The ECMWF Ensemble Prediction System

Transcription

The ECMWF Ensemble Prediction System
Design, Diagnosis and Developments
Martin Leutbecher
European Centre for Medium-Range Weather Forecasts
MPIPKS Dresden, January 2010
Acknowledgements: Renate Hagedorn, Linus Magnusson
M Leutbecher
Ensemble Prediction: D3
ECODYC10
1 / 47
Ensemble forecasting of weather
A probabilistic forecast for Dresden
The ECMWF Ensemble
Prediction System (EPS)
50 perturbed forecasts
forecasts start from slightly
different initial conditions.
Perturbations are based on
singular vectors of 2-day
propagator of the model.
model tendencies are
stochastically perturbed
2 ensembles per day at 00
and 12 UTC
M Leutbecher
ECODYC10
2 / 47
Comparison with other global ensembles in TIGGE
850 hPa Temperature, Northern Hemisphere, DJF 2008/09
0.8
TIGGE-4
CMC
ECMWF
MetOffice
NCEP
0.6
CRPSS
EC-CAL
0.4
CRPS: Continuous
Ranked Probability
Score ≡ Mean Squared
error of the cumulative
distribution
Converted to skill with
CRPS of climatological
distribution (1 perfect,
0 as good as climate)
0.2
0.0
2
4
6
8
10
Lead Time / days
12
14
Hagedorn et al. (2010)
Quasi-independent analysis (ERA-Interim) used for
verification
Symbols indicate that differences to TIGGE-4 are
statistically significant at the 1% level.
M Leutbecher
EC-CAL: Calibrated
ECMWF EPS as good
as multi-model
TIGGE-4
(4 best ensembles in
TIGGE including
ECMWF)
ECODYC10
3 / 47
What determines the skill of the EPS?
Accurate centre of pdf of initial conditions: 4D-Var assimilation
scheme using millions of observations every 12 hours
Accurate forecast model: efficient and accurate dynamics, advanced
parametrisations. Spatial resolution: global NWP model with 50 km
(32 km from 26 Jan 2010) horizontal resolution and 62 levels up to
5 hPa
Efficient representation of sources of uncertainty
I
I
Initial uncertainties: Singular vectors (SVs)
Model uncertainties: Stochastically perturbed parametrisation
tendencies (SPPT)
Decisions about upgrades subject to detailed diagnostics . . .
M Leutbecher
ECODYC10
4 / 47
EPS Design: Representation of Initial Uncertainties
not all initial condition perturbations grow vigorously
perturb only those directions of the state space that are dynamically
the most sensitive in a linear sense
initial time t0
t1 > t0
p.d.f. with covariance
probability density
function (p.d.f.)
with covariance C0
C1 = M C0 MT ,
where M is the propagator
a suitable singular-value-decomposition of the propagator of the NWP
model yields such perturbations (“singular vectors (SVs)”)
M Leutbecher
ECODYC10
5 / 47
EPS Design: Initial Uncertainties (2)
Initial pdf represented by a Gaussian in the space spanned by the
leading O(100) SVs in a state space with dimension O(107 )
Norms (linear transformations of state space) are required to define a
physically meaningful SVD
The appropriate initial-time norm is based on the initial error
covariance matrix
I
I
I
I
If initial error cov. matrix Pa was used in SVD, the SVs evolve into
leading eigenvectors of (a linear and perfect model estimate of) the
forecast error covariance matrix
If we had access to Pa we could use it directly to define the pdf.
In the operational system the so-called total energy norm is used as
proxy
A more sophisticated estimate of the analysis error covariances based
on the Hessian of the 4D-Var cost function as been tried
. . . (Barkmeijer et al 1998,1999, Lawrence et al 2009).
M Leutbecher
ECODYC10
6 / 47
Singular vectors of the propagator
1/2
Consider the SVD of the scaled propagator D1/2 MC0
