localization-induced filter instability and a simple ... · localization-induced filter instability...

Localization-induced filter instability and a simple adaptive localization method

Lars Nerger

Alfred Wegener Institute for Polar and Marine Research Bremerhaven, Germany

with

Paul Kirchgessner, Angelika Bunse-Gerstner

MFO, Oberwolfach, Germany, October 6, 2016

Application Aspects of Ensemble Methods L. Nerger

Outline

!  Instability of serial observation processing filter in case of localization

! Adaptive localization following the degrees of freedom given by the ensemble


Localization


Localization: Why and how?

!  Combination of observations and model state based on ensemble estimates of error covariance matrices

!  Finite ensemble size leads to significant sampling errors

•  errors in variance estimates

!  usually too small

•  errors in correlation estimates !  wrong size if correlation exists !  spurious correlations when true correlation is zero

!  Assume that long-distance correlations in reality are small

!  damp or remove estimated long-range correlations

0 10 20 30 40−2

−1

0

1

2

3

4Example: Sampling error and localization

truesampledlocalized

0 10 10 20 20 distance


Covariance localization

Covariance localization !  Applied to forecast covariance matrix

!  Element-wise product with correlation matrix C of compact support to reduce covariances

!  Only possible if forecast covariance matrix is computed (not in ETKF or SEIK)

E.g.: Houtekamer & Mitchell (1998, 2001), Whitaker & Hamill (2002)

Kloc

=�C �Pf

�HT

�H

�C �Pf

�HT +R

��1


Domain & Observation localization Domain localization (local analysis) !  Perform local filter analysis with

observations from surrounding domain

Observation localization (Hunt et al. 2007)

!  Use non-unit weight for observations

!  reduce weight for remote observations by increasing variance estimate

!  Localization effect similar to covariance localization

!  equivalence to covariance localization only shown for single observation (Nerger et al. QJRMS, 2012)

S: Analysis region D: Corresponding data region

Domain Localization

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

distance

wei

ght

Weight of observation

GC99EXP

R�1� = C̃� �R�1

E.g.: Brankart et al. (2003), Evensen (2003), Ott et al. (2004), Hunt et al. (2007)


Instability of serial observation processing filters in case of localization

(EnSRF, EAKF)


Serial observation processing

Synchronous assimilation

ETKF, SEIK, ESTKF, (EnKF)

•  Assimilation all observation at a given time at once

•  Usually using ensemble-space transformations

•  Possible for arbitrary observation error covar. matrices

Serial observation processing

EnSRF, EAKF

•  Perform a loop assimilating each single observation

•  Efficient: Avoids matrix-matrix operations

•  Requires diagonal observation error covar. matrix

Use

observation localization

Use

covariance localization

(EnSRF: Whitaker & Hamill, 2002; EAKF: Anderson, 2001)


EnSRF (Whitaker & Hamill 2002)

For obs. error=1.0: •  EnSRF and LESTKF almost identical

Test with Lorenz9[568] Model

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

LESTKF, obs. error=1.0

forg

ettin

g fa

ctor

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

EnSRF, obs. error=1.0fo

rget

ting

fact

or

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1


Test with Lorenz9[568] Model

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1


forg

ettin

g fa

ctor

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

EnSRF, obs. error=1.0fo

rget

ting

fact

or

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1


forg

ettin

g fa

ctor

support radius 0.0180.0185 0.0190.0195 0.020.0205 0.0210.0215 0.0220.0225 0.0230.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

EnSRF, obs. error=0.1

forg

ettin

g fa

ctor

support radius 0.0180.0185 0.0190.0195 0.020.0205 0.0210.0215 0.0220.0225 0.0230.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1


RMS error over number of observations

How does the RMS error develop during the loop over all observations?

At first analysis step: •  EnSRF: Compute RMS errors at each iteration •  LESTKF: Do 40 experiments with increasing number of obs.

0 10 20 30 400.5

1

1.5

2

2.5

3

3.5σR = 1.0

number of assimilated observations

rms

erro

r

EnSRF trueEnSRF estimateLESTKF trueLESTKF est.

0 10 20 30 400

1

2

3

4

5

6

7

8 σR = 0.1


rms

erro

r


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

40 obs.

