Data Assimilation- A06. Ensemble Kalman Filter Shunji Kotsuki
Environmental Prediction Science Laboratory
Center for Environmental Remote Sensing (CEReS), Chiba University(shunji.kotsuki@chiba-u.jp)

DA Lectures A (Basic Course)‣ (1) Introduction and NWP‣ (2) Deterministic Chaos and Lorenz-96 model‣ (3) A toy model and Bayesian estimation‣ (4) Kalman Filter (KF)
‣ (5) 3D Variational Method (3DVAR)‣ (6) Ensemble Kalman Filter (PO method)‣ (7) Serial Ens. Square Root Filter (Serial EnSRF)‣ (8) Local Ens. Transform Kalman Filter (LETKF)‣ (9) Innovation Statistics‣ (10) Adaptive Inflations‣ (11) 4D Variational Method (4DVAR)Today’s Goal
‣ Lecture: Ensemble Kalman Filter‣to introduce EnKF
‣to understand PO method‣ Training: Lorenz 96‣to implement PO method‣to implement localization

Ensemble Kalman Filter(EnKF)

Why EnKF?
Kalman Filternn

Background error covariance cannot be
stored on RAMfor high dimensional models
such as NWP (n~O(1012~1015)
Ex) if n=106  1012 x 8 byte = 8 TB

Ensemble Kalman Filtermn

An approximation of error covariancewith ensemble perturbation matrix

m: ensemble size

Conceptual Images
Kalman Filter

obs error cov: REnsemble Kalman Filter

obs error cov: REnsemble Forecasts
Analysis Ensemble=Ensemble Forecasts=for i = 1,…,mEnsemble Mean

Ensemble Perturbation

δ represents ensemble perturbation===

Approximation of Pb
Error Propagation in Ensemble Forecasts

Ensemble forecasts can be used for approximating
propagation of error covariance !

why m-1?
‣ Pls. study discussions on “unbiased variance”

Ensemble Kalman Filter
Ensemble Perturbation in Observation Spacei.e. no H is needed
Error Covariance Approximation

Kalman Gain

We can choose inverse computation
depending on p and m.

pxp※ usually R is diagonal (i.e., no obs error corr.)

𝟏

𝟏

𝟏

𝟏

𝟏

𝟏mxm

KF

EnKF

Prediction (state)

Ensemble Prediction (state))

for i = 1,…,mPrediction of Error Cov. (explicitly)

Prediction of Error Covariance (implicitly)

Kalman Gain

Kalman Gain

Analysis (state)

Analysis (state)

Analysis Error Covariance

Analysis Error Covariance(1) Stochastic: PO method(2) Deterministic: Square Root Filter (SRF)(e.g., serial EnSRF, EAKF, LETKF)

PO Method (stochastic)
Analysis of Ensemble

randomly drawn perturbation
 perturbed observation

Why do we need perturbation?
if w/o perturbation (to take ave. from both sides)

Analysis error covarianceis underestimatedif without perturbation!

Burgers et al. (1998)

Analysis error covarianceshould be (cf. 4th lecture)EnKF (PO) Algorithm
next time step
ensemble
Prediction(t-1  t)
for i=1,…,m
Analysis
BackgroundObservations

Compute Kalman Gain

Update Analysis Ensemble
member loop for i=1,…,m (for updating ith member)

Ensemble Kalman Filter
Forecast error
covariance)
Ensemble FCST

Analysis error
covariance=Ensemble Kalman Filter)

Forecast error
covariance

Analysis error
covarianceObservation error
covariance R

Ensemble Kalman Filter)

Forecast error
covariance

Ensembleupdate

Analysis error
covarianceObservation error
covariance REnsemble Kalman Filter)

Forecast error
covariance

Ensemble
updateAnalysiserror
covariance

Forecast error
covarianceBasic Task 5

Basic Task 5

EnKF (PO) w/o Localization

m=10

m=100

m=180

m=500

no inflation is used here

Treatments(1) Perturbed Observations

Random number should be different
for each member

for i=1,…,m
this error should be modified so that(2) Variance Inflation

Specifically, generate random numbers 𝛆′and compute their average 𝛆 = ∑Then, perturb with 𝛆(3) Localization

()

=𝛆′()

()

𝑓𝑜𝑟 𝑖 = 1, , , 𝑚()𝛆′.

− 𝛆 that satisfies ∑𝛆()=0Assimilating surrounding local obs.
to update centered grid x1 (★)- to limit impacts of obs far from analysis grid points
for erroneous error “co”variance due to sampling errors- several localizations have been proposed- K localization (in PO or serial EnKF
- R localization (in LETKF)- B localization (usually complex for high-dim modes)★

Kekem and Leendert (2018)

covariance localization (EnKF)
Empirical treatment for(1) reducing sampling noise(2) increasing the rank◦: Schur productlocalization

Sampled error covariance(ensemble approximation)

Error Cov. w/ LocalizationLocalization Function

Localization Function
Gaspari Cohn Function

tuning parameter: distance b/w grids: localization length scale

Gaspari and Cohn (1999)
usually used in PO and serial EnSRF

Gaussian Function: distance b/w grids: localization length scaletuning parameter

usually used in LETKF

Localization Function
Localization Function

Localization Matrix (Gaussian Function)

Localization in PO method
Kalman Gain

: Localization Matrix:Shur productalso known as Hadamard productor, element-wise productImpacts of Localization (PO)m=8, σ=1, δ=10%Inflation :Localization should be applied for all “co”varianceEnKF (PO) w/ Localization

ensemble m = 40
localization σ = 40
inflation α =0.05

ensemble m = 40
localization σ = 10
inflation α =0.05

ensemble m = 40
localization σ = 3
inflation α =0.05

ensemble m = 40
localization σ = 1
inflation α =0.05

Thank you for your attention!
Presented by Shunji Kotsuki(shunji.kotsuki@chiba-u.jp)
Further information is available athttps://kotsuki-lab.com/