The ECMWF Hybrid 4D-Var and Ensemble of Data Assimilations Lars - - PowerPoint PPT Presentation

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 1

The ECMWF Hybrid 4D-Var and Ensemble of Data Assimilations

Lars Isaksen, Massimo Bonavita and Elias Holm Data Assimilation Section ECMWF lars.isaksen@ecmwf.int

Acknowledgements to: Mike Fisher and Marta Janiskova

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 2

The ECMWF Hybrid 4D-Var and Ensemble of Data Assimilations

4D-Var Ensemble of Data Assimilations (EDA) Hybrid 4D-Var & EDA

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 3

Innovations are computed at the observation time using the high-resolution non-linear forecast model 4D-Var finds the 12-hour forecast that take account of the observations in a dynamically consistent way Based on a tangent linear and adjoint forecast models, that works in an 80,000,000 dimensional model subspace

ECMWF will continue to use deterministic 4D-Var

• All observations within a 12-hour period (~17,000,000) are used

simultaneously in one global (iterative) estimation problem

9Z 12Z 15Z 18Z 21Z

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 4

Regularizations remove the most important threshold processes in physical parametrizations improving the validity of the tangent linear approximation

Accuracy of Tangent-Linear and adjoint important: linearity issues

• 12
• 8
• 4
• 2
• 1
• 0.5

0.5 1 2 4 8 12

Non-linear finite difference TL integration

u-wind increments fc t+12, ~700 hPa

• 12
• 8
• 4
• 2
• 1
• 0.5

0.5 1 2 4 8 12

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 5

The Ensemble of Data Assimilations (EDA)

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 6

The Ensemble of Data Assimilations at ECMWF (EDA)

• 10 (25 from November 2013) ensemble members using 4D-Var

assimilations

• T399 (50km) outer loop, T95/T159 inner loops. (Deterministic

4D-Var: T1279 (16km) outer loop, T159/T255/T255 inner loops)

• Observations randomly perturbed according to their estimated

errors

• SST perturbed with climatological perturbations 
• Model error represented by stochastic methods (SPPT,

Leutbecher, 2009)

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 7

The hybrid approach used at ECMWF: EDA&4D-Var

Hybrid approach: Use cycled, flow-dependent background error estimates from an Ensemble of Data Assimilations in a deterministic 4D-Var analysis. This hybrid formulation has many benefits:

• Introduces flow-dependent background errors into 4D-Var system
• Maintain the full rank representation of B and its implicit

evolution inside the assimilation window

• More robust than pure EnKF for limited ensemble sizes
• Ensemble perturbations are used in 4D-Var control vector space,

beneficial for e.g. assimilation of radiance observations

• Allows for flow-dependent Quality Control of observations

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 8

The EDA&4D-Var hybrid implementation at ECMWF

Variance post-process

xa+εi

a

Analysis Forecast

SST+εi

SST

y+εi

• xb+εi

b

xf+εi

f

i=1,2,…,10 (25)

EDA Cycle

εi

f raw

variances Variance Recalibration Variance Filtering EDA scaled variances

4DVar Cycle

xa

Analysis Forecast

EDA scaled Var

xb xb

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 9

• We performs an online calibration (Ensemble Variance

Calibration; Kolczynsky et al., 2009, 2011; Bonavita et al., 2011)

• Calibration factors depend on latitude bands and parameter
• Calibration factors also depend on the size of the expected error

Raw EDA variance estimates needs to be calibrated to become statistically consistent

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 10

Sampling noise due to small ensemble size is a problem

Noise filtering method used until June 2012:

• Use a spectral filter to disentangle noise from signal
• Truncation wavenumber is determined by maximizing signal-to-noise ratio of

filtered variances (Raynaud et al., 2009; Bonavita et al., 2011)

A more direct strategy applied now, based on two 50-member EDAs: 1. Sampling noise assumed a random process 2. Time average sampling noise spectrum samples 3. Compute raw filters and time average to smooth out noise (based on Berre et al., 2010)

j i e

S S P n S P 2 1

raw e

S P S P n 1 1

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 11

In the ECMWF 4D-Var, the B matrix is defined implicitly in terms of a transformation from the background departure (x-xb) to a control variable χ: (x-xb) = Lχ So that the implied B=LLT. In the current wavelet formulation (Fisher, 2003), the variable transform can be written as: T is the balance operator Σb is the gridpoint variance of background errors Cj(λ,φ) is the vertical covariance matrix for wavelet index j ψj are the set of radial basis function that define the wavelet transform.

Introducing flow-dependent background errors in 4D-Var

j j j j b b

,

2 / 1 2 / 1 1

C Σ T x x

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 12

Cj(λ,φ) are full vertical covariance matrices, function of (λ,φ). They determine both the horizontal and vertical background error correlation structures; In standard 4D-Var T and Cj are computed off-line using a climatology

• f EDA perturbations.

