Covariance Modeling

“Where America’s Climate, Weather, Ocean and Space Weather Services Begin”

JCSDA Summer Colloquium July 27, 2015

Presented by John DerberNational Centers for Environmental Prediction

Covariance matrices provide the weighting of the various

terms in the objective function and determine how the

information is distributed.

Covariance matrices

• Will discuss– Observational error covariance– Background error covariance

• Will not discuss covariance matrices in other terms of objective function (if they exist). Similar but usually smaller impact.

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Observational error covariance matrix

• Observation term ½(H(x)-y)TO-1(H(x)-y)• Includes errors due to

– Errors in observations• Can be correlated/uncorrelated (off-diagonal/diagonal matrix)

due to non-random or random errors

– Errors in forward model• Example: Not including effect of aerosols in radiative transfer when using

satellite radiances. Error in simulating observation due to inadequacy of forward model – may be correlated from one observation to next.

• Example: Thunderstorm within grid-box where observation is taken. Thunderstorm cannot be resolved by analysis or background. It would take a very fancy downscaling scheme to do this. Resolution/physics dependent.

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Diagonal of O

• Have upper bound (difference between observation and background) – includes both observation and background error– First step in inclusion of new data is producing

these statistics (for error specification and QC).• Often have lower bound (manufacturer’s or

producers values)• Further refinement

– Desroziers, G., Berre, L., Chapnik, B. and Poli, P., 2005: Diagnosis of observation, background and analysis-error statistics in observation space. Q.J.R. Meteorol. Soc., 131: 3385–3396.

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Estimates of σo for AMSU-A on NOAA-18 from the Hollingsworth/Lönnberg (purple) and Desroziers (red) diagnostics, together with estimates of the instrument noise (black), the standard deviation of background departures (dashed grey), and the observation error assumed in 2008 (grey). From Bormann, Bonavita and McNally, 2015. Seminar on Use of Satellite Observations in Numerical Weather Prediction, 8-12 September 2014.

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Forecast impact of reducing the AMSU-A observation errors in terms of the normalised difference in the root mean square error for the 500 hPa geopotential for the Northern Hemisphere (left) and Southern Hemisphere extra-tropics (right). Vertical bars indicate 95% significance intervals. The results are based on 120 forecasts obtained during December 2009–January 2010 and May–July 2010. ). From Bormann, Bonavita and McNally, 2015. Seminar on Use of Satellite Observations in Numerical Weather Prediction, 8-12 September 2014.

Off-diagonal components of O(correlated error)

• Known to exist from both observation and representativeness error – but often assumed small.

• Two traditional (flawed but still used) methods to deal with this correlated error– Thinning – Satellite winds– Inflate diagonal component

• Most work with off-diagonal components with satellite data

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Estimates of observation error correlations obtained with the Hollingsworth/Lönnberg diagnostic for IASI. From Bormann, Bonavita and McNally, 2015. Seminar on Use of Satellite Observations in Numerical Weather Prediction, 8-12 September 2014.

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Change in mean percentage forecast root mean squared error (RMSE) and weighted skill against (a) observations and (b) analysis for winter and summer trials of accounting for correlated errors for IASI. The verification metrics used are pressure at mean sea level (PMSL), geopotential height at 500 hPa (H500) and winds at 250 and 850 hPa (W250 and W850) for the Tropics (TR) and Northern and Southern Hemispheres (NH, SH). From Weston, Bell and Eyre(2014), Accounting for correlated error in the assimilation of high-resolution sounder data. Q.J.R. Meteorol. Soc., 140: 2420–2429.

Off-diagonal components of O(correlated error)

• Satellite Radiances– Correlated error used operationally at Met

Office since 2013 for AIRS and IASI. – Some conditioning problems had to be

addressed. Several proposed/used techniques.

– Significant impact when used for hyperspectral IR sounders

– Cloudy radiances?11

Background error covariance - B

• Background term ½(x-xb)TB-1 (x-xb)

– x is analysis variable • Does not have to be same as model variables, but must

be able to convert to model variables.• Choice of analysis variable can impact complexity of B,

quality of analysis and level of balance.– Streamfunction, unbalanced temperature, unbalanced velocity

potential, unbalanced surface pressure removes off diagonal components between winds and temperature

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Covariance structure – unbalanced t,d,psz1000 observation

From: DERBER, J. and BOUTTIER, F. (1999), A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus A, 51: 195–221.

Covariance structure – unbalanced t,d,psz1000 observation

Background error covariance - B

– xb is the Background • Usually short term forecast• Can have as much (or more) information in it as

observations – but different error structure.• Quality of forecast model and previous analysis is very

important!

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Background error covariance - B

– Background error covariance• Determines how information is distributed spatially and

among analysis variables• Until the last few years, B was usually assumed to be

static (i.e., not changing with time) or quasi-static. Of course, everyone acknowledged that this was sub-optimal.

• Situation dependent errors area of current significant development

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Kalman Filter in VariationalSetting

KFKF BKHIA

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1TKF

TKF

HHBRHBK

QMAMB TKFKF

Forecast Step

Analysis

• Analysis step in variational framework (cost function)

Extended Kalman Filter

• BKF: Time evolving background error covariance

abt1t M xx

boba HxyKxx

yxHRyxHxBxx 1T1KF

TKF 2

1

2

1J

HRHBA 1T1KF

1KF

Important things to note from previous slide

• The background error from one analysis time to the next evolves due to model dynamics/physics and model error

• Analysis step should always reduce error• Background error should be very inhomogeneous since

depends on– Previous analysis error (distribution of obs, etc.)– Model error (different in highs and lows)– Model dynamics

• Real forecast model is nonlinear – not linear as assumed here.

• Use of full Kalman Filter not really practical18

Motivation for ensemble based Background error from [Ensemble] Kalman Filter

TbbKF 1

1XXB

N

ab XX

ba XX

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• Problem: dimensions of AKF and BKF are huge, making this practically impossible for large systems (GFS for example)

• Solution: sample and update using an ensemble instead of evolving AKF and BKF explicitly

TaaKF 1

1XXA

N

Ensemble Perturbations

Forecast Step:

Analysis Step:

N is ensemble size

bf & be: weighting coefficients for fixed and ensemble covariance respectively

xt’: (total increment) sum of increment from fixed/static B (xf’) and ensemble B

ak: extended control variable; :ensemble perturbations

- analogous to the weights in the LETKF formulation

L: correlation matrix [effectively from the localization of ensemble perturbations]

Hybrid Variational-Ensemble

kex

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• Incorporate ensemble perturbations directly into variational cost function through extended control variable– Lorenc (2003), Buehner (2005), Wang et. al. (2007), etc.

yxHRyxH

LxBxx

t1T

t

1

1T

ef1

fT

fff

2

1

2

1

2

1 N

n

nnββ,J ααα

N

n

nn

1eft xxx α

Hybrid assimilation

• Solution combination of static term and ensemble term. Static term includes degrees of freedom not described by ensembles.

• Localization included to exclude spurious correlations which are due to small ensemble size.

• Localization results in imbalance.• Use of different resolution also results in

imbalance, especially for small scale.• More realistic increments – better forecasts.

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Single Temperature Observation

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3DVAR

bf-1=0.0 bf

-1=0.5

Final comments

• Covariance matrices provide information on weighting and structures in observations and analyses.

• Until recently fairly simple structures for these covariances were used (e.g., uncorrelated observation errors, homogeneous static background errors).

• Early results using more complex covariance matrices show significant improvement.

• There is room for very significant additional improvement.

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