Successes and Challenges in 4D-Var
Third THORPEX International Science Symposium
Andrew Lorenc, Monterey, Sept 2009.
© Crown copyright Met Office
Successes and Challenges in 4D-Var
1. Success: 4D-Var has made a significant contribution to
the “day per decade” improvement in NWP skill over my
career, but it comes behind model improvements (esp.
resolution) and statistical allowance for errors (VAR).
2. Current challenge: getting full information from remote
sensing (time-sequences of high-resolution images) is a
multi-scale, nonlinear problem. 4D-Var can tackle this.
3. Longer-term challenge: the atmosphere is nonlinear, with
an attractor of recognisable “meteorological” features, and
non-Gaussian PDFs. But NWP models are so large that
only quasi-linear data assimilation methods are affordable.
Perhaps an ensemble of spun-up 4D-Vars can help?
© Crown copyright Met Office Andrew Lorenc 2
1. Success:
Causes of improvements to NWP
NWP systems are improving by 1 day of predictive
skill per decade. This has been due to:
1. Model improvements, especially resolution.
2. Careful use of forecast & observations, allowing
for their information content and errors.
Achieved by variational assimilation e.g. of
satellite radiances.
3. 4D-Var.
4. Better observations.
© Crown copyright Met Office Andrew Lorenc 3
Performance Improvements
“Improved by about a day per decade”
Met Office RMS surface pressure error over the N. Atlantic & W. Europe
© Crown copyright Met Office Andrew Lorenc 4
Peak Flops
60 Years of Met Office Computers
1.E+15
Moore’s Law
1.E+14
18month doubling
time
1.E+13
1.E+12
1.E+11
1.E+10
1.E+09
1.E+08
1.E+07
1.E+06
1.E+05
KDF 9
1.E+04
1.E+03
Mercury
LEO 1
1.E+02
1950
1960
1970
IBM Power -Phase 1&2
Cray T3E
NEC SX6/8
Cray C90
Cray YMP
Cyber 205
IBM 360
1980
1990
Year of First Use
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2000
2010
Ratio of global computer costs:
ratio of supercomputer costs:
11 day’s
(total incl. /FC)
/ 1forecast
day’s forecast.
day's DA
assimilation
1 day
100
Only 0.04% of the Moore’s Law increase over
this time went into improved DA algorithms,
rather than improved resolution!
10
31
20
4D-Var
with
outer_loop
simple
4D-Var
on SX8
8
3D-Var on
T3E
5
AC scheme
1 day of MOGREPS (24 member LETKF) / 1 day’s forecast : 56.
1 day of MOGREPS / 1 day’s ensemble: 2.3
1
1985
1990
© Crown copyright Met Office Andrew Lorenc 6
1995
2000
2005
2010
CBS N-hem Pmsl T+24
RMSE v analysis
Rectangles show 12-month running mean
impact period of 4D-Var implementation
© Crown copyright Met Office Andrew Lorenc 8
Change in OSE results 2001-2007.
N-hem 500hPa height ACC.
Not the same period,
so only make qualitative comparisons!
© Crown copyright Met Office Andrew Lorenc 16
Richard Dumelow
2. Current challenge:
Multi-scale assimilation of image sequences
• Getting information from the perceived
movement of a detailed tracer field is a
multi-scale nonlinear problem.
• Incremental 4D-Var with an outer-loop
can tackle it, at a cost which is becoming
affordable.
© Crown copyright Met Office Andrew Lorenc 18
Statistical, incremental 4D-Var
Adjoint of PF model is needed
PF model evolves any simplified perturbation,
and hence covariance of PDF
Simplified
Gaussian
PDF t1
Simplified
Gaussian
PDF t0 Full
N.B.
model evolves mean of PDF
Statistical 4D-Var approximates entire PDF by a Gaussian.
PF model need not be tangent-linear to full model
and in all NWP implementions, is not.
