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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 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

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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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**Analysis = Interpolation + Filtering**

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**

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**Observations currently in use at CMC**

Canadian Meteorological Centre – Centre Météorologique Canadien Observations currently in use at CMC Maps of data used in assimilation on July 1, Z

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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 Cannot do X=f(Y), must do Y=f(X)**

X = state vector Z = observation vector Model Lat x long x lev x variables CMC global oper. 800x600x58x4 =1x108 CMC meso-strato 800x600x80x4 =1.5x108 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 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

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**Optimal Interpolation**

N×M M×1 M×N N×1 Analysis vector Background or model forecast Observation vector Observation operator Weight matrix NxN MxM Can’t invert! NxM

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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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m x 1 n x 1 n x m

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representativeness measurement

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n x 1 m x 1

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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)

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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

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