and final time norm
kxk2i = xT C−1
0 x,
for the initial time norm
kxk2f = xT Dx
The singular value decomposition of the scaled propagator is
1/2
D1/2 MC0
e V
eT
= US
(1)
Here, S is the diagonal matrix containing the decreasing singular values
e and V
e contain the non-dimensional
σ1 ≥ σ2 ≥ . . . ≥ σN . Orthonormal matrices U
left and right singular vectors, respectively (as column vectors). In the usual
physical coordinates, we refer to the singular vectors as
initial SVs
normalised evolved SVs
1/2 e
V = C0 V
e
U = D−1/2 U
The leading SVs evolve into the leading eigenvectors of the fc error cov. matrix C1
C1 = MC0 MT = US2 UT .
M Leutbecher
(2)
ECODYC10
7 / 47
Singular vectors in the operational ECMWF EPS
topt ≡ t1 − t0 = 48 h
resolution: T42 (300 km)
Extra-tropics: 50 SVs for N.-Hem. (30◦ –90◦ N)
+ 50 for S.-Hem.(30◦ –90◦ S). Tangent-linear model with vertical diffusion
and surface friction only.
Tropical cyclones: 5 singular vectors per region targeted on active tropical
depressions/cyclones. Up to 6 such regions. Tangent-linear model with
representation of diabatic processes (large-scale condensation, convection,
radiation, gravity-wave drag, vert. diff. and surface friction).
Localisation is required to avoid that too many leading singular vectors are
located in the dynamically more active winter hemisphere. Also required to
obtain (more slowly growing) perturbations associated with tropical cyclones.
In order to optimise perturbations for a specific region simply replace the
propagator M in the equations by PM, where P denotes the projection
operator which sets the state vector ( T , u, v , ln psfc in grid-point space) to
zero outside the region of interest and is the identity inside it.
M Leutbecher
ECODYC10
8 / 47
1
1
0
0
10
20
30
Total Wave Number
0
40
0
10
20
30
Total Wave Number
40
Upward and upscale growth of singular vectors
average energy of the leading 50 singular vectors
initial time (× 50),
final time t =48 h (× 1)
: total energy;
: kinetic energy
Northern hemisphere extra-tropics, 2006032100
0001 2006032100 MEAN N= 50 FAC= 5.00E+01- 1.00E+00
0001 2006032100 MEAN N= 50 FAC= 5.00E+01- 1.00E+00
vertical profile
10
20
30
40
50
60
spectrum
5
Energy
Level N
4
3
2
0
1
2
3
Energy
4
1
0
5
0
10
20
30
Total Wave Number
wave number
5
200 hPa ↔ level 20 300 hPa ↔ level 27
10
MAGICS500
6.10 hPa
tamlane
Thu 35
Apr 13700
11:44:54
1 ERROR(S) FOUND *
↔- nel
level
hPa2006
↔ 0001
level* 42
20
850 hPa ↔ level 48 925 hPa ↔ level 52
40
40
wave length
8000 km
4000 km
2000 km
1000 km
see also Buizza and Palmer (1995) and Lawrence et al (2009)
M Leutbecher
ECODYC10
9 / 47
Regional distribution of Northern Hem. SVs
square root of vertically integrated total energy of SV 1–50 (shading)
500 hPa geopotential (contours)
ECMWF Analysis VT:Tuesday 21 March 2006 00UTC 500hPa geopotential height (Units: dam Exp: 0001 )
initial
singular
vectors,
211March
2006,
00total
UTC
ECMWF SV VT:Tuesday 21 March 2006 00UTC
Model Level
**Vertically
integrated
energy (Units: Exp: 0001
60°W
0°
540
540
530
520
L
540
L
L
L
0 L
H
570
H
L
L
L
L
550
540
570
L
H
L
H
L
550
570
540
560
30°N
580
580
H
L
L
L
0
56
580
580
60°N
0
H
560
H
LH
540
530
53
L
L
30°N
520L
L
550
H
60°N
52
550
540
520
520
510
0
530
)
120°E
550
550
550
L
60°E
57
120°W
H
H
L
H
L
L
H
H
H
L
L
0.2321
2
4
6
8
10
11.34
ECMWF Analysis VT:Thursday 23 March 2006 00UTC 500hPa geopotential height (Exp: 0001 )
evolved singular vectors, 23 March 2006, 00 UTC
ECMWF Evolved SV FC VT:Tuesday 21 March 2006 00UTC Model Level 1 **Vertically integrated total energy (Exp: 0001 )
