Hans Huang: Variational Data Assimilation, SNU lecture. June 9th, 20091 DA Variational Data Assimilation Hans Huang NCAR Acknowledge: Dale Barker, The

Hans Huang: Variational Data Assimilation, SNU lecture. June 9th, 2009 1

DA

Variational Data Assimilation

Hans Huang

NCAR

Acknowledge: Dale Barker, The WRFDA team, MMM and DATC staff

AFWA, NSF, KMA, CWB, BMB, AirDat


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• Introduction to data assimilation.• Basics of variational data assimilation.• Demonstration with a simple system.• The WRFDA approach.

Outline


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Modern weather forecast (Bjerknes,1904)

• Analysis: using observations and other information, we can specify the atmospheric state at a given initial time: “Today’s Weather”

• Forecast: using the equations, we can calculate how this state will change over time: “Tomorrow’s Weather”

Vilhelm Bjerknes (1862–1951)

• A sufficiently accurate knowledge of the state of the atmosphere at the initial time

• A sufficiently accurate knowledge of the laws according to which one state of the atmosphere develops from another.

(Peter Lynch)


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initial state forecast

Analysis

Model

Model state x, ~107

Observationsyo, ~105-106


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Assimilation methods

• Empirical methods– Successive Correction Method (SCM)

– Nudging

– Physical Initialisation (PI), Latent Heat Nudging (LHN)

• Statistical methods – Optimal Interpolation (OI)

– 3-Dimensional VARiational data assimilation (3DVAR)

– 4-Dimensional VARiational data assimilation (4DVAR)

• Advanced methods– Extended Kalman Filter (EKF)

– Ensemble Kalman Filter (EnFK)


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Outline


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The analysis problem for a given time

Consider a scalar x.

The background (normally a short-range forecast):

xb=xt+b.

The observation:

xr=xt+r.

The error statistics are assumed to be known: < b > = 0, mean error (unbiased), < r > = 0, mean error (unbiased), <b2> = B, background error variance, <r2> = R, observation error variance, <br> = 0, no correlation between b and r ,

where <⋅> is ensemble average.


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BLUE: the Best Liner Unbiased Estimate

The analysis: xa =xt +a.Search for the best estimate: xa=αxb+βxr

Substitute the definitions, we have: α+β=1. <a>=0The variance: A=<a2>=B−2βB+β2 B+R( )

To determine β: dAdβ

=−2B+2β B+R( )=0

we have β= BB+R

The analysis: xa=xb+B

B+Rxr−xb( )

The analysis error variance: A−1=B−1+R−1


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limB→0

xa =xb

The analysis: xa =xb+B

B+Rxr−xb( )

0 <

BB+R

<1

The analysis error variance: A−1 =B−1 + R−1

limR→0

xa =xr

A < B

A < R

The analysis value should be

between background and

observation.

Statistically, analyses are better than

background.Statistically, analyses are better than

observations!

If B is too small,

observations are less useful.If R can be tuned, analysis

can fit

observations as close as one

wants!


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3D-Var

The analysis is obtained by minimizing the cost function J , defined as :

J=12

x-xb( )

T

B-1 x-xb( )+

12

x-xr( )

T

R-1 x-xr( )

The gradient of J with respect to x:

′J =B-1 x-xb( )+R-1 x-xr

( )

At the minimum, ′J =0, we have:

xa=xb+B

B+Rxr−xb( )

the same as BLUE.


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Sequential data assimilation (I)

True states : ..., xi−1t , xi

t , xi+1t , ...

Observations : ..., xi−1r , xi

r , xi+1r , ...

Forecasts : ..., xi−1f , xi

f , xi+1f , ...

Analyses : ..., xi−1a , xi

a, xi+1a , ...


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Sequential data assimilation (II)

Forecast model:

xi+1t =M xi

t( )+qi

Where qi is the model error.

