Convective-scale data assimilation in the Weather Research...

1/20

Convective-scale data assimilation in theWeather Research and Forecasting model

using a nonlinear ensemble filter

Jon Poterjoy, Ryan Sobash, and Jeffrey Anderson

National Center for Atmospheric Research†

ASP/MMM/DAReS

25th May 2016

†The National Center for Atmospheric Research is sponsored by the National Science Foundation.

2/20

Data assimilation in weather models

Filters and smoothers are applied regularly for dataassimilation in geophysics.

Current methods are based on variational approaches (3DVarand 4DVar), Ensemble Kalman filters (EnKFs), orcombinations of the two.

Assumptions:

The model dynamics are linear.

Observations relate linearly to the model state variables.

The model state and observation errors are Gaussian.

3/20

Example problem

100

150

200

250

300

*

Dis

tance (

km

)

140 160 180 200 220 240

5

10

15

Distance (km)

Heig

ht (k

m)

10 20 30 40 50 60

*

Given:

100-member ensemble forecastsare samples from prior errordistribution p(x)

Radar reflectivity measurement at? (denoted y).

Top and bottom panels show crosssections through true storm atobservation location.

Reflectivity and storm-relative windsare plotted.

6/20

Particle filter update at ?

0

5

10x 10

−4

|y

qic

e

(g k

g−

1)

2

4

6

8

|y

qra

in

(g k

g−

1)

0

0.1

0.2

|y

qsn

ow

(g k

g−

1)

35 40 45 50 55

0

2

4

Reflectivity (dBZ)

|y

qg

rau

pe

l

(g k

g−

1)

|y

|y

|y


|y







7/20

Particle filters as a practical solution?

Challenges:

PFs have known limitations for high-dimensional systems(e.g., Bengtsson et al. 2008; Bickel et al. 2008; Snyder et al.2008).

They may also be inappropriate for models containing errorsources that are represented poorly or ignored.

Several attempts have been made to circumvent these issues:van Leeuwen (2010), Frei and Kunsch (2013), Majda et al.(2014), Cheng and Reich (2015), etc.

8/20

The local PFPoterjoy (MWR, 2016) and Poterjoy and Anderson (MWR, 2016)introduce the local PF for data assimilation.

Local PF vs. EAKF (Anderson, 2001) using 40-variable Lorenz (1996) model:

Linearmeasurement

operator:H(x) = x

101

102

103

1

2

3

||

|||| || || || || ||

||

||

|

|

|| || || ||

Prio

r e

rro

r

Ensemble size

Average EAKF RMSEAverage PF RMSE

Nonlinearmeasurement

operator:H(x) = ln(|x |)

101

102

103

2

3

4

|

|| ||

|

|

||

| |

|

||

||

|

|

|||

| |

Prio

r e

rro

r

Ensemble size

Average EAKF RMSEAverage PF RMSE

9/20

Prior with ob located 2 km lower

0

5

10x 10

−4

|y

qic

e

(g k

g−

1)

2

4

6

8

|y

qra

in

(g k

g−

1)

0

0.1

0.2

|y

qsn

ow

(g k

g−

1)

35 40 45 50 55

0

2

4

Reflectivity (dBZ)

|y

qg

rau

pe

l

(g k

g−

1)

|y

|y

|y


|y





10/20

PF update

0

5

10x 10

−4

|y

qic

e

(g k

g−

1)

2

4

6

8

|y

qra

in

(g k

g−

1)

0

0.1

0.2

|y

qsn

ow

(g k

g−

1)

35 40 45 50 55

0

2

4

Reflectivity (dBZ)

|y

qg

rau

pe

l

(g k

g−

1)

|y

|y

|y


|y







11/20

Local PF update

0

5

10x 10

−4

|y

qic

e

(g k

g−

1)

2

4

6

8

|y

qra

in

(g k

g−

1)

0

0.1

0.2

|y

qsn

ow

(g k

g−

1)

35 40 45 50 55

0

2

4

Reflectivity (dBZ)

|y

qg

rau

pe

l

(g k

g−

1)

|y

|y

|y


|y







12/20

Case study: idealized MCS

Distance (km)

Dis

tan

ce

(km

)

a) 0 min.

50 100 150 200 250

50

100

150

200

250

300

350

Distance (km)

b) 30 min.

50 100 150 200 250

Distance (km)

c) 90 min.

100 150 200 250 300

Distance (km)

d) 180 min.

150 200 250 300 3500

10

20

30

40

50

60

Model: NCAR Weather Research and Forecasting model(3-km grid spacing with 40 vertical levels)Observations: radar velocity and reflectivity every 5 minutesData assimilation: Local PF and EAKF in DART frameworkusing 100 members

See Sobash and Stensrud (2013) for details

13/20

EAKF members (180 min)

Distance (km)

Dis

tance (

km

)

Truth

100 200 300

100

200

300

0

10

20

30

40

50

60

14/20

Local PF members (180 min)

Distance (km)

Dis

tance (

km

)

Truth

100 200 300

100

200

300

0

10

20

30

40

50

60

15/20

Probabilistic verification

0.02

0.04

0.06

0.08

qice

EAKF

Fre

quency q

ice

PF

0.02

0.04

0.06

0.08

qrain

Fre

quency q

rain

0.02

0.04

0.06

0.08

qsnow

Fre

quency q

snow

0.02

0.04

0.06

0.08

qgraupel

Fre

quency q

graupel

20 40 60 80 100

0.02

0.04

0.06

0.08

qcloud

Rank

Fre

quency

20 40 60 80 100

qcloud

Rank

Rank histograms calculatedfrom prior members every10 min in convective cellswithin leading edge of squallline.