60°W
0°
550
540
530
L
H
L
L
30°N
L
0
56
570
L
520
53
0
L
H
550
30°N
580
580
L
H
L
560
570
570
H
H
M Leutbecher
L
550
L
570
580
580
L
60°N
L
540
0
L
H
520
56
L
H
H
H
530
510
L
530
540
540
530
520
550
H
120°E
550
L
540
H
60°N
60°E
550
540
540
120°W
H
L
Ensemble Prediction: D
L
L
3
H
H
H
H
1.279 8
16
24
32
40
48
56
64
ECODYC10
72
L80 81.30
10 / 47
Initial condition perturbations
• Initial condition uncertainty is represented by a (multi-variate) Gaussian
distribution in the space spanned by the leading singular vectors
• The perturbations based on a set of singular vectors v1 , . . . , vm are of
the form
m
X
xj =
αjk vk
(3)
k=1
• The αjk are independent draws from a truncated Gaussian distribution.
0.50
0.45
0.40
0.35
p( α )
• The width of the distribution is set
so that the spread of the ensemble
matches the root-mean square error
in an average over many cases (β ≈
10).
• The Gaussian is truncated at ±3
standard deviations to avoid numerical instabilities for extreme values
(α = 10σ is unlikely but possible).
0.30
0.25
β
0.20
0.15
0.10
0.05
0.00
−4
−3
−2
−1
0
α
M Leutbecher
1
2
ECODYC10
3
4
11 / 47
Initial condition perturbations (2)
• For the extra-tropical perturbations, the leading 50 initial singular vectors
and the leading 50 evolved singular vectors are combined (in each hemisphere)
evolved SVs
−2
initial SVs
0
+2
t (d)
EPS forecasts
• For each of the (up to 6) optimisation regions targeted on a tropical cyclone,
the leading 5 initial singular vectors are combined.
• To make sure that the ensemble mean is centred on the unperturbed analysis a
plus-minus symmetry has been introduced:
coefficients for members 1, 3, 5, . . . , 49 are sampled,
the perturbation for members 2, 4, 6, . . . 50 is set to minus the perturbation
of the member j − 1 (xj = −xj−1 ).
Note: The sign of a singular vector itself is arbitrary.
M Leutbecher
ECODYC10
12 / 47
Initial condition perturbation for member 5
Temperature (every 0.2 K);
21 March 2006, 00 UTC
at ≈ 700 hPa
120°W
60°W
0°
60°E
120°E
-0.2
0.2
60°N
60°N
-0.4
30°N
30°N
0°
0°
TC Wati
30°S
30°S
0.2
60°S
60°S
0.2
-0.2
120°W
-0.2
60°W
0°
60°E
120°E
Cross section of temp at
20060321
50◦ N 00 step 0 Expver 0001
200
0.2
300
400
500
600
800
1000
-0
.2
0.2
0.4
-0.4
0.
2
150 W
100 W
O
O
50 W
0
O
M Leutbecher
50 E
O
50.0N
Ensemble Prediction:
0.2
100 E
O
D
-0
.2
150 E
O
O
3
ECODYC10
13 / 47
Initial condition perturbation for member 50
Temperature (every 0.2 K);
21 March 2006, 00 UTC
at ≈ 700 hPa
120°W
60°W
0°
60°E
120°E
0.2
-0.2
60°N
60°N
30°N
30°N
0°
0°
30°S
30°S
-0.2
60°S
60°S
0.2
0.4
0.2
120°W
60°W
0°
60°E
120°E
Cross section of temp at
20060321
50◦ N 00 step 0 Expver 0001
200
-0.2
.2
.2
-0
0.
4
-0
.2
-0
.4
-0
-0.2
0.2
0.2
0.2
2
0.
M Leutbecher
0.2
0.2
300
400
500
600
800
1000
150 W
O
100 W
O
50 W
O
0
50 E
O
50.0N D
Ensemble Prediction:
O
3
100 E
O
150 E
O
ECODYC10
14 / 47
EPS Design: Representation of Model Uncertainties
Model uncertainties are represented by stochastically perturbed
parametrisation tendencies (SPPT)
Original scheme developed by Buizza et al (1999, “stochastic
physics”)
Revised scheme outlined below (see Palmer et al, 2009, for details)
M Leutbecher
ECODYC10
15 / 47
Perturbed parametrized tendencies
Tendency perturbation: ∆Xp = (1 + r )∆X0 ,
where ∆X0 denotes the unperturbed tendency of u, v , T , q
Random pattern r given by AR(1) processes in spectral space
t=0h
t=3h
t=6h
Decorrelation scales: 500 km (horizontal), 6 h
Distribution of r is Gaussian with stdev 0.5 in grid point space
No perturbations in stratosphere and close to the surface
