Data Assimilation- A04. 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: Kalman Filter (KF)

‣to introduce background error covariance‣to introduce analysis error covariance‣to introduce Kalman gain

‣ Training: Lorenz 96‣to develop Tangent Linear Model (TLM)‣to implement Kalman filter into L96

Review:

Minimum Variance Estimation(復習: 最小分散推定)

Minimum Variance Estimation: truth: random error

forecast
observation<・>: expectation

Assumption (1) : unbiased error

Assumption (2) : uncorrelated error

Minimum Variance Estimation
forecast(1) unbias

observation(2) uncorr.
& minimize variance of analysis (a)definition of variance: standard deviation: varianceMinimum Variance Estimation
forecast(1) unbias

observation(2) uncorr.

weighted average by variance (σ2; =accuracy)

first guessincrementKalman Filter

Exercise‣ to introduce Kalman gain w/ following
EquationsAssumption & Definition
Assumption (1) : unbiased error

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

εerroryobservationM( )nonlinear model

M

Jacobian of M

∈ℝ ×

K

Kalman gain

∈ℝ ×

H( )

nonlin. obs. operator

H

Jacobian of H∈ℝ ×

Pmodel error covariance∈ℝ ×

Robs. error covariance

∈ℝ ×

n

# of model vars.

p# of observationsm# of ensembletru

truthbbackgroundaanalysisttimeoobservation<>

expectation∈ℝ∈ℝ

Error Covariance
Variance, Standard Deviation
Covariance
Correlation
Error Covarianceerrorvarianceerror
covariance
P and R are symmetric matrices by definition.

symmetric

Linear Approximations
Tayler expansion (scalar)

Tangent Linear Model (TLM)

whereattractorof analysis

t-1t

attractorof truthtime

M is the linearized modelfor propagating errors.
(M cannot be used to x)

Forecast Error CovarianceState Prediction

suppose that M=Mtru
Error Prediction

Error Covariance Prediction

Tangent Linear Model (Jacobian of M)

Analysis Error Covariance

No correlation b/w

andKalman Gain
‣ KF minimizes analysis error variance to find K that minimizes trace(Pa)Eq. (１)

Eq. (2)

Analysis Error Covariance
SubstituteintowhereKalman Filter
Prediction (state))

nonlinear model

Prediction (error covariance)

Prior Error Cov. Pb(flow-dependent!)

linearized model

Kalman gain

Analysis (state)

Analysis (error covariance)
obs error cov: RKalman Filter Algorithm
next time step

Prediction(t-1  t)

Background

Observations

AnalysisSequential Kalman Filter
Prediction

Assimilation)true state (unknow)
t: forecast: observation

t+1: analysis (DA)time: error cov. (accuracy)

Tangent Linear Model (Numerical)
Require: to get M such thatjth column
jth column of M describes
how error of jth variable
propagatesjthvariable

where

jth elementof M

jthvariable

computable‣ repeat these steps for j=1,…,n (e.g. n=40 for L96)

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 3

Basic Task 3This means the experiments assume R to be I (i.e., identity matrix)

OSSE: Observing Sys. Sim. Experimentalso known as Idealized Twin Experiment
forecast③ independent
simulation④ DataAssimilation

analysis

observation

Simulation Model① nature run② to generate obs
w/ Gaussian noise
True state⑤ validation(e.g. RMSE)

Basic Task 4

Basic Task 4

An additional treatment
will be needed.
Let’s think about by your self.

KF (also known as Extended KF)

Initial Condition‣‣‣  randomly chosen from nature run in spin up‣  should be large (e.g.

)

Variance Inflation (KF)
Empirical treatment for variance underestimation due to(1) limited ensemble size(2) model nonlinearity(3) model imperfection
inflation factor (a tuning parameter)no inflation

10% inflation

First Variable X(1) as a func. of timeAnalysis RMSE

Ave RMSE (from 10th day to 300th day)δ=0.00  RMSE=3.970
δ=0.03  RMSE=3.970
δ=0.05  RMSE=0.204
δ=0.10  RMSE=0.211

Sensitivity to Infl. Factor
too large inflation
degrades gradually

too small inflation
causes filter divergenceFCST Error CovarianceTips

(1) Splitting M into sub Ms(a) Construct M once(b) Construct M separately=00.05

time00.05

splitting M
improves slightlytime(2) Alternative Way of M
(1)(2)(2) – (1) & ignore second order terms gives=

(2) Alternative Way of M
For example=M(2) KF Comparison of M
Numerical Method

Mathematical Approx.

Numerical method
shows better RMSE
slightly

Splitting M (i.e., =)is necessary for this method to include
impacts beyond neighboring grids.

We would be appreciated if you obtained different results

Mao追試・なんでインフレーションが要らない?
 非対角成分が理由?

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