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Errors, Uncertainties in Data Assimilation

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Errors, Uncertainties in Data Assimilation

François-Xavier LE DIMET

Université Joseph Fourier+INRIA

Projet IDOPT, Grenoble, France

- Pierre Ngnepieba ( FSU)
- Youssuf Hussaini ( FSU)
- Arthur Vidard ( ECMWF)
- Victor Shutyaev ( Russ. Acad. Sci.)
- Junqing Yang ( LMC , IDOPT)

- Model
- law of conservation mass, energy
- Laws of behaviour
- Parametrization of physical processes

- Observations in situ and/or remote
- Statistics
- Images

- 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

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

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

- 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)

- 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.

- 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.

- 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 ?

- 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?

- 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.

- 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.

- 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.

- 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?

- 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

- With Burger’s equation
- Laplacian and covariance matrix have considered separately then jointly
- The number of vectors considered in the correctin term varies

- Shallow water on a square domain with a flat bottom.
- An bias term is atted into the equation and controlled

- Retrieve the evolution of a river
- With transport+sedimentation

Physical phenomena

- fluid and solid transport
- different time scales

2D sedimentation modeling

1. Shallow-water equations

2. Equation of constituent concentration

3. Equation of the riverbed evolution

- Bed load function :

- Suspended sediment transport rate :

are empirical constants

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)

- 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

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

- 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

- 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