Numerical methods for Data Assimilation

It is the process of estimating

• The true state of a system at a given time

• Possibly other model parameters

It is based on

• Observational data (e.g. physical measurements)

• A model of the physical system 

• Some ”a priori” information on the system

• Possibly additional constraints (desired properties of the solution that are not present in the model)

Based on 

• Synchronous observations on-board ships (surface pressure)

• Model parameters : temperature, wind and pressure

• An evolution model defined on a spatial grid

This analysis was the following

1. Estimation of a first guess field obtained by extrapolation (time integration) of the model

2. Interpolation to get a predicted value at the observation location

3. Computation of the observed minus predicted quantities (a misfit)

4. Interpolation back on the model grid point of the misfit

5. Correction of the model parameters to reduce the misfit

This led to the optimal interpolation process still in use.

In the domain of the numerical weather forecasting,

• The analysis prepares intial conditions for weather forecasts

• Observations of various types are available (data from satellites, ground stations, commercial flights, sounding balloons)

• The discrepency between observed and predicted values is minimized : optimization problem

We will focus on the formulation and solution of the optimization problem. Note however that to have a rea- sonable solution, the data must be of a good quality in terms of coverage, and the models for the data have to be accurate.

• Solving a linear problem may be chalenging if it is large and/or very ill-conditioned

• Nonlinearity introduces additional difficulties.

• For instance on the Duffing's equation sensitivity with respect to initial conditions. Take for as truth , and consider the perturbed conditions , and , .

• Comparison of the solutions obtained by time integration and interpret error in intial condition as analysis error.

• The analysis error reduces the time period for which the forecast is close to the truth.

• Until , the forecast seems reasonable with a perturbation

• With a perturbation the two trajectories diverges already for

• The reduction of the nonlinear term enlarges the time validity for the forecast

The domain can be discussed from many angles :

1. Variational analysis

2. Optimization/Control theory

3. Estimation theory

4. Probability theory

5. Numerical optimization/linear algebra

6. High performance computing on parallel machines

Goal of this course show the connections between all this fields in the framework of Data Assimilation

1. Goal : find the initial state of a dynamical system to perform forecasts

2. Use observations and a model of the system

3. Determine the initial state by solving an optimization problem (here, a control problem). Minimize the

   discrepancy between observations and model.

• A priori knowledge on values of

• Observations :

• Observation model + noise

• Dynamical model + noise

• Find

• The simplest instance of a Data Assimilation problem is a linear least squares problem

• Typical sizes would be for this problem  unknowns and observations (including a priori infomation)

• The problem is not sparse

• If no particular structure taken into account, the solution of the problem on a modern operations/s

  computer would take 200 centuries of computation by the normal equations

• In terms of memory, no available computer (2006) is able to store in core memory the matrix

• Therefore parallel iterative methods are sought a are run on parallel computers