Craig H. BishopNaval Research Laboratory, Monterey

(with many slides taken from Mike Fisher’s ECMWF lecture on the same subject)

JCSDA Summer ColloquiumJuly 2012

Santa Fe, NM

Background Error Covariance Modeling

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Overview• Covariances of what, precisely?• Static error covariances from observations• Error covariances from proxies

– Bayes’ theorem– Perturbed observation ensembles– Kalman filters and EnKF ensembles

• Computationally efficient static error covariances.• Inclusion of balance constraints• Conclusions

What is the true error distribution ?

• Imagine an unimaginably large number of quasi-identical Earths.

Each Earth has one true state and one prediction but these differ from one Earth to another.

Collect all Earths having the same true state but differing forecasts of this state to

defin . T|e

t f

f t x x

x x

he error distribution is the distribution of differences

between individual forecasts and single truth within this set.

Collect all Earths having the same historical observati

fixed-tr

ons b

u

u

th

f

t diy

1 2

fering true atmospheric

states to define . The differences between individual truths and the

mean truth within this

| , ,...

fixed-obs set define the forecast error distribution. i

t ti x x y y

(Slartibartfast – Magrathean designer of planets, D. Adams, Hitchhikers …)

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What is an error covariance?

12 1

2

12 1 2 1 21

Covariance between forecast error of forecast of variable 1 with

forecast error of forecast of variable 2 is defined by1lim

where indexes the replicate Earth t

f

f

nf f f f

i ii

P

P nn

i

12 12

12 1 12 2

2 2

1 1 2 2

o which the errors pertain. Oftenwe represent in terms of its correlation ; i.e.

where a d

n

f f

f f f f

P C

P C

1. The temperature forecast for Albuquerque is much colder than the verifying observation by 5K. Does this mean that the forecast for Santa Fe was also to cold?

2. What if the forecast error was associated with an approaching cold front?

3. How would the orientation of the cold front change your answer to question 1?

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Why do error covariances matter?

• We don’t know the true state and hence cannot produce error samples.

• We can attempt to – infer forecast error covariances from

innovations (y-Hxf) (e.g. Hollingsworth and Lohnberg, 1986, Tellus)

• And/or – create proxies of error from first principals

(e.g. ensemble perturbations)6

Problem

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Static forecast error covariances from innovations

Assume that the jth observation where is the true

value of the ob. and is the true atmospheric state. Assume that the

forecast of the observation then the innovati

t o t tj j j j j

t

f t fj j j

y H H y

H H

x x

x

x x

on

associated with the jth observation is given by

;

in other words, the innovation is just the difference between the observation errorand the forecast error (for unb

f t o t f o fj j j j j j j j jv y H H H x x x

iased obs/fcsts).

Assume that 0.

Now consider the covariance of innovations at nearby observation sites and .

o fj j

f f o f f o o oi j i j i j i j i j

i j

v v

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Static forecast error covariances from innovations

2 2

Would observation errors be correlated with forecast errors?

Assuming that 0,

= when . However,

, when .

In other words, for

o o f o o f o f o fj i j i j i i i j j

f fi j i j ij

f oi i i i ii ii

v v p i j

v v P R i j

uncorrelated observation errors,innovation covariance equals forecast error covariance, while innovation variance equals the forecast error variance plus the observation error variance.

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Hollingsworth-Lönnberg Method(Hollingsworth and Lönnberg, 1986)

Innovation covariances binned by separation distance

Extrapolate green curve to zero separation, and compare with innovation variance

Fcst error variance Pii

Static forecast error covariances from innovations

Ob error variance RiiIncludes uncorrelated error

of representation

Desroziers’ Method(Desroziers et al 2005) TO O

FAE

d d R

TO O TF FE

d d R HBH

TA O TF FE

d d HBHFrom O-F, O-A, and A-F statistics, the observation error covariance matrix R, the representer HBHT, and their sum can be diagnosed

An attractive property of the HL method is that its estimates are entirely independent of the estimates of P and R that are used in the data assimilation scheme.

Desrozier’s method depends on differences between analyses and observations. These differences are entirely dependent on the assumptions made in the DA scheme.

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Bauer et al., 2006, QJRMS

Static forecast error covariances from innovations

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C/O Mike Fisher

Why does the correlation function for the AIREP data look qualitatively different to that from the SSMI radiances?

• Pros– Ultimately, observations are our only means to

perceive forecast error.– Innovation based approaches enable both forecast

error covariances and observation error variances to be simultaneously estimated.

• Cons– Only gives error estimates where there are

observations (what about the deep ocean, upper atmosphere, cloud species, etc)

– Provides extremely limited information about multi-variate balance.

– Limited flow dependent error covariance information.12

Pros and cons of error covariances from binned innovations

1. Parish and Derber’s (1992, MWR)“very crude 1st step” using the difference between a 48 hr and 24 hr fcsts valid at the same time as a proxy for 6 hr fcst error has been widely used.

