umcp chaos weather project - comparison of hybrid ensemble … · 2013. 12. 23. · deterministic...

Page 1

Comparison of hybrid ensemble-4DVar with EnKFand 4DVar for regional-scale data assimilation

Jon Poterjoy and Fuqing Zhang

Department of MeteorologyThe Pennsylvania State University

Wednesday 18th December, 2013

Page 2

Introduction

Zhang and Zhang(MWR, 2012)

Zhang, Zhang,and Poterjoy

(MWR, 2013)

A prototype hybrid ensemble-4DVar (E4DVar) systemoutperformed EnKF and 4DVar in a month-long experimentover the continental United States. Experiments used theWRF model with a 90-km grid spacing.

Page 3

Introduction

The current study examines the performance of a newlydeveloped E4DVar system that is based on multi-incrementalWRF-4DVar.

EnKF, 4DVar, and E4DVar are applied for a case of tropicalcyclogenesis to compare the three methods at the mesoscale.

Page 4

Coupled EnKF-Var data assimilation

x1,0f , x2,0

f ,... , xN ,0f

x0f

!x1,0f , !x2,0

f ,... , !xN ,0f

3D/4DVar - is used as first guess - are used in the cost function

x0f

!x1,0f , !x2,0

f ,... , !xN ,0f

EnKF - is used as first guess - are used to estimate the covariance

x0f

!x1,0f , !x2,0

f ,... , !xN ,0f

Ensemble Forecast

Separate ensemble into mean and perturbations

Run 3D/4DVar and EnKF separately

x0a

!x1,0a , !x2,0

a ,... , !xN ,0a

3D/4DVar analysis becomes posterior

mean for EnKF perturbations

x1,0a , x2,0

a ,... , xN ,0a

Ensemble Analysis

Zhang et al. (AAS, 2009)

Page 5

Hybrid cost function

Composition of hybrid increments:

δx0 = δxc0 +

1√N − 1

N∑n=1

(an ◦ x′f n,0)

The N weighting vectors an are concatenated to form the αcontrol variables for the ensemble perturbations

αT = [aT1 , aT

2 , ..., aTN ].

Page 6

Hybrid cost function

J(δx0) = βcJb(δxc0) + βeJe(α) + Jo(δx0)

= βc12δx

cT0 B−1δxc

0

+ βe12α

T A−1α

+12

τ∑t=0

[HtMtδx0 − dt ]T R−1

t [HtMtδx0 − dt ].

dt is calculated with respect to a model trajectory from x̄f0:

dt = yt − Ht [Mt(x̄f0)].

Lorenc (QJRMS, 2003), Wang et al. (MWR, 2008)

Page 7

Tropical cyclogenesis case

100oW 90

oW 80

oW 70

oW 60

oW 50

oW

0o

10oN

20oN

30oN

00 UTC 9 Sept.00 UTC 18 Sept.

Track

− tropical wave

− tropical storm

− hurricane

4DVar, EnKF, and E4DVarare compared for 10 daysduring the genesis, rapidintensification and decay ofHurricane Karl (2010).

Karl developed from a tropical disturbance that initiated on 8Sept., and formed a tropical depression on 18 UTC 14 Sept.

Experiments are initialized from GFS/GDAS analyses on 18UTC 07 Sept.

Analyses are performed every six hours until 00 UTC 18 Sept.

Page 8

ObservationsConventional observations

100oW 90

oW 80

oW 70

oW 60

oW 50

oW

0o

10oN

20oN

30oN

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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00 UTC 9 Sept.00 UTC 18 Sept.

Track

− tropical wave

− tropical storm

− hurricane

Observations~ sat winds below 500 mb

~ sat winds above 500 mb

o soundings

+ buoys

x METARs

Field observations

100oW 90

oW 80

oW 70

oW 60

oW 50

oW

0o

10oN

20oN

30oN

o o o o o o

o

o

oooo

oo o o o o

ooo

o o o o o o

o

o

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o o o o

oooo

o

oo

o

o ooo

o o

o

o

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o

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o

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o

o o

o

o o

o

o o

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o o

o

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o ooo

o ooo o

oo

oo o o

oo

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oo o

o

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oo

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oo

oo

o o

oo

oo

o

ooo

oo

oooo

ooooo

oooooo ooo

oooo o

oo

ooo

oooo

ooo

o oooooooo

ooooo oooooo

00 UTC 9 Sept.

00 UTC 18 Sept.

Track

− tropical wave

− tropical storm

− hurricane

PREDICT dropsondeso 10 Sept. flight missions (2)

o 11 Sept. flight mission

o 12 Sept. flight mission

o 13 Sept. flight mission

o 14 Sept. flight mission

GRIP dropsondeso 12 Sept. flight mission

o 13 Sept. flight mission

o 14 Sept. flight mission

o 16 Sept. flight mission

o 17 Sept. flight mission

The assimilated data includeconventional observations anddropsonde measurementscollected during the NSFPredepression Investigation ofCloud Systems in the Tropics(PREDICT) field campaign.

