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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

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)


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..

  • 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

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

  • J cost function

  • U* and V* minimizes J


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

  • 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:

  • 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

  • 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.

  • 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


Second order adjoint


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.

  • 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

  • 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

  • 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.


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


Control of the error


Choice of the base


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

  • 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 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


An application in hydrology(Yang Junqing )

  • Retrieve the evolution of a river

  • With transport+sedimentation


Physical phenomena

  • fluid and solid transport

  • different time scales


N

2D sedimentation modeling

1. Shallow-water equations

2. Equation of constituent concentration

3. Equation of the riverbed evolution


Semi-empirical formulas

  • Bed load function :

  • Suspended sediment transport rate :

are empirical constants


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)


  • Model error estimation controlled system

  • model

  • cost function

  • optimality conditions

  • adjoint system(to calculate the gradient)


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

If P is thesolution ofadjoint system, we search for

optimal values of to minimize J :


  • 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

  • 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


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


Experiments without size reduction (1083*48) :

the discrepancy of models at the end of integration

before optimization

after optimization


Experiments with size reduction (380*48) :

the discrepancy of models at the end of integration

before optimization

after optimization


Experiments with size reduction (380*8) :

the discrepancy of models at the end of integration

before optimization

after optimization


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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