a)(�a − �̄a)T

analysis error covariancematrix.

�o : y − H(xt) observationerrors.

R: (�o − �̄o)(�o − �̄o)T

observation errorcovariance matrix.

Linear combination of xb and y: optimal interpolation

Formulae

xa = xb + K(y − H(xb)),K = BHT (HBHT + R)−1.

Study

B: using Balgovind correlationfunction.

f (r) =(1 + rL

)exp

(− rL

)vb

Perturbs ICs (upper).

Biased model (lower).

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xb generated by model dynamics M : (extended)Kalman filter

Formulae

Model error:�m,k−1 = Mk−1→k(xt ,k−1) − xt ,k

Qk−1: model error covariance matrix.

Forecast formula:xfk = Mk−1→k(x

ak−1)

Pfk = Mk−1→kPak−1M

Tk−1→k + Qk−1

Analysis formula:xak = x

fk + Kk(yk − H(x

fk)),

Kk = PfkHTk (Hk P

fk H

Tk + Rk )

−1,Pak = (I − Kk Hk)P

fk

Reduced rank square root Kalman filter: RRSQRTHeemink et al. 2001

Formulae

Initialization: xa0, La0 = [l

a,10 , . . . , l

a,q0 ]

Forecast formula:xfk = Mk−1→k(x

ak−1)

lf ,ik =1�{Mk−1→k(xak−1 + �l

a,ik−1) − Mk−1→k(x

ak−1)}, � = 1

L̃fk = [lf ,1k , . . . , l

f ,qk , Q

12k−1], L

fk = Π

fk L̃

fk

Analysis formula:Pfk = L

fkL

f ,Tk

Kk = PfkHTk (Hk P

fk H

Tk + Rk )

−1,xak = x

fk + Kk(yk − H(x

fk)),

L̃ak = [(I − KkHk )Lfk , Kk R

12k ], L

ak = Π

ak L̃

ak

Reduced rank square root Kalman filter: RRSQRT

Study

The column{

Q12k−1

}

i=

(Mk−1→k(xak−1, d + ε · wi) − Mk−1→k(x

ak−1, d)

)/ε, ε = 1

Experiment settingsAssimilation: 00H00 - 10H00, July 5Prediction: 11H00, July 5 - 10H00, July 9

Number of mode q = 10, column number of Q12k set to 10,

column number of R12k set to 10.

Perturbed fields are attenuation, deposition velocities,photolysis rates, surface emissions, and boundaryconditions for O3.

RRSQRT: Preliminary results

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Ensemble Kalman filter: EnKFEvensen 1994

Formulae

Initialization: given initial pdf p(xt0), an ensemble of rmembers are generated randomly,

{xa,i0 , , i = 1, . . . , r}

x̄a0 =1r

r∑

i=1xa,i0

P̃a0 =1

r−1

r∑

i=1

(

xa,i0 − x̄a0

) (

xa,i0 − x̄a0

)T

Forecast formula:xf ,ik = Mk−1[x

a,ik−1] + η

ik−1

P̃fk =1

r−1

r∑

i=1

(

xf ,ik − x̄fk

)(

xf ,ik − x̄fk

)T

where x̄fk is the mean of ensemble {xf ,ik , i = 1, .., r}

Ensemble Kalman filter: EnKF

Formulae

Analysis formula:

xa,ik = xf ,ik + K̃k

(

yik − Hk [xf ,ik ]

)

xak =1r

r∑

i=1xa,ik

P̃ak =1

r−1

r∑

i=1

(

xa,ik − xak

)(

xa,ik − xak

)T

Study

Assimilation: 00H00 - 06H00, July 5

Prediction: 07H00, July 5 - 06H00, July 9

Number of ensemble members r = 30.

Perturbed fields are same as those of RRSQRT.

EnKF: Preliminary results

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Four-Dimensional Variational Assimilation: 4D-VarBouttier and Courtier 1999

A cost function J(x) that deals with a set of obs.:

12

(x − xb)T B−1(x − xb)︸ ︷︷ ︸

Jb

+12

N∑

k=0

Joi︷ ︸︸ ︷

(yk − Hk (xk ))T R−1k (yk − Hk(xk ))

︸ ︷︷ ︸

Jowhere the assimilation window is 0 − N,

xk = M0→k(x) = Mk Mk−1 . . . M1x

The gradient is calculated by the backward intergration ofadjoint model

x̃N = 0For k = N, . . . , 1, calculates x̃k−1 = MTk

(x̃k − HTk dk

), where

dk = R−1k (yk − Hk (xk ))x̃0 := x̃0 − HT0 (d0) gives the gradient of Jo with respect to x

Implementation: Adjoint Coding and GradientChecking

Adjoint model obtained by automatic differentiation(O∂yssee, TROPICS team, INRIA Sophia-Antipolis)

Checks gradient calculation by finite difference

ϕ(α) =Jo(x + αh) − Jo(x)

α 〈∇xJo, h〉, α → 0

Checking case: synthetic observations, Rk , Hk identitymatrix

α ϕ(α) α ϕ(α)

10−1 0.97169086073122756808 10−6 1.000021038656205396610−2 0.99714851925111402942 10−7 0.9999271202068579222910−3 0.999714847771238313 10−8 0.9987313992082194058510−4 0.99997151406647999394 10−9 0.9861437181498253767810−5 0.99999664594783310712 10−10 0.87396334809574471869

Conclusion

Platform readyEasily extended to new features

Aerosol modelAdvanced data assimilation methods for ozone columnobservationsNonlinear data assimilation methods, i.e. Maximum entropyfilter (M.Bocquet), particle filter (Project ASPI)

Open scientific issuesEnsemble initializationBackground and model error modeling and itsparameterization

ApplicationsLong-run objective - operational platform (ensembleprediction already operational)Open to other models / teams (GNU GPL), educationpurpose

Quick Introduction to PolyphemusPurpose and Overall StructureQuick Review of its Content

Data AssimilationIntroductionData Assimilation System Within PolyphemusAssimilation Algorithms and Preliminary Results