Bill Campbell and Liz Satterfield JCSDA Summer Colloquium on Satellite Data Assimilation 27 Jul - 7 Aug 2015 Accounting for Correlated Satellite Observation.
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Bill Campbell and Liz Satterfield
JCSDA Summer Colloquium on Satellite Data Assimilation
27 Jul - 7 Aug 2015
Accounting for Correlated Satellite Observation Error in NAVGEM
1
Everything the data assimilation (DA) system knows about the error of every
observation everywhere is contained in R
• Observation error is defined as the difference between a measured quantity and the truth, which is the true state of the continuous atmosphere discretized, mapped into model space, then mapped into observation space (via interpolation, radiative transfer, etc.)
• Observation error variance is the time mean squared observation error, which is not observed directly, but computed statistically
• Observation error covariance is the time mean of the product of the errors of different observations (e.g. a temperature measurement error in Fort Collins and a wind speed measurement error in Denver)
• All covariance matrices are symmetric and positive definite• Covariance has physical units; correlation is normalized covariance, with
values between -1 and 1• Observation error covariance matrix R should contain the best estimates
of variances on the diagonal, covariances everywhere else3
Observation Error Defined
4
Sources of Observation Error
IMPERFECTOBSERVATIONS
True Temperature in Model Space
T=28° T=38° T=58°
T=30° T=44° T=61°
T=32° T=53° T=63°
T=44°
1) Instrument error (usually, but not always, uncorrelated)2) Mapping operator (H) error (interpolation, radiative transfer)3) Pre-processing, quality control, and bias correction errors4) Error of representation (sampling or scaling error), which can
lead to correlated error:
5
Correlated Error in Operational DA
• Until recently, most operation DA systems assumed no correlations between observations at different levels or locations (i.e., a diagonal R)
• To compensate for observation errors that are actually correlated, one or more of the following is typically done:– Discard (“thin”) observations until the remaining ones are
uncorrelated (Bergman and Bonner (1976), Liu and Rabier (2003))– Local averaging (“superobbing”) (Berger and Forsythe (2004))– Inflate the observation error variances (Stewart et al. (2008, 2013)
• Theoretical studies (e.g. Stewart et al., 2009 & 2013) indicate that including even approximate correlation structures outperforms diagonal R with variance inflation
• *In January, 2013, the Met Office went operational with a vertical observation error covariance submatrix for the IASI instrument, which showed forecast benefit in seasonal testing in both hemispheres (Weston et al. (2014))
6
Error Covariance Estimation Methods
• There are four statistical methods we are aware of for observation error covariance estimation: Hollingsworth & Lönnberg (1986) (H-L), Fisher(2003), Desroziers (2005), and Daescu (2013)
• H-L, Fisher (aka background error method), and Daescu can diagnose observation error variances (diag(R)), and H-L can diagnose vertical correlation, but none of these can diagnose horizontally-correlated observation error
• The background error method assumes that the forecast error covariance matrix B is essentially correct, which is difficult to validate
• Desroziers accounts for all types of correlated error, but the diagnosed R depends on the initial specification of the observation and forecast error covariance matrices (R and B, respectively)
• The diagnosed R must be made symmetric and positive definite, and it may be necessary to adjust its eigenvalue spectrum to address convergence issues
• None of these methods is perfect
7
Hollingsworth-Lönnberg Method(Hollingsworth and Lönnberg, 1986)
• Use innovation statistics from a dense observing network• Assume horizontally uncorrelated observation errors• Calculate a histogram of background innovation covariances binned by
horizontal separation• Fit an isotropic correlation model, extrapolate to zero separation to estimate
the correlated (forecast) and uncorrelated (observation) error partition
Correlation between observation error and forecast error: possibly induced by QC methods or bias correction methods
Observation error variance
Bias terms
Extrapolate red curve to zero separation, and compare with innovation variance (purple dot)
Mean of ob minus forecast (O-F) covariances, binned by separation distance
u2
c2
Assumes no spatially-correlated observation error
2 2 2o f innovation
22 ' '
22 2 2 ' '2
innovation o o f f
innovation o f o f o f
8
• From O-F, O-A, and A-F statistics from any model (e.g. NAVGEM), the observation error covariance matrix R, the representer HBHT, and their sum can be diagnosed
• This method is sensitive to the R and HBHT that is prescribed in the DA system
• An iterative approach may be necessary• Diagnose R1 , which will be different from the original R • Symmetrize R1 , possibly adjusting its eigenvalue spectrum• Implement R1 and run NAVGEM • Diagnose R2 , which we hope will be closer to Rtrue
Desroziers Method(Desroziers et al. 2005)
4DVar Primal Formulation
1
11 1 1 1
1 1 1
T Tf f
T T T Tf
T Tf
w x x BH HBH R y H x
B H R H w B H R H BH HBH R y H x
B H R H w H R y H x
Scale by B-1/21 2
1 2
s B w
w B s
1 2 1 1 1 2 1 2 1
1 2 1 1 2 1 2 1
T Tf
T Tf
B B H R H B s B H R y H x
I B H R HB s B H R y H x
4D-var iteration is on this problem -- We need to invert R!
