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Errors, Uncertainties in Data Assimilation. François-Xavier LE DIMET Université Joseph Fourier+INRIA Projet IDOPT, Grenoble, France. Acknowlegment. Pierre Ngnepieba ( FSU) Youssuf Hussaini ( FSU) Arthur Vidard ( ECMWF) Victor Shutyaev ( Russ. Acad. Sci.) Junqing Yang ( LMC , IDOPT).

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Errors, Uncertainties in Data Assimilation

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Errors uncertainties in data assimilation l.jpg

Errors, Uncertainties in Data Assimilation

François-Xavier LE DIMET

Université Joseph Fourier+INRIA

Projet IDOPT, Grenoble, France


Acknowlegment l.jpg

Acknowlegment

  • Pierre Ngnepieba ( FSU)

  • Youssuf Hussaini ( FSU)

  • Arthur Vidard ( ECMWF)

  • Victor Shutyaev ( Russ. Acad. Sci.)

  • Junqing Yang ( LMC , IDOPT)


Prediction what information is necessary l.jpg

Prediction: What information is necessary ?

  • Model

    • law of conservation mass, energy

    • Laws of behaviour

    • Parametrization of physical processes

  • Observations in situ and/or remote

  • Statistics

  • Images


Forecast l.jpg

Forecast..

  • Produced by the integration of the model from an initial condition

  • Problem : how to link together heterogeneous sources of information

  • Heterogeneity in :

    • Nature

    • Quality

    • Density


Basic problem l.jpg

Basic Problem

  • U and V control variables, V being and error on the model

  • J cost function

  • U* and V* minimizes J


Optimality system l.jpg

Optimality System

  • P is the adjoint variable.

  • Gradients are couputed by solving the adjoint model then an optimization method is performed.


Remark on statistical information l.jpg

Remark on statistical information

  • Statistical information is included in the assimilation

  • In the norm of the discrepancy between the solution of the model ( approximation of the inverse of the covariance matrix)

  • In the background term ( error covariance matrix)


Remarks l.jpg

Remarks:

  • This method is used since May 2000 for operational prediction at ECMWF and MétéoFrance, Japanese Meteorological Agency ( 2005) with huge models ( 10 millions of variable.

  • The Optimality System countains all the available information

  • The O.S. should be considered as a « Generalized Model »

  • Only the O.S. makes sense.


Errors l.jpg

Errors

  • On the model

    • Physical approximation (e.g. parametrization of subgrid processes)

    • Numerical discretization

    • Numerical algorithms ( stopping criterions for iterative methods

  • On the observations

    • Physical measurement

    • Sampling

    • Some « pseudo-observations », from remote sensing, are obtained by solving an inverse problem.


Sensitivity of the initial condition with respect to errors on the models and on the observations l.jpg

Sensitivity of the initial condition with respect to errors on the models and on the observations.

  • The prediction is highly dependant on the initial condition.

  • Models have errors

  • Observations have errors.

  • What is the sensitivity of the initial condition to these errors ?


Optimality system including errors on the model and on the observation l.jpg

Optimality System : including errors on the model and on the observation


Second order adjoint l.jpg

Second order adjoint


Models and data l.jpg

Models and Data

  • Is it necessary to improve a model if data are not changed ?

  • For a given model what is the « best » set of data?

  • What is the adequation between models and data?


A simple numerical experiment l.jpg

A simple numerical experiment.

  • Burger’s equation with homegeneous B.C.’s

  • Exact solution is known

  • Observations are without error

  • Numerical solution with different discretization

  • The assimilation is performed between T=0 and T=1

  • Then the flow is predicted at t=2.


Partial conclusion l.jpg

Partial Conclusion

  • The error in the model is introduced through the discretization

  • The observations remain the same whatever be the discretization

  • It shows that the forecast can be downgraded if the model is upgraded.

  • Only the quality of the O.S. makes sense.


Remark 1 l.jpg

Remark 1

  • How to improve the link between data and models?

