forecasts & their errors ross bannister 7th october 2008 1 / 23...

Forecasts & their Errors Ross Bannister 7th October 2008 1 / 23

ChallengesRefiningModellingMeasuringAssimilationPDFs

Forecasts & their ErrrorsRoss Bannister

National Centre for Earth Observation (the Data Assimilation Research Centre)

Thanks to: Stefano Migliorini (NCEO), Mark Dixon (MetO), Mike Cullen (MetO), Roger Brugge (NCEO)

Forecast Possible error in forecast

Horiz. winds and pressure, at 5.5 kmMet Office North Atlantic/European LAM



Forecast errorsForecast errors (from a numerical model): are a fact of life! depend upon the model formulation, synoptic situation (‘flow dependent’), model’s initial conditions, length of the forecast. are impossible to calculate in reality, δx = xf - xt.

Of interest: forecast error statistics - the probability density fn. of xt , Pf(xt).

Applications: probabilistic forecasting. model evaluation/monitoring. state estimation (data assimilation).



Seminar structure

Probability density functions (PDFs) of the state, Pf(xt).

The use of Pf(xt) in data assimilation problems.

Measuring Pf(xt).

Modelling Pf(xt) for large-scale data assimilation.

Refining Pf(xt) for large-scale data assimilation.

Challenges for small-scale Meteorology.



PDF of state, Pf(xt)

Pf(xt)

xtxf

Impossible state Probable state

Possible but unlikely state

0

Forecast comprising a single number

σ = √var(δx)

2

2fttf

2

)(exp~)(

xx

xP

xf = xt + δx



Two-component state vector

Forecast comprising two numbers

t2

t1t

f2

f1f

2

1

x

x

x

x

x

x

x

x

x

Pf(xt)

0

xf = xt + δx

σx1 = √var(δx1)

σx2 = √var(δx2)

cov(δx1,δx2)

t1x

t2x

)()(exp~

2

)(exp~)( variable1

ft12ft21

2

2fttf

xxxx

xxxP

)var(),cov(

),cov()var(

)()(exp~)(

221

211

ftTft21tf

xxx

xxx

P

B

xxBxxx 1



Geophysical error covariances – B

The B-matrix

specifies the PDF of errors in xf (Gaussianity assumed)

describes the uncertainty of each component of xf and

how errors of elements in xf are correlated

is important in data assimilation problems

107 – 108 elements

107 –

108

ele

men

ts

structure function associated (e.g.) with pressure at a location

δu δv δp δT δq

δu

δ

v

δp

δ

T

δq

xf =

u––v

––p––T––q

B =



Example standard deviations (square-root of variances)

From Ingleby (2001)



Example geophysical structure functions (covariances with a fixed point)

Univariate structure function

Multivariate structure functions



Covariances are time dependent

Structure function for tracer in simple transport model

1.0

0.9

0.7

0.8

t = 0

t > 0

1.00.9

0.70.8



Pf(xt) \ B in data assimilation

Data assimilation combines the PDFs of

i. forecast(s) from a dynamical model, Pf(xt) and

ii. measurements, Pob(y|xt)

to allow an ‘optimal estimate’ to be found (Bayes’ Theorem).

Maximum likelihood solution (Gaussian PDFs)

])[(])[(exp )()(exp~

)|( )( ~)|(t1Tt

21ft1Tft

21

tobtfta

xhyRxhyxxBxx

xyxyx

PPP

x

hH

xhyRHBHBHxx

where

])[()( f1TTfa

forecast = prior knowledge

Solved e.g. by direct inversion or by variational methods

PDF of combination of

forecast and observational

information



Pseudo satellite tracks

Tracerassimilation

Data assimilation example(for inferred quantities)

x(0)

initial conditions

y(t1)

y(t2)

y(t3)

y(t4)

y(t5)

T, q, O3 satellite radiancesInitial conditions inferred from measurements made at a later timeSources/sinks of tracer, r measurements of tracer r

–– sources/

sinks

Tracer +source/sinkassimilation

30-day assimilation



Dangers of misspecifying Pf(xt) \ B in data assimilation?

Example 1: Anomalous correlations of moisture across an interface

Example 2: Anomalous separability of structure functions around tilted structures

Normally dry air

Normally moist air



Ensembles

Measuring Pf(xt) \ BForecast errors are impossible to measure in reality, δx = xf - xt.

All proxy methods require a data assimilation system.

