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Transcript of ECMWF Training Course 2008 slide 1 Influence matrix diagnostic to monitor the assimilation system...
ECMWFTraining Course 2008 slide 1
Influence matrix diagnostic to monitor the assimilation
system
Carla Cardinali
ECMWFTraining Course 2008 slide 2
Monitoring Assimilation System
ECMWF 4D-Var system handles a large variety of space and surface-
based observations. It combines observations and atmospheric state a
priori information by using a linearized and non-linear forecast model
Effective monitoring of a such complex system with 107 degree of
freedom and 106 observations is a necessity. No just few indicators but a
more complex set of measures to answer questions like
How much influent are the observations in the analysis?
How much influence is given to the a priori information?
How much does the estimate depend on one single influential obs?
ECMWFTraining Course 2008 slide 3
Influence Matrix: Introduction
Unusual or influential data points are not necessarily bad observations
but they may contain some of most interesting sample information
In Ordinary Least-Square the information is quantitatively available in
the Influence Matrix
Tuckey 63, Hoaglin and Welsch 78, Velleman and Welsch 81
Diagnostic methods are available for monitoring multiple regression
analysis to provide protection against distortion by anomalous data
y = Sy
ECMWFTraining Course 2008 slide 4
Influence Matrix in OLS
The OLS regression model is
y = Xβ + ε
y = Sy
T -1 Tβ (X X) X y
Y (mx1) observation vector
X (mxq) predictors matrix, full rank q
β (qx1) unknown parameters
(mx1) error2( ) 0, ( )E Var ε ε I
1T TS X(X X) X
The fitted response is
OLS provide the solution
ECMWFTraining Course 2008 slide 5
Influence Matrix Properties
y = Sy
S (mxm) symmetric, idempotent and positive definite matrix
It is seen
The diagonal element satisfy 0 1iiS
ˆ
y
Sy
( )Tr qS
ˆiij
j
yS
y
Cross-Sensitivity
ˆiii
i
yS
y
Self-Sensitivity
Average Self-Sensitivity=q/m
ECMWFTraining Course 2008 slide 6
Outline
Conclusion
Observation and background Influence
Generalized Least Square method
Findings related to data influence and information content
Toy model: 2 observations
ECMWFTraining Course 2008 slide 7
Hxb
y
HK
I-HK
Hxb
y
HKI-HK
Solution in the Observation Space
The analysis projected at the observation location
ˆ ( )a b y Hx HKy I HK Hx
The estimation ŷ is a weighted mean
Hxb y
HKI-HK
1 1 1 1T T K = (B + H R H) H R
B(qxq)=Var(xb)R(pxp)=Var(y)
K(qxp) gain matrixH(pxq) Jacobian matrix
( )a q b x Ky I KH x
ECMWFTraining Course 2008 slide 8
Influence Matrix
ˆ( )T T T Observation Influence
y
S HK K Hy
ˆ
b
Background Influence
yI - S
Hx
Observation Influence is complementary to Background Influence
ˆ ( ) b y Sy I S Hx
ˆ ( )a b y Hx HKy I HK Hx
ECMWFTraining Course 2008 slide 9
Influence Matrix Properties
The diagonal element satisfy 0 1iiS
1
N
iii
S Total Information Content
1
. .
N
iii
SAverage Influence
Tot Obs Number
ECMWFTraining Course 2008 slide 10
Synop Surface Pressure Influence
ECMWFTraining Course 2008 slide 11
Airep 250 hPa U-Comp Influence
ECMWFTraining Course 2008 slide 12
QuikSCAT U-Comp Influence
ECMWFTraining Course 2008 slide 13
Global and Partial Influence
ECMWFTraining Course 2008 slide 14
AMSU-A channel 13 Influence
ECMWFTraining Course 2008 slide 15
Global and Partial Influence
Global Influence = GI = 1
N
iii
S
N
100% only Obs Influence
0% only Model Influence
Partial Influence = PI = ii
i I
IpS Type
Variable
Area
Level
ECMWFTraining Course 2008 slide 16
Global and Partial Influence
Level
1 1000-850 2 850-700 3 700-500 4 500-400 5 400-300 6 300-200 7 200-100 8 100-70 9 70-50 10 50-30 11 30-0
Type
SYNOPAIREPSATOBDRIBUTEMPPILOTAMSUAHIRSSSMIGOESMETEOSATQuikSCAT
Variable
uvTqps
Tb
Area
Tropics S.HemN.Hem
Europe US N.Atl …
ECMWFTraining Course 2008 slide 17
Toy Model: 2 Observations
1 1 1 1( )T T S R H B + HR H H
2oR IH I
2 1
1b
B2
2o
b
r
x1
x2y2
y1