Σb is computed by random sampling of the static B matrix (randomization procedure, Fisher and Courtier, 1995) How do we make this error covariance model flow-dependent? We look for flow-dependent EDA estimates of Σb and Cj(λ,φ)

j j j j b b

,

2 / 1 2 / 1 1

C Σ T x x

Introducing flow-dependent background errors in 4D-Var

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 13

Improved static background-error covariance statistics based on the latest EDA (implemented June 2012)

Resolution upgrades and more observations since last update resulted in sharper structure functions: reduced correlation length scales both horizontally and vertically

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 14

EDA Variances for the Unbalanced Control Vector (ηu, (T,ps)u).

u s T u T s u T

p T Var Var Var p T Var Var Var Var Var Var , , P P N N M M

u s u s u

p T p T , , P N M

EDA-based flow-dependent background errors for unbalanced control variables (Tu,Du,LNSPu) - June 2013

Derber, Bouttier, Fisher (1997) Similar plot for the 2013 ECMWF DA system

Explained variance Ratio for divergence and temperature

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 15

Previous bg error model for unbal. temp. EDA bg error for unbal. temp.

EDA-based flow-dependent background errors for unbalanced control variables (Tu,Du,LNSPu) - June 2013

Average unbalanced temperature (st.dev. in Kelvin) 90N 90S 90N 90S Surface Top of atmosphere

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 16

Variance estimation needs an EDA sample size of ~10 Covariance estimation needs an EDA sample size of ~600 a) Background error covariances (JB) are computed in a post- processing step of 25 member EDA b) EDA perturbations from the past 12 days are used for a weighted running mean (Sample size: 25*12*2=600) c) Continuously updated JB is used in deterministic 4D-Var

Flow-dependent covariance estimation from an EDA

Similar activities on-going at Météo-France (Varella et al. 2011)

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 17

EDA-based flow-dependent variances are computed for each analysis cycle - sufficient with 10 EDA members to estimate error-of-the day.

St.dev of vorticity errors and Z at 500hPA

Flow-dependent covariance estimation from an EDA

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 18

Covariance estimation requires a large sample size (order 600). This is computed with a lagged 12 days running average. Correlations are representative of prevailing weather patterns, not distinct weather features!

St.dev of vorticity errors and Z at 500hPA

Flow-dependent covariance estimation from an EDA

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 19

Vertical correlations at (30N,140W) at 850hPa The 12-day averaging allows JB to cater for flow-regime changes The vertical correlations between 850hPa and the boundary layer change significantly from 10th of January 2012 to 10th of February 2012.

Why is flow-dependent JB better?

Surface 850 hPa 400 hPa 500 hPa 700 hPa 950 hPa

Static JB 20120110 JB 20120210 JB

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 20

a) Reduce the time window used to compute the online JB (use more EDA forecast steps; hybrid with static JB; filtering of correlations) b) Introduce EDA errors and JB in the EDA members analysis (interactive EDA) c) Further extend the use of EDA errors for observation Quality Control d) Test EDA perturbations as EPS initial conditions – towards a unified EDA and EPS

The next steps to improve the hybrid EDA & 4D-Var

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 21

• Use more EDA forecast steps
• Hybrid with static JB; filtering of correlations

Reduce the time window used to compute the online JB

Online JB with 12 day window, t+3h perturbations Online JB with 4 day window, t+0/3/6h perturbations

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 22

Improved statistical noise filtering of EDA variances Static background errors updated, using latest EDA Impact of EDA based variances in hybrid 4D-Var

The introduction of flow-dependent background error variance and covariance estimated from the EDA has by far been the largest source

• f improvement in recent years in analysis and forecast skill at ECMWF

Hybrid EDA & 4DVAR improves forecast skill

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 23

Impact of online JB

Reduction in Z RMSE - 95% confidence

Impact of EDA-based unbalanced control vector

Reduction in Z RMSE (95% confidence, RAOBs)

The introduction of flow-dependent background error variance and covariance estimated from the EDA has by far been the largest source

• f improvement in recent years in analysis and forecast skill at ECMWF

Hybrid EDA & 4DVAR improves forecast skill

NH NH SH SH

200 hPa 1000 hPa 500 hPa

NH NH SH SH

50 hPa 100 hPa 200 hPa 500 hPa 1000 hPa

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 24

The background error model at ECMWF has evolved over the past 30

• years. The new flow-dependent wavelet-based B has many advantages:

 Compact model  Good spectral resolution  Non-separable (vertical and horizontal scales of background error

covariance are non-separable: large horizontal scales tend to have deeper vertical correlations than small horizontal scales. It is important for a B model to retain this property)

 Non-homogeneous, based on wavelet formulation (Fisher,2003)

 Before: Isotropic  Now: Partially anisotropic (variances)  Before: Static

 Now: Flow-dependent

Conclusions and summary

ECMWF

Lars Isaksen, 6th WMO Symposium on Data Assimilation, Washington DC, Oct 2013 Assimilation at ECMWF 25

Conclusions and summary

The EDA and hybrid EDA&4D-Var developments have been central to the recent analysis and forecast skill improvements at ECMWF In general to estimate analysis uncertainty

 Improve the initial perturbations in the Ensemble Prediction June 2010  To estimate flow-dependent background error from the EDA for the balanced part of the control vector in 4D-Var May 2011  To improve observation QC decisions and usage in 4D-Var May 2011  To improve filtering of EDA sampling noise June 2012  To update the static background-error covariance statistics based on the latest EDA June 2012  To estimate flow-dependent background error estimates from the EDA for the unbalanced part of the control vector in 4D-Var June 2013  To estimate flow-dependent background correlations from the EDA variances in 4D-Var

• Nov. 2013