Analysis of innovationsDifferences between varying length forecasts

xHyxhy

xHxhxxh

xxx

yxhy

][

][][

][

f

ff

ft

t

2212

212121

2

1T

)(

xxx

xxxxx

x

x

xxB

t

x

√2 δx

Canadian ‘quick covs’

x

t

√2 δx

t

x



Modelling Pf(xt) \ B with transforms for data assimilation

PDF in model variables

)()(exp~)( ft1Tft21tf xxBxxx P

107 – 108 elements

107 –

108

ele

men

ts

δu δv δp δT δq

δu

δ

v

δp

δ

T

δq

B

χKxx ft

(multivariate) model variable

control variable transform

(univariate) control variable

3

2

1

χ

χ

χ

χ

q

T

p

v

u

x

Transform to new variables that are assumed to be univariate

321 χχχ

1χ

2χ

3χ

BT

1T21fff exp~)(~)(

KKBB

χBχχχKx

PP



Ideas of ‘balance’ to formulate K (and hence Pf(xt) \ B)

rft

ft

ft

ft

ft

0

10

0//

0//

p

yx

xy

TT

pp

vv

uu

TH T

H

χKxx

← streamfunction (rot. wind) pert. (assume ‘balanced’)

← velocity potential (div. wind) pert. (assume ‘unbalanced’)

← residual pressure pert. (assume ‘unbalanced’)

H geostrophic balance operator (δψ → δpb)T hydrostatic balance operator (written in terms of temperature)

Approach used at the ECMWF, Met Office, Meteo France, NCEP, MSC(SMC), HIRLAM, JMA, NCAR, CIRAIdea goes back to Parrish & Derber (1992)

kurelation Helmholtz

these are not the same(clash of notation!)

TKKBB

Implied f/c error covariance matrix



Assumptions

This formulation makes many assumptions e.g.:

A. That forecast errors projected onto balanced variables are uncorrelated

with those projected onto unbalanced variables.

B. The rotational wind is wholly a ‘balanced’ variable (i.e. large Bu regime).

C. That geostrophic and hydrostatic balances are appropriate for the motion

being modelled (e.g. small Ro regime).



A. ‘Non-correlation’ test

),cor( rp

latitude

vert

ica

l mo

de

l le

vel

rp

χ



rft

ft

ft

ft

ft

0

1 0

0 //

0 //

p

yx

xy

TT

pp

vv

uu

TH T

H

χKxx

u

b

ft

ft

ft

ft

0

10

///

///

p

xyx

yxy

TT

pp

vv

uu

TH T

H

H

H

Modified transform

B. Rotational wind is not wholly balanced

Standard transform

Could there be an unbalanced component of δψ?

H geostrophic balance operatorT hydrostatic balance operatorH anti-geostrophic balance operator



Non-correlation test for refined model

),cor( rp

latitude

vert

ica

l mo

de

l le

vel

),cor( ub p

Modified transform



C. Are geostrophic and hydrostatic balances always appropriate?

Uf

g

z

p

UHf

P

dt

dw

U

WRo

y

p

ULf

Pu

f

f

dt

dvRo

x

p

ULf

Pv

f

f

dt

duRo

00

00

00

etc. , , , , LxxPppUvvUuu

)10( )10( 21

0

OU

WO

Lf

URo

from Berre, 2000

E.g. test for geostrophic balance



What next? Hi-resolution forecasts need hi-resolution Pf(xt) \ B

High impact weather! The Reading/MetO HRAA Collaboration

www.met.rdg.ac.uk/~hraa

Can forecast error covariances at hi-resolution be successfully modelled with the transform approach?

What is an appropriate transform at hi-resolution? At what scales do hydrostatic and geostrophic balance become

inappropriate?

There is little known theory to guide us at hi-res.

→ What is the structure of forecast error covariances in such cases?



Hi-resolution ensemblesEarly results from Met Office 1.5 km LAM (a MOGREPS-like system)

Thanks to Mark Dixon (MetO), Stefano Migliorini (NCEO), Roger Brugge (NCEO)



Summary

All measurements are inaccurate and all forecasts are wrong!

Accurate knowledge of forecast uncertainty (PDF) is useful:

» to allow range of possible outcomes to be predicted,

» to give allowed ways that a forecast can be modified by observations (data assimilation).

For synoptic/large scales the forecast error PDF is modelled with a change of variables and

balance relations.

For hi-res (convective scales) the forecast error PDF is still important but there is no formal

theory to guide PDF modelling:

» hydrostatic/geostrophic balance less appropriate,

» non-linearity/dynamic tendencies may be more important.

forecasts & their errors ross bannister 7th october 2008 1 / 23...

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