4DVar Dual Formulation
Scale by R-1/2
1/2 1/2
,{ }i j
R R
R
R C
diag
C 4D-Var iteration is on this problem – No need to invert C!
Iteration is done on partial step and then mapped back with BHT
• An advantage of the dual formulation is that correlated observation error can be implemented directly
• No matrix inverse is required, which lifts some restrictions on the feasible size of a non-diagonal R
• In particular, implementing horizontally correlated observation error is significantly less challenging
11
Correlated Observation Error for the ATMS Microwave Sounder
ATMSMicrowave Sounder
13 temperature channels9 moisture channels
Channel Number
Cha
nnel
Num
ber
4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22
22
21
20
19
18
15
14
13
12
11
10
9
8
7
6
5
40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Current observation error correlation matrix used for ATMS,
and for ALL observations
12
Error Covariance Estimationfor the ATMS
Statistical Estimate
Chan
nel N
umbe
r
4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22 Channel Number
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 4
5
6
7 8
9
10
11
12 1
3 1
4 15
18
19
20
21
22
Desroziers’ method estimate of interchannel portion of observation
error correlation matrix for ATMS
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 4
5
6
7
8 9
10
11
12 1
3 14
15
18 1
9 20
21
22 4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22
Chan
nel N
umbe
r
Current Treatment
Channel Number
Temperature Temperature
Moisture Moisture
Current observation error correlation matrix used for ATMS,
and for ALL observations
13
Iterating Desroziers
Old Statistical Estimate
Chan
nel N
umbe
r
4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22 Channel Number
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 4
5
6
7 8
9
10
11
12 1
3 1
4 15
18
19
20
21
22
Desroziers’ method estimate of interchannel portion of observation
error correlation matrix for ATMS
4
5
6
7
8 9
10
11
12 1
3 14
15
18 1
9 20
21
22 4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22
Chan
nel N
umbe
r
New Statistical Estimate
Channel Number
The change is not large, which is (weak) evidence that the procedure may converge
14
Practical Implementation: What about Convergence?
• The condition number of a matrix X is defined by σmax(X)/σmin(X), which is the ratio of the maximum singular value of X to the minimum one (same as the eigenvalue ratio if X is symmetric and positive definite).
• It is a commonly-used barometer for the numerical stability of matrices.
• How to improve conditioning: 1. Increase the diagonal values (additively) of the matrix until
the desired condition number is reached (e.g. Weston et al. (2014)).
2. Preconditioning 3. Find a positive definite approximation to the matrix such
that the condition number is constrained.
Condition Number Constrained Correlation Matrix Approximation
We want to find a positive definite approximation to the matrix ,
ˆp k
X X Xis minimized. In the trace norm, simply set all of the smallest singular values equal to the value that gives the desired condition number, and then reconstruct the matrix with the singular vectors to obtain the approximate matrix.
Another method, used by Weston et al. 2014, is to increase the diagonal values of the matrix until the desired condition number is reached. We believe there is better theoretical justification for the first method using the Ky Fan p-k norm (Tanaka, M. and K. Nakata, 2014)
1
,1
k pp
ip ki
X XThe Ky Fan p-k norm of m n X where
i X denotes the ith largest singular value of X
When p=2 and k=n, it is called the Frobenius norm;when p=1 and k=n, it is called the trace norm.