Grid points where truereflectivity > 0 dBZ areused for verification,assuming no spatial andtemporal correlationsbetween variables.

16/20

RMSE and bias (5-min cycling)

−2 0 2 4

2

4

6

8

10

12

He

igh

t (k

m)

m s−1

u

0 1 2 3

m s−1

v

0 1 2

m s−1

w

0 0.5 1K

T

−0.5 0 0.5 1 1.5

g kg−1

qvapor

−0.1 0 0.1 0.2

2

4

6

8

10

12

He

igh

t (k

m)

g kg−1

qcloud

−0.02 0 0.02 0.04

g kg−1

qice

−0.2 0 0.2 0.4

g kg−1

qrain

0 0.2 0.4

g kg−1

qsnow

EAKF RMSE

EAKF biasPF RMSE

PF bias

−0.5 0 0.5 1

g kg−1

qgraupel

Horizontal mean RMSEs (solid) and bias (dashed) from 60-minforecasts.

Values are averaged in vicinity of squall line.

17/20

RMSE and bias (20-min cycling)

−2 0 2 4

2

4

6

8

10

12

Heig

ht (k

m)

m s−1

u

0 1 2 3

m s−1

v

0 1 2

m s−1

w

0 0.5 1K

T

−0.5 0 0.5 1 1.5

g kg−1

qvapor

−0.1 0 0.1 0.2

2

4

6

8

10

12

Heig

ht (k

m)

g kg−1

qcloud

−0.02 0 0.02 0.04

g kg−1

qice

−0.2 0 0.2 0.4

g kg−1

qrain

0 0.2 0.4

g kg−1

qsnow

EAKF RMSE

EAKF biasPF RMSE

PF bias

−0.5 0 0.5 1

g kg−1

qgraupel

Horizontal mean RMSEs (solid) and bias (dashed) from 60-minforecasts.

Values are averaged in vicinity of squall line.

18/20

Forecast error evolution

Mean forecast RMSEs as afunction of time.

Values are averaged in vicinityof squall line from 20-min obsfrequency experiment.

Initial error growth in EAKFforecasts is much more rapidthan in PF forecasts.

0 10 20 30 40 50 602.5

3

3.5

4

m s

−1

u

EAKF RMSE

PF RMSE

0 10 20 30 40 50 601.5

2

2.5

3

m s

−1

v

0 10 20 30 40 50 601.4

1.5

1.6

1.7

1.8

1.9

m s

−1

w

0 10 20 30 40 50 600.8

0.85

0.9

0.95

1

K

T

0 10 20 30 40 50 600.52

0.54

0.56

0.58

0.6

g k

g−

1

qvapor

0 10 20 30 40 50 600.075

0.08

0.085

0.09

0.095

0.1

g k

g−

1

qcloud

0 10 20 30 40 50 600.012

0.013

0.014

0.015g k

g−

1

qice

0 10 20 30 40 50 600.1

0.12

0.14

0.16

0.18

g k

g−

1

qrain

0 10 20 30 40 50 600.1

0.12

0.14

0.16

0.18

Forecast time (min)

g k

g−

1

qsnow

0 10 20 30 40 50 600.4

0.5

0.6

0.7

0.8

Forecast time (min)g k

g−

1

qgraupel

19/20

Summary

Particle filters provide a means of assimilating observations forapplications that are difficult for linear/Gaussian filters.

A new data assimilation system is developed thatapproximates the particle filter within local neighborhoods ofobservations (Poterjoy 2016).

The local PF is computationally affordable—with a costcomparable to the NCAR DART EAKF.

Recent testing of the local PF in the WRF model provide anincentive to explore real applications.

20/20

ReferencesAnderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129,2884–2903.

Bengtsson, T. and P. Bickel and B. Li, 2008: Curse-of-dimensionality revisited: Collapse of the particle filter invery large scale systems. Probability and Statistics: Essays in Honor of David A. Freedman, D. Nolan and T.Speed, Eds., 2, 316–334.

Bickel, P. and B. Li and T. Bengtsson, 2008: Sharp failure rates for the bootstrap particle filter in highdimensions. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3,318–329.

Cheng, Y. and S. Reich, 2015: A McKean optimal transportation perspective on Feynman-Kac formulae withapplication to data assimilation. Frontiers in Applied Dynamical Systems., URLhttp://arxiv.org/abs/1311.6300.

Frei, M. and H. R. Kunsch, 2013: Bridging the ensemble Kalman and particle filters. Biometrika, 1–20.

Majda, A. J., D. Qi, and T. P. Sapsis, 2014: Blended particle filters for large-dimensional chaotic dynamicalsystems. Proc. Natl. Acad. Sci. U.S.A., 111, 7511–7516.

Poterjoy, J., 2016: A localized particle filter for high-dimensional nonlinear systems. Mon. Wea. Rev., 144,59–76.

Poterjoy, J. and J. L. Anderson, 2016: Efficient assimilation of simulated observations in a high-dimensionalgeophysical system using a localized particle. Mon. Wea. Rev., accepted.

Snyder, C., T. Bengtsson, P. Bickel, and J. Anderson, 2008: Obstacles to High-Dimensional Particle Filtering.Monthly Weather Review, 136, 4629–4640.

Sobash, R. A. and D. J. Stensrud, 2013: The Impact of covariance localization for radar data on EnKFanalyses of a developing MCS: Observing system simulation experiments. Mon. Wea. Rev., 141, 3691–3709.

van Leeuwen P. J., 2010: Nonlinear data assimilation in geosciences: an extremely efficient particle filter.Quart. J. Roy. Meteor., 136, 1991–1999.

Convective-scale data assimilation in the Weather Research...

Documents

Transcript of Convective-scale data assimilation in the Weather Research...