© Crown copyright Met Office Successes and Challenges in 4D-Var Third THORPEX International Science...

© Crown copyright Met Office

Successes and Challenges in 4D-Var

Third THORPEX International Science SymposiumAndrew Lorenc, Monterey, Sept 2009.

© Crown copyright Met Office Andrew Lorenc 2


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?


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.


Performance Improvements

Met Office RMS surface pressure error over the N. Atlantic & W. Europe

“Improved by about a day per decade”


60 Years of Met Office Computers

1.E+021.E+031.E+041.E+05

1.E+061.E+071.E+081.E+091.E+101.E+11

1.E+121.E+131.E+141.E+15

1950 1960 1970 1980 1990 2000 2010

Year of First Use

Pea

k F

lop

s

LEOMercury

LEO 1

KDF 9

IBM 360

Cyber 205

Cray YMP

Cray C90

Cray T3E

NEC SX6/8

IBM Power -Phase 1&2

Moore’s Law 18month doubling

time


ratio of supercomputer costs: 1 day's assimilation / 1 day forecast

5

8

20

31

1

10

100

1985 1990 1995 2000 2005 2010

AC scheme

3D-Var on T3E

simple 4D-Var on SX8

4D-Var with

outer_loop

Ratio of global computer costs: 1 day’s DA (total incl. FC) / 1 day’s forecast.

Only 0.04% of the Moore’s Law increase over this time went into improved DA algorithms, rather than improved resolution!

1 day of MOGREPS (24 member LETKF) / 1 day’s forecast : 56.

1 day of MOGREPS / 1 day’s ensemble: 2.3


CBS N-hem Pmsl T+24 RMSE v analysis

Rectangles show 12-month running mean impact period of 4D-Var implementation


Change in OSE results 2001-2007.N-hem 500hPa height ACC.

Not the same period, so only make qualitative comparisons!

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.


Statistical, incremental 4D-Var

SimplifiedGaussian

PDF t1SimplifiedGaussian

PDF t0 Full model evolves mean of PDF

PF model evolves any simplified perturbation,and hence covariance of PDF

Statistical 4D-Var approximates entire PDF by a Gaussian.

Adjoint of PF model is needed

N.B. PF model need not be tangent-linear to full model and in all NWP implementions, is not.


Perturbation Forecast model for Incremental 4D-Var

Cloud fraction

(RHtotal-1)/(1-RHcrit)

• Minimise:

• Designed to give best fit for finite perturbations

• Not Tangent-Linear

• Filters unpredictable scales and rounds IF tests

• Requires physical insight – not just automatic differentiation

Tim Payne

cloud fraction

I E M M x x x M x x


Incremental 4D-Var with Outer Loop

y y yy

FULL FORECAST MODEL

PERTURBATION FORECAST MODEL

ADJOINT OF P.F. MODEL

DES

CEN

TA

LGO

RIT

HM

Inner low-resolution incremental variational iteration

back

grou

nd

Outer, full-resolution iteration

INCREMENTAL 4-DIMENSIONAL VARIATIONAL ASSIMILATION

U

U

T

xg

xb

δx +δη

+ η

Optional model error terms


What spread to assume in regularisation?

• If guess=background, need to approximate whole of PDFf

• In final outer-loop, only need to approximate PDFa

Observation

x b2

x2a

x1bx

1a

PDFf

PDFa

y o=(x1 +x

2 )/2


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.


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.


Comparison of observed and modelled cloud

9Z 13-10-2002

Observed Simulated

Samatha Pullen


12Z 13-10-2002


15Z 13-10-2002


18Z 13-10-2002


21Z 13-10-2002


0Z 14-10-2002


3Z 14-10-2002


6Z 14-10-2002


9Z 14-10-2002


12Z 14-10-2002


15Z 14-10-2002


18Z 14-10-2002


21Z 14-10-2002


Equations for tracer advection

DmS

Dt

mm S

t

u

mm m S

t

uu

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.

Determining u & m simultaneously is a nonlinear problem.


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 spun-up, balanced model state.


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).


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 Andrew Lorenc 52© 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 53© Crown copyright Met Office

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 high-resolution model and stochastic perturbations are used to generate an ensemble to sample this detail.


Questions and answers



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?


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.


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.


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 spun-up, balanced model state.

© Crown copyright Met Office Successes and Challenges in 4D-Var Third THORPEX International Science...

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