M Leutbecher
ECODYC10
16 / 47
EPS skill and SPPT
CRPSS of Meridional Wind Component at 850 hPa
Northern Extratropics 20◦ –90◦ N
Tropics 20◦ S–20◦ N
v850hPa, Northern Extra-tropics
v850hPa, Tropics
ContinuousRankedProbabilitySkillScore
2007112300-2008083100 (40)
2007112300-2008083100 (40)
0.48
0.7
0.42
0.6
0.36
CRPSS
0.54
0.8
CRPSS
0.9
0.5
0.4
0.3
0.24
0.3
0.18
0.2
0.12
0.1
0.06
0
0
2
4
6
8
10
12
0
14
0
2
4
6
fc-step (d)
8
10
12
fc-step (d)
No tendency perturbations
Buizza et al (1999)
revision (2009)
M Leutbecher
ECODYC10
17 / 47
SPPT revision and the tail of the precipitation distribution
CDF of 6-hourly precipitation estimated from 2000 10-day forecasts
precipitation frequency in forecasts with SPPT
precip. frequency in fcs. without tendency pertns.
Northern Extratropics 20◦ –90◦ N
Tropics 20◦ S–20◦ N
n.hem
tropics
N40/2007112300to2008083100
N40/2007112300to2008083100
1.6
3.2
3
1.5
freq(pert.) / freq(unpert.)
freq(pert.) / freq(unpert.)
2.8
1.4
2.6
2.4
1.3
2.2
1.2
2
1.8
1.1
1.6
1
1.4
1.2
0.9
1
0.8
0.50.6 0.8 1
2
3
4
5 6 7 8 10
20
30 40 50
0.8
0.50.6 0.8 1
rate (mm/6h)
3
4
5 6 7 8 10
20
30 40 50
rate (mm/6h)
Buizza et al (1999)
M Leutbecher
2
revision (2009)
ECODYC10
18 / 47
14
Diagnosis
Comparison of SV-based perturbations with other initial perturbations
Focus on 3 recent studies
Model and unperturbed initial state based on operational NWP
system
Comparison using the same forecast model (IFS of ECMWF) and the
same unperturbed initial state (operational ECMWF analysis)
Comparisons
I
I
I
Bred vectors ↔ singular vectors
Ens. Transform Pertns. ↔ Random States ↔ SVs
Short-range forecast errors ↔ SVs
M Leutbecher
ECODYC10
19 / 47
Comparison of ensembles using bred vectors and singular
vectors
Magnusson, Leutbecher and Källén (2008, MWR)
Bred vectors:
I
I
I
I
rescaling every 6-hours
2 flavours: global rescaling, regionally varying rescaling (“masked”)
18 perturbed ICs from adding/subtracting 9 independent BVs to
analysis
ensemble spread tuned to get same spread at Day 3 as SV ensemble
Singular vectors: operational ECMWF configuration
TL 255L40 (80 km, 40 levels up to 10 hPa)
18 members, model cycle 31r1
Buizza et al (1999) tendency perturbations
46 cases; period: 1 December 2005 – 15 January 2006
M Leutbecher
ECODYC10
20 / 47
Ensemble standard deviation and Ens. Mean RMS error
500 hPa height
← RMSE
← Spread
M Leutbecher
ECODYC10
21 / 47
ECODYC10
22 / 47
Discrete Ranked Probability Skill Score
500 hPa height
M Leutbecher
Scaled differences of random states and Ens. Transform
Magnusson, Nycander and Källén, 2009, Tellus
Random Perturbation Fields
I
I
scaled difference between randomly selected states
scaling factor (∼ 0.1) tuned to get similar spread as SV ensemble at
Day 3 (Z500, N-Hem)
Ensemble Transform with rescaling (NCEP’s current method)
I
I
I
Ensemble Transformation every 6-hours
20 perturbed ICs from adding/subtracting 10 ET perturbations to
analysis
ensemble spread tuned to get same spread at Day 3 as SV ensemble
Singular vectors: operational configuration as described earlier
TL 255L40 (80 km, 40 levels up to 10 hPa)
20 members, model cycle 31r1
Buizza et al (1999) tendency perturbations
90 cases; period: 1 December 2005 – 28 February 2006
M Leutbecher
ECODYC10
23 / 47
Ensemble standard deviation and Ens. Mean RMS error
500 hPa height
← RMSE
← Spread
Singular vectors: solid
Ens. Transform: dashdotted
Random Field Perturbations: dashed
M Leutbecher
ECODYC10
24 / 47
Discrete Ranked Probability Skill Score
850 hPa temperature
Singular vectors: solid
Ens. Transform:
dash-dotted
Random Field
Perturbations: dashed
However, SVs better
initially in terms of Z500
(overdispersion of ET
and RPF).
M Leutbecher
ECODYC10
25 / 47
Nonmodal perturbation growth in the atmosphere
MARCH 2004
704
TRIBBIA AND BAUMHEFNER
Tribbia & Baumhefner (2004)
MONTHLY WEATHER REVIEW
Lorenz (1969, 1984)
FIG. 1. Growth of errors initially confined to smallest scales, according to a theoretical model Lorenz (1984). Horizontal scales are
on the bottom, and the upper curve is the full atmospheric motion
spectrum.
gorinsky (1969), Williamson and Kasahara (1971), and
later by Daley (1981), gave progressively shorter estimates of the time, beginning with 1 month and culminating at about 10 days. Throughout, the conceptual
model of a gradual inverse cascade has remained, and
with the extension to three-dimensional flow through
the concept of quasigeostrophic (QG) turbulence developed by Charney (1971), the supposition of longtime
M Leutbecher
Ensemble
predictability of planetary scales has remained part of
the conventional wisdom of atmospheric predictability.
It should be noted that cascade rates are intimately
VOLUME 132
sparked by programs such as the North American Observing System (NAOS), the U.S. Weather Research
Program (USWRP), and the Navy Predictability Directed Research Initiative, the quantification of propagation rates of error variance across scales and of scale
interactions is extremely relevant. Such estimates will
permit a rational choice of observational strategy that
may be targeted to allow the accurate prediction of phenomena of a chosen scale. Since it has been two decades
since such quantities were computed in the works cited
above, and forecast and climate models have continued
to develop over that time, this work will reexamine the
questions of interscale influence in predictability error
growth. In the following sections we develop an experimental design that can test the validity of this conventional wisdom regarding scale interactions and predictability error growth in large and small scales. We
also test directly the effects of scale limitations of errors
and unresolved scales and processes to examine anew
the relevance of QG turbulence ideas in the context of
a modern atmospheric GCM, useful both for numerical
weather prediction and climate simulation.
2. Experimental design
Prediction: D3
ECODYC10
There are two components of the experimental design
that are to be specified: the models used and the types
of initial errors specified. The interrelationship of these
26 / 47
Initial perturbations based on short-range forecast errors
Previous work
Mureau, Molteni & Palmer (1993)
I
I
I
I
assimilation method: Optimum Interpolation
model T63
initial perturbations based on 6-hour errors from past 30 days
& Gram-Schmidt-orthonormalisation
conclusion: SV perturbations are superior
revisit with a state-of-the-art system
methodology here slightly different
M Leutbecher
ECODYC10
27 / 47
Methodology
simple: use what is in the archive, avoid interpolation
remove systematic component of error
define set X of short-range forecast errors or lagged forecast
differences valid for the season (00 and 12 UTC control forecast fields)
compute mean error(s) µ00 , µ12 from set X
sample 25 realisations j from X , subtract mean, scale
xj = α(j − µ)
add and subtract xj from unperturbed analysis → 50 perturbed ICs
Here: x includes the dry upper air model state: vorticity, divergence, T, log(psfc )
M Leutbecher
ECODYC10
28 / 47
Experiments
Experiments TL 255L62, cycle 32r3