Instability of serial obs. processing with localization

More detailed view:

•  State estimate for different numbers of observations

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

35 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

30 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

25 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

20 obs.


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

40 obs.


More detailed view:


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

35 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

30 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

25 obs.


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

40 obs.


More detailed view:


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

35 obs.

0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

30 obs.


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

40 obs.


More detailed view:


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

35 obs.


0 5 10 15 20 25 30 35 40

−20

0

20

grid point

stat

e

40 obs.


More detailed view:



Inconsistent matrix updates

The Kalman filter updates the covariance matrix according to

With the Kalman gain

this simplifies to

(1) and (3) yield same result only with gain (2)!

Pa = (I�KH)Pf (I�KH)T +KRKT

K = PfHT�HPfHT +R

��1

Pa = (I�KH)Pf

Not fulfilled with localization:

!  Update of P is inconsistent in localized EnSRF (already noted by Whitaker & Hamill (2002), but never further examined)

Kloc

=�C �Pf

�HT

�H

�C �Pf

�HT +R

��1

(1)

(3)

(2)


Inconsistent matrix updates (2)

The inconsistency also occurs in LETKF, LESTKF, LSEIK ...

•  But here: update is only done once followed by ensemble forecast

•  LETKF with serial observation processing also shows instability

0 10 20 30 400

1

2

3

4

5

6

7

8σR = 0.1


rms

erro

r Blue: LETKF with serial

observation processing


Simple Example

x

f =

✓11

◆; P

f =

✓1 0.80.8 1

◆

y =

✓00

◆; R =

✓0.1 00 0.1

◆; H =

✓0 11 0

◆

C =

✓1 0.25

0.25 1

◆

State estimate & covariance matrix

Observation

Localization matrix


Simple Example

Bulk update (all observations at once)

Serial update

Pa(sym.) =

✓0.089 0.0070.007 0.089

◆; Pa

(1sided) =

✓0.080 0.0580.058 0.080

◆

Pa(sym.) =

✓0.088 0.0090.009 0.088

◆; Pa

(1sided) =

✓0.089 0.0550.055 0.076

◆

x

a(bulk) =

✓0.0770.077

◆

x

a(sym.) =

✓0.0970.073

◆; x

a(1sided) =

✓0.0910.046

◆


Effect of observation reordering

•  Before: Assimilated observation from grid point 1 to 40 with increasing index

•  What is the effect when we re-order the observations?

0 10 20 30 400

1

2

3

4

5σR = 0.1

EnSRF with re−ordered observations


rms

erro

r

0 10 20 30 400

1

2

3

4

5σR = 0.1


L−EnSRF and sorted observations

rms

erro

r

„maximum distance“ (1, 21, 11, 31, 6, 26,...)

local observation sorting (Whitaker et al. 2008)


Maximum-distance reordering

•  practically no effect on final results

Observation reordering

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

EnSRF re−ordered obs., obs. error=0.1

forg

ettin

g fa

ctor

support radius 0.0180.0185 0.0190.0195 0.020.0205 0.0210.0215 0.0220.0225 0.0230.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1

2 6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

L−EnSRF sorted obs., obs. error=0.1

forg

ettin

g fa

ctor

support radius 0.0180.0185 0.0190.0195 0.020.0205 0.0210.0215 0.0220.0225 0.0230.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1

Full experiment over 50000 analysis steps, N=10

EnSRF with local observation sorting

•  improves stability •  But not minimum error

L. Nerger, Mon. Wea. Rev. 143 (2015) 1554-1567


Optimal Localization Radius


Domain & Observation localization

Localization radius can depend on

!  Ensemble size

!  Model dynamics & resolution

!  Field

Optimal localization radius

!  Typically determined experimentally (very costly)

!  Some authors proposed adaptive methods (e.g. Anderson, Bishop/Hodyss, Harlim)

