Introduction to data assimilation: Lecture 3

PIMS Institute, Victoria, 14-18 July 2008

Saroja PolavarapuMeteorological Research Division

Environment Canada

OUTLINE

1. Covariance modelling – 2,3

2. 4D-Variational assimilation

3. Nonlinear dynamics

4. Constrained variational data assimilation

Covariance Modelling

1.Innovations method2.NMC-method3.Ensemble method

2. NMC-method

• Need global statistics

• N. American radiosonde network is only 4000 km in extent defining only up to wavenumber 10. Vertical and horizontal resolution is too coarse.

• A posteriori justification: compare resulting statistics with those obtained using other methods

• Compares 24-h and 48-h forecasts valid at same time• Provides global, multivariate corr. with full vertical and spectral

resolution• Not used for variances• Assumes forecast differences approximate forecast error

4824 xx

• 24-h start forecast avoids “spin-up” problems• 24-h period is short enough to claim similarity with 0-6 h forecast

error. Final difference is scaled by an empirical factor• 24-h period long enough that the forecasts are dissimilar despite

lack of data to update the starting analysis• 0-6 h forecast differences reflect assumptions made in OI

background error covariances

Why 24 – 48 ?

-48 -24 0

?664824truexxxx

The NMC-method

A posteriori justification: compare NMC results to innovation-method results

Horizontal correlation length scale

Rabier et al. (1998) Hollingsworth and Lonnberg (1986)

NMCInnovations

Different horizontal correlationlengths for different vertical levels

Different vertical correlationlengths for different wavenumbers

Rabier et al. (1998) Rabier et al. (1998)

Properties of the NMC-methodBouttier (1994)

• For linear H, no model error, 6-h forecast difference, can compare NMC P calc. to what Kalman Filter suggests.

• NMC-method breaks down if there is no data between launch of 2 forecasts. With no data P is under-estimated

• For dense, good quality hor. uncorr. obs, P is over-estimated

• For obs at every gridpoint, where obs and bkgd error variances are equal, the NMC-method P estimate is equivalent to that from the KF.

Center Region Reference

NCEP U.S.A. Parrish & Derber 1991

ECMWF* Europe Rabier et al.1998

CMC* Canada Gauthier et al. 1999

Met Office U.K. Ingleby et al. 1996

BMRC Australia Steinle et al. 1995

Meteo-Fr.* France Desroziers et al. 1995

NMC-method usage

*Later replaced by ensemble-based methods

3. Ensemble-based methods of covariance estimation

Generate ensemble ofN background states

These methods attempt to simulate error of actual assimilation systems by perturbing obs and background states with specified errors and computing ensemble spread

Belo Pereira and Berre (2006)

Comparing NMC and ensemble-based method results

Belo Pereira and Berre (2006)

Horizontal correlation length scales are longer with NMC method

vorticity

temperature

Belo Pereira and Berre (2006)

Vertical correlations are too deep with NMC method

Ensemble method NMC method

Vertical correlations of temperature background error (at level 21, ~500 hPa)

Buehner (2005)

Specified NMC STD are independent of longitude

Ensemble-based STD show reduced error in data dense regions

Time averaged background errors from actual EnsKF is used as reference

at 250 hPa T at 500 hPa Background error standard deviations

Center Region Reference

ECMWF Europe Fisher 2003

CMC Canada Buehner 2005

Meteo-Fr. France Berre et al. 2006

Ensemble-method usage

2. Four-Dimensional variational data assimilation

Extension to the time dimension

3D DA schemes make sense when all obs are taken at the same time (e.g. radiosondes).

But they don’t take full advantage of measurements which have high temporal resolution (satellite obs, profilers, aircraft, etc.).