Buizza et al (1999) stochastically perturbed parametrisation
tendencies
initial perturbations:
I
I
operational singular vector configuration
sampling of (unscaled) 24-hour forecast errors
50 cases in NDJF2008 (every other day)
Additional experiments:
I
I
I
12-hour, 48-hour forecast errors
lagged fc differences (48 h – 24 h)
spectrally filtered forecast errors
M Leutbecher
ECODYC10
29 / 47
Time mean spread vs. RMSE of Ens. mean
Meridional wind component (m s−1 ) at 850 hPa, t=48 h
singular vector init. perts.
24-hour fc. error init. perts.
Friday 23 November 2007 00UTC ECMWF EPS Perturbed Forecast t+48 VT: Sunday 25 November 2007 00UTC
850hPa V velocity (Exp: ezh1) - Ensemble member number 1 of 51
850hPa V velocity (Exp: f0u0) - Ensemble member number 1 of 51
8
8
5.6
5.6
4
4
2.8
2.8
2
2
1.4
1.4
1
1
0.7
850hPa V velocity (Exp: ezh1) - Ensemble member number 1 of 51
top: ens. stdev.;
M Leutbecher
0.5
0.7
850hPa V velocity (Exp: f0u0) - Ensemble member number 1 of 51
0.5
8
8
5.6
5.6
4
4
2.8
2.8
2
2
1.4
1.4
1
1
0.7
0.7
0.5
0.5
bottom: ens. mean RMS error; 50 cases: 23 Nov ’07–29 Feb ’08
TL 255, 32r3, unscaled 24-hour FCEs
ECODYC10
30 / 47
−1
Meridional
wind
component
(m
s
) at 850 hPa, t=48 h
Friday 23 November 2007 00UTC ECMWF EPS Control Forecast t+48 VT: Sunday 25 November 2007 00UTC
850hPa **V velocity (Exp: f0u0)
0.1743
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.3771
-0.4
CRPS (Continuous Ranked Probability Score ≡ mean squared error of the
cumulative distribution)
Blue means EPS based on short-range forecast errors is more skilful.
50 cases: 23 Nov 2007 – 29 Feb 2008
M Leutbecher
ECODYC10
31 / 47
Ensemble Mean RMSE & Ensemble Standard Deviation
500 hPa geopotential, Northern Mid-latitudes 35◦ –65◦ N
z at 500hPa
sample of 50 cases; 2007112300 - 2008022900, area n.hem.mid
symbols: RMSE of Ens. Mean; no sym: Spread around Ens. Mean
900
800
700
RMS
600
500
400
300
oper. conf.
200
fce24 x 1.00
100
0
0
M Leutbecher
1
2
3
4
5
fc-step (d)
6
7
8
9
10
ECODYC10
32 / 47
Continuous Ranked Probability Skill Score
500 hPa geopotential, Northern Mid-latitudes 35◦ –65◦ N
z at 500hPa
Cont. Ranked Prob. Skill Score
0.88
0.8
0.72
0.64
0.56
0.48
0.4
oper. conf.
0.32
fce24 x 1.00
0.24
0.16
0
1
2
3
4
5
fc-step (d)
6
7
8
9
10
M Leutbecher
ECODYC10
33 / 47
Ensemble Mean RMSE & Ensemble Standard Deviation
850 hPa temperature, Northern Mid-latitudes 35◦ –65◦ N
t at 850hPa
symbols: RMSE of Ens. Mean; no sym: Spread around Ens. Mean
4
RMS
3
2
oper. conf.
1
0
fce24 x 1.00
0
M Leutbecher
1
2
3
4
5
fc-step (d)
6
7
8
9
10
ECODYC10
34 / 47
Continuous Ranked Probability Skill Score
Lower spread of
operational
configuration is not
the cause of lower
skill at ∼ D1–3 as
experiments with
20% inflated SV
initial perturbations
prove.
t at 850hPa
0.8
0.72
0.64
0.56
0.48
0.4
oper. conf.
0.32
fce24 x 1.00
0.24
0.16
0
1
2
3
4
5
fc-step (d)
6
7
8
9
10
M Leutbecher
ECODYC10
35 / 47
Spread-reliability
t850hPa, t=+24h, N.hem.mid
N50/2007112300TO2008022900
.
t850hPa, t=+48h, N.hem.mid
N50/2007112300TO2008022900
.
1.6
2
RMS error (K)
RMS error (K)
1.4
1.2
1
0.8
1
0.6
0.4
0.2
0.2 0.4 0.6 0.8
1
SV (oper)
SV (oper)
FCE24 x 1.00
FCE24 x 1.00
1.2 1.4 1.6
RMS spread (K)
1
2
RMS spread (K)
stratify pairs of RMSE and spread by predicted spread
1 pair (spread, RMSE) for each grid point and each initial time
compute RMSE and spread in 20 equally populated bins
M Leutbecher
ECODYC10
36 / 47
Which scales are the most important?
Initial perturbation variance spectra
Temperature
T at 700hPa,
700
hPat=0h
Rotational wind component
Rot.
