Representation Error in Ocean Data Assimilation

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Representation Error in Ocean Data Assimilation
Representation Error in Ocean
Data Assimilation
Robert N. Miller
James G. Richman
College of Oceanic and Atmospheric Sciences
Oregon State University
Corvallis, OR 97330
Representation Error in Ocean Data Assimilation – p.1/30
Data Assimilation: Assumptions
Given
• A model: ut − Lu = f
•
Chosen to mimic the “true” state u(t) which
evolve according to:
(t)
ut
•
•
− Lu(t) = f + b; b random
Observations z = Hu(t) + eobs ; H defines the
relation between the state vector and the observed
quantities
Question for Today: What, precisely, is u(t) ?
Representation Error in Ocean Data Assimilation – p.2/30
In Search of the True State
•
•
•
The ocean measured by instruments doesn’t
know about physical approximations, coarse
resolution or their consequences
It is not subject to the limitations in computing
power that restrict models to coarse resolution
Measurements are not subject to the same
requirements for approximate physical
parameterizations
So ask: What quantity in nature is the “true” value of
the model state?
No specific answers today; Rather a suggestion for
what to do while we are waiting.
Representation Error in Ocean Data Assimilation – p.3/30
Representation Error
•
•
•
Data assimilation makes use of data misfits, aka
innovations: z − Hu(f )
u(f ) is the forecast state
Let ũt be the “true” ocean, as the instruments
measure it.
Representation Error in Ocean Data Assimilation – p.4/30
Representation Error
Write the innovation:
z − Hu(f ) = z − z(t) + z(t) − Hu(f )
= ǫ0 + H(ũ(t) − u(t) ) + H(u(t) − u(f ) )
•
•
•
•
ǫ0 = z − z(t) , the instrument error
H(ũ(t) − u(t) ) is the representation error
Estimates of its statistics must appear in the terms
reserved for instrument error
u(t) − u(f ) is the forecast error
Representation Error in Ocean Data Assimilation – p.5/30
Estimating Representation Error
Our method for estimating the representation error for
SST:
1. Generate a long model run
2. Calculate EOFs of the model run, considered as a
matrix whose (i, j) element is the value of state
element j at time i
3. Determine the number of meaningful degrees of
freedom
4. Project the innovations on the meaningful EOFs
5. Subtract the result from the innovations.
6. The difference is an estimate of the representation
error
Representation Error in Ocean Data Assimilation – p.6/30
Pacific Circulation Model
•
•
•
•
•
Parallel Ocean Program (POP)
Domain:
• 105o E to 85o W
• 30o S to 64o N
Resolution
• 1o at the Equator, Mercator projection
• 0.5o average resolution
• 50 vertical levels, 25 in top 500m
25 years (1978-2002), forced by NCEP/DOE
reanalysis
Initialized from Levitus, 30 year spinup
Representation Error in Ocean Data Assimilation – p.7/30
EOFs of SST Anomalies
EO F AnalysisofSea Surface
Tem perature Anom alie
s
Representation Error in Ocean Data Assimilation – p.8/30
Maps of Variance Described by
SST EOFs
Representation Error in Ocean Data Assimilation – p.9/30
EOF Analysis of HOT SST
Anomalies
EO F AnalysisofSea Surface
Tem perature Anom alie
s
¥ The second EOF of the
SST with 4% of the total
variance described is
dominated by variability in
the strength of the
subtropical gyre. In the
subtropical gyre, this
mode describes 30-50%
of the SST variance. The
SST anomaly at HOT
(blue) is dominated by the
second mode (red) with
little contribution by the
other two modes (black)
Representation Error in Ocean Data Assimilation – p.10/30
Model and AVHRR Seasonal
Anomalies: First EOF
Representation Error in Ocean Data Assimilation – p.11/30
The Kalman Filter
Given a forecast state uf and observations z, correct
according to:
ua = uf + K(z − Huf )
f
T
f
T
−1
K = P H (HP H + R)
•
•
•
Pf is the forecast error covariance matrix; K is
the Kalman Gain Matrix
f
T
HP H + R is an estimate of the covariance of
the model-data misfit
In the Kalman filter, Pf evolves according to
model dynamics
Representation Error in Ocean Data Assimilation – p.12/30
A Static Reduced State Space
Filter
1. Compute multivariate EOFs of the 23 year time
series
2. Determine significant degrees of freedom: in this
case the Preisendorfer (1988) test indicates 35
significant modes, accounting for 59% of the total
variance
3. Estimate Pf by fitting the EOFs of the SST
misfits with the temperature portions of the
multivariate model EOFs
4. Estimate a static Kalman gain and assimilate SST
data
Representation Error in Ocean Data Assimilation – p.13/30
Results of Data Assimilation
Model-data correlations, before and after assimilation:
Representation Error in Ocean Data Assimilation – p.14/30
Why Not Better?