"  still with tunable parameters


Relation between ensemble size and localization radius

!  Test runs with Lorenz-96 model

!  Vary ensemble size and localization radius

!  White: Filter divergence

5 10 15 20 25 28

0

10

20

30

40

50

Ensemble size

Lo

cali

zati

on

rad

uis

Observation localization

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

0.3

0.35

0.4

0.7

1

5 10 15 20 25 28

0

5

10

15

20

Lo

cali

zati

on

rad

uis

Ensemble size

Domain localization

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

0.3

0.35

0.4

0.7

1

5 10 15 20 25 28

0

10

20

30

40

50

Ensemble size

Lo

cali

zati

on

rad

uis


0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

0.3

0.35

0.4

0.7

1

5 10 15 20 25 28

0

5

10

15

20

Locali

zati

on

radu

is

Ensemble size

Domain localization

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

0.3

0.35

0.4

0.7

1

divergence minimum RMSe


Optimal localization radius

!  Optimal localization radius as function of ensemble size

!  Linear dependence for domain and observation localization

!  Radius larger for OL than DL

5 10 15 20 252

4

6

8

10

12

14

16

18

20

22

Ensemble size

Loca

lizat

ion

radi

us

Domain localization

loptldiv

5 10 15 20 25 305

10

15

20

25

30

35

40

45

50

Ensemble size

Loca

lizat

ion

radi

us


loptldiv


Relate domain and observation localizations

!  Define: Effective observation dimension dW = sum of observation weights

!  Minimum RMS errors when effective obs. dimension slightly larger than ensemble size

!  When dW=N, errors are almost as small (optimal use of degrees of freedom from ensemble?)

P. Kirchgessner et al. Mon. Wea. Rev. 142 (2014) 2165-2175

5 10 15 200

5

10

15

20

25

30

35

40

Ensemble size

Effe

ctiv

e ob

s. d

im.

Divergence: Effective obs. dimension

DLOL

5 10 15 200

5

10

15

20

25

30

35

40

Ensemble size

Effe

ctiv

e ob

s. d

im.

Optimal effective obs. dimension

DLOL


2D Shallow Water Model

!  Shallow water model simulating a double gyre in a box !  Assimilate sea surface height at each grid point

!  For DL: steps due to addition of observations

!  dW optimal if about or slightly lower than ensemble size

!  relation holds for different weight functions


5 10 15 20 25 30 35 40

20

50

100

150

180

Loca

liza

tion r

aduis

(km

)

Ensemble size

Domain localization

0.260.28 0.30.320.340.360.38 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2

5 10 15 20 25 30 35 40

20

100

200

300

340

Loca

liza

tion r

aduis

(km

)

Ensemble size


0.260.28 0.30.320.340.360.38 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2

5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

50Optimal effective observation dimension

Ensemble size

Effe

ctiv

e ob

s. d

im

DLOL


PDAF: A tool for data assimilation

PDAF - Parallel Data Assimilation Framework "  a software to provide assimilation methods "  an environment for ensemble assimilation "  for testing algorithms and real applications "  useable with virtually any numerical model "  also:

•  apply identical methods to different models •  test influence of different observations

"  makes good use of supercomputers (Fortran and MPI; tested on up to 17000 processors)

"  first public release in 2004; continued development

More information and source code available at

http://pdaf.awi.de

L. Nerger & W. Hiller, Computers & Geosciences 55 (2013) 110-118


Extending a Model for Data Assimilation

Aaaaaaaa

Aaaaaaaa

aaaaaaaaa

Start

Stop

Do i=1, nsteps

Initialize Model generate mesh Initialize fields

Time stepper consider BC

Consider forcing

Post-processing

Aaaaaaaa

Aaaaaaaa

aaaaaaaaa

Start

Stop

Do i=1, nsteps



Consider forcing

Post-processing

Model Extension for data assimilation

ensemble forecast enabled by parallelization

Aaaaaaaa

Aaaaaaaa

aaaaaaaaa

Start

Stop



Consider forcing

Post-processing

init_parallel_PDAF

Do i=1, nsteps

Init_PDAF

Assimilate_PDAF

plus: Possible

model-specific adaption.


2D Shallow Water Model

!  Sparser observations !  1/4 and 1/9 of observations

!  Still linear dependence between effective obs. dimension and N

!  Effective obs. dimension has to be scaled by obs. density


normalized by the observational density. This espe-cially becomes an issue, if the spatial distribution of theobservations is not regular. This case will be examinedin future studies.Figure 5 also allows us to estimate the optimal local-

ization radius as a function of the ensemble size. Therelationship is approximately

lopt ’ 8

ffiffiffiffiffiN

40

rdx , (11)

where dx denotes the grid spacing. At this localizationradius, the effective observation dimension is approxi-mately equal to the ensemble size. This relation shouldhold in general for dense observations that are distrib-uted in two dimensions and a regular orthogonal modelgrid.