2. Oke et al. (2008, Ocean Modelling) use deviations of state about 3 month running average as a proxy for forecast error.

3. Both 1 and 2 can be made to be somewhat consistent with innovations 13

Covariances of proxies of forecast error

How could we produce better proxies of forecast error?

• Forecast error distributions depend on analysis error distributions and model error distributions.

• Analysis error distributions depend on the data assimilation scheme used and the location and accuracy of the observations assimilated.

• Estimation of these distributions is difficult in practice but there is theory for it.

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Covariances of proxies of forecast error

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pdf of truth given prior information (the prior)

| likelihood density of observations given a particular

| posterior distribution of truth given prior information and

Ideal DA would

t t

t t

t

L

x x

y x y x

x y y

use

||

|

(Bayes' theorem) to turn a prior pdf of truth into an observation informed posterior pdf.

t tt

t t

V

L

L dV

y x xx y

y x x

The effect of observations on errors,Bayes’ theorem

Prior pdf of truth

prior tx

Ensemble forecasts are used to estimate this distribution. They are a collection of weather forecasts started from differing but equally plausible initial conditions and propagated forward using a collection of equally plausible dynamical or stochastic-dynamical models.

Probability density

Value of truth

Likelihood density function

prior tx

22

2 0

0

1 1Likelihood | exp , 22

,

~ 0,

1 and 1/ 4in this example.

t

t

t

y xL y x

RR

y x

N R

y R

Value of truth

In interpreting the likelihood function (red curve) note that y is fixed at y=1.

The red curve describes how the probability density of obtaining an error prone observation of y=1 varies with the true value xt.

Posterior pdf

prior tx

2 0

0

Likelihood | ,

,

~ 0,1 in this example

t

t

L y x

y x

N Ry

Posterior

||

|

t tt

t t t

L y x xx y

L y x x dx

Value of truth

Probability density No operational or near operational data assimilation schemes are capable of accurately representing such multi-modal posterior distributions.

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Works for Gaussian forecast and observation errors.

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Ensemble of perturbed obs 4DVARS does not solve Bayes’ theorem

Green line is pdf of ensemble of converged perturbed obs 4DVARs having the correct prior and correct observation error variance.

Blue line is the pdf of ensemble of 4DVARS after 1st inner loop (not converged)

Black line is the true posterior pdf.

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EnKF doesn’t solve Bayes’ theorem either

Cyan line is posterior pdf from EnKF

Black line is the true posterior pdf.

1. Ensembles of 4DVARs and/or EnKFs provide accurate flow dependent analysis and forecast error distributions provided all error distributions are Gaussian and accurately specified.

2. In the presence of non-linearities and non-Gaussianity, the 4DVAR/EnKF proxies are inaccurate but probably not as inaccurate as proxies for which 1 does not hold.

3. One can use an archive of past flow dependent error proxies to define a static or quasi-climatological error covariance. (Examples follow) 23

Recapitulation on proxy error distributions

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Computationally Efficient Quasi-Static Error Covariance Models

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Boer, G. J., 1983: Homogeneous and Isotropic Turbulence on the Sphere. J. Atmos. Sci., 40, 154–163.

Pointed out that isotropic correlation functions on the sphere are obtained from EDE^T where E is a matrix listing spherical harmonics and D is a diagonal matrix whose values (variances) only depend on the total wave number.

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1

1

1

1

0 0...

0 0

0...

,0

.

0 0...

0 0

so and define the

0 0 0

0 0 00 0

0 0

0 0 0

0 0 0

n

n

n

n

h

h

h

h

VH V

V

V V

vertical correlation matrix of total wavenumber = 1and total wavenumber modes.n

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Wavelet transforms permit a compromise between these two extremes. ECMWF currently has a wavelet transform based background error covariance model.May have time to touch on this tomorrow.

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Divergence without omega equation

Divergence with omega equation

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Sophisticated balance operators impart a degree of flow dependence to both the error correlations

and the error variances!

Recapitulation on today’s lecture• Differences between forecasts and observations can be used to

infer aspects of spatio-temporal averages of – Observation error variance– Forecast error variance– Quasi-isotropic error correlations

• Monte Carlo approaches (Perturbed obs 3D/4D VAR, EnKF) and deterministic EnKFs (ETKF, EAKF, MLEF) provide compelling error proxies for both flow-dependent error covariance models and flow-dependent error covariance models.

• In variational schemes, the need for cost-efficient matrix multiplies has led to elegant idealizations of the forecast error covariance matrix– sophisticated balance constraints can be built into these models.

• There were many approaches I did not cover (Recursive filters, Wavelet Transforms, etc).

• Tomorrow: Ensemble based flow dependent error covariance models

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