Introduction to Data Assimilation: Lecture 2

Saroja Polavarapu

Meteorological Research Division Environment Canada

PIMS Summer School, Victoria. July 14-18, 2008

Outline of lecture 2

1. Covariance modelling – Part 1

2. Initialization (Filtering of analyses)

3. Basic estimation theory

4. 3D-Variational Assimilation (3Dvar)

Background error covariance matrix filters analysis increments

)(1TT bba H xzRHBHBHxx

Bvxx ba

Analysis increments (xa – xb) are a linear combination of columns of B

Properties of B determine filtering properties of assimilation scheme!

K

A simple demonstration of filtering properties of B matrix

2

1exp

2

0.2

d

d

x yx y

L

L

cos(x)

cos(2x)

Choose a correlation function and obs increment shape then compute analysis increments

obs/b = 0.5

cos(3x)

cos(4x)

cos(5x)

cos(6x)

cos(7x)

cos(8x)

cos(9x)

cos(10x)

1. Covariance Modelling

1. Innovations method2.2. NMC-methodNMC-method3.3. Ensemble methodEnsemble method

Background error covariance matrix

Ttbtbb xxxxP

•If x is 108, Pb is 108 x 108. •With 106 obs, cannot estimate Pb.•Need to model Pb.•The fewer the parameters in the model, •the easier to estimate them, but•less likely the model is to be valid

• Historically used for Optimal Interpolation • (e.g. Hollingsworth and Lonnberg 1986,• Lonnberg and Hollingsworth 1986, Mitchell et al. 1990)• Typical assumptions:

•separability of horizontal and vertical correlations

•Homogeneity

•Isotropy

1. Innovations method

),(),,,( jibxjjii

bH ryxyx CC

),(),,,(),,,,,( jibVjjii

bHjjjiii

b zzyxyxzyxzyx CCC

)(),,,( ibxjjii

bH ryxyx CC

r

i

j

r

l

m

lr

i

j

m

r

Instrument+representativeness

Background error

Choose obs s.t. these terms =0

Assume homogeneous, isotropiccorrelation model. Choose acontinuous function (r) which hasonly a few parameters such as L, correlation length scale. Plot allinnovations as a function of distanceonly and fit the function to the data.

Dec. 15/87-Mar. 15/88radiosonde data. Model: CMC T59L20

Mitchell et al. (1990)

Mitchell et al. (1990)Mitchell et al. (1990)

Obs and Forecast error variances

Vertical correlationsof forecast error

Height

Non-divergent wind

Lonnberg and Hollingsworth (1986)

Hollingsworth and Lonnberg (1986)

Multivariate correlations

Mitchell et al. (1990)

Bouttier and Courtierwww.ecmwf.int 2002

If covariances are homogeneous,variances are independent of space

If correlations are homogeneous,correlation lengths are independentof location

Covariances are not homogeneous

Correlations are not homogeneous

Gustafsson (1981)

Daley (1991)

Correlations are not isotropic

Are correlations separable?

If so, correlation length should beIndependent of height.

Mitchell et al. (1990)Lonnberg and Hollingsworth (1986)

Covariance modelling assumptions:1. No correlations between background and

obs errors

2. No horizontal correlation of obs errors

3. Homogeneous, isotropic horizontal background error correlations

4. Separability of vertical and horizontal background error correlations

None of our assumptions are really correct. Therefore Optimal Interpolation is not optimal so it is often called Statistical Interpolation.

2. Initialization

1. Nonlinear Normal Mode (NNMI)2. Digital Filter Initialization (DFI)3. Filtering of analysis increments

Daley 1991

Balance in data assimilation

The “initialization” step

• Integrating a model from an analysis leads to motion on fast scales

• Mostly evident in surface pressure tendency, divergence and can affect precipitation forecasts

• 6-h forecasts are used to quality check obs, so if noisy could lead to rejection of good obs or acceptance of bad obs

• Historically, after the analysis step, a separate “initialization” step was done to remove fast motions

• In the 1980’s a sophisticated “initialization” scheme based on Normal modes of the model equations was developed and used operationally with OI.