As qi is unknown and xia is the best estimate of xi

t the forecast

model usually takes the form:

xi+1f =M xi

a( )

OI (and 3DVAR): xb=xif

xia=xi

f +B

B+Rxi

r−xif⎛

⎝⎜⎞⎠⎟


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Sequential data assimilation (III)

4DVAR analysis is obtained by minimizing the cost function J , defined as:

J xi( )=12

xi−xif⎛

⎝⎜⎞⎠⎟

T

B−1 xi−xif⎛

⎝⎜⎞⎠⎟

+12

Mk−1 xi( )−xi+kr⎡

⎣⎢⎤⎦⎥T

R−1 Mk−1 xi( )−xi+kr⎡

⎣⎢⎤⎦⎥

k=0

K∑

where, K is the assimilation window and

M−1 xi( ) = xi

M0 xi( ) = M xi( )

Mk−1 xi( ) = M (M (...M

k

1 24 34 (xi )...))


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Sequential data assimilation (IV)

4DVAR (continue)

The gradient of J with respect to x:

′J =B−1 xi−xif⎛

⎝⎜⎞⎠⎟ + M i+ j

T R−1 Mk−1 xi( )−xi+kr⎡

⎣⎢⎤⎦⎥

j=0

k−1∏

k=0

K∑

where, M i+ jT is the adjoint model of M at time step i+ j .


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Scalar to Vector

From scalar to vector:

Number of grid points ≈ 107 :

x→ x xb → xb

Dimension of B, P ≈ 107x 107Number of observations, 106 :

xr → yo

x−xr → H(x) −yo

Dimension of R ≈ 106x 106

Observation operator H


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H - Observation operatorH maps variables from “model space” to “observation space” x y

– Interpolations from model grids to observation locations

– Extrapolations using PBL schemes

– Time integration using full NWP models (4D-Var in generalized form)

– Transformations of model variables (u, v, T, q, ps, etc.) to “indirect” observations (e.g.

satellite radiance, radar radial winds, etc.)• Simple relations like PW, radial wind, refractivity, …

• Radar reflectivity

• Radiative transfer models

• Precipitation using simple or complex models

• …

!!! Need H, H and HT, not H-1 !!!

Z =Z(T, IWC,LWC,RWC,SWC)

0

( )( ) ( , ( ))

dTRL B T z dz

dz

νν ν∞ ⎡ ⎤≈ ⎢ ⎥⎣ ⎦∫


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Level of preprocessing and the observation operator

Phase and amplitude

Ionosphere correctedobservables

Refractivity profiles

Bending angle profiles

Temperature profiles

Raw data:

Frequency relations

Geometry

Abel trasformor ray tracing

Hydrostatic equlibriumand equation of state

H =I

H =H

N

H =H

RH

N

H =H

GH

RH

N

H =H

FH

GH

RH

N


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Sequential data assimilation

Model: xi+1

f = M x ia( )

OI: xia=xi

f +BHT HBHT + R( )-1

yo - H xi

f( )⎡⎣

⎤⎦

4D-Var: J xi( )=1

2xi -xi

f( )

T

B-1 xi -xif

( )

+1

2H Mk-1 xi( )( )-yi+k

o⎡⎣

⎤⎦

TR -1 H Mk-1 xi( )( )-yi+k

o⎡⎣

⎤⎦

k=0

k∑

′J =B-1 xi -xif

( ) + M i+ jT HTR−1 H Mk−1 xi( )[ ]−yi+k

o⎡⎣

⎤⎦

j=0

k−1∏

k=0

K∑


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Issues on variational data assimilation

• Observations yo

• Observation operator H

• Observation errors R

• Backgound xb

• Size of B: statistical model and tuning

• Minimization algorithm Quasi−Newton; Conjugate Gradient; …( )

• M and MT : development and validity


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• Introduction to data assimilation.• Basics of variational data assimilation.• Demonstration with a simple system.

• The WRFDA approach.

Outline


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Nonlinear equation (NL):

xi+1= axi−xi2=M xi( )

Depending on a,three types of solution are found; steady state; limited cycle; chaotic.

Tangent Linear equations (TL):

xi+1tl = a−2xi

bs( )xitl=Μixi

tl

(linearized around basic state xibs)

Adjoint equation (AD):

xiad= a−2xi

bs( )xi+1ad =Μi

T xi+1ad

Note here for this simple case we have

Μi= ΜiT = a−2xi

bs( )

The Lorenz 1964 equation


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Issues on data assimilation

for the system based on the Lorenz 64 equation

• Observation operator H =Η =ΗT =1

• Estimate of the true states - generated (with model error!)

