Outline

NOAA/NCEP/EMC, Camp Springs, MD 3 November 2005

The Maximum Likelihood Ensemble Filter development at the Colorado State University

Milija Zupanski

Cooperative Institute for Research in the AtmosphereColorado State University

Fort Collins, CO 80523-1375E-mail: [email protected]

In collaboration with: Colorado State University: D. Zupanski, S. Fletcher, D. Randall, R. Heikes

G. Carrio, W. Cotton Florida State University: I.M. Navon, B. Uzunoglu NOAA/NCEP: Zoltan Toth, Mozheng Wei, Yucheng SongComputational Support: NCEP IBM SP (frost) ,NCAR SCD (bluesky)

Outline

Motivation

Hessian preconditioning

Maximum Likelihood Ensemble Filter (MLEF)

Preliminary results

Double-resolution MLEF

Future research directions

Motivation Uncertainties

- Assign a degree of confidence in the produced analysis/forecast

- Transport in time of the forecast/analysis state vector + uncertainty

Universality of assimilation/prediction

- Same system can be used in wide range of applications

- Portability

Single assimilation/prediction system- Complete feed-back between uncertainties

- Easy to maintain and upgrade

Fewer assumptions/restrictions

- Non-differentiable operators (discontinuity)

- Highly nonlinear operators (microphysics, clouds)

User-friendly: Non-experts can use DA and EF

- Allow more people to enjoy the benefit of new research

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1][][

2

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obsf-

fTf P

Hessian Preconditioning

Variational cost function

HHPx

112

2

RT-f

J

Hessian

Ideal preconditioning is a square-root of the Hessian matrix

Inverse Hessian

2/12/112/2/1

1

2

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TTff

JPHPHPIP

x

R

TJEE

2

2xzTf Exx

IEEEEEE22

TTTJJ 1

21

2 xz

Perfect preconditioning: Hessian in minimization space is an identity matrix !

Hessian condition number in variational data assimilation ~ 60-100 !

Hessian preconditioning

J=const.

min

x0

xmin

J=const.

Physical space

Preconditioning space

-g

-gx

One-iteration minimization for quadratic cost function !

IMPACT OF MATRIX Z TZ IN HESSIAN PRECONDITIONING

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1 6 11 16 21 26 31 36 41 46 51

Number of iterations

Co

st fu

nct

ion

21/f

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ff-T HPRHIE PP

Nonlinear observation operators

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Two strategies for nonlinear observation operators:

(1) Use linear KF solution, combined with nonlinear operators in covariance calculation

- EnKF algorithms

- EKF

(2) Directly search for nonlinear solution by minimizing non-quadratic cost function

- Maximum Likelihood Ensemble Filter (MLEF) – conditional mode

- Iterative KF – conditional mean

The KF, EKF, EnKF solution identical to the minimization of a quadratic cost-function

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fT

ffa xH yRHHPHPxx

Maximum Likelihood Ensemble Filter (MLEF)(Zupanski 2005, MWR; Zupanski and Zupanski 2005, MWR)

- Estimate of the conditional mode of the posterior PDF

- Ensembles used to estimate the uncertainty of the conditional mode

- Non-differentiable minimization with Hessian preconditioning:

(Generalized conjugate-gradient and BFGS quasi-Newton algorithms)

- Augmented control variable: initial conditions, model bias, empirical parameters, boundary conditions

- Related to: (i) variational data assimilation,

(ii) Iterative Kalman filters, and

(iii) Ensemble Transform Kalman Filter – ETKF

- Not sample based• Reduces to Kalman filter for linear operators and Gaussian PDF

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1][][

2


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MLEF Framework

Minimize cost function in the subspace spanned by ensemble perturbations

][ 2121 f

Nff

f EpppP

Use the forecast (prior) error covariance square-root

Similar to variational, however:

Non-differentiable iterative minimization with superior preconditioning Solution in ensemble subspace (reduced rank) Analysis uncertainty estimate

Ensemble size

fiS

fi

fi

fi

p

p

p

,

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State-space dimension

)()( xpxp MM ai

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Hessian preconditioning in MLEF

Change of variable (Hessian preconditioning)

k

/-bTb

ff

k ζZZIPxx21

2/1

bN

bbb

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ifb

i xpxRz HH

kkkk dζζ 1 k – iteration index

Ensemble-size matrix

Background (first guess)