Dropsondes from the NASAGenesis and RapidIntensification Processes(GRIP) experiment are used forverification.

Page 9

Experiment configuration

The data assimilation is performed on a 13.5-km grid spacingdomain with 34 vertical levels.

Deterministic forecasts use a 4.5-km storm-following nesteddomain.

Inner-loop iterations (4DVar/E4DVar) use a 40.5-km gridspacing.

Ensemble experiments use 60 members with a 900-km ROIfor localization and 80% relaxation to prior perturbations

The hybrid cost function uses 80% of the ensemble incrementand 20% of the climatological increment.

Page 10

Deterministic forecast results

0 h 24 h 48 h 72 h1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

a) u (m s−1)

RM

SD

(observ

ations)

0 h 24 h 48 h 72 h1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

b) v (m s−1)

0 h 24 h 48 h 72 h0.6

0.7

0.8

0.9

1

1.1

c) T (K)

0 h 24 h 48 h 72 h1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

d) q (g kg−1)

0 h 24 h 48 h 72 h1.8

2

2.2

2.4

2.6

2.8

3

e) u (m s−1)

RM

SD

(G

DA

S)

Forecast lead time0 h 24 h 48 h 72 h

1.6

1.8

2

2.2

2.4

2.6

2.8

3

f) v (m s−1)

Forecast lead time0 h 24 h 48 h 72 h

0.45

0.5

0.55

0.6

0.65

0.7

g) T (K)

Forecast lead time0 h 24 h 48 h 72 h

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

h) q (g kg−1)

Forecast lead time

EnKF

4DVar

E4DVar

RMSD to routinesoundings and fieldobservations within 800km of storm center

RMSD to GDAS datawithin 2500 km of stormcenter

Page 11

Deterministic track and intensity forecasts

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xx

xx

xxxxxxxxx

EnKF

a)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

x x

x

x

x x

x

x

x

x

x x

x

Ma

x w

ind

sp

ee

d (

m s

−1)

d)

Analysis time

00 UTC 12 Sept.

06 UTC 12 Sept.

12 UTC 12 Sept.18 UTC 12 Sept.

00 UTC 13 Sept.

06 UTC 13 Sept.12 UTC 13 Sept.

18 UTC 13 Sept.

00 UTC 14 Sept.06 UTC 14 Sept.

12 UTC 14 Sept.

18 UTC 14 Sept.

00 UTC 15 Sept.

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xxx

x

xx

xx

xxxxx

4DVar

b)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

xx

x

xx

x x

x

x

xx

x x

e)

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xx

xxx

x

xxxxx

xx

E4DVar

c)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

x xx

x

xx

x x

x

x

x x x

f)

Forecasts to 00 UTC 18 Sept. using analysis times leading upto genesis.

Page 12

Deterministic track and intensity forecasts

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xxxx

EnKF

a)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

x

x

x x

Ma

x w

ind

sp

ee

d (

m s

−1)

d)

Analysis time

00 UTC 14 Sept.

06 UTC 14 Sept.

12 UTC 14 Sept.

18 UTC 14 Sept.

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xxxx

4DVar

b)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

x

xx

x

e)

100oW

95oW 90

oW 85

oW 80

oW 75

oW 70

oW 65oW

10oN

12oN

14oN

16oN

18oN

20oN

22oN

24oN

xxx

x

E4DVar

c)

12 Sept. 14 Sept. 16 Sept. 18 Sept.

10

20

30

40

50

x

x

x x

f)

Forecasts to 00 UTC 18 Sept. using analysis times leading upto genesis.

Page 13

4DVar and E4DVar structure functions

Page 14

Balance

0 2 4 6 8 10 1210

−6

10−5

10−4

10−3

Domain root mean square ∂2p

s/∂t

2

Me

an

(h

Pa

s−

2)

a)

0 2 4 6 8 10 1210

−7

10−6

10−5

10−4

Sta

nd

ard

de

via

tio

n (

hP

a s

−2)

Forecast lead time (h)

b)

EnKF4DVarE4DVar

Domain-averaged ∂2Ps∂t2 is

used to quantify gravitywave activity afterinitialization. Value areaveraged over alldeterministic forecasts.