•
•
The model does pretty well at what it can do.
Most of the signal variability outside of the
tropics comes from eddies, and this can’t be
usefully assimilated
Representation Error in Ocean Data Assimilation – p.15/30
Representation Error
•
•
•
Project model-data misfits on multivariate EOFs.
This is the portion of the data that is compatible
with the model
Subtract the result from the model-data misfits.
This is an estimate of the error of representation
Calculate the EOFs of the error of representation
Representation Error in Ocean Data Assimilation – p.16/30
EOFs of SST Error of Representation
Representation Error in Ocean Data Assimilation – p.17/30
Next: The NCEP Climate Forecast System
•
•
•
•
Climate models are often run past their forecast
horizons
Climate models can only produce forecasts
consistent with their internal physics
Representation errors could take on crucial
importance
Maybe a statistical characterization of
representation errors can be used to devise a
stochastic model to simulate them
Representation Error in Ocean Data Assimilation – p.18/30
Ocean Component, NCEP CFS
•
•
•
Basically global MOM/POP
Resolution 1o over most of the ocean, tapering to
0.33o from 30o N/S to 8.5o
24 years (1982-2005), 9-month forecasts, from
the restart files
Representation Error in Ocean Data Assimilation – p.19/30
Spectral Analysis of the CFS
Restart Records
Temperature EOFs and Time Series of Amplitudes
4
50
11%
2
0
0
−2
−50
100
200
300
−4
1980
1990
2000
2010
1990
2000
2010
1990
2000
2010
4
50
4.3%
2
0
0
−2
−50
100
200
300
−4
1980
5
50
3.5%
0
0
−50
100
200
300
−5
1980
Representation Error in Ocean Data Assimilation – p.20/30
Lead EOF, SSH
Lead EOF, SSH
10
60
8
6
40
4
20
2
0
0
−2
−4
−20
−6
−40
−8
−10
−60
−12
50
100
150
200
250
300
350
Representation Error in Ocean Data Assimilation – p.21/30
Second EOF, SSH
EOF No. 2, SSH
25
60
20
40
15
10
20
5
0
0
−5
−20
−10
−40
−15
−20
−60
−25
50
100
150
200
250
300
350
Representation Error in Ocean Data Assimilation – p.22/30
Third EOF, SSH
EOF No. 3, SSH
10
60
40
5
20
0
0
−5
−20
−40
−10
−60
−15
50
100
150
200
250
300
350
Representation Error in Ocean Data Assimilation – p.23/30
Amplitude of the Lead EOF
SOI and First PC
4
2
0
−2
−4
−6
−8
1985
1990
1995
Year
2000
2005
Blue=lead PC; Red=SOI. Correlation ≈ 0.4
Representation Error in Ocean Data Assimilation – p.24/30
The Preisendorfer Test
Spectrum of Model Covariance Matrix
0.12
Proportion of Total Variance
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150
Eigenvalue Number
200
250
300
61 modes ≈ 60% of the total variance
Representation Error in Ocean Data Assimilation – p.25/30
EOFs of SST Observations
First EOF of Pathfinder SST
60
0.2
40
0
20
-0.2
0
-0.4
-20
-0.6
-40
-0.8
-60
50
100
150
200
250
300
350
-1
Representation Error in Ocean Data Assimilation – p.26/30
Significance
EOFs
Test
for
SSTA
Preisendorfer Test, AVHRR SSTA
35
30
25
20
15
10
5
0
−5
0
50
100
150
200
250
Representation Error in Ocean Data Assimilation – p.27/30
Amplitude of Lead EOF of
SSTA
Amplitude of Pathfinder SST EOF
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
1980
1985
1990
1995
2000
2005
2010
Representation Error in Ocean Data Assimilation – p.28/30
Lead EOF of Residuals
First EOF of SST Residuals
60
0.2
40
0
20
-0.2
0
-0.4
-20
-0.6
-40
-0.8
-60
50
100
150
200
250
300
350
-1
Representation Error in Ocean Data Assimilation – p.29/30
Amplitude of Lead EOF of
Residuals
Amplitude of SST Residuals EOF
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
1980
1985
1990
1995
2000
2005
2010
...This is going to be harder than we thought.
Representation Error in Ocean Data Assimilation – p.30/30

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