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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
ApproachApproach
• 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• 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 AnalysisGoal: 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• Collect, receive, format and process the data• quality control the data
2. Analysis• Use data to obtain a spatial representation of the atmosphere
3. Initialization• Filter noise from analysis
4. Forecast • Integrate initial state in time with full PE model and
parameterized physical processes
Dat
a A
ssim
ilatio
n
Data Assimilation Cycles
http://www.wmo.ch/web/www/OSY/GOS.html
The Global Observing System
Observations currently in use at CMC
Maps of data used in assimilation onJuly 1, 2008 12Z
Canadian Meteorological Centre – Centre Météorologique Canadien
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
75Z
X
N
N
Underdeterminacy
• Cannot do X=f(Y), must do Y=f(X)• Problem is underdetermined, always will be• Need more information: prior knowledge, time evolution, nonlinear
coupling
Data Reports x items x levels
Sondes,pibal 720x5x27
AMSU-A,B 14000x12
SM, ships, buoys 7000x5
aircraft 19000x3x18
GOES 5000x1
Scatterometer 7000x2
Sat. winds 21000x2
TOTAL 1.3x106
Model Lat x long x lev x variables
CMC global oper. 800x600x58x4
=1x108
CMC meso-strato 800x600x80x4
=1.5x108
X = state vector Z = observation vector
Optimal Interpolation
)( bba H xzKxx Analysis vector
Background or model forecast
Observation vector
Observation operator
Weight matrix
N×1 N×1 M×1N×M M×N N×1
1 RHBHBHK TT
NxN MxM
Can’t invert!
NxM
Bvxx ba
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 errorBackground errorObservation error
Obs 1 analysis
Daley (1991)
m x 1n x 1
n x m
n x 1 m x 1
representativeness measurement
n x 1
m x 1n x 1
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 multivariateOI 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