Find the expression for S as function of r and the expression of for α=0 and 1given the assumptions:
ˆ ( ) b y Sy I S x
y
ECMWFTraining Course 2008 slide 18
Toy Model: 2 Observations
1 1 1 1( )T T S R H B + HR H H
2oR IH I
2 1
1b
B
2
2 2 2 2
2
2 2 2 2
1
2 1 2 1
1
2 1 2 1
r r
r r r r
r r
r r r r
S
2
2o
b
r
Sii
r
1
1
1/2
1/3
1/2
=0
=1
x1
x2y2
y1
ECMWFTraining Course 2008 slide 19
Toy Model: 2 Observations
ˆ ( ) b y Sy I S x
2
1 1 1 2 22 2 2
2 2ˆ ( )
4 4 4y y x y x
2
21o
b
r
=0 1 1 1
1 1ˆ
2 2y y x
=1 1 1 1 2 2
1 2 1ˆ ( )
3 3 3y y x y x
x1
x2y2
y1
ECMWFTraining Course 2008 slide 20
Average Influence and Information Content
Global Observation Influence GI=15%
Global Background Influence I-GI=85%
0
5
10
15
20
25
Synop
Dribu
Airep
Satob
Tem
pPilo
t
SCAT
AMSU-A
AMSU-B
AIRS
HIRS
TCW
V
SSM/I
Goes
Met
eo
Ozone
Info
rma
tio
n c
on
ten
t %
0
0.1
0.2
0.3
0.4
0.5
0.6
Synop
Dribu
Airep
Satob
Temp
Pilot
SCAT
AMSU-A
AMSU-B
AIRS
HIRS
TCWV
SSM/I
Goe
s
Met
eo
Ozo
ne
Mea
n I
nfl
uen
ce
2003 ECMWF Operational
N.HemispherePI = 15%
TropicsPI = 17.5%
S.HemispherePI = 12%
GI = 15.3%
0 10 20 30 40 50 60
SynopDribuPaob
QuikSCATAirepSatobTempPilot
AmsuaHirs
SsmiGoes
MeteoOzone
0 10 20 30 40 50 60
SynopDribuPaob
QuikSCATAirepSatobTempPilot
AmsuaHirs
SsmiGoes
MeteoOzone
0 10 20 30 40 50 60
SynopDribuPaob
QuikSCATAirepSatobTempPilot
AmsuaHirs
SsmiGoes
MeteoOzone
2003 ECMWF Operational
ECMWFTraining Course 2008 slide 22
2007 ECMWF Operational
ECMWFTraining Course 2008 slide 23
DFS % first bar Global, second North Pole and third South Pole Global Information Content 20% North Pole 5.6%South Pole 1.1%
ECMWFTraining Course 2008 slide 24
DFS % first bar North Pole and second South Pole Global Information Content 20%North Pole 5.6%South Pole 1.1%Tropics 5.5%
ECMWFTraining Course 2008 slide 25
ConclusionsThe Influence Matrix is well-known in multi-variate linear regression. It is
used to identify influential data. Influence patterns are not part of the
estimates of the model but rather are part of the conditions under which the
model is estimated
Disproportionate influence can be due to:incorrect data (quality control)legitimately extreme observations occurrence
to which extent the estimate depends on these data
Sii=1Data-sparse Single observation
Model under-confident (1-Sii)
Sii=0Data-dense
Model over-confident tuning (1-Sii)
ECMWFTraining Course 2008 slide 26
Conclusions
Observational Influence pattern would provide information on different
observation system
New observation systemSpecial observing field campaign
Thinning is mainly performed to reduce the spatial correlation but also to
reduce the analysis computational cost
Knowledge of the observations influence helps in selecting
appropriate data density
Diagnose the impact of improved physics representation in the linearized
forecast model in terms of observation influence
ECMWFTraining Course 2008 slide 27
Background and Observation Tuning in ECMWF 4D-Var
Observations
Model
ECMWFTraining Course 2008 slide 28
Influence Matrix Computation
1
1 1
11( '') ( )( ) ( )( )
N MT Ti
i i i ii i i
u uN
J L L L L
1 1( '') T S R H J H
B
A sample of N=50 random vectors from (0,1) Truncated eigenvector expansion
with vectors obtained through the combined Lanczos/conjugatealgorithm. M=40
1 1 1( )T A B + HR H
ECMWFTraining Course 2008 slide 29
Hessian Approximation B-A
1
1( )( )
MTi
i ii i
L L
1
1( )( )
NT
i ii
u uN L L 500 random vector to represent B
ECMWFTraining Course 2008 slide 30
Ill-Condition Problem
A set of linear equation is said to be ill-conditioned if small variations in
X=(HK I-HK) have large effect on the exact solution ŷ, e.g matrix close
to singularity
max
min
( )
X =K
A Ill-conditioning has effects on the stability and solution accuracy . A
measure of ill-conditioning is
A different form of ill-conditioning can results from collinearity: XXT
close to singularity
Large difference between the background and observation error
standard deviation and high dimension matrix
ECMWFTraining Course 2008 slide 31
Flow Dependent b: MAMT+Q
DRIBU psInfluence
b
o