Data Assimilation- A07. Serial EnSRF 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: serial EnSRF‣to introduce serial EnSRF‣ Training: Lorenz 96‣to implement serial EnSRF 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)

Serial EnSRF

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 SRF (EnSRF)
EnSRF assumes the following ensemble update

This equation was solved by Andrew (1968) such thatp×p matrix

p×p matrix

However, solving a pxp matrix is computationally expensive…

Serial EnSRF assimilates observation serially (i.e., p=1)

Whitaker &Hamill (2002)

scalar since now assimilating one observation

Serial EnSRF Algorithm
next time stepensemble
Prediction(t-1  t)

Analysis
Update Analysis Ensemble
obs loop for i=1,…,p (for assimilating ith observation)(1) Compute Kalman Gain by
for i=1,…,mBackground(2) Mean update(3) Perturbation update
Observations

whereSerial EnSRF Algorithm
FCST

Serial Data Assimilation (i=1.,,,p)
DA of 1st obs

DA of 2nd obs

FCST

DA of 3rd obs

updated3rdobsupdated1stobsupdatedRepeat for
all obs

localization
function

2ndobslocalization
function
localization
function

…

…

…

…

…

…

…

Point!: Regarding the updated state as the background for assimilating 2nd obs.

Serial Assimilation

・To assimilate obs sequentially・To update Xb (& HXb) within
localization radius.

・: model grid points
×: observing station

Limitations for parallelization

Basic Task 5

Basic Task 5Techniques for serial EnSRF
Gaspari Cohn Function

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

Gaspari and Cohn (1999)

Localization

Because of assimilating observation serially, analysis equation isKalman Gain () ith obs (scalar)

obs ope for ith obs

Localize k depending on distance b/w grid and obs using the GC function.
Inflation

Note: this inflation should be applied
only once at assimilating 1st obs!
(i.e. no inflation for assimilating 2nd-pth obs)

Y axis: inflation factor (δ)

Kotsuki (m=10, p=40)

Y axis: inflation factor (δ)

Analysis RMSE (Serial EnSRF)

Kotsuki (m=8, p=20)

Miyoshi (2005)

Miyoshi (2006)
X axis: localization length scale (σ)

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

homogeneous
dense

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/