Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division...

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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

Transcript of Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division...

Page 1: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Introduction to Data Assimilation: Lecture 1

Saroja Polavarapu

Meteorological Research Division Environment Canada

PIMS Summer School, Victoria. July 14-18, 2008

Page 2: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 3: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 4: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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)

Page 5: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 6: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

• 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

Page 7: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 8: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 9: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Data Assimilation Cycles

Page 10: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

http://www.wmo.ch/web/www/OSY/GOS.html

The Global Observing System

Page 11: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Observations currently in use at CMC

Maps of data used in assimilation onJuly 1, 2008 12Z

Canadian Meteorological Centre – Centre Météorologique Canadien

Page 12: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Radiosonde observations used

U,V,T,P,ES profiles at 27 levels

Page 13: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Pilot balloon observations used

U,V profiles at 15 levels

Page 14: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Wind profiler obs used

U,V (speed, dir) profiles at 20 levels

Page 15: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

SYNOP and SHIP obs used

U,V,T,P,ES at surface

Page 16: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Buoy observations used

U,V,T,P,ES at surface

Page 17: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Aircraft observations used

T,U,V single level (AIREP,ADS) or up to 18 levels (BUFR,AMDAR)

Page 18: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Cloud motion wind obs used

U,V (speed, dir) cloud level

Page 19: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

AMSU-A observations used

Brightness temperatures ch. 3-10

Page 20: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

AMSU-B observations used

Brightness temperatures ch. 2-5

Page 21: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

GOES radiances used

Brightness temperature 1 vis, 4 IR

Page 22: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Quikscat used

U,V surface

Page 23: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

SSM/I observations used

Related to integrated water vapour, sfc wind speed, cloud liquid water

Page 24: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 25: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 26: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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!

Page 27: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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

Page 28: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Simple examples of data assimilation

Page 29: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Analysis errorBackground errorObservation error

Page 30: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
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Page 32: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 33: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 34: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 35: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
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Page 37: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 38: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

Obs 1 analysis

Daley (1991)

Page 39: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

m x 1n x 1

n x m

n x 1 m x 1

Page 40: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

representativeness measurement

Page 41: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

n x 1

m x 1n x 1

Page 42: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 43: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 44: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,
Page 45: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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)

Page 46: Introduction to Data Assimilation: Lecture 1 Saroja Polavarapu Meteorological Research Division Environment Canada PIMS Summer School, Victoria. July 14-18,

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