Addressing the nonlinear problem of low order clustering ... · Nonlinear case with obs/assim...

Addressing the nonlinear problem

of low order clustering in

deterministic filters by using

mean-preserving non-symmetric

solutions of the ETKF

Javier Amezcua, Dr. Kayo Ide, Dr. Eugenia Kalnay

3rd annual PSU-UMD joint DA workshop, 12/19/2011

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Outline

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Mean-Preserving Non-Symmetric Ensemble Transform

Kalman Filter (MPNS-ETKF)

Ensemble Square Root Filters (EnSRFs)

Solutions of the ETKF

The low order clustering problem

Using the MPNS-ETKF to solve this problem

Simple univariate nonlinear model

Lorenz 1963 model

Lorenz 1996 and the importance of symmetry in R-localization


Stochastic vs. deterministic filters

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Stochastic EnKF (Evensen, 1994; Burgers et al, 1996;

Houtekamer and Mitchell, 1998)

Perturbed observations, the covariance equation fulfilled only

statistically.

Deterministic (EnSRF: Tippett et al, 2003)

Serial (Whitaker and Hamill, 2002)

EAKF (Anderson, 2001)

ETKF (Bishop et al, 2001), LETKF (Hunt et al, 2007)

𝐗𝑎 = 𝐗𝑏𝐖𝑎 ,𝐖𝑎 ∈ ℜ𝑀×𝑀


Ensemble Transform Kalman Filter

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The original formulation (Bishop et al, 2001)

The ‘one-sided’ ETKF fulfills the covariance equation, but we also

require the mean of the perturbations to be zero.

This solution doesn’t fulfill this in general (Livings et al, 2007) and

leads to bad performance (Sakov and Oke, 2008).

𝐖𝑎 = 𝐂 𝐈 + 𝚪 −12 𝐂𝚪𝐂T =

𝐘𝑏T𝐑−1𝐘𝑏

𝑀 − 1

Is a diagonal matrix

with eigenvalues.

The columns of this matrix contains eigenvectors.

Hence, it is an orthonormal matrix.

𝐗𝑎𝟏 = 𝟎


Where 𝐂𝚪𝐂T = 𝐘𝑏T𝐑−1𝐘𝑏 𝑀 − 1 . In this [symmetric] case

𝐖𝑎 is the ‘closest’ to 𝐈; hence, 𝐗𝑎 is the closest to 𝐗𝑏 (Ott et

al, 2004).

ETKFs that preserves the mean

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Spherical simplex

(Wang et al, 2004)

𝐖𝑎 = 𝐂 𝐈 + 𝚪 −

12𝐂T

Symmetric square root in the

LETKF (Hunt et al, 2007)

𝐖𝑎 = 𝐈 + 𝐂𝚪𝐂T−12


A general ETKF is:

where 𝐒 ∈ ℜ𝑀×𝑀 must be orthonormal and ‘mean-

preserving’. There are cheap ways to construct 𝐒 (Bishop, pers.

comm.).

A particular form (which will be important in R-localization) is

to rotate the symmetric solution:

where 𝚺T𝚺 = 𝐈 and 𝚺T𝟏 = 𝟏

General solution to the ETKF

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𝐖𝑎 = 𝐂 𝐈 + 𝚪 −12𝐒T


𝐖𝑎 = 𝐂 𝐈 + 𝚪 −12𝐂T𝚺T

Some choices:

𝐒 = 𝐈 gives the one sided

𝐒 = 𝐂 gives LETKF

Outline

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Lorenz 1963 model



Kalman filtering

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The filtering problem

The conditions aren’t usually perfectly fulfilled. How well

they are approximated depends upon:

The length of the assimilation window.

The magnitude of the model error covariance and the

observational error covariance.

𝐱t = 𝑓 𝐱t−1 +𝐰t

𝐲t𝑜 = ℎ 𝐱t + 𝐯t Gaussian

linear


The nonlinear effect of ensemble

clustering

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Using the Ikeda model, Lawson and Hansen (2004) realized that

the performance of the EnSRF (they used the serial) is more

sensitive to nonlinearity. M-1

members

1 member

Background ensemble, Analysis ensemble Lawson and Hansen (2004)

EnKF EnSRF


The nonlinear effect of ensemble

clustering

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The higher order moments are the most affected.

The analysis RMSE was not affected much. The spread is conserved.

The stochastic EnKF is more robust, but it introduces more sampling error (Whitaker and Hamill, 2002).

Lawson and Hansen (2004)

EnKF linear EnSRF linear

EnKF nonlinear EnSRF nonlinear


Outline

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Lorenz 1963 model



A univariate quadratic model

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Consider the following nonlinear ODE (similar to

Anderson, 2010):

An Euler Forward discretization leads to the following map:

Let’s consider the unstable fixed point 𝑥∗ = 0 to be the

truth.

