data assimilation - european space agency · 2013. 10. 23. · data assimilation alan o’neill...
TRANSCRIPT
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Data Assimilation
Alan O’NeillData Assimilation Research Centre
University of Reading
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Contents
• Motivation• Univariate (scalar) data assimilation• Multivariate (vector) data assimilation
– Optimal Interpolation (BLUE)– 3d-Variational Method– Kalman Filter– 4d-Variational Method
• Applications of data assimilation in earth system science
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Motivation
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What is data assimilation?
Data assimilation is the technique whereby observational data are combined with output from a numerical model to produce an optimal estimate of the evolving state of the system.
DARC
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Why We Need Data Assimilation
• range of observations• range of techniques• different errors• data gaps• quantities not measured• quantities linked
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DARC
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Some Uses of Data Assimilation
• Operational weather and ocean forecasting
• Seasonal weather forecasting• Land-surface process• Global climate datasets• Planning satellite measurements• Evaluation of models and observations
DARC
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Preliminary Concepts
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What We Want To Know
csx
)()(
tt atmos. state vector
surface fluxes
model parameters
)),(),(()( csxX ttt =
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What We Also Want To Know
Errors in models
Errors in observations
What observations to make
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Numerical ModelDAS
DATA ASSIMILATION SYSTEM
O
Data Cache
A
A
B
F
model
observations
Error Statistics
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The Data Assimilation Process
observations forecasts
estimates of state & parameters
compare reject adjust
errors in obs. & forecasts
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X
t
observation
model trajectory
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Data Assimilation:an analogy
Driving with your eyes closed: open eyes every 10 seconds and correct trajectory
DARC
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Basic Concept of Data Assimilation
• Information is accumulated in time into the model state and propagated to all variables.
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What are the benefits of data assimilation?
• Quality control• Combination of data• Errors in data and in model• Filling in data poor regions• Designing observational systems• Maintaining consistency• Estimating unobserved quantities
DARC
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Methods of Data Assimilation
• Optimal interpolation (or approx. to it)
• 3D variational method (3DVar)
• 4D variational method (4DVar)
• Kalman filter (with approximations)
DARC
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Types of Data Assimilation
• Sequential• Non-sequential• Intermittent• Continous
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Sequential Intermittent Assimilation
analysis analysisanalysismodel model
obsobsobsobsobs obs
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Sequential Continuous Assimilation
obs obs obs obs obs
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Non-sequential Intermittent Assimilation
analysis + model analysis + model analysis + model
obs obs obs obs obs obs
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Non-sequential Continuous Assimilation
analysis + model
obsobs
obsobsobs
obs
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Statistical Approach to Data Assimilation
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DARC
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Data Assimilation Made Simple(scalar case)
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Least Squares Method(Minimum Variance)
eduncorrelat
are tsmeasuremen twothe,0 )(
)(
0
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>=<>=<
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>=>=<<+=+=
εεσε
σε
εεεε
t
t
TT
TT
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1
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:nsobservatio theof
ncombinatiolinear a as Estimate
21
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t
=+⇒
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aa
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TaTaT
T
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a
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Least Squares Method Continued
1 constraint thesubject to
)(
))()(()(
:error squaredmean its minimizingby Estimate
21
22
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=+
+>=+=<
>−+−>=<−=<
11
aa
aaaa
TTaTTaTT
T
tttaa
a
σσεε
σ
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Least Squares Method Continued
22
2
2
1 11
1
σσ
σ+
=
1
1a2
22
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2 11
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σσ
σ+
=
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a
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111σσσ
+=⇒a
The precision of the analysis is the sum of the precisions of the measurements. The analysis therefore has higher precision than any single measurement (if the statistics are correct).
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Variational Approach
0for which of value theis
)()(21
)( 22
22
2
21
=∂∂
−+
−=
1
TJTT
TTTTTJ
a
σσ
)(TJ
T
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Maximum Likelihood Estimate
• Obtain or assume probability distributions for the errors
• The best estimate of the state is chosen to have the greatest probability, or maximum likelihood
• If errors normally distributed,unbiased and uncorrelated, then states estimated by minimum variance and maximum likelihood are the same
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Maximum Likelihood Approach (Baysian Derivation)
pdf.prior the2
)(exp )(
:is statisticserror Gaussian for truth theofon distributiy probabilit Then the
on).assimilati datain forecast background the(
,n observatioan madealready have weAssume
2
21
1
−−∝
1σTT
Tp
T
T
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Maximum Likelihood Continued
).statisticsGaussian (for case ldimensiona-multifor holds eEquivalencfunction.cost theminimizing asanswer same get the We
truth. theestimate toln)(or pdfposterior theMaximizeT. oft independenfactor gnormalisin is )( since
2)(
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)(exp
)()()¦(
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2
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Tp
TTTTTp
TpTTpTTp
T
−−
−−∝=
σσ
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Simple Sequential Assimilation
22
1222
21
)1(
:is anceerror vari analysis theand ,)(
:bygiven is weight optimal The
".innovation" theis ) ( where) (
Let
ba
obb
boboba
ob
W
W
W
TTTTWTT
TTTT
σσ
σσσ
−=
+=
−−+=
==
−
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Comments
• The analysis is obtained by adding first guess to the innovation.
