A Local Ensemble Transform Kalman Particle Filter

A Local Ensemble Transform Kalman Particle Filter

Sylvain Robert and Hans R. KünschETH Zürich, Seminar for Statistics

Seminar for StatisticsSfS Sylvain Robert and Hans R. Künsch 18. July 2016 1

Outline

• Introduction

• Algorithm

• Numerical experiments

• Conclusions


Motivation

• Target application: Convective Scale Data Assimilation

• Challenges:

– high resolution (≈1km)

– non-linear forecasting step xat → xb

t+1.

– non-Gaussian background distribution πbt (x).

– non-Gaussian analysis distribution πat (x) (even if likelihood is

linear and Gaussian).


Introduction

• We assume that y |x ∼ N (Hx , R) and skip time index t.

• xb and xa: background and analysis ensembles.

• πb(x) and πa(x): background and analysis distributions.

Analysis:

• xb "+" y → xa

• Bayes’ formula: πa(x) ∝ πb(x) · `(y |x)


Introduction

different assumptions = different solutions

• πb(x) Gaussian + unlimited computation→ Kalman Filter (KF)

• πb(x) non-Gaussian + unlimited computation→ Particle Filter (PF)

• πb(x) Gaussian + limited computation→ EnKF

• πb(x) non-Gaussian + limited computation→ ???


The EnKPF in a nutshell

Frei and Künsch (Biometrika, 2013).

• πa(x) ∝ πb(x) · `(y |x) = πb(x) · `(y |x)γ︸︷︷︸∝πγ (x)

·`(y |x)1−γ

• Two steps:πb(x) EnKF−−−−−→

γπγ(x) PF−−−−−→

1−γπa(x)

• γ = 1→ EnKF

• γ = 0→ PF


The EnKPF in a nutshell

●

●

●

●

●

Analysis distribution πaEnKPF (x):

xa,j ∼k∑

i=1

αγ,i N (µγ,i , Pa,γ)


Outline

• Introduction

• Algorithm

– Ensemble space

– Localization


• Conclusions


Ensemble space

• Split ensembles into mean and deviations:

xb = x̄b1′ + X b and xa = x̄a1′ + X a

• Use the empirical covariance Pb = 1k−1X b(X b)′

• Analysis mean :

x̄a = x̄b + X bm, m = k × 1

• Analysis deviations:

X a = X bW , W = k × k


EnKPF in ensemble space

m = mµ + 1k W µW α1

W = W µW α + W ε − 1k W µW α11′

• mµ and W µ: mixture components µγ,i

• W α: particle resampling

• 1k W µW α1: correction of the mean due to resampling.

• W ε: individual perturbations to ensure correct covariance:

– stochastic

– deterministic: transform filter


Step 1: analysis mean

●

●

First we move the ensemble mean

towards the observation:

x̄a = x̄b + X bmµ


Step 2: mixture components

(x̄b + X bmµ)1′ + X bW µ

●

●

0.0

0.2

0.4

0.6

0.8

1.0


Step 3: weights and resampling

(x̄b + X bmµ)1′ + X bW µWα

●

1

1

3

0.0

0.2

0.4

0.6

0.8

1.0


Step 4: individual perturbations

X bW ε ∼ N (0, Pa,γ), Transform version

0.0

0.2

0.4

0.6

0.8

1.0

Remark: W ε is not just a square-root of Pa,γ → solve for Cov (xa).


All together

(x̄b + X bmµ)1′ + X bW µWα + W ε

●

●

●

●

●

0.0

0.2

0.4

0.6

0.8

1.0


LocalizationThe curse of dimensionality:

• PF: necessary number of particles increases exponentially.

• EnKPF: a bit better but not immune to the problem.

Possible remedies:

• Carefully chosen proposal distribution.

• Localization.

Problem:

• Not easy to apply to PF methods.


Localization: step 1

Local weights, resample locally and glue together→ discontinuities

longitude

field

val

ue

Each line is an analysis particle with three cases highlighted in color.



Permute indices locally→ remove some discontinuities.

longitude

field

val

ue



Interpolate W on finer grid→ smooth out remaining discontinuities.

longitude

field

val

ue


Summary of new algorithm

Local Ensemble Transform Kalman Particle Filter: LETKPF

• Hybrid between EnKF and PF: handles non-Gaussian distributionswhile maintaining sample diversity.

• Transform: guarantees exact covariance with deterministic scheme.

• Local: uses local weights and local resampling while avoiding prob-lems with discontinuities.


