Introduction to Data Assimilation: Lecture 1

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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
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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

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Approach

• Cant 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

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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)

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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
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• 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
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Why do people do data assimilation?

• To obtain an initial state for launching NWP
  forecasts
• To make consistent estimates of the atmospheric
  state for diagnostic studies.
• reanalyses (eg. ERA-15, ERA-40, NCEP, etc.)
• For an increasingly wide range of applications
  (e.g. atmospheric chemistry)
• To challenge models with data and vice versa
• UKMO analyses during UARS (1991-5) period

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Producing a Numerical Weather Forecast

• Observation
• Collect, receive, format and process the data
• quality control the data
• Analysis
• Use data to obtain a spatial representation of
  the atmosphere
• Initialization
• Filter noise from analysis
• Forecast
• Integrate initial state in time with full PE
  model and parameterized physical processes

Data Assimilation
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Data Assimilation Cycles
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The Global Observing System
http//www.wmo.ch/web/www/OSY/GOS.html
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Observations currently in use at CMC
Canadian Meteorological Centre Centre
Météorologique Canadien
Maps of data used in assimilation on July 1, 2008
12Z
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Radiosonde observations used
U,V,T,P,ES profiles at 27 levels
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Pilot balloon observations used
U,V profiles at 15 levels
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Wind profiler obs used
U,V (speed, dir) profiles at 20 levels
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SYNOP and SHIP obs used
U,V,T,P,ES at surface
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Buoy observations used
U,V,T,P,ES at surface
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Aircraft observations used
T,U,V single level (AIREP,ADS) or up to 18 levels
(BUFR,AMDAR)
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Cloud motion wind obs used
U,V (speed, dir) cloud level
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AMSU-A observations used
Brightness temperatures ch. 3-10
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AMSU-B observations used
Brightness temperatures ch. 2-5
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GOES radiances used
Brightness temperature 1 vis, 4 IR
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Quikscat used
U,V surface
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SSM/I observations used
Related to integrated water vapour, sfc wind
speed, cloud liquid water
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Underdeterminacy
X state vector
Z observation vector
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

• Cannot do Xf(Y), must do Yf(X)
• Problem is underdetermined, always will be
• Need more information prior knowledge, time
  evolution, nonlinear coupling

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Optimal Interpolation
N1
N1
M1
NM
MN
N1
Weight matrix
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Analysis increments (xa xb) must lie in the
subspace spanned by the columns of B
Properties of B determine filtering properties of
assimilation scheme!
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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

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Simple examples of data assimilation
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Analysis error Background error Observation error
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Obs 1
analysis
Daley (1991)
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representativeness
measurement
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OI was the standard assimilation method at
weather centres from the early 1970s to the
early 1990s.
Canada was the first to implement a
multivariate OI scheme.
Gustafsson (1981)
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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