• Observation errors xo−xt=σoG

• Size of Β is 1, but B=σb2 still needs attention

• Μ and ΜT : no eort in development but their validity is still a major problem

(model, assimilation window length, ...)


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Too simple?

Try: Lorenz63 modelTry: 1-D advection equationTry: 2-D shallow water equation…Try: ARW!!!


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WRFDA is a Data Assimilation system built within

the WRF software framework, used for application in both research and operational environments….

Outline


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WRF-Var (WRFDA) Data Assimilation Overview

• Goal: Community WRF DA system for • regional/global, • research/operations, and • deterministic/probabilistic applications.

• Techniques: • 3D-Var• 4D-Var (regional)• Ensemble DA, • Hybrid Variational/Ensemble DA.

• Model: WRF (ARW, NMM, Global)• Support: WRFDA team

• NCAR/ESSL/MMM/DAG• NCAR/RAL/JNT/DATC

• Observations: Conv.+Sat.+Radar


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WRFDA Observations In-Situ:

- Surface (SYNOP, METAR, SHIP, BUOY).- Upper air (TEMP, PIBAL, AIREP, ACARS,

TAMDAR). Remotely sensed retrievals:

- Atmospheric Motion Vectors (geo/polar).- Ground-based GPS Total Precipitable Water.- SSM/I oceanic surface wind speed and TPW.- Scatterometer oceanic surface winds.- Wind Profiler.- Radar radial velocities and reflectivities.- Satellite temperature/humidities.- GPS refractivity (e.g. COSMIC).

Radiative Transfer:- RTTOVS (EUMETSAT).- CRTM (JCSDA).

2004082600 ~ 2004092812

Threshold = 5.0mm

TIME

3 6 9 12 15 18 21 240.0

0.2

0.4

0.6

0.8

1.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Th

reat

Sco

re

Bias

KMA Pre-operational Verification:

(with/without radar)


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Assume background error covariance estimated by model perturbations x’ :

Two ways of defining x’:

The NMC-method (Parrish and Derber 1992):

where e.g. t2=24hr, t1=12hr forecasts…

…or ensemble perturbations (Fisher 2003):

Tuning via innovation vector statistics and/or variational methods.

Model-Based Estimation of Climatological Background Errors

€

€

B0 = (xb − x t)(xb − x t)T ≈ x' x'T

€

B0 = x'x'T ≈ A(x t 2 − x t1)(x t 2 − x t1)T

€

B0 = x'x'T ≈ C(x k − x )(x k − x )T


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Single observation experiment- one way to view the structure of B

The solution of 3D-Var should be

€

xa = xb + BHT HBHT + R[ ]−1

y − Hxb[ ]

Single observation

€

xa − xb = Bi σ b2 + σ o

2[ ]

−1yi − x i[ ]


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Pressure,Temperature

Wind Speed,Vector,

v-wind component.

Example of B 3D-Var response to a single ps observation


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-20 -15 -10 -5 0 5 10 15 20

X GRID

5

10

15

20

25

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91

-20 -15 -10 -5 0 5 10 15 20

X GRID

5

10

15

20

25

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91

CV5-ENS

-20 -15 -10 -5 0 5 10 15 20

X GRID

5

10

15

20

25

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91

CV5-NMC

-20 -15 -10 -5 0 5 10 15 20

X GRID

5

10

15

20

25

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91

B from ensemble methodB from NMC method

Examples of BT increments : T Observation (1 Deg , 0.001 error around 850 hPa)


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Sensitivity to Forecast Error Covariances in Antarctica(Rizvi et al 2006)

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

May 2004

bias/RMSE (K)

60km Verification (vs.

radiosondes)T+24hr Temperature

GFS-based

errors

WRF-based

errors

20km Domain2

60km Domain 1


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Define preconditioned control variable v space transform

where U transform CAREFULLY chosen to satisfy Bo = UUT .

Choose (at least assume) control variable components with uncorrelated errors:

where n~number pieces of independent information.

