Data Assimilation- A09. Innovation Statistics 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: innovation statistics‣to introduce innovation statistics‣to understand adaptive inflation‣ Training: Lorenz 96‣to implement adaptive inflation into L96

Y axis: inflation factor (δ)

Motivation
How can we optimize
these parameters ?

X axis: localization length scale (σ)

Innovation statistics
b: backgrounda: analysis
o: observation
Desroziers’ innovation statistics (Desroziers et al. 2005)

Innovation Statistics(Derivations)Innovation Statistics
 : Statistical Expectations

Definition，，Derivation from Kalman Gain

matrices used in DA(usually imperfect)Derivations௢ି௕௢ି௕ ்௢ି௕௢ି௕ ்௔ି௕௢ି௕ ்௔ି௕௢ି௕ ்௕

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Derivations (cont’d)௔ି௕௢ି௔ ்்௔ି௕௢ି௔ ்௢ି௕௢ି௔்௘ିଵ்ିଵ ௢ି௕ ்்்்்்்ିଵିଵ்ିଵିଵ ௢ି௕Innovation Statistics
Innovation Statistics

Their Relationship

NOTE: We need to consider
DA uses imperfect B & R
in the real-world applications: Truth: DA (maybe imperfect): Estimatedo, R: obs
b, B: backgrounda, A: analysis

Geometric Interpretations ofData Assimilation
For simplicity, following discussion assumes a scalar problem

Pythagorean theorem
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Innovation Statistics(1) OMB-OMB Statistics(2) AMB-OMB Statistics(3) OMA-OMB Statisticsanalysis minimizesits variancecross terms correspond to right angle:(4) AMB-OMA Statistics௕௢ ்௢ ்௕+Kalman Gain
(1) Kalman gain gives weights

Pi.e.,(2) Also, Kalman gain isi.e.,

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Namely, Kalman gain is uncertainty ofposterior error cov. w.r.t. obs. error cov.
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Analysis Error Covariance(1) Analysis minimizes variance of estimation
 verticali.e.,(2) Also, analysis isi.e.,Analysis State(1) a standard update equation(2) Also, analysis is given byReviewమమమమమమమఈఉఈమ ାఉమ

Analysis State (cont’d)Geometric Interpretations
Focusing on the weights gives:

Focusing on the AN cov. gives:

To understand DA via two ways
(1) Kalman gain gives the optimal weight
From the view point of analysis states(2) Kalman gain is the analysis error cov. normalized by R
From the view point of analysis uncertainty

To discuss information (or entropy), (2) is more important
For example, DFS (Degrees of Freedom for the Signal) is given by:i.e., DFS is obs-space A normalized by R

DFS=0 : no information from obs, DFS=1 : all information from obs

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