Transcript Introduction to Data Assimilation

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.
2.
3.
4.
Covariance modelling – Part 1
Initialization (Filtering of analyses)
Basic estimation theory
3D-Variational Assimilation (3Dvar)
Background error covariance
matrix filters analysis increments

x  x  BH HBH  R
a
b
T
T
 z  H (x )
1
x  x  Bv
a
b
Analysis increments (xa – xb) are a
linear combination of columns of B
Properties of B determine filtering
properties of assimilation scheme!
b
A simple demonstration of filtering properties of B matrix
K
Choose a correlation function
and obs increment shape then
compute analysis increments
cos(x)
 1  x  y  2 
  x  y   exp  
 
 2  Ld  
sobs/sb = 0.5
Ld  0.2
cos(2x)
cos(3x)
cos(4x)
cos(5x)
cos(6x)
cos(7x)
cos(8x)
cos(9x)
cos(10x)
1. Covariance Modelling
1. Innovations method
2. NMC-method
3. Ensemble method
Background error covariance matrix


P  x x x x
b
b
t
b

t T
•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
1. Innovations method
• 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
Cb ( xi , yi , zi , x j , y j , z j )  CbH ( xi , yi , x j , y j )CVb ( zi , z j )
•Homogeneity
•Isotropy
m
CbH ( xi , yi , x j , y j )  Cbx (ri )
r
i
j
CbH ( xi , yi , x j , y j )  Cbx (ri , j )
j
l
r
m
r
i

r
l

Instrument+
representativeness
Dec. 15/87-Mar. 15/88
radiosonde data.
Model: CMC T59L20
Background error
Choose obs s.t. these terms =0
Assume homogeneous, isotropic
correlation model. Choose a
continuous function (r) which has
only a few parameters such as
L, correlation length scale. Plot all
innovations as a function of distance
only and fit the function to the data.
Mitchell et al. (1990)
Obs and Forecast error variances
Mitchell et al. (1990)
Mitchell et al. (1990)
Vertical correlations
of forecast error
Height
Lonnberg and Hollingsworth (1986)
Non-divergent wind
Hollingsworth and Lonnberg (1986)
Multivariate correlations
Bouttier and Courtier
www.ecmwf.int 2002
Mitchell et al. (1990)
If covariances are homogeneous,
variances are independent of space
Covariances are not
homogeneous
If correlations are homogeneous,
correlation lengths are independent
of location
Correlations are not
homogeneous
Correlations are not
isotropic
Gustafsson (1981)
Daley (1991)
Are correlations separable?
If so, correlation length should be
Independent of height.
Lonnberg and Hollingsworth (1986)
Mitchell et al. (1990)
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
Balance in data assimilation
Daley 1991
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.
Nonlinear Normal Mode Initialization (NNMI)
Consider model
Determine modes
Separate R and G
Project onto G
Define
balance
Solution
dx
 iAx  N ( x )
dt
T
A  E E
E  E R | EG 
c G  EGT x, c R  ETR x
dc G
 ic G  EGT N ( EG c G , E R c R )  0
dt
c G  i1EGT N ( EG c G , E R c R )
The slow manifold
dc G
0
dt
G
A
S
N
R
L
Daley 1991
Digital Filter Initialization (DFI)
Lynch and Huang (1992)
x0I 
N
u
h
x
kk
k  N
N=12, Dt=30 min
Tc=6 h
Tc=8 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.
Gravity modes
N
Rossby modes
Daley (1986)
Variational Normal model initialization
Daley (1978), Tribbia (1982), Fillion and Temperton (1989), etc.
I 
S
Minimize I such that uI, vI, fI stays on M.
~
~
~
~
(u I  u A ) 2 wV  (vI  v A ) 2 wV  ( I   A ) 2 w dS ,   gh


Daley (1986)
Some signals in
the forecast e.g.
tides should NOT
be destroyed by
NNMI!
So filter analysis
increments only
Semi-diurnal mode has
amplitude seen in free
model run, if anl
increments are filtered
Seaman et al. (1995)
3. A bit of Estimation theory
(will lead us to 3D-Var)
a posteriori p.d.f.
Data Selection
From: ECMWF training
course available at
www.ecmwf.int
Bouttier and Courtier (2002)
The effect of data selection
PSAS
OI
Cohn et al. (1998)
The effect of
data selection
Cohn et al. (1998)
Advantages of 3D-var
J (x)  (x  xb )T B1 (x  xb )  (z  H (x))T R1 (z  H (x))
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.
3D-Var Preconditioning
J (x)  (x  xb )T B1 (x  xb )  (z  H (x))T R1 (z  H (x))
(1)
• Hessian of cost function is B-1 + HTR-1H
• To avoid computing B-1 in (1), change control
variable to dx=Lc so first term in (1) becomes
½ ccT and we minimize w.r.t. c. Here dx=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
With covariances in spectral space,
longer correlation lengths scales are
permitted in the stratosphere
Andersson et al. (1998)
With flexibility of choice of obs,
can assimilate many new types
of obs such as scatterometer
Andersson et al. (1998)
To assimilate radiances directly, H includes an
instrument-specific radiative transfer model
b T
1
b
T
1
J (x)  (x  x ) B (x  x )  (z  H (x)) R (z  H (x))
Normalized AMSU
weighting functions
14
13
12
11
10
9
8
7
6
5
Impact of Direct Assimilation of Radiances
Anomaly = difference between forecast and climatolgy
Anomaly correlation = pattern correlation between forecast anomalies and
verifying analyses
1974 – improved
NESDIS VTPR
Retrievals
1978 – TOVS
retrievals
Kalnay et al. (1998)
Operational weather centers used 3D-Var from1990’s
*Later replaced by 4D-Var
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
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.