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