𝑥 = 𝑥 + 𝑏𝑥2

𝑥𝑡+1 = 1 + ∆ 𝑥𝑡 + 𝑏∆𝑥𝑡2

Time step for the integration: 0.05.

Nonlinearity

coefficient 0,0.5


Progressive deformation of the

ensemble due to quadratic evolution

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b=0 b=0.2

b=0.5 b=0.7

T

I

M

E

𝑥 𝑥

𝑥 𝑥


Experimental settings

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Several combinations of settings were chosen, the

results shown here use:

𝑀 = 20 ensemble members.

𝜎2 = 1, observational error.

The initial ensemble was centered in 𝑥 = 0 with 𝑝0 = 1.

The observation/assimilation frequency was varied.

The degree on nonlinearity was varied.


𝑥𝑡+1 = 1 + ∆ 𝑥𝑡 + 𝑏∆𝑥𝑡2

Observation/assimilation window: 𝟐∆

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ETKF 𝑏 = 0

ETKF 𝑏 = 0.2

𝑥

𝑥

time


When using the

symmetric ETKF, the

clustering appears as a

consequence of the

strong nonlinearity

Observation/assimilation window: 𝟐∆

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MPNS-ETKF 𝑏 = 0

MPNS-ETKF 𝑏 = 0.2

𝑥

𝑥

time


With the non-

symmetric ETKF,

the clustering

doesn’t appear

even in the

presence of strong

nonlinearity.

WHY?

What happens in the assimilation?

Nonlinear case with obs/assim window: 𝟓∆

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There is a constant ‘scrambling’ of the ensemble that prevents any deformation (due to nonlinearities) in the ensemble to remain.

ETKF

MPNS-ETKF

ense

mble

val

ues

ense

mble

val

ues

Background ensemble

Analysis ensemble


assimilation cycle

Experiments with Lorenz 1963

The system:

Solved with RK4

Identical twin experiment observing all variables

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𝑥 1 = 𝜎 𝑥 2 − 𝑥 1

𝑥 2 = 𝑥 1 𝑟 − 𝑥 3 − 𝑥 2

𝑥 3 = 𝑥 1 𝑥 2 − 𝑏𝑥 3

∆𝑡 = 0.01


Higher order moments: skewness

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The figure is very similar for background and analysis.

The figure for kurtosis is omitted since it is redundant.

R = 2𝐈, 8 steps

S S S S NS NS NS NS

R = 2𝐈, 24 steps

skew

ness

𝐌 = 3 𝐌 = 10 𝐌 = 25 𝐌 = 40

S S S S NS NS NS NS

𝐌 = 3 𝐌 = 10 𝐌 = 25 𝐌 = 40


How important is symmetry in R-

Localization?

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In the LETKF, the symmetric solution guarantees a smooth transition in the analysis values among neighboring gridpoints.

Using a different S (rotation matrix) for each gridpoint introduces noise to the system.

Will using a fixed 𝐒𝑭 be enough? NOT AUTOMATICALLY!

𝐒𝟏 𝐒𝟐 𝐒𝟑 ⋯ Each local analysis

would be ‘oriented’

in a different

direction.


Symmetry and R-localization

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How smooth is the transition for the weights among neighboring gridpoints? Let’s see for gridpoint 15.

The symmetric solution automatically guarantees smoothness!


One-sided ETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2

MPNSETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2𝐒𝐅T MPNSETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −

1

2𝐒𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒T

LETKF: 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2𝐂T

The same is true for all gridpoints.

Symmetry and R-localization

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One-sided ETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2

MPNSETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2𝐒𝐅T MPNSETKF 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −

1

2𝐒𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒T

LETKF: 𝐖𝑎 = 𝐂 𝐈 + 𝚪 −1

2𝐂T

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Nonsymmetry and R-Localization

An R-localized analysis must be locally symmetric.

It can be globally rotated afterwards.

This rotation can still bring benefits in the higher order

moments.

Skewness reduction


𝐖𝑎 = 𝐂 𝐈 + 𝚪 −12 𝐂T𝚺T

𝐒𝐅T

𝐗𝑎𝑔𝑙𝑜𝑏𝑎𝑙 = 𝐗𝑏𝑔𝑙𝑜𝑏𝑎𝑙𝚺T

Conclusions

Non-symmetric, in particular randomly rotated EnSRFs

are a good alternative to stochastic filters when

nonlinearity causes ensemble clustering.

The process can be considered a type of resampling.

Special care must be paid when using R-Localization. The

symmetry is needed to form the local analyses, but the

global analysis can be rotated afterwards.

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