# Presentation: 875387

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

_**9. Impact of Time Sale on Ω**_

- When all EMs are completely uncorrelated,

- When all EMs produce the exact same time series,

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

- Tomohito YAMADA1), Shinjiro KANAE2),  Taikan OKI1), and Randal D. KOSTER3)1) Institute of Industrial Science, The University of Tokyo2) Research Institute for Humanity and Nature3) 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

**(1)**

**(2)**

**(3)**

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

- 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)**

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

**(7)**

- where, *k*, *l*: arbitrary number of EM

- : anomaly correlation coefficient

**(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**_

_**6. Grid Scale**_

_**7. Zonal Scale**_

- 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**_

- 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 correlation
- coefficient (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 of
- similarity 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.

**Day**

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

- ※ Yamada et al in preparation

- ※ Koster et al 2002