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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
Application Aspects of Ensemble Methods L. Nerger
Localization
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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)
Application Aspects of Ensemble Methods L. Nerger
Instability of serial observation processing filters in case of localization
(EnSRF, EAKF)
Application Aspects of Ensemble Methods L. Nerger
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)
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
2 6 10 14 18 22 26 30 340.9
0.92
0.94
0.96
0.98
1
LESTKF, 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
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
Application Aspects of Ensemble Methods L. Nerger
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
number of assimilated observations
rms
erro
r
Application Aspects of Ensemble Methods L. Nerger
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.
Application Aspects of Ensemble Methods L. Nerger
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.
Application Aspects of Ensemble Methods L. Nerger
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.
Application Aspects of Ensemble Methods L. Nerger
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.
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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)
Application Aspects of Ensemble Methods L. Nerger
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
number of assimilated observations
rms
erro
r Blue: LETKF with serial
observation processing
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
◆
Application Aspects of Ensemble Methods L. Nerger
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
number of assimilated observations
rms
erro
r
0 10 20 30 400
1
2
3
4
5σR = 0.1
number of assimilated observations
L−EnSRF and sorted observations
rms
erro
r
„maximum distance“ (1, 21, 11, 31, 6, 26,...)
local observation sorting (Whitaker et al. 2008)
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
Optimal Localization Radius
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
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
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
Application Aspects of Ensemble Methods L. Nerger
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
Observation localization
loptldiv
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
P. Kirchgessner et al. Mon. Wea. Rev. 142 (2014) 2165-2175
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
Observation 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 35 400
5
10
15
20
25
30
35
40
45
50Optimal effective observation dimension
Ensemble size
Effe
ctiv
e ob
s. d
im
DLOL
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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
Initialize Model generate mesh Initialize fields
Time stepper consider BC
Consider forcing
Post-processing
Model Extension for data assimilation
ensemble forecast enabled by parallelization
Aaaaaaaa
Aaaaaaaa
aaaaaaaaa
Start
Stop
Initialize Model generate mesh Initialize fields
Time stepper consider BC
Consider forcing
Post-processing
init_parallel_PDAF
Do i=1, nsteps
Init_PDAF
Assimilate_PDAF
plus: Possible
model-specific adaption.
Application Aspects of Ensemble Methods L. Nerger
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
P. Kirchgessner et al. Mon. Wea. Rev. 142 (2014) 2165-2175
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
Application Aspects of Ensemble Methods L. Nerger
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)
Application Aspects of Ensemble Methods L. Nerger
Model mesh at the equator
Drake passage
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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)
P. Kirchgessner et al. Mon. Wea. Rev. 142 (2014) 2165-2175
Application Aspects of Ensemble Methods L. Nerger
Weight Functions
Application Aspects of Ensemble Methods L. Nerger
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
Application Aspects of Ensemble Methods L. Nerger
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!