Quality of model and Error Analysis in Variational Data Assimilation

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Quality of model andError Analysis in
Variational Data Assimilation

• François-Xavier LE DIMET
• Victor SHUTYAEV
• Université Joseph FourierINRIA
• Projet IDOPT, Grenoble, France
• Russian Academy of Sciences
• Institute of Numerical Mathematiques

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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

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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

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Basic Problem

• U and V control variables, V being and error on
  the model
• J cost function
• U and V minimizes J

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Optimality System

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

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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.

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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 ?

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Optimality System including errors on the model
and on the observation
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Second order adjoint
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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?

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A simple numerical experiment.

• Burgers equation with homegeneous B.C.s
• Exact solution is known
• Observations are without error
• Numerical solution with different discretization
• The assimilation is performed between T0 and T1
• Then the flow is predicted at t2.

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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.

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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?

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Because D.A. is a non linear process then the
initial condition is no longer gaussian
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Control of the error
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Choice of the base
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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

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Numerical experiment

• With Burgers equation
• Laplacian and covariance matrix have considered
  separately then jointly
• The number of vectors considered in the correctin
  term varies

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With the eigenvectors of the Laplacian
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• Model error estimation controlled system
• model

• cost function

• optimality conditions

• adjoint system(to calculate the gradient)

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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
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Optimality conditions for the estimation of
model errors after size reduction
If P is the solution of adjoint system, we
search for optimal values of
to minimize J
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• Problem how to choose the spatial base
  ?
• Consider the fastest error propagation direction
• Amplification factor
• Choose as leading eigenvectors of
• Calculus of
• - Lanczos Algorithm

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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

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The difference of potential field between two
models after 8 hours integration
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Experiments without size reduction (108348)
the discrepancy of models at the end of
integration
before optimization
after optimization
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Experiments with size reduction (38048)
the discrepancy of models at the end of
integration
before optimization
after optimization
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Experiments with size reduction (3808)
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 its cheap.
• In term of computational cost it could be
  expensive.
• It is a powerful tool for the analysis and
  identification of errors