Spatial correlation of sea ice observation errors in - ESA

Effect of spatially correlated observation
errors in the ensemble (sea ice) data
assimilation
Anna Shlyaeva, Mark Buehner
Data Assimilation and Satellite Meteorology Research,
Environment Canada
Sea ice data assimilation in CMC/CIS
• Collaboration between Canadian Meteorological Center
(CMC) and Canadian Ice Service (CIS)
• Global and regional ice prediction systems analysis:
Alain Caya, Mark Buehner, Michael Ross and Anna Shlyaeva
(Meteorological Research Division)
Lynn Pogson, Tom Carrieres, Jack Chen and Yi Luo
(Marine and Ice Services Division)
Manon Lajoie
(Prediction Development Division)
Page 2 – May 2, 2014
The Regional Ice Prediction System
(RIPS) analysis
• Main use: provides input for generation
•
•
•
•
•
of CIS operational products (both
manual and automated)
The system is based on a variational
approach to data assimilation
Four analyses per day of ice
concentration at 5 km resolution on
rotated lat-lon grid
Domain chosen to include new
METAREAs and meet the needs of
North American Ice Service
(USA/Canada)
Includes the Great Lakes and many
other lakes (those for which CIS
already produces analyses)
Also serves as the test-bed for
evaluating upgrades for all
implementations (global and regional)
Page 3 – May 2, 2014
The Regional Ice Prediction System
(RIPS) analysis
• Observations:
–
–
–
–
–
–
SSMI NT2 retrievals
SSMI/S NT2 retrievals
ASCAT anisotropy
Ice charts
AMSR2 NT2 retrievals (research)
AVHRR (research)
• 3D-Var data assimilation scheme for sea ice
concentration assimilation
• Research on EnVar data assimilation
Page 4 – May 2, 2014
Assimilated data: Typical data coverage
SSM/I ice concentration
SSMI/S ice concentration
CIS daily ice charts
ASCAT anisotropy
Page 5 – May 2, 2014
Passive microwave ice concentration
retrievals
• SSMI, SSMI/S, AMSR2 observations
• Nasa Team 2 algorithm for retrieving ice concentration
• Retrievals are sensitive to atmospheric influence
(especially for low ice concentrations)
• Observation footprint (diameters)
– 58km SSMI/S
– 55km SSMI
– 22km AMSR2
Page 6 – May 2, 2014
Sea ice concentration assimilation
• Ice concentration observations are likely spatially
correlated
• Thinning data is not a good option:
– Land: can thin out the data in narrow channels, lose data close
to land, etc.
– Ice edge: can lose useful information on the location of ice edge
• How does introducing spatially correlated observation
errors affect the analysis?
Page 7 – May 2, 2014
Simple problem for assessing spatially
correlated observations errors effect
• 1D (or 2D) data assimilation problem
• Gridded observations
• Background error covariance is specified perfectly
• Background and observations have homogeneous,
isotropic errors
Page 8 – May 2, 2014
Notations
• Background error covariance matrix: B gs
• In spectral space: B  SBgs S T
• Observation error covariance matrix: R gs
gs T
R

SR
S
• In spectral space:
• Background and observations have homogeneous,
isotropic errors: B and R are diagonal in spectral space
Page 9 – May 2, 2014
B and R variances in the spectral space
Uncorrelated errors have same variances across all scales
Page 10 – May 2, 2014
B and R variances in the spectral space
Correlated errors have higher variances at larger scales,
lower variances at smaller scales
Page 11 – May 2, 2014
B and R variances in the spectral space
Correlated errors have higher variances at larger scales,
lower variances at smaller scales
Page 12 – May 2, 2014
Kalman gain in spectral space
For our simple problem:
– B and R in spectral space are diagonal
– H is identity
Thus, we can solve the problem separately for different
scales, and Kalman gain will be the observation weight
x a  x b  K ( y o  Hxb )
K  BH ( HBH  R )
T
T
xia  (1  Ki ,i ) xib  Ki ,i yio
Bi ,i
Ki ,i 
Bi ,i  Ri ,i
1
Page 13 – May 2, 2014
Kalman gain in spectral space:
uncorrelated errors
Uncorrelated errors: observations can mostly correct larger scales
Page 14 – May 2, 2014
Kalman gain in spectral space:
correlated errors
Correlated errors: observations can mostly correct smaller scales
Page 15 – May 2, 2014
Kalman gain in spectral space:
assuming uncorrelated errors
using correlated
errors
assuming
uncorrelated errors
Assuming uncorrelated errors (when they are correlated):
overfitting obs at larger scales, underfitting at smaller scales
Page 16 – May 2, 2014
How does introducing spatially correlated
observation errors affect the analysis?
• It can help to correct smaller scales
• It can also help to reduce overfitting to obs at the
larger scales
• Can be useful for the sea ice concentration data
assimilation: we can try to correct ice edge on smaller
scales
Page 17 – May 2, 2014
Ensemble data assimilation
Introducing spatially correlated observation errors in the
ensemble assimilation poses more questions:
• How does the analysis ensemble covariance
(spread) change if we introduce correlated observation
errors?
