Data Assimilation- A05. 3DVAR and OI 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 goals‣ Lecture‣ what is the 3D-Var?
‣ what is the cost function?
‣ maximum likelihood vs. minimum variance‣ how can we get a reasonable B?

‣ Training Course‣ to implement 3DVAR‣ hints to develop KF‣ some tips for KF

Review:Max. Likelihood Estimation(復習: 最尤推定)Maximum Likelihood Estimation
forecast(1) unbias

observation(2) uncorr.
Likelihood

Prior (uniform, i.e., no prior info)

Bayesian EstimatesPosterior

constant (since they are given)to maximize likelihood

Maximum Likelihood Estimation
forecast(1) unbias

observation(2) uncorr.

Suppose x1 & x2 follow
Gaussian PDF N(x, σ)Maximum Likelihood Estimation
forecast(1) unbias

observation(2) uncorr.

analysis of maximum likelihood estimates

Assumption & Definition
Assumption (1) : unbiased error

Assumption (2) : uncorrelated error
since background and obs errors are independentx

model state

εerroryobservationM( )nonlinear modelM

Jacobian of M

∈ℝ ×

K

Kalman gain

∈ℝ ×

H( )

nonlin. obs. operator

HJacobian of H∈ℝ ×

Pmodel error covariance∈ℝ ×

Robs. error covariance

∈ℝ ×

n

# of model vars.

p# of observationsm# of ensemble

tru

truth

b

backgroundaanalysisttimeoobservation<>

expectation∈ℝ∈ℝ

Multidimensional Extension
ScalarSuppose x1 & x2 follow
Gaussian PDF N(x, σ)Multi-dims.
Supposefollow N(x, B)

Supposefollow N(H(x), R)

Joint Probability=+

Variational DA=+==
gradient++&+=d: innovation, departurenecessarycondition

B, Pb: background error covariance
A, Pa: analysis error covarianceVariational DA (cont’d)
Proof of Kalman Gain

Proof of Analysis Error Cov.

Important Equations
Kalman Gain

Analysis Error Covariance

Analysis Update Equation

3DVAR
next time stepPrediction(t-1  t)

Optimal Interpolation
OI only assimilates retrieved vars
(can consider only H)

Background

Observations

Analysis

Training Course

DA Study w/ 40-variable Lorenz-96
Lorenz-96 model (Lorenz 1996)

Advection term

For j=1,…,N, Xj=Xj+N
Dissipation term Forcing termText Books① Training Description

pswd: ceres‣ https://kotsuki-lab.com/internal-pages/

Basic Task 5

Basic Task 5

3DVAR (Full Observations)when B is diagonal matrix.

 b11



0


B
bii

0


bnn 


Time-mean RMSE
minimized (0.3962) whenAve Bii = 0.225(analysis RMSE)

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

homogeneousdense

Full Observations

n=4020 Obs (Case1)

１…

n=40…

…

p=20p=40n=40…
20 Obs (Case2)

…

p=20…

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

homogeneousdense

Case 1Case 2

Case 1Case 2(analysis RMSE)

Hints to Develop
KF & 3DVAR

(1) Steps for KF & 3DVAR
KF
State Prediction
Background Error Cov.

3DVAR)B (static)

Kalman Gain
State Analysis
Analysis Error Cov.

Starting with 3DVAR
is a good strategy,
followed by KF

3DVAR

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