Data Assimilation- A08. LETKF 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: LETKF‣to introduce ETKF
‣to introduce LETKF‣ Training: Lorenz 96‣to implement LETKF into L96

Ensemble Kalman Filter(EnKF)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)

Square Root Filter (SRF)
SRF assumes the following update w/o adding perturbation in obs.
×: Ensemble Ptb. Transform Matrix

and compute W that satisfies

However, SRF cannot determine
deterministically.
For example, for that satisfies,a new matrix
can be also a ptb. transform matrix since=Question: how can we determine W?

Ensemble Transform
Kalman Filter

Data Assimilation
model space

projection

observation spacestandard Kalman filter

Kalman GainAnalysis Error Covariance

Analysis Update Equation

See Lecture 5 on 3DVAR for more details of equations

Ensemble Transform KF
model space

projection

observation spacestandard Kalman filter
ETKF

ensembletransformation

ensemble space

projection

Bishop et al. (2001; MWR)
Figure adopted from Kotsuki et al. (2020; QJRMS)ETKF (Ens. Trans. KF)
ETKF considers (m-1)-dimensional subspace (ensemble space) spanned by Zb① Background Error Cov.

② Analysis Error Cov.

③ Analysis Increment④ Analysis Equation

Eigenvalue Decomposition
Analysis Equations
meanperturbation

×: weight vector: weight matrix

Eigenvalue Decomposition=: eigenvectors
×: eigenvalues (diagonal)×

Hunt et al. (2007)’s approach requiring O(m3)Analysis Update Equation

whereis always m×because of adding I.
1≡[1,1,…,1] : row vector with ones.
×: transform matrix of the ETKF

ETKF  LETKF (Local ETKF)
localization

Adopted from
Kotsuki et al. (2020; QJRMS): observations

LETKF (i)LETKF ( j): analysis grid pointsof the LETKF‣ The LETKF computes the transform matrix T at
every model grid point by assimilating surrounding
obs within a prescribed localization cutoff radius.
‣ And the LETKF updates analysis ensemble at everymodel grid point

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

R-localization (to reduce impacts of obs far from anl. grid point)
Localized obs error varianceof ith observation
Localized R is used forEigenvalue decomposition

Assimilating surrounding local obs.
to update centered grid x1 (★)=★

mean update equationKekem and Leendert (2018)

Symmetric Square RootSRF including ETKF assumes the following update equation.
We cannot determine W uniquely/The symmetric square root matriccan be determined deterministically!

Symmetric Square Root
whereImportance‣ Since the LETKF generates analysis ens. perturbations as
at all model grid points independently, the smoothtransition ofin space is essential not to produce imbalanced
analysis ensemble. The symmetric ofensures aspatially smooth transition of
from one grid point tothe next (Hunt et al. 2007).
‣ The symmetric square root matrix also ensures the analysis
ensemble perturbations are consistent with the backgroundensemble perturbations because it minimizes the mean squaredistance b/wand I.

Kotsuki and Bishop (2022)

Characteristics
Eigenvalue decomposition

sum of ptb. = 0

Posterior perturbation
is given bylinear combination of prior members.

Characteristics (2)Eigenvalues & Eigenvectors
This equation means that if we add kI for matrix A,(1) eigenvalue increases λ λ+k, and(2) eigenvectors are unchanged.
(a) Regularization :: adding λincI for a matrix to reduce condition number (CN), and required condition number is(b) Large-ensemble ETKF

We can first solve eigenvalue decomposition ofand add m−1 to get eigenvalues of,. Eigenvectors are unchanged.

LETKF
next time stepensemble
Prediction(t-1  t)
for i=1,…,mAnalysis(1) Preparation Step
(2) Local Analysis
grid loop for i=1,…,n (for updating ith grid points)(2.1) Eigenvalue Decomposition=Background(2.2) Update Ensemble
Observations/(3) Collect analysis ens of all model grid points
to obtain
for subsequent forecasts

LETKF・: model grid points
×: observing station

LETKF can be parallelized for model grid points

Basic Task 5

Basic Task 5

Techniques for LETKF: distance b/w grids: localization length scale

Gaussian Function

tuning parameter

localized obs error variance of ith observation
Localization

where EVD is solved byInflation=

Y axis: inflation factor (δ)

Kotsuki (m=8, p=20)

Y axis: inflation factor (δ)

Analysis RMSE (Serial EnSRF vs. LETKF)

Kotsuki (m=8, p=20)

Serial EnsRF

LETKF
X axis: localization length scale (σ)

Miyoshi (2006)
Serial EnSRF

Miyoshi (2006)
LETKF

Sensitivity to Obs. Network
Case 1: Num. Obs. = X
Case 2: Num. Obs. = X

homogeneousdense

Case 1Case 2

Case 1Case 2(analysis RMSE)

Kotsuki et al. (2017)
Analysis RMSE with 40 observations w/ L96 (w/ best loc. scale)

LETKF

Serial EnSRF

Kotsuki, S., Greybush, S., and Miyoshi, T. (2017):
Can we optimize the assimilation order in the serial ensemble Kalman filter?
A study with the Lorenz-96 model. Mon. Wea. Rev., 145, 4977-4995.

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