Correlation length scales in background/observation error covariancesand how they influence filter performance

Yue (Michael) Ying

Fuqing ZhangJeffery Anderson

Groupmeeting, Aug 2017

Why is error covariance important?Consider a single-variable system, e.g. Lorenz (1996) model.

Covariance propagates observed information in space/time- covers unobserved variables- enhance observed variables

Ensemble-estimated covariance is noisy, therefore requires localization.

What is the appropriate localization distance? - Zhen and Zhang (2014) adaptive algorithm- Anderson and Lei (2013) empirical localization functions

Covariance is flow-dependent!

(also variable-, scale-dependent?)

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correlation length scale

Correlation length scale (L) in background errorL=0, white noise, all variables must be observed.Luckily, atmosphere has L>0, like a red noise. Its spectrum determines L.

x

y

k

+1

Correlation function Power spectrum

x

y

Error field

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x

y

k

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Correlation function Power spectrum

x

y

Error field

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Variable dependence for LTwo-layer QG model, different power-law for each variable:

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𝛻 𝛻

Cross-variable correlation functions

|corr(𝜓, 𝜓)| |corr(𝑇, 𝜓)|

|corr(𝑢, 𝜓)|

distance

Time evolution of LFree ensemble run: L increase over time as error saturates upscale.

With cycling data assimilation: L finds a quasi-steady value.

mean absolute correlation (u, 𝜓)

distance (grid points)

k

KE spectrum

Impact of background L on filter performance

8 16 32 64 128posterior mean error with fixed ROI:

reference energy from truthno DA ensemble mean errorobservation error

N=40 memberbackground L~50observation interval Δ=4

Best-performing localization ROI is matching L

Large scale favors larger ROI.

Complication in the codependence amongensemble size N, correlation scale L, observation interval Δ

Δ=2Δ=1

Δ=4

L~10L~20

L~40

N=1000

N=10

N=40

* * * *

distance

Δ

L| | | | | | |

L in correlated observation error

EnKF (square root algorithm) assumes uncorrelated observation error:instrument errors are white noise.

According to information theory:more (uncorrelated) observations -> less uncertainty

However, observation errors can be correlated:- preprocessing- error of representation- observation operator

Correlated error means additional observation does not increase information content as much.

Δx = 1 dxΔx = 2 dxΔx = 4 dx

white noise

red noise

Treating correlated error in ensemble filters

- Ignore it: suboptimal analysis + ensemble spread is reduced too much.

- inflate observation error variance

- account for correlation with a full-rank observation error covariance (in ETKF)

obs error is white noise obs error is red noise

(dotted line: prior spread)

4 8 16 32 64analysis error with ROI =

reference no DA obs error

Consider real atmosphere data assimilation…

Unlike QG, there are several state variables with different spectral slope coupled with dynamic equations

For example, a wind observation will be used to update p and was well as wind itself.

Errors in p and w will feedback into wind during forecast step.

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𝜓 (like p) u,v 𝜁 (like w)

noisy observationupdates smooth state

smooth observationupdates noisy state

Plans for further experiments

In QG, change state variable from 𝜓 to u, v,then test assimilating 𝜓 and 𝜁 to update u, v.

Assimilate u, v observation to update state variable 𝜓 and 𝜁, some how couple the two corresponding wind analysis (averaging) to form the final state.

Assimilate u, v to update u, v, but draw observation error with correlation length scale from 0 to 2L.