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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
Forecasts & their Errors Ross Bannister 7th October 2008 2 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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).
Forecasts & their Errors Ross Bannister 7th October 2008 3 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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.
Forecasts & their Errors Ross Bannister 7th October 2008 4 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 5 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 6 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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 =
Forecasts & their Errors Ross Bannister 7th October 2008 7 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
Example standard deviations (square-root of variances)
From Ingleby (2001)
Forecasts & their Errors Ross Bannister 7th October 2008 8 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
Example geophysical structure functions (covariances with a fixed point)
Univariate structure function
Multivariate structure functions
Forecasts & their Errors Ross Bannister 7th October 2008 9 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 10 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 11 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 12 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 13 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 14 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 15 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 16 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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).
Forecasts & their Errors Ross Bannister 7th October 2008 17 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
A. ‘Non-correlation’ test
),cor( rp
latitude
vert
ica
l mo
de
l le
vel
rp
χ
Forecasts & their Errors Ross Bannister 7th October 2008 18 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 19 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
Non-correlation test for refined model
),cor( rp
latitude
vert
ica
l mo
de
l le
vel
),cor( ub p
Modified transform
Forecasts & their Errors Ross Bannister 7th October 2008 20 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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
Forecasts & their Errors Ross Bannister 7th October 2008 21 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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?
Forecasts & their Errors Ross Bannister 7th October 2008 22 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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)
Forecasts & their Errors Ross Bannister 7th October 2008 23 / 23
ChallengesRefiningModellingMeasuringAssimilationPDFs
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.