Outline

• Introduction

• Algorithm

• Numerical experiments: COSMO-KENDA

– Case study (7th of June 2015 12 UTC)

– Cycled experiment (June 04-16)

– Forecast experiment (12 hours)

• Conclusions


COSMO-KENDA experiment

In collaboration with Daniel Leuenberger from Meteoswiss and with thehelpful support of DWD.

• Area surrounding Switzerland, high resolution (≈ 2.2km).

• 04-16th of June 2015 (period of intense convective activity).

• 40 ensemble members.

• Assimilation of conventional observations (no radar).

• Algorithms: LETKPF and LETKF.


7th of June 12 UTCZonal and meridional wind components: ensemble mean and std.

U

V

−15

−10

−5

0

5

10

15

MeanU

V

1

2

3

4

Std


Local particle weights: αγ,i

• Different particles fit the observations better in different places.

• 1/40 < α < 2/40→ particle resampled once or twice.

• 2/40 < α < 3/40→ particle resampled twice or thrice.

• . . .

1 2 3

12345678

1/40

α


Combination of particles

(xa,1 − x̄a) = (xb,1 − x̄b)W11 + (xb,2 − x̄b)W21 + (xb,3 − x̄b)W31 + ...

1 2 3

12345678

1/40

α

1 2 3

0

1

Wi1


Discontinuities

• Contribution of particle 1 to its own analysis: W11.

• Large continuous patches: good.

• Some discontinuous patterns could be fixed, but it would requiresome global communication (work in progress).

12345678

1/40

0

1


Adaptive choice of γ

• Chosen locally such that ESS=50% (≈ half of the mixture compo-nents µγ,i are used).

• Small γ means more PF, big γ more EnKF.

• Joint property of the background distribution and the observations.

0

0.5

1value


Cycled experiment

• Hourly assimilation of conventional observations.

• LETKPF vs LETKF.

• Vertical profiles of RMSE and spread for T, RH and WIND.

• Averaged over whole period 04-16.06.2015.

Parameters:

• Parameter γ chosen adaptively such that ESS=50%.

• Localization radius: chosen adaptively.

• Multiplicative covariance inflation.


Wind

300

400

500

600

700

850

925

1000

0.0 0.5 1.0 1.5 2.0 2.5error

hPa

RMSE

300

400

500

600

700

850

925

1000

0.0 0.5 1.0 1.5value

Spread

300

400

500

600

700

850

925

1000

−0.1 0.0 0.1 0.2 0.3 0.4 0.5error

typeanalysis

forecast

modelLEnKPF

LETKF

LETKPF

Bias


Temperature

300

400

500

600

700

850

925

1000

0.0 0.5 1.0 1.5error

hPa

RMSE

300

400

500

600

700

850

925

1000

0.0 0.1 0.2 0.3 0.4 0.5value

Spread

300

400

500

600

700

850

925

1000

−0.1 0.0 0.1 0.2 0.3error

typeanalysis

forecast

modelLEnKPF

LETKF

LETKPF

Bias


Relative humidity

300

400

500

600

700

850

925

1000

0.00 0.05 0.10 0.15 0.20error

hPa

RMSE

300

400

500

600

700

850

925

1000

0.00 0.02 0.04 0.06value

Spread

300

400

500

600

700

850

925

1000

−0.075−0.050−0.025 0.000 0.025 0.050error

typeanalysis

forecast

modelLEnKPF

LETKF

LETKPF

Bias


Outline• Introduction

• Algorithm


– Case study (7th of June 2015 12 UTC)

– Cycled experiment (June 04-16)

– Forecast experiment (12 hours)• Conclusions


Forecast: RMSE and bias

bias

PS

bias

WIND

rmse

PS

rmse

WIND

0

20

40

60

0.00

0.05

0.10

0.15

60

80

100

1.7

1.8

1.9

2.0

0 200 400 600 200 400 600

0 200 400 600 200 400 600lead time (min)

erro

r methodLETKF

LETKPF

SYNOP


Forecast: ETS of TOT_PREC

How well did the forecast predict rain > 0.1 mm/h (accounting for chance)?


Forecast: FBIFrequency Bias Index: 1: perfect, >1: over-, <1: under-forecasting.


Conclusions• New algorithm: LETKPF

– Hybrid EnKF and PF in ensemble space.

– transform and local.

• Positive results with COSMO-KENDA:

– Case study of 07.06.2015: reasonable behavior.

– Transform filter better than stochastic version.

– Results equivalent to LETKF on cycled experiment.

– Successful 12 hour forecasts.


Thank you!

Questions?


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A Local Ensemble Transform Kalman Particle Filter

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