  • C is the operator mapping the space of the state variable into the space of observations

  • We considered the liear case.


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Remark 2 : ensemble prediction

  • To estimate the impact of uncertainies on the prediction several prediction are performed with perturbed initial conditions

  • But the initial condition is an artefact : there is no natural error on it . The error comes from the data throughthe data assimilation process

  • If the error on the data are gaussian : what about the initial condition?


Because d a is a non linear process then the initial condition is no longer gaussian l.jpg

Because D.A. is a non linear process then the initial condition is no longer gaussian


Control of the error l.jpg

Control of the error


Choice of the base l.jpg

Choice of the base


Remark l.jpg

Remark .

  • The model has several sources of errors

  • Discretization errors may depends on the second derivative : we can identify this error in a base of the first eigenvalues of the Laplacian

  • The systematic error may depends be estimated using the eigenvalues of the correlation matrix


Numerical experiment l.jpg

Numerical experiment

  • With Burger’s equation

  • Laplacian and covariance matrix have considered separately then jointly

  • The number of vectors considered in the correctin term varies


An application in oceanography in a vidard s ph d l.jpg

An application in oceanographyin A. Vidard’s Ph.D.

  • Shallow water on a square domain with a flat bottom.

  • An bias term is atted into the equation and controlled


Rms ot the sea surface height with or without control of the bias l.jpg

RMS ot the sea surface height with or without control of the bias


An application in hydrology yang junqing l.jpg

An application in hydrology(Yang Junqing )

  • Retrieve the evolution of a river

  • With transport+sedimentation


Slide45 l.jpg

Physical phenomena

  • fluid and solid transport

  • different time scales


Slide46 l.jpg

N

2D sedimentation modeling

1. Shallow-water equations

2. Equation of constituent concentration

3. Equation of the riverbed evolution


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Semi-empirical formulas

  • Bed load function :

  • Suspended sediment transport rate :

are empirical constants


An example of simulation l.jpg

An example of simulation

Initial river bed

  • Domain :

  • Space step : 2 km in two directions

  • Time step : 120 seconds

Simulated evolution

of river bed (50 years)


Slide49 l.jpg

  • Model error estimation controlled system

  • model

  • cost function

  • optimality conditions

  • adjoint system(to calculate the gradient)


Reduction of the size of the controlled problem l.jpg

Reduction of the size of the controlled problem

  • Change the space bases

Suppose is a base of the phase space and

is time-dependent base function on [0, T], so that

then the controlled variables are changed to

with controlled space size


Optimality conditions for the estimation of model errors after size reduction l.jpg

Optimality conditions for the estimation of model errors after size reduction

If P is thesolution ofadjoint system, we search for

optimal values of to minimize J :


Slide52 l.jpg

  • Problem : how to choose the spatial base ?

  • Consider the fastest error propagation direction

  • Amplification factor

  • Choose as leading eigenvectors of

  • Calculus of

  • - Lanczos Algorithm


Numerical experiments with another base l.jpg

Numerical experiments with another base

  • Choice of “correct” model :

  • - fine discretization: domain with 41 times 41 grid points

  • To get the simulated observation

  • - simulation results of ‘correct’ model

  • Choice of “incorrect” model :

  • - coarse discretization: domain with 21 times 21 grid points


Slide54 l.jpg

The difference of potential field between two models after 8 hours’ integration


Slide55 l.jpg

Experiments without size reduction (1083*48) :

the discrepancy of models at the end of integration

before optimization

after optimization


Slide56 l.jpg

Experiments with size reduction (380*48) :

the discrepancy of models at the end of integration

before optimization

after optimization


Slide57 l.jpg

Experiments with size reduction (380*8) :

the discrepancy of models at the end of integration

before optimization

after optimization


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Conclusion

  • For Data assimilation, Controlling the model error is a significant improvement .

  • In term of software development it’s cheap.

  • In term of computational cost it could be expensive.

  • It is a powerful tool for the analysis and identification of errors


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