),(

0),(

,

|

)(

T1

T

TT

T

RRGGGG

RRGGGGG

RRGG

GR

Ni

Nidt

d

Nidt

d

cEcEEc

cEcEEcc

xEcxEc

EEE

EEA

xAxx

Nonlinear Normal Mode Initialization (NNMI)

Consider model

Determine modes

Separate R and G

Project onto G

Define balance

Solution

G

R

SA

N

L

The slow manifold

0dt

d Gc

Daley 1991

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)

Combining Analysis and Initialization steps

• Doing an analysis brings you closer to the data.• Doing an initialization moves you farther from the data.

Daley (1986)

N

Gravity modes

Rossby modes

Variational Normal model initializationDaley (1978), Tribbia (1982), Fillion and Temperton (1989), etc.

hgdSwwvvwuuIS AIVAIVAI

~~,)()(

~)(

~ 222

Minimize I such that uI, vI, I stays on M.

Daley (1986)

Some signals in the forecast e.g. tides should NOT be destroyed by NNMI!

So filter analysis increments only

Seaman et al. (1995)

Semi-diurnal mode has amplitude seen in free model run, if anl increments are filtered

3. A bit of Estimation theory(will lead us to 3D-Var)

a posteriori p.d.f.

Data Selection

Bouttier and Courtier (2002)

From: ECMWF training course available at www.ecmwf.int

The effect of data selection

Cohn et al. (1998)

OI

PSAS

The effect of data selection

Cohn et al. (1998)

Advantages of 3D-var

1. Obs and model variables can be nonlinearly related.

• H(X), H, HT need to be calculated for each obs type

• No separate inversion of data needed – can directly assimilate radiances

• Flexible choice of model variables, e.g. spectral coefficents

2. No data selection is needed.

))(())(()()()( 1 xzRxzxxBxxx b1b HHJ TT

3D-Var Preconditioning

• Hessian of cost function is B-1 + HTR-1H

• To avoid computing B-1 in (1), change control variable to x=Lso first term in (1) becomes ½ and we minimize w.r.t. . Herex=x-xb

• After change of variable, Hessian is I + term• If no obs, preconditioner is great, but with

more obs, or more accurate obs, it loses its advantage

(1)))(())(()()()( 1 xzRxzxxBxxx b1b HHJ TT

With covariances in spectral space,longer correlation lengths scales arepermitted in the stratosphere

With flexibility of choice of obs,can assimilate many new typesof obs such as scatterometer

Andersson et al. (1998) Andersson et al. (1998)

Normalized AMSU weighting functions

1413121110 9 8 7 6 5

To assimilate radiances directly, H includes an instrument-specific radiative transfer model

))(())(()()()( 1 xzRxzxxBxxx b1b HHJ TT

Kalnay et al. (1998)

Impact of Direct Assimilation of RadiancesAnomaly = difference between forecast and climatolgyAnomaly correlation = pattern correlation between forecast anomalies and

verifying analyses

1974 – improvedNESDIS VTPRRetrievals1978 – TOVSretrievals

Center Region Operational Ref.

NCEP U.S.A. June 1991 Parrish& Derber (1991)

ECMWF* Europe Jan. 1996 Courtier et al. (1997)

CMC* Canada June 1997 Gauthier et al. (1998)

Met Office* U.K. Mar. 1999 Lorenc et al. (2000)

DAO NASA 1997 Cohn et al. (1997)

NRL US Navy 2000? Daley& Barker (2001)

JMA* Japan Sept. 2001 Takeuchi et al. (2004) SPIE proceedings,5234, 505-516  

Operational weather centers used 3D-Var from1990’s *Later replaced by 4D-Var

Summary (Lecture 2)• Estimation theory provides mathematical basis

for DA. Optimality principles presume knowledge of error statistics.

• For Gaussian errors, 3D-var and OI are equivalent in theory, but different in practice

• 3D-var allows easy extension for nonlinearly related obs and model variables. Also allows more flexibility in choice of analysis variables.

• 3D-var does not require data selection so analyses are in better balance.

• Improvement of 3D-var over OI is not statistically significant for same obs. Systematic improvement of 3DVAR over OI in stratosphere and S. Hemisphere. Scores continue to improve as more obs types are added.