• Optimal weight is background error variance multiplied by inverse of total variance.
• Precision of analysis is sum of precisions of background and observation.
• Error variance of analysis is error variance of background reduced by (1- optimal weight).
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Simple Assimilation Cycle
• Observation used once and then discarded.
• Forecast phase to update and • Analysis phase to update and • Obtain background as
• Obtain variance of background as
bT 2bσ
aT 2aσ
)]([)( 1 iaib tTMtT =+
)()(ely alternativ )()( 21
221
2iaibibib tattt σσσσ == ++
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Simple Kalman Filter
22,
221,
21,
,
,,11,
221
)(M)()(
:is covarianceerror backgroundForecast
M whereM
)()()(Then
)biased!not (model ,)]([)(
before. as step Analysis
Q
TM
TMTMTT
QtTMtT
iaibib
mia
mitiaitbib
mmitit
+>==<
∂∂=+=
+−=−=
>=<−=
++
++
+
σεσ
εε
εε
εε
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Multivariate Data Assimilation
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Multivariate Case
=
nx
xx
t 2
1
)(x vector state
=
my
y
y
t 2
1
)( vector nobservatio y
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State Vectors
x
bxtx
ax
state vector (column matrix)
true state
background state
analysis, estimate of tx
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Ingredients of Good Estimate of the State Vector (“analysis”• Start from a good “first guess” (forecast
from previous good analysis)• Allow for errors in observations and first
guess (give most weight to data you trust)
• Analysis should be smooth• Analysis should respect known physical
laws
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Some Useful Matrix Properties
inversion) through conserved isproperty this(. unless 0,scalar the,
:matrix symmetrix for ssdefinitene Positive)()( : transposea of Inverse
)( :product a of Inverse
)( :product a of Transpose
T
T11T
111
TTT
0xxAxx
AAA
ABAB
ABAB
=>∀
=
=
=
−−
−−−
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Observations
• Observations are gathered into an observation vector , called the observation vector.
• Usually fewer observations than variables in the model; they are irregularly spaced; and may be of a different kind to those in the model.
• Introduce an observation operator to map from model state space to observation space.
y
)(xx H→
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Errors
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Variance becomes Covariance Matrix
• Errors in xi are often correlated– spatial structure in flow– dynamical or chemical relationships
• Variance for scalar case becomes Covariance Matrix for vector case COV
• Diagonal elements are the variances of xi
• Off-diagonal elements are covariances between xi and xj
• Observation of xi affects estimate of xj
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The Error Covariance Matrix
( )
><><><
><><><><><><
>==<
=
=
nnnn
n
n
n
n
eeeeee
eeeeeeeeeeee
eee
e
ee
.....................
...
...
...
.
.
.
21
22212
12111
T
21T
2
1
εε
εε
P
2iiiee σ>=<
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Background Errors
• They are the estimation errors of the background state:
• average (bias)• covariance
tbb xx −=ε>< bε
>><−><−=< Tbb ))(( εεεεB
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Observation Errors• They contain errors in the observation
process (instrumental error), errors in the design of , and “representativeness errors”, i.e. discretizaton errors that prevent from being a perfect representation of the true state.
H
tx
)( to xy H−=ε >< oε
>><−><−=< Toooo ))(( εεεεR
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Control Variables
• We may not be able to solve the analysis problem for all components of the model state (e.g. cloud-related variables, or need to reduce resolution)
• The work space is then not the model space but the sub-space in which we correct , called control-variable space
bxxxx δ+= ba
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Innovations and Residuals
• Key to data assimilation is the use of differences between observations and the state vector of the system
• We call the innovation
• We call the analysisresidual
)( bxy H−)( axy H−
Give important information
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Analysis Errors
• They are the estimation errors of the analysis state that we want to minimize.
taa xx −=ε
Covariance matrix A
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Using the Error Covariance Matrix
Recall that an error covariance matrix for the error in has the form:
T >=< εεC
If where is a matrix, then the error covariance for is given by:
Hxy = H
x
yTHCHC =y
C
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BLUE Estimator• The BLUE estimator is given by:
• The analysis error covariance matrix is:
• Note that:
1TT
bba
)(
))((−+=
−+=
RHBHBHK
xyKxx H
BKHIA )( −=
1T11T11TT )()( −−−−− +=+ RHHRHBRHBHBH
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Statistical Interpolation with Least Squares Estimation
• Called Best Linear Unbiased Estimator (BLUE).
• Simplified versions of this algorithm yield the most common algorithms used today in meteorology and oceanography.
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Assumptions Used in BLUE• Linearized observation operator:
• and are positive definite.• Errors are unbiased:
• Errors are uncorrelated:
• Linear anlaysis: corrections to background depend linearly on (background – obs.).
• Optimal analysis: minimum variance estimate.
)()()( bb xxHxx −=− HHB R
0)( ttb >=−>=<−< xyxx H
0))()(( Tttb >=−−< xyxx H
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Optimal Interpolation
))(( bba xyKxx H−+=
“analysis”
“background”(forecast)
observation
1RHBHBHK −+= )( TT
• linearity H H
• matrix inverse
• limited area
observation operator
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∆
∆
∆
∆
∆
∆∆
∆
)( bxy H−=∆ at obs. point
bx
data void
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END