Alfred Wegener Institute Helmholtz Center for Polar and
Marine Research
On sequential observation processing in localized ensemble Kalman filters
Lars Nerger
Alfred Wegener Institute, Helmholtz Center for Polar and Marine Research, Bremerhaven, Germany Contact: Lars.Nerger@awi.de
·
http://www.awi.deThe different variants of current ensemble square-root Kalman filters assimilate either all observations at once or perform a sequence in which batches of observations or each single observation is assimilated. The sequential ob- servation processing in filter algorithms like the EnSRF [1]
can result in computationally efficient algorithms because matrix inversions in the observation space are reduced to the inversion of single numbers.
Whitaker and Hamill [1] noted that the modification of the En- SRF for localization leads to an inconsistency of the update equation for the state error covariance matrix. Often, this in- consistency does not lead to a significant impact on the as- similation performance. However, using a simple model, we demonstrate with the localized EnSRF algorithm that the se- quential observation processing can significantly deteriorate the assimilation performance under some circumstances.
We perform assimilation experiments with the Lorenz- 96 model. Compared are the performances of the En- SRF with the LESTKF filter, both with localization. As- similation experiments are performed over 50000 time steps with an ensemble of 10 states. The support ra- dius of the localization and the inflation (forgetting fac- tor) are varied.
For the LESTKF the regulated localization weight func- tion [3] is used. In [3] it was shown that this method results in equal effective localization lengths for a single observation for covariance localization and observation localization.
The filter algorithms and the Lorenz96 model are im- plemented in the Parallel Data assimilation Framework (PDAF, [5, 6], http://pdaf.awi.de).
EnSRF
•
Ensemble square-root filter [1]•
Assimilate an observation vector as a sequence of single observations•
Localize with state error covariance matrix (“covariance localization”)LESTKF
•
Error subspace transform Kalman filter [2]•
Assimilate full observation vector at once•
Perform local analysis with observation weights com- puted from regulated localization [3]2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
EnSRF, obs. error=1.0
forgetting factor
support radius
0.19 0.195 0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25 0.3 0.4 0.5 0.6 0.8 1
2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
EnSRF, obs. error=0.5
forgetting factor
support radius
0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.15 0.2 0.25 0.3 0.35 0.4 0.5
2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
EnSRF, obs. error=0.1
forgetting factor
support radius
0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1
2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
LESTKF, obs. error=1.0
forgetting factor
support radius
0.19 0.195 0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25 0.3 0.4 0.5 0.6 0.8 1
2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
LESTKF, obs. error=0.5
forgetting factor
support radius
0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.15 0.2 0.25 0.3 0.35 0.4 0.5
2 6 10 14 18 22 26 30 34
0.9 0.92 0.94 0.96 0.98 1
LESTKF, obs. error=0.1
forgetting factor
support radius
0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024 0.025 0.03 0.04 0.06 0.08 0.1
Time-mean RMS errors
The figure compares mean RMS errors for different localiza- tion support radii and forgetting factors. The error in the ob- servations is reduced from left to right. In white fields, the filter diverges.
For decreasing observation error, the difference in RMS er- rors between both filters increases. The errors of the EnSRF are larger than for the LESTKF. Also, the region of filter con- vergence is smaller for the EnSRF than for LESTKF.
To get an idea about the reason for the different performance, we focus on the optimal configuration for LESTKF: observa- tion error 0.1; support radius: 26 grid points; forgetting factor 0.96 (see circle).
0 10 20 30 40
0.5 1 1.5 2 2.5 3 3.5
Observation error = 1.0
number of assimilated observations
RMS error
EnSRF true EnSRF estimate LESTKF true LESTKF est.
0 10 20 30 40
0 1 2 3 4 5
Observation error = 0.5
number of assimilated observations
RMS error
0 10 20 30 40
0 2 4 6 8 10 12 14
Observation error = 0.1
number of assimilated observations
RMS error
RMS errors at first analysis time
We perform a series of experiments varying the number of ob- servations. The EnSRF realizes each of the state estimates in its analysis sequence. The true and estimated errors at the first analysis time are different for both filters.
For the two smaller observation errors, the intermediate state estimates can have a larger RMS error than without assimilat- ing any data. After all observations are assimilated, the state estimate from EnSRF is still worse than for LESTKF.
Thus, for a strong impact of the assimilation, the sequential observation processing in the local EnSRF can have a large deteriorating influence.
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
20 obs.
observation true state EnSRF LESTKF
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
24 obs.
observation true state EnSRF LESTKF
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
28 obs.
observation true state EnSRF LESTKF
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
40 obs.
observation true state EnSRF LESTKF
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
36 obs.
observation true state EnSRF LESTKF
0 10 20 30 40
−40
−30
−20
−10 0 10 20 30 40
grid point
state
32 obs.
observation true state EnSRF LESTKF
State estimates
for different numbers of observations
To illustrate the reason for the large RMS errors for obser- vation error 0.1, the state estimates are shown when differ- ent numbers of observations are assimilated. In case of the EnSRF these estimates are realized in the assimilation se- quence.
20 observations: The wave is well estimated where obser- vations are present. In the other half of the domain, the esti- mates of both filters are similar.
24-32 observations: The estimated wave from EnSRF shows erroneously strong oscillations in the region where no observation are assimilated yet. In contrast, the estimate of the LESTKF is still of the correct magnitude.
36-40 observations: The strong oscillations are finally damped. The final estimate of the EnSRF show larger errors than that of the LESTKF.
[1] Whitaker, J. S. and T. M. Hamill (2002).
Ensemble data assimilation without per- turbed observations. Mon. Wea. Rev.
130, 1913–1927
[2] Nerger, L., T. Janji´c, J. Schr ¨oter, J., and W. Hiller (2012). A unification of en- semble square root Kalman filters. Mon.
Wea. Rev. 140, 2335–2345
[3] Nerger, L., T. Janji´c, J. Schr ¨oter, J., and W. Hiller (2012). A regulated localiza- tion scheme for ensemble-based Kalman filters. Q. J. Roy. Meteorol. Soc. 138, 802–812
[4] L. Nerger, S. Danilov, W. Hiller, and J. Schr ¨oter (2006). Using sea-level data to constrain a finite-element primitive- equation ocean model with a local SEIK filter. Ocean Dynamics 56: 634–649
[5] Nerger, L., W. Hiller, and J. Schr ¨oter (2005).
PDAF - The Parallel Data Assimilation Framework:
Experiences with Kalman Filtering, in Use of High Performance Computing in Meteorology - Proceed- ings of the 11th ECMWF Workshop / Eds. W. Zwiefl- hofer, G. Mozdzynski. World Scientific, pp. 63–83
[6] Nerger, L. and W. Hiller (2012). Software for Ensemble-based Data Assimilation Sys- tems – Implementation Strategies and Scal- ability. Computers & Geosciences. 55, 110–
118