ETKF transformation utilized in Hessian preconditioning

“Gradient” calculation in MLEF

• Not a gradient, rather a directional derivative in the direction of ensemble perturbation

• Important for nonlinear/non-differentiable operators

Observation component of the gradient is

an innovation vector projection onto the ensemble perturbations

)(12/1

k

Tkk

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“Gradient” calculation

)()()(-)( 2/12/12/1 fikkk

fik

ki pxyRxyRxpxRz HHHH

Innovation vector

Analysis (posterior) error covariance

21-)(2121 ZZIPP Tfa

)()( 21-21- afi

ai xRpxRz HH

• Analysis error covariance estimated from minimization algorithm

• At the minimum (xmin=xa) use

Inverse Hessian = Analysis error covariance

Justification for the assumption

• Accurate minimization implies small distance between the analysis and the truth

• Good Hessian preconditioning allows efficient and accurate minimization

• By monitoring minimization, assure the calculated solution is close to the true minimum

][ 2121 a

Naa

a EpppP

Analyisis (minimizer)

• Initial error covariance noisy, but quickly becomes spatially localized

• No need to force error covariance localization

Cycle No. 4 Cycle No. 7 Cycle No. 10Cycle No. 1

j

i

Analysis Error Covariance in KdVB model

Model dynamics forces adequate localization of uncertainties !

23

3

6x

u

x

u

x

uu

t

u

2

MLEF with CSU global shallow-water model(Heikes and Randall 1995, MWR; Zupanski et. al. 2005, Tellus)

(a) (b)

(b)

Cycle 3

Cycle 5

Cycle 1

Height analysis increment [xa-xf] Height RMS error [xa-xt]

Initially noisy random perturbations quickly become smooth: Consequence of error covariance localization by dynamics

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2/12/1

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NTN

TTNNa ZZIZZIZZIPMMMP

Lyapunov vector

Error covariance localization(linear framework)

2121

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2/1

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2121 ZZIPMMPMP T

af

2/1

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Possible explanation for error covariance localization:

Dynamic localization of Lyapunov vectors

Assimilation of real boundary-layer cloud observations using the LES RAMS model

23 2-h DA cycles: 18UTC 2 May 1998 – 00 UTC 5 May 1998(Mixed phase Arctic boundary layer cloud at Sheba site)

Experiments initialized with typical clean aerosol concentrations

May 4 was abnormal: high IFN and CCN above the inversion

x= 50m, zmax = 30m (2d domain: 50col, 40lev), t=2s, Nens=48

Sophisticated microphysics in RAMS/LES, Prognostic IFN, CCN

Control variables: _il, u, v, w, N_x, R_x (8 species), IFN, CCN (dim= 22 variables x 50 columns x 40 levels = 44,000)

Radar/lidar/aircraft observations (retrievals) of IWP, LWP

G. Carrio, W. Cotton

LIQUID WATER CONTENT ASSIMILATION

CONTROL

EXP

VERIF

Better timing of maximaG. Carrio, W. Cotton

Vertically integrated observations:

LWP

Vertical structure of the analysis:

LWC

ICE FORMING NUCLEI (IFN) CONCENTRATION

Independent observation

IFN above the inversion, as observed

IFN below inversion as cloud forms

G. Carrio, W. Cotton

22Z

Double-resolution MLEF framework(THORPEX)

Operational resolution forecast model for the control

Low resolution forecast model for the ensembles

Cost function defined in operational resolution

Minimization in ensemble subspace

Motivation

- 3DVAR/4DVAR operational systems- Computational savings- Fewer number of ensembles required than in the full operational setup- More adequate number of degrees of freedom in the ensembles

Double-resolution MLEF:Forecast step

Interpolation operator from the operational (high) to coarse (low) resolution

CC CC xx )(: xx

Forecast error covariance column vector (low resolution):

)()( aai

afi C xpxp MMC

MC – low resolution forecast model

M – operational resolution (control) forecast model

xC – low resolution state vector

x – operational resolution state vector

Double-resolution MLEF:Analysis step

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1][][

2


obsf-

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Operational resolution cost-function

Change of variable (low resolution)

k

/-bTb

ff

C C ζZZIPxx21

2/1)( )(-))((2/1 ffi

fbi C xpxRz HHC

Directional gradient (ensemble subspace)