6. Localization in a global ocean model

The experiments discussed above indicate that an op-timal localization radius is obtained when the effectiveobservation dimension is approximately equal to theensemble size. To assess whether this localization can beapplied in a realistic large-scalemodel, we apply it here intwin experiments using a global configuration of the

finite-element sea ice ocean model (FESOM; Danilovet al. 2004; Wang et al. 2008; Timmermann et al. 2009).The twin experiments are similar to an application ofFESOM by Janji!c et al. (2012) where real satellite dy-namic ocean topography data were assimilated.

a. Experimental setup

FESOM is an ocean general circulation model thatutilizes finite elements to solve the hydrostatic oceanprimitive equations. Unstructured triangular meshes areused, which allow for a varying resolution of the mesh.The configuration used here has a horizontal resolutionof about 1.38 with refinement in the equatorial region.The model uses 40 vertical levels.For the data assimilation, FESOMwas coupled to the

assimilation frameworkPDAF (Nerger et al. 2005;Nergerand Hiller 2013; http://pdaf.awi.de) into a single program.The state vector includes the sea surface height (SSH)and the three-dimensional fields of temperature, salinity,and the velocity components. The state vector has a sizeof about 10 million. For the twin experiments, the modelis initialized from a spinup run and a trajectory over 1 yr iscomputed. This trajectory contains the model fields ateach tenth day and represents the ‘‘truth’’ for the assim-ilation experiments. An ensemble of 32 members is used,which is generated by second-order exact sampling fromthe variability of the true trajectory (see Pham 2001). Theinitial state estimate is given by the mean of the true tra-jectory. Pseudo observations of the SSH at each surfacegrid point are generated by adding uncorrelated randomGaussian noise with a standard deviation of 5 cm to the

FIG. 6. The optimal and divergent observation dimensions for (top)DL and (bottom) OL for the shallow-water model.

FIG. 7. The optimal effective observation dimension with anobservation frequency of 1 (blue), 2 (green), and 3 (red). For eachobservation frequency, the optimal value depends linearly on theensemble size. The smaller the observation density, the smaller is theslope of the function.

2172 MONTHLY WEATHER REV IEW VOLUME 142


Large scale data assimilation: Global ocean model

•  Finite-element sea-ice ocean model (FESOM, Danilov et al. )

•  Global configuration (~1.3 degree resolution with refinement at equator)

•  State vector size: 107

•  Scales well up to 256 processor cores

Sea surface elevation

•  Assimilate synthetic sea surface height (SSH) data for ocean state estimation

•  Costly due to large model size (using up to 2048 processor cores)


Model mesh at the equator

Drake passage


Adaptive localization radius in global ocean model

•  Localization radius follows mesh resolution

•  Fixed 1000km radius leads to increasing errors in 2nd half of year

•  Lower RMS error in SSH than fixed 500km radius

truemodel state. The analysis step is computed after eachforecast phase of 10 days with an observation vectorcontaining about 68 000 observations. Overall, the ex-periments were conducted over a period of 360 days.The experiments use the ETKF with OL. Two ex-

periments with fixed localization radii of l5 500 km andl5 1000 km are performed. A third experiment uses thelocalization radius determined such that the effectiveobservation dimension is equal to the ensemble size.The inflation factor was set to r 5 1.1.

b. Assimilation performance

Figure 8 shows of the RMS errors of the sea surfaceheight over time relative to an experiment without dataassimilation for the three experiments. For the fixedradius of l 5 1000 km, the relative RMS error is quicklyreduced below 0.5, but increases again after day 150. Therelative RMS errors for the fixed radius of 500 km andthe experiment with the localization radius based on theeffective observation dimension are similar and the er-rors generally decrease over time.However, the variablelocalization results in smaller RMS errors than the fixedlocalization radius. In the second half of the experiment,the RMS errors obtained with the variable localizationradius are even smaller than those for the fixed locali-zation radius of 1000 km.Overall, the experiments show that the effective ob-

servation dimension can be used to specify a spatiallyvarying localization radius that yields estimates of similarquality than those produced by a fixed radius. However,while the fixed radius has to be tuned with several ex-periments, this is not required for the variable radius.