KE athPa
700hPa, t=0h
700
20 cases: 2007112300 . . . 2007123100
20 cases: 2007112300 . . . 2007123100
70
40
20
10
6
4
2
1
0.6
0.4
0.2
0.1
0.06
0.04
0.02
0.01
0.006
0.004
0.002
0.001
0.0006
0.0004
n × variance
n × variance
10
7
4
2
1
0.7
0.4
0.2
0.1
0.07
0.04
0.02
0.01
0.007
0.004
0.002
0.001
0.0007
0.0004
SV
FCE24 ×1.00
1
2
3
4
5 6 7 8 10
20
30 40 50 60 80 100
total wavenumber n
SV
FCE24 ×1.00
1
2
3
4
5 6 7 8 10
20
total wavenumber n
30 40 50 60 80 100
20 cases: 23 Nov – 31 Dec 2007
full fields (analyses, grey)
M Leutbecher
ECODYC10
37 / 47
Spectrally filtered forecast errors
Ensemble dispersion
t850hPa, Northern Extra-tropics
spread_em, rmse_em
2007112300-2007123100 (20)
RMS
3
2
FCE24
FCE24, N<=40
1
FCE24, N> 40
0
0
1
2
3
4
5
6
7
8
9
10
fc-step (d)
TL 255, 20 cases: 23 Nov – 31 Dec 2007
(unscaled) 24-hour forecast errors
M Leutbecher
ECODYC10
38 / 47
Spectrally filtered forecast errors
Probabilistic skill
t850hPa, Northern Extra-tropics
2007112300-2007123100 (20)
1
0.9
0.8
0.7
CRPSS
0.6
0.5
0.4
FCE24
0.3
FCE24, N<=40
0.2
FCE24, N> 40
0.1
0
0
1
2
3
4
5
6
7
8
9
10
fc-step (d)
TL 255, 20 cases: 23 Nov – 31 Dec 2007
(unscaled) 24-hour forecast errors
M Leutbecher
ECODYC10
39 / 47
other proxys for initial uncertainty
Experiments
12-hour forecast errors (×1.28)
48−24-hour forecast differences ( ×0.61, NMC-method)
I
I
I
I
Scaling factors: exponential growth model with error doubling time of
1.4 d (cf. Simmons and Hollingsworth, 2002)
Results based on 20 cases Nov–Dec 2007 (TL 255, cycle 32r3)
t at 850hPa
t at 850hPa
0.8
0.8
0.72
0.72
0.64
0.64
0.56
0.56
0.48
0.48
0.4
fce12 x 1.28
0.32
fce24 x 1.00
0.24
fce48 x 0.61
0.4
oper. conf.
0.32
fce24 x 1.00
0.24
48-24 x 0.61
0.16
0.16
0
1
2
M Leutbecher
3
4
5
fc-step (d)
6
7
8
9
10
0.08
0
1
2
3
4
5
fc-step (d)
6
7
8
9
10
ECODYC10
40 / 47
Projection of initial perturbations on singular vectors
Method
The singular vectors are orthonormal with respect to the total energy
metric
vjT Evk = δjk
Any initial perturbation x can be written as
x=
N
X
αj vj + x⊥ ,
where
j=1
αj = xT Evj
and xT
⊥ Evj = 0
For the operational EPS configuration the α-s are independent &
normally distributed.
What is the distribution of α-s for the short-range forecast errors?
M Leutbecher
ECODYC10
41 / 47
Projection of initial perturbations on singular vectors
Results
svifs, stdev=
7.16.9
SVs (γ =ezh1,
0.014),
σα =
50 cases, 2007112300 to 2008022900
f0u0,
svifs, stdev=
24-hour FCE
(×1.0),
σα3.9= 3.9
50 cases, 2007112300 to 2008022900
0.1
0.1
0.0875
0.0875
0.075
0.075
PDF
0.1125
PDF
0.1125
0.0625
0.0625
0.05
0.05
0.0375
0.0375
0.025
0.025
0.0125
0.0125
0
-25 -20 -15 -10 -5
0
alpha
5 10 15 20 25
0
-25 -20 -15 -10 -5
0
alpha
5 10 15 20 25
20 cases: 23 Nov – 31 Dec 2007
leading 50 northern extra-tropical SVs
50 × 50 × 20 = 5 × 104 coefficients
M Leutbecher
ECODYC10
42 / 47
A schematic of the initial uncertainty representations
model’s phase space
M Leutbecher
ECODYC10
43 / 47
Developments
Resolution upgrade . . . 50 km → 32 km (Jan 2010)
Evolved singular vectors −→ perturbations from a 10-member ensemble of
perturbed 4D-Vars
I
I
I
perturbed obs.
perturbed SSTs
perturbed tendencies (SPPT)
see Buizza et al. (2008)
Stochastically Perturbed Parameterization Tendency (SPPT) scheme upgraded
(Sep 2009)
Stochastic backscatter scheme to represent uncertainty due to missing variability
on the near-gridscale
P
Multi-scale version of SPPT r = Lk=1 rk where the rk differ in terms of variance,
spatial and temporal correlation scales
M Leutbecher
ECODYC10
44 / 47
Conclusions