)(-)(-))(( 2/12/1 fffi

fC xyRxpxRgT

HHHC

Interpolated from operational resolution

Low resolution observation operator

For double-resolution MLEF, need an interpolation operator C,

low resolution model MC, and low resolution observation operator HC

THORPEX related development of the MLEF

MLEF with T6228 GFS and SSI• Code development completed and debugged• GFS model + SSI (interpolation from model to observations)• Currently tested PREPBUFR observations, will include satellite/radar/lidar

Compare MLEF with other EnKF methods• Evaluate if dynamical localization holds• Ensemble size, robustness of the algorithm• Code efficiency: script driven algorithm, exploits the NCEP code structure

Model bias and parameter estimation• Capability included in the current MLEF/GFS version• Reduce the large number of degrees of freedom (by a projection operator)

Double-resolution MLEF• Test this capability, using two GFS resolutions• Evaluate the ensemble size issue

Other Research and Future

Development of a fully non-Gaussian algorithm• Allow for non-Gaussian state variable errors (initial conditions, empirical parameters)• Generalized algorithm with a list of PDFs

CloudSat assimilation• Observation information content • Relative information content from various observation types/groups

Microscale models• Boundary layer, 50m-500m horizontal resolution• Probabilistic transfer and interaction between scales

Carbon data assimilation • Exploit assimilation of new measurements (OCO-Orbiting Carbon Observatory)

MLEF with super-parameterization• Assimilation of clouds and precipitation observations – MMF (Multiscale Modeling Framework)• NASA GEOS + super-parameterization• Climate models and predictability

Thank you !

xmin

J=const

Starting point 1

Starting point 2

• For high Hessian condition numbers ~50-100, minimization success is unpredictable

• Low-rank assumption in ensemble DA may miss important perturbations (directions)

ISSUE A:

Hessian preconditioning vs. low-rank

x1

x2

L

Strong correlations

along frontal zone

Weak correlations

across frontal zone

ISSUE B: Error covariances

Dynamical localization vs. prescribed structure - Addressed by advanced methods

- Computational efficiency

- Moisture related variables (microphysics)

Forecast (prior) error covariance

• Control vector is the most likely forecast

• Square-root used in the algorithm (full covariance can be calculated)

• No sampling of error covariance

• Provides dynamic continuity between the analysis and forecast

)()( aai

afi xpxp MM

][ 2121 f

Nff

f EpppP

fiS

fi

fi

fi

p

p

p

,

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p

Non-Gaussian MLEF framework

Atmospheric observations have non-Gaussian statistics:

- precipitation

- specific humidity (moisture)

- ozone

- cloud droplet concentration

- microphysical variables

Atmospheric state variables have non-Gaussian statistics:

- specific humidity (moisture)

- ozone

- microphysical variables

- concentrations (clouds, aerosols)

Gaussian data assimilation framework is generally used

Need to evaluate the impact of this assumption


MLEF approach

• Define non-Gaussian cost function from the conditional PDFs

• Minimize such defined cost function – calculate the conditional mode

• Algorithmically, define a list of PDFs, follow the appropriate code branching

• Start with relatively well known Lognormal PDF

• Examine Gaussian assumptions using a non-Gaussian mathematical framework

• Fletcher and Zupanski (2005a-SIAM J.Appl.Math.; 2005b-J.Roy.Statist.Soc.B),

Mode, Mean, Median are identical in Gaussian (or any symmetric) PDF

Mode, Mean, Median are all different in Lognormal (or any skewed) PDF

mode

median

mean


Lognormal errors are multiplicative:

Gaussian PDFLog-Normal PDF

Gaussian errors are additive:

),LN(m)(

),LN(m~)(),,LN(m~ Given 233

222

211 sss

x

yxy

HH

),N(m)(),N(m~)(),,N(m~ Given 233

222

211 sss xyxy HH

Non-Gaussian MLEF experiment with SWM: Lognormal height observation errors

Assume:• Gaussian prior PDF• Lognormal observation PDF (height) • Lognormal observation operator H(x)=exp[a(x-b)]

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N

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Minimize mixed Normal-Lognormal cost function:

Higher nonlinearity of the cost function compared to the pure Gaussian

Gaussian-like lognormal observation term [for ln(x)]

