“Running in Place” and “OuterLoop” in EnKF

Shu-Chih Yang and and E Kalnay(Kalnay and Yang, subm. QJ, 2009,

Yang and Kalnay, subm. MWR)

No-cost smoother(Kalnay et al, 2007b, Yang et al, 2009)

It is based on the insight from Hunt that within an assimilation window a linearcombination of model trajectories is also a trajectory, and the trajectory that isclose to the truth (at tn) should be close to the truth throughout theassimilation window (assuming model errors are small and perturbations arelinear). Therefore, we can apply the weights at the beginning of the windowand obtain a smoothed analysis (cross) more accurate than the filter analysis(arrow) because it “knows” about the obs in the window.

No-cost smoother tested onQG model: it works

Outer Loop(also RIP using Wn)

Comparison of LETKF spin-up (with and w/o RIP)with 3D-Var and 4D-Var with 0.05*B3D-Var

a) Random Uniform IC perturbations

Comparison of LETKF spin-up: RandomUniform IC perturbations, and 3D-Var random

IC perturbations

Comparison of LETKF spin-up: 3D-Var randomIC perturbations and random Gaussian

perturbations: RIP is very robust!

Number of iterations for RIP:after spin-up it turns itself off

Outer loop and RIP can be used to handlenonlinearity and non-Gaussianity.

Both 4D-Var and LETKF optimized.

4D-Var

LETKF

(3 members )

LETKF

+ outer loop

LETKF

+ RIP

Obs every 8 time-steps

(linear window ) 0.31 0.30 0.27 0.27

Obs every 25 time-steps

(nonlinear window )

0.53

(assim window=75)

0.68

( =1.22)

0.47

( =1.08)

0.38

=1.08)

Test of how Gaussian are theperturbations.

Kurtosis should be zero and IQR 1.57

Kurtosis Standard LETKF With outer-loop With RIP

median IQR median IQR median IQR

Forecast ensemble 2.27 4.72 0.09 2.88 -0.61 1.57

Analysis ensemble 1.98 4.55 0.19 2.85 -0.09 1.38

Example of Lorenz 3-variable model dataassimilation with and without outer loop

Summary• The “no-cost” (and “no-equations”!) smoother uses the

weights computed in the LETKF and applies them tothe ensemble at the beginning of the window.

• It can be used to create an “outer loop” like in 4D-Varwhich re-centers the ensemble around a more accuratenonlinear solution, but keeping the perturbations.

• We can also smooth the perturbations: “running inplace” (RIP). We add small random noise to initialensemble to avoid ending with the same final analysis.RIP requires K integrations, outer loop only one.

• RIP is very efficient in accelerating spin-up.• Outer loop can make perturbations more Gaussian in

long assimilation windows. RIP is even better inhandling non-linearities and non-Gaussianities but it ismore expensive.