Background trajectory

Analysis trajectory

))(())((2

1)()(

2

1)( 1

00

100 kk

Tkk

N

kb

Tb HHJ xzRxzxxBxxx

4D-Variational assimilation

4D-Var experiment with obs every time step at only 1 of 128 grid points

Initial guess field misplaces front

With time series of obs from 1 station only, the frontal position is corrected

The benefit of temporal information

Dotted red line is 3D-Var solution

1. Run model with initial conditions xi0 from t0 to tN

2. Compute

3. Compute

4. Find step size: i

5. Modify initial state: 1

0 0i i

i ix x d

0 0( )ix J x0( )iJ x

Background trajectoryAnalysis trajectory

))(())((2

1)()(

2

1)( 1

00

100 kk

Tkk

N

kb

Tb HHJ xzRxzxxBxxx

4D-Varalgorithm

TLM

ADJ

TLM

ADJ

Minimization algorithm

• M1QN3• Gilbert & Lemaréchal 1989• limited memory quasi-

Newton technique (the L-BFGS method of J. Nocedal)

• designed for very large scale problems

Minimization of a quadratic cost function J(x). The gradient of the cost function and the cost function itself are supplied to a minimization algorithm which determines how to change x to get a lower cost.

http://www-rocq.inria.fr/estime/modulopt/optimization-routines/m1qn3/m1qn3.html

4D-Var as described1. Assumes NWP model is perfect

– Complex nonlinear relationships between analysis variables are permitted

– Aids in reducing underdeterminacy problem

2. Needs TLM and ADJ models for NWP model

– DA scheme now intimately tied to NWP model

3. Is expensive– Adjoint model about 1-2 times CPU of NWP

model. One iteration=NWP run + adj run. Typically 50 iterations.

Predictability error

Term in ( ) is a scalar

Circled term is 1 column of B matrix, i.e. a vectorLHS is a vector

Geopotential height analysis increments at the end of a 24-h assimilation period due to 1 obs

3D-Var: 1 height obs at(42N,180E,500 hPa)No change of shape with height

4D-Var: 1 height obs at(42N,170.6E,850 hPa)Changes shape with height

Thépaut et al. (1996)

500 hPa

1000 hPa

4D-Var single obs experiments show: • The shape of analysis increments depends on location of obs• The spreading of information is flow dependent

Why does 4D-Var beat 3D-Var?

4D-Var:• uses obs at their actual

time of measurement• Uses all temporally

continuous obs available within window

• evolves error covariances in time

3D-Var:• Treats obs as if valid at

00,06,12 or 18Z• Uses temporally

continuous obs only close to synoptic times

• Uses static error covariances

3. Complications due to nonlinear dynamics

Highly nonlinear dynamics

( )

10, 28, 8 / 3.

x y x

y x y xz

z xy z

Lorenz (1963) equations:

for

Miller et al. (1994)

If assimilation window is too long, 4D-Var fails

t=7

Miller et al. (1994)

t=8

t=10

t=15

Length of 4D-Var assimilation window

The longer the assimilation window, the greater the number of local minina in the cost function

Optimal assimilation period• examine ability to “fill in” small scales through downscale energy cascade

• barotropic vorticity equation

• Perfect model, observations

• Initial guess for trajectory is completely decorrelated from truth

Tanguay et al. (1995)

~3 days ~12 days

Nonlinear time scale is TNL=9

Obs at large scales only

~3 days

~6 days ~9 days

~1.5 days

Tanguay et al. (1995)

Downscale transfer of information to unobserved scales

Upscale propagation of error to observed scales

Incremental ApproachCourtier et al. (1994)

• TLM will be valid for large scales but not for some smaller scales

• So, solve for analysis increments at lower resolution. Write 4D-Var cost in terms of increments (departures from background).– Use of lower resolution filters scales and processes

not well forecast by TLM– Forecast model in cost function is then TLM model– Cost function is purely quadratic– Use of lower resolution reduces cost of 4D-Var– Compute the innovation (z-H(x)) at full resolution– Solve a series (2-3) 4D-Var problems, updating the

background between each one

4. Constrained variational data assimilation

Does 4D-Var inherently produce balanced analyses?