Gaussian prior PDF Additional Lognormal observation term

Impact of Lognormal observation errors: Analysis RMS errors

Logn= 1.0 ; = 3.15 Gauss= 2.5 m

Success of the Gaussian MLEF depends on the observation statistics

Logn= 1.0 ; = 1.27 Gauss= 1.5 m

Height RMS Error(SW model, 520 ensembles, Height obs 1_1)

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1 6 11 16

Cycles

Hei

gh

t E

rro

r(m

) GaussianMLEF

LognormalMLEF

Height RMS Error(SW model, 520 ensembles, Height obs)

0

2

4

6

8

10

12

1 6 11 16

Cycles

Hei

gh

t E

rro

r(m

) GaussianMLEF

LognormalMLEF

U-wind RMS Error(SW model, 520 ensembles, Height obs 1_1)

0

0.1

0.2

0.3

0.4

1 6 11 16

Cycles

U-w

ind

Err

or(

m/s

)

GaussianMLEF

LognormalMLEF

U-wind RMS Error(SW model, 520 ensembles, Height obs)

0

0.1

0.2

0.3

0.4

1 6 11 16

Cycles

U-w

ind

Err

or(

m/s

)

GaussianMLEF

LognormalMLEF

Impact of Lognormal observation errors: Innovation histogram

Logn= 1.0 ; = 3.15 Gauss= 2.5 m

Innovation statistics is significantly impacted by the PDF framework

Logn= 1.0 ; = 1.27 Gauss= 1.5 m

PDF HISTOGRAM - GAUSSIAN MLEF(SWM - 520 ens, Height obs 1_1)

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

Bin Categories

PDF

PDF HISTOGRAM - LOGNORMAL MLEF(SWM - 520 ens, Height obs 1_1)

0

0.1

0.2

0.3

0.4

0.5

-5 -4 -3 -2 -1 0 1 2 3 4 5

Bin Categories

PDF

PDF HISTOGRAM - GAUSSIAN MLEF(SWM - 520 ens, Height obs 3_1)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

-5 -4 -3 -2 -1 0 1 2 3 4 5

Bin Categories

PD

F

PDF HISTOGRAM - LOGNORMAL MLEF(SWM - 520 ens, Height obs 3_1)

0.00E+005.00E-021.00E-011.50E-012.00E-012.50E-013.00E-013.50E-014.00E-014.50E-01

-5 -4 -3 -2 -1 0 1 2 3 4 5

Bin Categories

PD

F

N(0,1)

]([]([2

1][][

2


obsf-

fTf P

MLEF Analysis Step

Minimize cost function in the subspace spanned by ensemble perturbations pif

][ 2121 f

Nff

f EpppP

Use the forecast (prior) error covariance square-root

Similar to variational, however: Non-differentiable iterative minimization with preconditioning• No differentiability assumption: works for all bounded operators• Generalized gradient, generalized Hessian• Perfect preconditioning for quadratic cost function

Solution in ensemble subspace• Reduced dimensions of the analysis correction subspace

• Focus on unstable, growing perturbations in the analysis

• Search for the vectors (ensemble perturbations) that span the attractor subspace

fNN

ff/f

fa www pppwPxx 221121

Dusanka Zupanski, CIRA/[email protected]

Information Content AnalysisNASA GEOS-5 Single Column Model

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is trd

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2

21

CCI

i i

iTTs trd

2

2

1)(

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ds measures effective DOF of an ensemble-based data assimilation system (e.g., MLEF). Useful for addressing DOF of the model error.

DOF for signal (ds)impact of ensemble size

0

10

20

30

40

50

60

70

1 6 11 16 21 26 31 36 41 46

Analysis cycle

ds

ds_10_ensds_20_ensds_40_ensds_80_ens

-300 -180 -60 60 180 300-300

-180

-60

60

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-300 -180 -60 60 180 300-300

-180

-60

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180

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0

0.01

0.05

0.1

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0.5

-300 -180 -60 60 180 300-300

-180

-60

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-300 -180 -60 60 180 300-300

-180

-60

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-300 -180 -60 60 180 300-300

-180

-60

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-300 -180 -60 60 180 300-300

-180

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60

180

300

cycle 1 cycle 2 cycle 3

Bayesian

MLEF450 ens

A : Reduction of uncertainty (0-)

MLEF (full rank) successfully reproduces Bayesian inversion results

Outline

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