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solid: 40～50° N

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9. Impact of Time Sale on Ω

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When all EMs are completely uncorrelated,

When all EMs produce the exact same time series,

Predictability of Ensemble Weather Forecasts Predictability of Ensemble Weather Forecasts with a Newly Derived Similarity Indexwith a Newly Derived Similarity Index

2 2b m

klR

Tomohito YAMADATomohito YAMADA11)), , Shinjiro KANAEShinjiro KANAE22)), , Taikan OKITaikan OKI11)), and, and Randal D. KOSTERRandal D. KOSTER33))

1) Institute of Industrial Science, The University of Tokyo1) Institute of Industrial Science, The University of Tokyo2) Research Institute for Humanity and Nature2) Research Institute for Humanity and Nature3) NASA Goddard Space Flight Center3) NASA Goddard Space Flight Center

1. Introduction

2. Existing Evaluation Method of Ensemble Forecast

3. Similarity Index Ω

Fig. 3-1. 2 types of variances for Ω calculation.

Tomohito Yamada (tomohito@iis.u-tokyo.ac.jp)

Discovery of atmospheric chaotic behavior (Lorenz 1963).

Subtle perturbation of initial condition or computational error grows large discrepancy of prediction (Fig.1-2). Chaotic behavior constrains the use of individual forecasts of instantaneous weather patterns to about 10 days (Lorenz 1982).

Ensemble forecast that includes several initial conditions for which values have been slightly perturbed can gauge and reduce the numerical errors that arise from chaotic behavior.

(a). Anomaly correlation Coeff.

(b). Standard deviation

Evaluate time series of anomalycorrelation coeff. among ensemblemembers (EMs).

Evaluate time series of standarddeviation among EMs.

m: ensemble members, n: time periods

2 2

1

1 ( )n

b jj

b xn

2 2

1 1

1 ( )m n

iji j

x xmn

2 2

2( 1)bm

m

1

1 m

j iji

b xm

1

1 n

jj

x bn

(1) (2)

(3)

Ω has been introduced as a similarity index (Koster et al. 2000)

2 2b

0

Koster et al (2000) and some Ω related studies have not revealed the detail mathematical structure of Ω.

(4)

4. Mathematical Structure of Similarity Index Ω Derived Ω is expressed as

Ω can be expressed as

(5)

(6)

2 2 2i ii amp mean

Here in Eq.(6), Eq.(7) can be written as

(7)

where, k, l: arbitrary number of EM: anomaly correlation coefficient

1

1 1

2( 1)

m m

klk l k

ACCC Rm m

1

1 1

2( 1)

k lm m

amp amp

k l k

AVRm m

(A) (B)

Mathematical characteristics of Ω, which are related to similarity in both the phase (A) and shape (B) of the ensemble time series, show the index to be more robust than other statistical indices. Ω is the index to quantify that all EMs have identical time series or not.To clarify the impact of phase and shape similarity on Ω, we introduce 2 new statistical indices, shown in below.

(A) Average value of Anomaly Correlation Coefficient

(B) Average value of Variance Ratio

5. Experimental Design

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Black: 1day (n=1)Blue: 3days (n=12)Green: 5days (n=20)Red: 7days(n=28)

6. Grid Scale

7. Zonal Scale

1

1 1

2( 1)

k lm m

amp ampkl

k l k

Rm m

•Climate Model ： CCSR/NIES AGCM5.6•Period ： Dec. 1994 – Jan. 1995•SST ： AMIP2•Ensemble member ： 16 •Initial Conditions: Every 1 hour data on December 1st.　　　　　　

Fig. 5-1. 16 time series of temperatureat 500hPa height (57°N, 135°E).

Fig. 5-2. Evaluation methodfor predictability using Ω

(A)

(B)

(A)(B)

Decrease of phase similarity mainly induces decrease of Ω.

Decrease of both phase and shape similarity induces decrease of Ω.

8. Global Scale

10. Summary

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Fig. 1-1. Butterfly attractor of Lorenz model.

Individual method with (a) or (b) can only evaluate one aspect of predictability. Therefore, the predictability with each method is not practical. However, unified or comprehensive method has not been suggested.

(Mean values or amplitudes aresame among EMs.)

(There is no phase discrepancyamong EMs.)

※ Murphy, 1988

Fig 2-1. Time series of predictability of ensemble forecast.

There are mainly 2 types for the decrease of predictability.

In cases of long time scale for recognition (5, 7 days),Ω shows large increase on around 22nd and 32nd . Time scale of 5 and 7 days includes low-frequencyatmospheric variation in the mathematical concept ofsimilarity.

Mathematical structure of Ω was revealed. Ω is the average value of anomaly correlationcoefficient (ACCC) among ensemble members weighted by average value of variance ratio (AVR).

Ω is the statistical index to show both phase (correlation) and shape (mean value and amplitude) similarities.

We proposed a new method to evaluate predictability of ensemble weather forecast using Ω. 2 types of predictability, such phase and shape was introduced from the mathematically derived Ω.

We introduced the low-frequency atmospheric variation in the mathematical concept ofsimilarity by changing the time scale for recognition.

Fig. 1-2. Chaotic behavior of atmosphere in ensemble simulations.

According to Fig. 3-1 (a), According to Fig. 3-1 (b),

When the time scale for recognition is small (Black line), Ω rapidly loses its value. This shows the difficulty of daily weather forecast.

Fig. 8-1. Global distributions of Ωon Dec. 10th. (Similarity predictability)

•　 At middle latitude, the predictability is the highest for Ω, ACCC, and AVR.•　 Global distributions of Ω is similar as ACCC.

Fig. 7-1. Predictability of temperature at500hPa height in 3 latitudes.

AVR becomes almost stable after decrease of predictability. This is the climatologically value of AVR after losing the impact of initial condition.

Fig. 9-1. Time series of Ω of temperature at 500hPaover a grid cell for 4 cases of time periods.

Predictability： Large

(a)

(b).

Fig. 4-1. 2 types of similarity (phase and shape) in Ω.(A): Phase (correlation), (B): Mean value and Amplitude

Fig. 8-2. Global distributions of ACCCon Dec. 10th. (Phase predictability)

Fig. 8-3. Global distributions of AVRon Dec. 10th. (Shape predictability)

(A) (B)

※ 0.05 is 92% significance level.

(8)

(9)

Fig. 7-2. Reliable forecast day in 3 latitudes.

Reliable forecast day is about10 to 13 days in the mathematicalconcept of similarity.

Ω is smaller than ACCC for all latitudes. This is caused by the increase of shape discrepancy among EMs.

1 D

ay

Ω: Similarity predictability, ACCC: Phase predictability, AVR: Shape Predictability

※ Yamada et al in preparation

※ Koster et al 2002