• 4D-Var tries to find the model state which best fits the observations in a time window

• The model contains many modes at its disposal, for use in fitting observations: Rossby waves, gravity waves, …

• If the obs contain high frequency signals (which they will), the model will use as many gravity waves as needed to fit the obs

def. pos. is)~()~(ˆ)~(.3

0)~()~(.2

0)~(ˆ.1

01

00

00

0

xZxGHxZ

xxZ

x

t

iii

T

T J

c

Strong ConstraintsMinimize J(x0) subject to the constraints: .,...,1,0)(ˆ 0 tici x

Necessary and sufficient conditions for x0 to be a minimum are:

Gill, Murray, Wright (1981)

Projection onto constraint tangent

Hessian of constraints

)(ˆ)(ˆ2

T4 xx ccJJ DVARweak

Penalty Methods: Minimize

Weak Constraints

Small Large

4DVAR with NNMI: strong constraint

…owing to the iterative and approximate character of the initialization algorithm, the condition || dG/dt || = 0 cannot in practice be enforced as an exact constraint.

Courtier and Talagrand (1990)

4DVAR with NNMI: weak constraint2

4

ˆ

dt

dJJ GDVAR

c

Courtier and Talagrand (1990)

Thépaut and Courtier (1991)

A’

Digital Filter Initialization (DFI)

N

Nk

ukk

I xhx0

Lynch and Huang (1992)

N=12, t=30 min

Tc=8 hTc=6 h

Fillion et al. (1995)

4DVAR with DFI: Strong Constraint

• Because filter is not perfect, some inversion of intermediate scale noise occurs, but DFI as a strong constraint suppresses small scale noise.

• Introduced by Gustafsson (1993)

• Weak constraint can control small scale noise (Polavarapu et al. 2000)

• Implemented operationally at Météo-France (Gauthier and Thépaut 2001)

4DVAR with DFI: Weak Constraint

Polavarapu et al. (2000)

Disadvantages of 4D-Var

• Model specific (Needs TLM and ADJ)– The U.K. Met Office uses Perturbation Forecast

Model and its Adjoint

• Assumes NWP model is perfect.– Weak constraint formulations relax this assumption.

Already under investigation at ECMWF* (see Tremolet QJ papers)

• Expensive. 2-3 x CPU of NWP model per iteration, with ~50 iterations per outer loop– Computing power keeps increasing

*European Centre for Medium Range Weather Forecasting

4D-Var Challenges• Obtaining fast, efficient large-scale

optimization routines• Extracting analysis error covariance

A-1 = B-1 + HTR-1H• Want to know MAMT to learn about

forecast error levels• Cycling 4D-Var (Using evolved covariance

at end of one assimilation window to start next assimilation cycle.)

• Estimating and incorporating model error

Center Region Opera-tional

Ref.

ECMWF Europe Jan. 1999 Rabier et al. (1999)

Météo-

France

France 2000 Gauthier and Thepaut (2001)

JMA Japan Mar. 2002 http://www.jma.go.jp/jma/jma-eng/jma-center/nwp/NAPS-8_DDB_spec.txt

Met Office U.K. Mar. 2003 Rawlins et al. (2007)

CMC Canada Mar. 2005 Gauthier et al. (2007)

Weather centers using 4D-var operationally

Uppala et al. (2005, QJ)

ERA-40 reanalyses• model, DAS fixed in time• observing system changes with time• Little improvement over 25 years

Operational system • model, DAS changes with time• observing system changes with time• Big improvement in skill in 25 years must be due to model, DAS improvements.

Exciting but missed topics• Ensemble Kalman Filter

– Operational at CMC for Ensemble prediction system

• Combining variational and Ensemble techniques – WWRP/THOPEX workshop on 4D-Var and

Ensemble Kalman Filter Inter-comparisons, Buenos Aires, Argentina, 10-13 Nov. 2008 http://4dvarenkf.cima.fcen.uba.ar/

– Operational ensemble/variational assimiliation system at Météo-France on July 1, 2008. Ref: Berre et al. (2007)

Final Summary

• The atmospheric data assimilation problem is characterized by huge, nonlinear systems and insufficient observations.

• Because the math of the linear estimation problem is well known, the key to progress is using atmospheric physics to make the right approximations

• There has been considerable improvement in forecast skill in the past 2.5 decades, partly due to improvements in data assimilation systems.

The End