© Crown copyright Met Office Andrew Lorenc 19
Perturbation Forecast model for
Incremental 4D-Var
• Minimise: I
E
 M  x   x  M  x
 M x x

• Designed to give
best fit for finite
perturbations
• Requires physical
insight – not just
automatic
differentiation
Tim Payne
© Crown copyright Met Office Andrew Lorenc 20
Cloud fraction
• Filters unpredictable
scales and rounds IF
tests
cloud fraction
• Not Tangent-Linear
(RHtotal-1)/(1-RHcrit)
Incremental 4D-Var with
INCREMENTAL 4-DIMENSIONAL VARIATIONAL ASSIMILATION
Outer Loop
Inner low-resolution incremental variational iteration
ADJOINT OF P.F. MODEL
T
DESCENT
xb
ALGORITHM
background
U
y
y
y
y
U
δx
xg
PERTURBATION FORECAST MODEL+δη
FULL FORECAST MODEL
Outer, full-resolution iteration
© Crown copyright Met Office Andrew Lorenc 21
+η
Optional model error terms
What spread to assume in
regularisation?
• If guess=background,
need to approximate
whole of PDFf
er
bs
O
va
n
• In final outer-loop, only
need to approximate PDFa
PDFf
tio
xb
2
x 2a
PDFa
o
y
x 2)
x 1+
=(
/2
x 1a
© Crown copyright Met Office Andrew Lorenc 22
x 1b
Information content of
imagery sequences
• Humans can make reasonable forecasts based
on imagery alone (satellite or radar):
information scarcely used in NWP.
• Time-sequences aid the interpretation of
images.
• Some important information is multi-scale;
details at high-resolution are used to recognise
patterns whose larger-scale movements are
significant.
© Crown copyright Met Office Andrew Lorenc 23
AMVs
• I am not suggesting we could replace AMVs by 4DDA in the near future!
• However they provide an example of demonstrated useful information from
imagery sequences, which a method should in principle be able to extract.
• 4DDA methods could, in theory, improve on current AMV techniques in
allowing for development and dynamical coupling of features.
© Crown copyright Met Office Andrew Lorenc 25
Comparison of observed and modelled cloud
9Z 13-10-2002
Observed
Samatha Pullen
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Simulated
12Z 13-10-2002
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15Z 13-10-2002
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18Z 13-10-2002
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21Z 13-10-2002
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0Z 14-10-2002
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3Z 14-10-2002
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6Z 14-10-2002
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9Z 14-10-2002
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12Z 14-10-2002
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15Z 14-10-2002
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18Z 14-10-2002
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21Z 14-10-2002
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Equations for tracer advection
Dm
S
Dt
m
  u m  S
t
Determining u & m simultaneously is a nonlinear problem.
m
 u  m  u  m  S 
t
In the linearised equations,
changes to the wind depend on the gradient of the linearisation state m,
biases in observations or model S′ can change the wind.
© Crown copyright Met Office Andrew Lorenc 39
3. Longer-term challenge:
Nonlinearity, attractor, non-Gaussianity.
The atmosphere is nonlinear, but the best NWP models are so
large that only quasi-linear quasi-Gaussian assimilation methods
are affordable.
• Nonlinearity helps: Without it small scale perturbations would grow rapidly and
we would be swept away! Coherent, predictable features like inversions, fronts,
cyclones are maintained by nonlinear processes.
• Current NWP is already non-Gaussian: the ensemble-mean “best estimate” is
not a plausible meteorological state – it lacks small scales and give a poor
precipitation forecast. In practice an ensemble is needed to represent the
correct power and uncertainty in small scales.
• Theoretically, a particle filter can solve the nonlinear non-Gaussian assimilation
problem, for a perfect model with the correct attractor. But it is completely
unaffordable for NWP.
• Linear Kalman filter methods cannot constrain states to a nonlinearly defined
attractor. But nonlinear 4D-Var using an outer-loop and a long time-window
might do so, via an additional constraint that the analysis must be near a spunup, balanced model state.
© Crown copyright Met Office Andrew Lorenc 48
Error growth v scale
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. (from Tribbia & Baumhefner 2004).