TIGGE and calibrated ECMWF ensemble
I
I
Multi-model based on four best ensembles can improve on the best
single-model, the ECMWF EPS
Reforecast-calibrated ECMWF EPS comparable or superior these
multi-model predictions
Representing model uncertainty can improve the skill of ensemble
predictions (in particular in the tropics)
Probabilistic skill of various flow-dependent initial perturbation
methodologies is very similar:
I
bred vectors ≈ Ens. Transform ≈ singular vectors
M Leutbecher
ECODYC10
45 / 47
Conclusions (II)
Flow-independent initial perturbations based on past short-range
forecast errors lead to an ensemble that is as skilful as or better than
SV-based system in terms of traditional probabilistic skill measures
Short-range forecast errors have a significant projection on the space
of the leading singular vectors; in addition, they perturb also in the
107 − 50 other directions
However,
I
I
initially somewhat overdispersive
unrealistic initial perturbations can occur due to flow-independence.
Technique not applicable without some prior filtering
Expected that ensemble data assimilation techniques will be
(eventually) superior to a simple flow-independent perturbation
technique (work in progress)
M Leutbecher
ECODYC10
46 / 47
References
Barkmeijer, Jan, Martin Van Gijzen and François Bouttier, 1998: Singular vectors and estimates of the analysis-error
covariance metric. Quart. J. Royal Met. Soc., 124, 1695–1713
Barkmeijer, J., R. Buizza and T. N. Palmer, 1999: 3D-Var Hessian singular vectors and their potential use in the
ECMWF Ensemble Prediction System. Quart. J. Royal Met. Soc., 125, 2333–2351
Buizza, R. and T. N. Palmer, 1995: The Singular-Vector Structure of the Atmospheric Global Circulation. J. Atmos.
Sci., 52, 1434-1456
Buizza, R., M. Miller and T. N. Palmer, 1999: Stochastic representation of model uncertainties in the ECMWF
Ensemble Prediction System. Quart. J. Royal Met. Soc., 125, 2887–2908
Buizza, Roberto, Martin Leutbecher and Lars Isaksen, 2008: Potential Use of an Ensemble of Analyses in the ECMWF
Ensemble Prediction System. Quart. J. Royal Met. Soc., 134, 2051–2066
Hagedorn, Renate, Roberto Buizza, Thomas M. Hamill, Martin Leutbecher and T. N. Palmer, 2010: Comparing TIGGE
multi-model forecasts with reforecast-calibrated ECMWF ensemble forecasts. submitted to Mon. Wea. Rev.
Magnusson, Linus, Martin Leutbecher and Erland Källén, 2008: Comparison between Singular Vectors and Breeding
Vectors as Initial Perturbations for the ECMWF Ensemble Prediction System. Mon. Wea. Rev., 136, 4092–4104
Magnusson, Linus and Jonas Nycander and Erland Källén, 2009: Flow-dependent versus flow-independent initial
perturbations for ensemble prediction. Tellus, 61A, 194–209
Lawrence, A. R., M. Leutbecher and T. N. Palmer, 2009: The characteristics of Hessian singular vectors using an
advanced data assimilation scheme. Quart. J. Royal Met. Soc., 135, 1117–1132
Palmer, T.N., R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G.J. Shutts, M. Steinheimer and A Weisheimer,
2009: Stochastic parametrization and model uncertainty. ECMWF Tech. Memo., 598
M Leutbecher
ECODYC10
47 / 47

The ECMWF Ensemble Prediction System

Transcription

Similar documents

SUNY Buffalo State College

Tales from the Forgotten Kingdom One Sheet

Interior_Panic Press Release August-amd

Ensemble Travel Members Meet in Dublin

After obtaining three 1st prizes at the Paris

Three Sides to Every Story

Photo Gallery of the Tour

ENGINE REPAIR STRUCTURAL INTEGRITY PROGRAM

SILVA Igor C.

Sealy Crown Jewel Fortune Firm Ensemble