Incremental WRFDA Jb Preconditioning

€

Jb δx t0( )[ ] =1

2δx t0( ) − xb t0( ) − xg t0( )[ ]{ }

TBo

−1 δx t0( ) − xb t0( ) − xg t0( )[ ]{ }

€

δx t0( ) = Uv

€

Jb δx t0( )[ ] =1

2vn

2

n∑


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WRFDA Background Error Modeling

€

δx t0( ) = Uv = UpUvUhv

Uh: RF = Recursive Filter,

e.g. Purser et al 2003

Uv: Vertical correlations

EOF Decomposition

Up: Change of variable,

impose balance.


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100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500

Error correlation lengthscale (km)

Pressure (hPa)

u

v

T

p

q

€





EOF Decomposition


impose balance.


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100

200

300

400

500

600

700

800

900

1000-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Eigenvector Amplitude

Pressure (hPa)Mode 1Mode 2Mode 3Mode 4Mode 5

€





EOF Decomposition


impose balance.


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Define control variables:

€

ψ '

€

r' = q'/ qs Tb,qb, pb( )

χu ' = χ '− χ b ' ψ '( )

Tu ' =T '−Tb ' ψ '( )

psu ' =ps '−psb ' ψ '( )

€





EOF Decomposition


impose balance.


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WRFDA Statistical Balance Constraints

• Define statistical balance after Wu et al (2002):

• How good are these balance constraints? Test on KMA global model data. Plot correlation between “Full” and balanced components of field:

€

€

χb • χ / χ • χ

€

Tb • T / T • T

€

psb • ps / ps • ps

€

χb' = cψ '

€

Tb' (k) = G(k,k1)ψ ' (k1)

k1∑

€

psb' = W (k)ψ ' (k)

k∑


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WRF 4D-Var milestones

2003: WRF 4D-Var project.

2004: WRF SN (simplified nonlinear model).

Modifications to WRF 3D-Var.

2005: TL and AD of WRF dynamics.

WRF TL and AD framework.

WRF 4D-Var framework.

2006: The WRF 4D-Var prototype.

Single ob and real data experiments.

Parallelization of WRF TL and AD.

Simple physics TL and AD.

JcDF

2007: The WRF 4D-Var basic system.

2008: Further optimization and testing


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Basic system: 3 exes, disk I/O, parallel, full dyn, simple phys, JcDF

NL(1),…,NL(K)

TL(1),…,TL(K)

AF(K),…,AF(1)

AD00

WRF

WRF_NL

WRF_TL

WRF_AD

VAR

Outerloop

Mk

dk

Innerloop

U

UT

call

call

call

TL00

Ry1

…

yK

xb

I/O

WRFBDY B

WRF+ BS(0),…,BS(N)

MkT

HkT

Mk

Hk

xn

TLDF

WRFINPUT


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Why 4D-Var?• Use observations over a time interval,

which suits most asynoptic data and use tendency information from observations.

• Use a forecast model as a constraint, which enhances the dynamic balance of the analysis.

• Implicitly use flow-dependent background errors, which ensures the analysis quality for fast developing weather systems.

• NOT easy to build and maintain!


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Radiance Observation Forcing at 7 data slots

€

H iTRi

−1(HiM iδx0 − di)


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Single observation experimentThe idea behind single ob tests:


€

xa = xb + BHT HBHT + R[ ]−1

y − Hxb[ ]

Single observation

€

xa − xb = Bi σ b2 + σ o

2[ ]

−1yi − x i[ ]

3D-Var 4D-Var: H HM; H HM; HT MTHT


€

xa = xb + BMTHT H MBMT( )HT + R[ ]−1

y − HMxb[ ]

Single observation, solution at observation time

€

M xa − xb( ) = MBMT

( )i

σ b2 + σ o

2[ ]

−1yi − x i[ ]


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Analysis increments of 500mb from 3D-Var at 00h and from 4D-Var at 06h

due to a 500mb T observation at 06h

++

3D-Var 4D-Var


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500mb increments at 00,01,02,03,04,05,06h to a 500mb T ob at 06h

OBS+


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500mb difference at 00,01,02,03,04,05,06h from two nonlinear runs (one from background; one from 4D-Var)

OBS+


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500mb difference at 00,01,02,03,04,05,06h from two nonlinear runs (one from background; one from FGAT)

OBS+


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Real Case: Typhoon HaitangExperimental Design (Cold-Start)

• Domain configuration: 91x73x17, 45km

• Period: 00 UTC 16 July - 00 UTC 18 July, 2005• Observations from Taiwan CWB operational database.• 5 experiments are conducted:

FGS – forecast from the background [The background fields are 6-h WRF forecasts from National Center for Environment Prediction (NCEP) GFS analysis.]