© Crown copyright Met Office
Limits to deterministic 4D-Var
with turbulence model
Tanguay and Gauthier (1995) showed deterministic
4D-Var does not work for a wide range of scales.
© Crown copyright Met Office
Tephigram of sounding, global
model background and analysis,
for the mean of 136 UK
soundings with layer cloud top
diagnosed at level 5 in the
background.
© Crown copyright Met Office Andrew Lorenc 52
3. Longer-term challenge:
Nonlinearity, attractor, non-Gaussianity.
As far as I know (research may prove me wrong!):
• For the most accurate forecasts and the best assimilation
NWP models will resolve detail which we cannot always observe.
• Linear Gaussian methods will not work. The minimum variance best
estimate is not meteorological, and likely to “head off into the bushes”.
• Full nonlinear methods (e.g. particle filters) are too expensive for NWP.
We need simple linear equations to have computationally feasible methods for
models with a billion degrees of freedom.
• We cannot define the “attractor” of meteorological states in practice
without relying on an NWP model. (But models will have biases.)
• Any method must be a compromise, only partially addressing all the above
problems.
• Could try long-window 4D-Var, so that any analysis is close to the model’s
attractor and the observations, while unobserved detail is generated by the highresolution model and stochastic perturbations are used to generate an ensemble
to sample this detail.
© Crown copyright Met Office Andrew Lorenc 53
Questions and answers
© Crown copyright Met Office
Successes and Challenges in 4D-Var
1. Success: 4D-Var has made a significant contribution to
the “day per decade” improvement in NWP skill over my
career, but it comes behind model improvements (esp.
resolution) and statistical allowance for errors (VAR).
2. Current challenge: getting full information from remote
sensing (time-sequences of high-resolution images) is a
multi-scale, nonlinear problem. 4D-Var can tackle this.
3. Longer-term challenge: the atmosphere is nonlinear, with
an attractor of recognisable “meteorological” features, and
non-Gaussian PDFs. But NWP models are so large that
only quasi-linear quasi-Gaussian methods are affordable.
Perhaps an ensemble of spun-up 4D-Vars can help?
© Crown copyright Met Office Andrew Lorenc 55
1. Success:
Causes of improvements to NWP
NWP systems are improving by 1 day of predictive
skill per decade. This has been due to:
1. Model improvements, especially resolution.
2. Careful use of forecast & observations, allowing
for their information content and errors.
Achieved by variational assimilation e.g. of
satellite radiances.
3. 4D-Var.
4. Better observations.
© Crown copyright Met Office Andrew Lorenc 56
2. Current challenge:
Multi-scale assimilation of image sequences
• Getting information from the perceived
movement of a detailed tracer field is a
multi-scale nonlinear problem.
• Incremental 4D-Var with an outer-loop
can tackle it, at a cost which is becoming
affordable.
© Crown copyright Met Office Andrew Lorenc 57
3. Longer-term challenge:
Nonlinearity, attractor, non-Gaussianity.
The atmosphere is nonlinear, but the best NWP models are so
large that only quasi-linear quasi-Gaussian assimilation methods
are affordable.
• Nonlinearity helps: Without it small scale perturbations would grow rapidly and
we would be swept away! Coherent, predictable features like inversions, fronts,
cyclones are maintained by nonlinear processes.
• Current NWP is already non-Gaussian: the ensemble-mean “best estimate” is
not a plausible meteorological state – it lacks small scales and give a poor
precipitation forecast. In practice an ensemble is needed to represent the
correct power and uncertainty in small scales.
• Theoretically, a particle filter can solve the nonlinear non-Gaussian assimilation
problem, for a perfect model with the correct attractor. But it is completely
unaffordable for NWP.
• Linear Kalman filter methods cannot constrain states to a nonlinearly defined
attractor. But nonlinear 4D-Var using an outer-loop and a long time-window
might do so, via an additional constraint that the analysis must be near a spunup, balanced model state.
© Crown copyright Met Office Andrew Lorenc 58