16
Condition Number Constrained Correlation Matrix Approximation
0
0.05
0.1
0.15
0.2
Cond. Number = 18 Cond. Number = 18
Chan
nel N
umbe
r
5 7 9 11 13 15 19 21
Channel Number
0
0.05
0.1
0.15
0.2 5 7 9 11 13 15 19 21
Channel Number
5
7
9
11
1
3
15
1
9
21
5
7
9
11
1
3
15
1
9
21
Difference between original and conditioned correlation matrix
Trace Norm Increase Diagonal
Convergence and the Cauchy Interlacing Theorem
Cauchy interlacing theoremLet A be a symmetric n × n matrix. The m × m matrix B, where m ≤ n, is called a compression of A if there exists an orthogonal projection P onto a subspace of dimension m such that P*AP = B. The Cauchy interlacing theorem states:
Theorem. If the eigenvalues of A are α1 ≤ ... ≤ αn, and those of B are β1 ≤ ... ≤ βj ≤ ... ≤ βm, then for all j < m + 1,
Notice that, when n − m = 1, we have αj ≤ βj ≤ αj+1, hence the
name interlacing theorem.
j j n m j
What happens when radiance profiles are incomplete (i.e., at a given location, some channels are missing, usually due to failing QC checks)?
18
Conjugate Gradient Convergence
Goal
C
19
Using the Desroziers Diagnostic for IASI Channel Selection
• Water vapor channels 2889, 2994, 2948, 2951, and 2958 have very high error correlation (>0.98)
• The eigenvectors corresponding to the 4 smallest eigenvalues project only on to these 5 channels
• It makes sense to use the Desroziers diagnostic to do a posteriori channel selection, which has the bonus of improving the condition number of the correlation matrix, and thus solver convergence
20
Conjugate Gradient Convergence
Goal
21
• The Desroziers error covariance estimation methods can quantify correlated observation error
• Minimal changes can be made to the estimated error correlations to fit operational time constraints
• After accounting for correlations, variances could potentially be reduced (assuming correct background)
• Correctly accounting for vertically correlated observation error in data assimilation has already yielded superior forecast results at multiple operational NWP centers without a large computational cost
• What about horizontally correlated error?
Main Conclusions
Correlated Error
RedSat
140 180 220 260 300 °K
Uncorrelated Error
WhiteSat
140 180 220 260 300 °KTruth (No Error)
True Atmosphere
• Imagine two hypothetical satellite instruments looking down on Hurricane Sandy• WhiteSat sees a fuzzy version of the truth; current DA systems handle uncorrelated error well• RedSat sees a warped version of the truth; current DA systems handle correlated error poorly• High resolution (i.e. future) instruments are more likely to have correlated error, which can be
much more subtle than this example 22
Visualizing Horizontally Correlated Observation Error
140 180 220 260 300 °K
23
• How can we best estimate errors in Desroziers/Hollingsworth- Lönnberg diagnostics?– Should we expect agreement between different methods?– Will the Desroziers diagnostic converge if both R and B are
incorrectly specified?– Amount of data required to estimate covariances? Seasonal
dependence?– Best methods to symmetrize the Desroziers matrix?
• How to gauge improvement?– Do we also need to adjust to see overall improvement to the
system? – How do we maintain the correct ratio for DA?
• What about convergence?– Should we do an eigenvalue scaling to improve the condition
number?
Discussion
24
References
IMPERFECTOBSERVATIONS
Hollingsworth, A. and P. Lonnberg, 1986. The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A, pp 111-136.
Desroziers, G., et al., 2005. Diagnosis of observation, background and analyis-error statistics in observation space. Quart. J. Roy. Meteor. Soc., 131, pp. 3385-3396.
Bormann , N. and P. Bauer, 2010: Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. I: Methods and application to ATOVS data. Quart. J. Roy. Meteor. Soc., 136, pp. 1036-1050.
Gorin, V. E., and M. D. Tsyrulnikov, 2011. Estimation of Multivariate Observation-Error Statistics for AMSU-A data. Mon. Wea. Rev., 139, pp. 3765-3780.
Stewart, L. M. et al., 2013. Data assimilation with correlated observation errors: experiments with a 1-D shallow water model. Tellus, 65.
Weston, P. P. et al., 2014. Accounting for correlated error in the assimilation of high-resolution sounder data. Quart. J. Roy. Meteor. Soc., 140, pp. 2420-2429.
Tanaka, M., and K. Nakata, 2014. Positive definite matrix approximation with condition number constraint. Optim. Lett., 8, pp. 939-947.
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