AVN- forecast from the NCEP AVN analysis 3DVAR – forecast from WRF-Var3d using FGS as background FGAT - forecast from WRF-Var3dFGAT using FGS as

background

4DVAR – forecast from WRF-Var4d using FGS as background


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Typhoon Haitang 2005


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A KMA Heavy Rain Case

Period: 12 UTC 4 May - 00 UTC 7 May, 2006

Assimilation window: 6 hours

Cycling (6h forecast

from previous cycle as

background for analysis) All KMA operational data

Grid : 60x54x31

Resolution : 30km

Domain size: the same as the

KMA operational 10km domain.


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Precipitation Verification

0.1 mm Precipitation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

3 6 9 12 15 18 21 24

FCST Time (hours)

CSI3dvar4dvar

5mm Precipitation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3 6 9 12 15 18 21 24

FCST Time (Hours)

CSI3dvar4dvar

15 mm Precipitation

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

3 6 9 12 15 18 21 24

FCST Time (Hours)

CSI3dvar4dvar

25 mm Precipitation

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

3 6 9 12 15 18 21 24

FCST Time (Hours)

CSI3dvar

4dvar


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Test domain: AFWA T8 45-km

Domain configurations:• 2007-08-15-00 to 2007-08-21-00 • Horizontal grid: 123 x 111 @ 45 km grid spacing, 27

vertical levels• 3/6 hours GFS forecast as FG • Every 6 hours

Observations used:sound sonde_sfc

synop geoamv

airep pilot

ships qscat

gpspw gpsref


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00 h 12 h 24 h

RMSE Profiles at Verification Times (U,V)


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00 h 12 h 24 h

RMSE Profiles at Verification Times (T,Q)


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The first radar data assimilation experiment using WRF 4D-Var (OSSE)

TRUTH ----- Initial condition from TRUTH (13-h forecast initialized at 2002061212Z from AWIPS 3-h analysis) run cutted by ndown,

boundary condition from NCEP GFS data. NODA ----- Both initial condition and boundary condition from NCEP

GFS data.3DVAR ----- 3DVAR analysis at 2002061301Z used as the initial

condition, and boundary condition from NCEP GFS. Only Radar radial velocity at 2002061301Z assimilated (total # of data points = 65,195).

4DVAR ----- 4DVAR analysis at 2002061301Z used as initial condition, and boundary condition from NCEP GFS. The radar radial velocity at 4 times: 200206130100, 05, 10, and 15, are assimilated (total # of data points = 262,445).


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TRUTH NODA

4DVAR3DVAR

Hourly precipitation ending at 03-h forecast


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TRUTH NODA

3DVAR 4DVAR

Hourly precipitation ending at 06-h forecast


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Real data experiments

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• Introduction to data assimilation.• Basics of variational data assimilation.• Demonstration with a simple system.• The WRFDA approach:

• y, H, H, HT

• B• M, M, MT

• 4D-Var

Summary


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A short list of

variational data assimilation issues

• yo observations, preprocessing, quality control

• H new observation types, improving operators

• R observational errors, tuning, correlated observational errors

• xb improve models (an important part of DA!!)

• B Statistical models, flow-dependent features

• Minimization algorithm Quasi−Newton; Conjugate Gradient; …( )

• M and MT : development and validity (and ensemble approaches)


DA

www.mmm.ucar.edu/wrf/users/wrfda

Documents

Hans Huang: Variational Data Assimilation, SNU lecture. June 9th, 20091 DA Variational Data Assimilation Hans Huang NCAR Acknowledge: Dale Barker, The