• How is error covariance of the analysis ensemble
mean (error) affected by introducing correlated
observation errors? (same question applies to the
deterministic assimilation)
Page 18 – May 2, 2014
Ensemble assimilation
(EnDA, EnKF with perturbed observations)
Ensemble forecast
step
Forecast step
Analysis, x
a
b
Background, x
Analysis step
o
Observations, y
Ensemble analysis
step
B
R
Backgrounds
x b (i ) , i  1 : N ens
Analyses
x , i  1 : N ens
a (i )
Observations
y o(i ) , i  1 : Nens
Page 19 – May 2, 2014
Obs, y
o
Ensemble
B
R
EnDA, EnKF with perturbed obs
• Observation error covariance in EnDA/EnKF with
perturbed obs:
– in assimilation (gain matrix)
– in observations’ perturbations
• How does the analysis spread change if we introduce
•
correlated observation errors?
How is mean analysis error affected by introducing
correlated observation errors?
• How does using different R in assimilation and
perturbations affect spread and error?
Page 20 – May 2, 2014
EnDA, EnKF with perturbed obs
R is the true observation error covariance, with spatial
correlations
Rd is the observation error covariance matrix that ignores
spatial correlations
1. Use R both in assimilation and perturbations
2. Use R in assimilation, but Rd in perturbations (unlikely
scenario)
3. Use Rd both in assimilation and perturbations (currently
used)
4. Use Rd in assimilation, but R in perturbations (relatively
easy to implement)
Page 21 – May 2, 2014
Case 1: Use true R
Using true R in assimilation
and perturbations:
• Analysis spread equals the
mean analysis error
• Mean analysis error is always
smaller then background error
Page 22 – May 2, 2014
Case 2: Use true R in assimilation,
diagonal Rd in perturbations
Using true R in assimilation and
diagonal Rd in perturbations:
• Mean analysis error is the same
as in previous case
• Analysis spread is different from
the error:
– Underestimated at larger scales
– Overestimated at smaller scales
Page 23 – May 2, 2014
Case 3: Use diagonal Rd
Using diagonal Rd in
assimilation and perturbations:
• Mean analysis error is higher
than in previous cases (that
have correct R in assimilation)
• Analysis spread is different
from the error:
– Way underestimated at larger
scales
– Overestimated at mid scales
Page 24 – May 2, 2014
Case 4: Use true R in perturbations,
diagonal Rd in assimilation
Using diagonal Rd in
assimilation and true R in
perturbations:
• Mean analysis error is same as
case 3 (that has diagonal Rd in
assimilation) and higher than in
previous cases (that have
correct R in assimilation)
• Analysis spread is equal to the
mean analysis error!
Page 25 – May 2, 2014
Error covariance of analysis ensemble
mean
• Error covariance of analysis ensemble mean depends on
true R and Ra used in the assimilation, and doesn’t
depend on Rp used in the perturbations
E  B  BH T ( HBH T  Ra ) 1 HB 
 BH T ( HBH T  Ra ) 1 ( R  Ra )( HBH T  Ra ) 1 HB
E  ( I  KH ) B  K ( R  Ra ) K T
R is the true error covariance
Ra is the error covariance used in the assimilation
Page 26 – May 2, 2014
Analysis ensemble covariance
• Analysis ensemble covariance depends on Ra used in
the assimilation and Rp used in the perturbations
A  B  BH T ( HBH T  Ra )1 HB 
 BH T ( HBH T  Ra )1 ( R p  Ra )( HBH T  Ra )1 HB
A  ( I  KH ) B  K ( R p  Ra ) K T
Ra is the error covariance used in the assimilation
R p is the error covariance used for observations perturbations
Page 27 – May 2, 2014
Analysis error covariance and analysis
ensemble covariance
• Error covariance of analysis ensemble mean:
E  B  BH T ( HBH T  Ra )1 HB 
 BH T ( HBH T  Ra )1 ( R  Ra )( HBH T  Ra ) 1 HB
• Analysis ensemble covariance:
A  B  BH T ( HBH T  Ra ) 1 HB 
 BH T ( HBH T  Ra )1 ( R p  Ra )( HBH T  Ra )1 HB
• Specifying “true” R in perturbations allows for analysis error
covariance and analysis ensemble covariance be close/equal
• For the ensemble system this gives us a background spread at the
next assimilation step consistent with the background error
Page 28 – May 2, 2014
Summary
Introducing spatially correlated observation errors:
• Can correct smaller scales
• Can reduce overfitting to obs at the larger scales
• Introducing them in the observation perturbations can
avoid ensemble covariance diverging from error
covariance of the ensemble mean
• Introducing them both in observation perturbations and
assimilation scheme will also reduce the error
Page 29 – May 2, 2014
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