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Errors, Uncertainties in Data AssimilationPowerPoint Presentation

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. 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 ?

Second order adjoint observation

Models and Data observation

- 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 observation.

- 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 observation

- 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 observation

- 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 observation

- 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 condition is no longer gaussian

Choice of the base condition is no longer gaussian

Remark condition is no longer gaussian.

- 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 condition is no longer gaussian

- 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 condition is no longer gaussianin 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

An application in hydrology bias(Yang Junqing )

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

Physical phenomena bias

- fluid and solid transport
- different time scales

N bias

2D sedimentation modeling

1. Shallow-water equations

2. Equation of constituent concentration

3. Equation of the riverbed evolution

Semi-empirical formulas bias

- Bed load function :

- Suspended sediment transport rate :

are empirical constants

An example of simulation bias

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 bias
- model

- cost function

- optimality conditions

- adjoint system(to calculate the gradient)

Reduction of the size of the controlled problem bias

- 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 biasof 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 ? bias
- Consider the fastest error propagation direction
- Amplification factor
- Choose as leading eigenvectors of
- Calculus of
- - Lanczos Algorithm

Numerical experiments with another base bias

- 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) : 8 hours’ integration

the discrepancy of models at the end of integration

before optimization

after optimization

Experiments with size reduction (380*48) : 8 hours’ integration

the discrepancy of models at the end of integration

before optimization

after optimization

Experiments with size reduction (380*8) : 8 hours’ integration

the discrepancy of models at the end of integration

before optimization

after optimization

Conclusion 8 hours’ integration

- 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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