Standard deviations in mean∂2Ps∂t2 show amount of

variability between cycles.

Page 15

Cost

Approximating a minimum to the 4DVar/E4DVar costfunction requires many iterations of the tangent linear andadjoint model (can be expensive).

Inner-loop iterations are averaged over the 40 dataassimilation cycles:

4DVar E4DVarMean iterations 37.5 25.6

Mean analysis time (256 cores) 2236 s 1566 s

Page 16

Conclusions

A two-way coupled hybrid method (E4DVar) is found tooutperform the benchmark EnKF and 4DVar systems for a10-day tropical cyclogenesis and rapid intensification event.

E4DVar does not require a long assimilation window todevelop flow-dependent information.

It may also use observations at the beginning of the timewindow more affectively than 4DVar.

The ensemble information improves initial condition balanceover 4DVar and EnKF and reduces the number of iterationsrequired to minimize the 4-D cost function.

Future research will focus on optimal window length,conditioning, and comparison with 4D-ens-Var.

Page 17

Particle filtering

Approximate moments of the posterior error distribution using

f (xt) =∫

f (xt)p(xt |yt)dxt ,

≈N∑

n=1wn

t f (xnt ).

For the simplest particle filter, the weights are given by

wnt =

p(yt |xnt )∑N

n=1(yt |xnt ).

Page 18

Local likelihood particle filter (LLPF)

Problem: unless the ensemble is relatively large, it willeventually lose track of the signal, in which case, the weightsbecome concentrated on a small number of particles. Thisproblem occurs faster when the dimensions of x and y arelarge (Snyder et al. 2008, MWR).

Possible solution: use a likelihood function that decaysexponentially away from observation locations. Filterdegenerecy is then restricted to local regions of the domain.

Page 19

Local likelihood particle filter (LLPF)The Lorenze-96 model is used to test this idea and comparewith EnKF:

dxt+1,i

dt = (xt,i+1 − xt,i−1)xt,i−1 − xt,i + 8.

Experiment details:

100 variables

Observations are taken from a “truth” run at every third gridpoint, with added Gaussian noise (σ = 1).

These observations are assimilated every 24 h (dt = 0.2) for1000 cycles.

Relaxation coefficient and localization radius for EnKF aretuned for each ensemble size.

Page 20

Local likelihood particle filter (LLPF)

Ne = 50

0 5 10 15 20 25 300.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

30−

cycle

mean R

MS

E

30−day averaging period

EnKF

LLPF

RMSEs are averaged every 30 cycles.

Page 21

Local likelihood particle filter (LLPF)

Ne = 100

0 5 10 15 20 25 300.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

30−

cycle

mean R

MS

E

30−day averaging period

EnKF

LLPF

RMSEs are averaged every 30 cycles.

Page 22

Local likelihood particle filter (LLPF)

Ne = 200

0 5 10 15 20 25 300.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

30−

cycle

mean R

MS

E

30−day averaging period

EnKF

LLPF

RMSEs are averaged every 30 cycles.

Page 23

Local likelihood particle filter (LLPF)

Ne = 300

0 5 10 15 20 25 300.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

30−

cycle

mean R

MS

E

30−day averaging period

EnKF

LLPF

RMSEs are averaged every 30 cycles.

Page 24

Conclusions

A localized likelihood approach to particle filtering is beinginvestigated for systems that contain a large spatialdimension.

This method makes use of a distance-dependent likelihoodfunction that can be applied to prevent filter degeneracy(much like localization in EnKF).

Much more work needs to be done; e.g., resampling, inflation,better choices for likelihood function, etc.

Page 25

Extra

A simple localization example is to use a function that decaysexponentially away from the location of an observation, sothat the likelihood of the the i th variable given the j th

observation is written

p(yt,j |xnt,i) = [p(yt,j |xn

t )−1

NeNy]exp(−di ,j

R ) +1

NeNy,

where di ,j is the physical distance between the observation andmodel grid point, and R is a tunable localization radius.

Page 26

Extra

EnKF(N = 60)

4DVar E4DVar(N = 60)

Mean number of observations 26140 26140 26140Mean inner-loop iterations N/A 37.4 25.6

Number of NLM runs 60 2 62Mean TLM time (s) N/A 17.3 17.3Mean ADM time (s) N/A 39.5 39.5Mean NLM time (s) 13.4 13.4 13.4

Mean IO (s) N/A 2.2 2.2Mean analysis time (s) 840 2236 1566

TOTAL TIME (s): 1644 2263 2397

Computing time for 40.5 km grid spacing, 34 vertical levelsand 256 nodes.