7. Conclusions

In this study, the optimal value for the localizationradius in domain localization and observation localiza-tion was examined using numerical experiments. Usingthe Lorenz-96model and a nonlinear shallow-watermodelallowed for the assessment the localization behavior withtwo simple nonlinear models with different dynamics.The main focus was on dense observations with uniformobservational error, which are used in real assimilationapplications (e.g., as gridded satellite observations of theocean surface temperature or sea surface height). For thistype of observations, it was possible to assess the relationof the localization radius to the ensemble size over thewhole model domain.The localization radius is optimal if the estimation errors

are minimal. It depends on the ensemble size and variesfor different weight functions. Typically, the optimalradius is determined by experimentation. Yet, one candefine an effective observation dimension given as thesum of the observation weights involved in a local anal-ysis. The optimal localization radius was obtained, ifthe effective observation dimensionwas about equal to thesize of the ensemble. Moreover, the optimal value of theeffective observation dimension is constant for differentweighting functions. This situation can be explained bythe fact that the degrees of freedom for the analysis aredetermined by the rank of the ensemble. The degrees offreedom are optimally utilized if the ensemble sizeequals the effective observation dimension. In the caseof constant observation errors, the degrees of freedomare distributed over different numbers of observationsfor different weight functions. If the observation networkis less dense, other effects, like sampling error for distantobservations, become more important so that this re-lation is weaker. For multivariate data assimilation in theshallow-water model, the optimal effective observationdimension was the same for all three model fields. If theobservation density is reduced, the linear relation in theshallow-water model was still conserved, but the slopewas different. For both models, the optimal value of theeffective observation dimension was roughly equal to theensemble size if a field was completely observed. Fordense observations that are distributed in two dimensions,a simple relation between the ensemble size and the op-timal localization radius was deduced from the experi-ments. This relation can be used to define an adaptivelocalization radius that ensures that the effective obser-vation dimension is equal to the number of ensemblemembers. The relation was tested using a global oceanmodel where synthetic observations of the sea surfaceheight were assimilated. With the adaptive localization,without tuning, a similar error reduction as using an

FIG. 8. RMS errors for the assimilation experiment using FESOMrelative to the errors from an experiment without assimilation.Shown are the relative RMS errors for a fixed localization radius of1000km (black), 500km (red), and the variable localization derivedfrom the effective observation dimension (blue).

JUNE 2014 K IRCHGES SNER ET AL . 2173Relative RMS error of SSH


Discussion on localization radius

!  Findings:

!  Effective observation dimension dW relates to degrees of freedom

!  dW close to ensemble size a good choice

!  No dependence on model dynamics

!  Limitations

!  Observations at each grid point (optimal dW smaller for incomplete observations)

!  Uniform observation error

!  Ignoring information content of observations (e.g. Migliorini, QJRMS 2013)



Weight Functions


Weight function

•  Why 5th-order Gaspari/Cohn polynomial?

•  Covariance function not required for OL

•  Furrer/Bengtsson (2007) indicate best sampling error reduction in Pf for exponential covariances

•  For Lorenz96, some other functions give similar errors – but not significantly lower ones

0 5 10 15 200

0.2

0.4

0.6

0.8

1

distance

wei

ght

Weight of observation

GC99EXPlinear

6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

Linear

forg

ettin

g fa

ctor

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

5th−order polynomial (Gaspari/Cohn)

forg

ettin

g fa

ctor

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

6 10 14 18 22 26 30 340.9

0.92

0.94

0.96

0.98

1

Exponential decay

forg

ettin

g fa

ctor

support radius 0.190.195 0.20.205 0.210.215 0.220.225 0.230.235 0.240.245 0.25 0.3 0.4 0.5 0.6 0.8 1

Weight function at optimal radius


Summary

!  Serial observation processing filters can be unstable when used with localization

!  Update of state error covariance matrix P inconsistent when localization is applied (all filters except classical EnKF)

!  Estimation of adaptive localization radius dependent on ensemble size possible for “dense” observations

•  luckily a usual situation for ocean models assimilating satellite data

Thank you!

localization-induced filter instability and a simple ... · localization-induced filter instability...

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