Transcript Introduction to Data Assimilation

Introduction to Data
Assimilation: Lecture 1
Saroja Polavarapu
Meteorological Research Division
Environment Canada
PIMS Summer School, Victoria. July 14-18, 2008
Goals of these lectures
• Basic idea of data assimilation (combining
measurements and models)
• Basic processes of assimilation
(interpolation and filtering)
• How a weather forecasting system works
• Some common schemes (OI, 3D, 4D-Var)
• Progress over the past few decades
• Assumptions, drawbacks of schemes
• Advantages and limitations of DA
Approach
• Can’t avoid equations– but there are only a few
(repeated many times)
• Deriving equations is important to understanding
key assumptions
• Introduce standard equations using common
notation in meteorological DA literature
• Introduce concepts and terminology used by
assimilators (e.g. forward model, adjoint model,
tangent linear model…)
• Introduce topics using a historical timeline
Outline of lectures 1-2
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General idea
Numerical weather prediction context
Fundamental issues in atmospheric DA
Simple examples of data assimilation
Optimal Interpolation
Covariance Modelling
Initialization (Filtering of analyses)
Basic estimation theory
3D-Variational Assimilation (3Dvar)
Atmospheric Data Analysis
Goal: To produce a regular, physically consistent,
four-dimensional representation of the state of
the atmosphere from a heterogeneous array of
in-situ and remote instruments which sample
imperfectly and irregularly in space and time.
(Daley, 1991)
analysis
• Approach: Combine information from past observations,
brought forward in time by a model, with information from
new observations, using
– statistical information on model and observation errors
– the physics captured in the model
• Observation errors
– Instrument, calibration, coding, telecommunication errors
• Model errors
– “representativeness”, numerical truncation, incorrect or missing
physical processes
Analysis = Interpolation + Filtering
Why do people do data assimilation?
1. To obtain an initial state for launching NWP
forecasts
2. To make consistent estimates of the atmospheric
state for diagnostic studies.
• reanalyses (eg. ERA-15, ERA-40, NCEP, etc.)
3. For an increasingly wide range of applications
(e.g. atmospheric chemistry)
4. To challenge models with data and vice versa
• UKMO analyses during UARS (1991-5) period
Producing a Numerical Weather
Forecast
1. Observation
•
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Collect, receive, format and process the data
quality control the data
Data Assimilation
2. Analysis
•
Use data to obtain a spatial representation of the atmosphere
3. Initialization
•
Filter noise from analysis
4. Forecast
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Integrate initial state in time with full PE model and
parameterized physical processes
Data Assimilation Cycles
The Global Observing System
http://www.wmo.ch/web/www/OSY/GOS.html
Canadian Meteorological Centre – Centre Météorologique Canadien
Observations currently
in use at CMC
Maps of data used in assimilation on
July 1, 2008 12Z
Radiosonde observations used
U,V,T,P,ES profiles at 27 levels
Pilot balloon observations used
U,V profiles at 15 levels
Wind profiler obs used
U,V (speed, dir) profiles at 20 levels
SYNOP and SHIP obs used
U,V,T,P,ES at surface
Buoy observations used
U,V,T,P,ES at surface
Aircraft observations used
T,U,V single level (AIREP,ADS) or up to 18 levels (BUFR,AMDAR)
Cloud motion wind obs used
U,V (speed, dir) cloud level
AMSU-A observations used
Brightness temperatures ch. 3-10
AMSU-B observations used
Brightness temperatures ch. 2-5
GOES radiances used
Brightness temperature 1 vis, 4 IR
Quikscat used
U,V surface
SSM/I observations used
Related to integrated water vapour, sfc wind speed, cloud liquid water
Underdeterminacy
X = state vector
Z = observation vector
Model
Lat x long x lev x
variables
Data
Reports x items x
levels
CMC global oper.
800x600x58x4
=1x108
Sondes,pibal
720x5x27
AMSU-A,B
14000x12
800x600x80x4
=1.5x108
SM, ships, buoys
7000x5
aircraft
19000x3x18
GOES
5000x1
Scatterometer
7000x2
Sat. winds
21000x2
CMC meso-strato
NX
 75
NZ
6
1.3x10
• Cannot do X=f(Y), must do Y=f(X) TOTAL
• Problem is underdetermined, always will be
• Need more information: prior knowledge, time evolution, nonlinear
coupling
Optimal Interpolation
N×1
N×1
N×M

M×1
M×N
N×1
x  x  K z  H (x )
a
Analysis
vector
b
Background or
model forecast
b
Observation
vector
Observation
operator
Weight matrix
NxM
NxN

MxM
K  BH HBH  R
T
T
Can’t invert!

1

x  x  Bv
a
b
Analysis increments (xa – xb) must lie
in the subspace spanned by the
columns of B
Properties of B determine filtering
properties of assimilation scheme!
The fundamental issues in
atmospheric data assimilation
• Problem is under-determined: not enough
observations to define the state
• Forecast error covariances cannot be
determined from observations. They must be
stat. modelled using only a few parameters.
• Forecast error covariances cannot be known
exactly yet analysis increments are composed of
linear combination of columns of this matrix
• Very large scale problem. State ~ O(108)
• Nonlinear chaotic dynamics
Simple examples of
data assimilation
Analysis error
Background error
Observation error
Obs 1 analysis
Daley (1991)
nxm
nx1
nx1
mx1 mx1
representativeness
measurement
nx1
nx1
mx1
OI was the standard assimilation
method at weather centres from the
early 1970’s to the early 1990’s.
Canada was the first to
implement a multivariate
OI scheme.
Gustafsson (1981)
Summary (Lecture 1)
• Data assimilation combines information of
observations and models and their errors
to get a best estimate of atmospheric state
(or other parameters)
• The atmospheric DA problem is
underdetermined. There are far fewer
observations than is needed to define a
model state.
• Optimal Interpolation is a variance
minimizing scheme which combines obs
with a background field to obtain an
analysis