Predictability of Ensemble Weather Forecasts with a Newly Derived Similarity Index

1
Predictability of Ensemble Weather Forecasts
with a Newly Derived Similarity Index
Tomohito Yamada (tomohito_at_iis.u-tokyo.ac.jp)
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
6. Grid Scale
1. Introduction
O Similarity predictability, ACCC Phase
predictability, AVR Shape Predictability
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)
Fig. 1-1. Butterfly attractor of Lorenz model.
PredictabilityLarge
Fig. 1-2. Chaotic behavior of atmosphere in
ensemble simulations.
? 0.05 is 92 significance level.
2. Existing Evaluation Method of Ensemble Forecast
(B)
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.
? Murphy, 1988
(a). Anomaly correlation Coeff.
(b).
Evaluate time series of anomalycorrelation
coeff. among ensemblemembers (EMs).
(a)
(b). Standard deviation
Evaluate time series of standarddeviation among
EMs.
Fig 2-1. Time series of predictability of
ensemble forecast.
There are mainly 2 types for the decrease of
predictability.
(A)
Decrease of phase similarity mainly induces
decrease of O.
3. Similarity Index O
(B)
Decrease of both phase and shape similarity
induces decrease of O.
O has been introduced as a similarity index
(Koster et al. 2000)
(1)
(2)
7. Zonal Scale
m ensemble members, n time periods
Reliable forecast day is about10 to 13 days in
the mathematicalconcept of similarity. O is
smaller than ACCC for all latitudes. This is
caused by the increase of shape discrepancy among
EMs.
According to Fig. 3-1 (a),
According to Fig. 3-1 (b),
Day
(3)
(4)
Fig. 3-1. 2 types of variances for O calculation.
O can be expressed as
Fig. 7-2. Reliable forecast day in 3 latitudes.
When all EMs produce the exact same time series,
When all EMs are completely uncorrelated,
(5)
AVR becomes almost stable after decrease of
predictability. This is the climatologically
value of AVR after losing the impact of initial
condition.
? Koster et al 2002
Fig. 7-1. Predictability of temperature at500hPa
height in 3 latitudes.
Koster et al (2000) and some O related studies
have not revealed the detail mathematical
structure of O.
8. Global Scale
4. Mathematical Structure of Similarity Index O
(A)
(B)
Derived O is expressed as
(6)
(A)
(B)
Fig. 8-1. Global distributions of Oon Dec. 10th.
(Similarity predictability)
Fig. 8-2. Global distributions of ACCCon Dec.
10th. (Phase predictability)
Fig. 8-3. Global distributions of AVRon Dec.
10th. (Shape predictability)
? Yamada et al in preparation
where, k, l arbitrary number of EM

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

anomaly correlation coefficient
Here in Eq.(6), Eq.(7) can be written as
9. Impact of Time Sale on O
Fig. 4-1. 2 types of similarity (phase and shape)
in O.(A) Phase (correlation), (B) Mean value
and Amplitude
(7)
When the time scale for recognition is small
(Black line), O rapidly loses its value. This
shows the difficulty of daily weather forecast.
Mathematical characteristics of O, 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. O is the index to quantify that all EMs
have identical time series or not.
In cases of long time scale for recognition (5, 7
days),O shows large increase on around 22nd and
32nd .
To clarify the impact of phase and shape
similarity on O, we introduce 2 new statistical
indices, shown in below.
Time scale of 5 and 7 days includes
low-frequencyatmospheric variation in the
mathematical concept ofsimilarity.
(A) Average value of Anomaly Correlation
Coefficient
Fig. 9-1. Time series of O of temperature at
500hPaover a grid cell for 4 cases of time
periods.
(8)
(Mean values or amplitudes aresame among EMs.)
10. Summary
(B) Average value of Variance Ratio
Mathematical structure of O was revealed. O is
the average value of anomaly correlation coefficie
nt (ACCC) among ensemble members weighted by
average value of variance ratio (AVR). O 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 O. 2 types of
predictability, such phase and shape was
introduced from the mathematically derived O.
We introduced the low-frequency atmospheric
variation in the mathematical concept
of similarity by changing the time scale for
recognition.
(There is no phase discrepancyamong EMs.)
(9)
5. Experimental Design

• 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 (57N, 135E).
Fig. 5-2. Evaluation methodfor predictability
using O