III. Data assimilation on the model.

Two models were used in order to show the impact of data assimilation.

The first model directly integrates the speed at the date t, in order to obtain the position at date t+dt. The speed variation is calculated using the difference of angles between the speed and the line joining the center if the earth to the sattelite. The satelites thus moves in a linear way from observation to observation, and the model can only diverge.
The second model is based on the first model, but using it many time. The integration is nearer to the real solution and should need less observations.

The assimilation model is 3D-VAR : at given observation dates, the model is compared to these observations, having as much confidence in the model as in the observations. The data given contains all 4 parameters ( position and speed), so the matrix used in the cost function is a dimension 4 identity matrix.

The first model :

The solution shows that the model would diverge if it was not constantly corrected. This shows how a wrong model can be lead to give an acceptable solution by assimilating the observations during the simulation.

The solution obtained using data assimilation is quite close to the real solution, although the model is completly false. This can help a model converge, but only allows simulation of states on wich data has been collected.
Even though this case shows the use of data assimilation as a powerful tool, the scientific interest of this use is very small. The second model, being closer to the given data can be used as a better way to use data assimilation.

The second model

This model diverges, but is really more acceptable than the first model. A view of the real solution, compared to the given solution shows the error made by the model :

Using data assimilation on the same observations as on the first model leads to the following solution :

The solution is now really close to what the real solution is, usin the base of a model that would diverge.. The aim of data assimilation is to have a model able to predict what will happen, using disponble data. With as many observations as this, the predictions are not done on long term. The case is studied with 4 observation, on one revolution around the earth :

The solution still is quite close to what it should be.A study of what the trajectories would give without the correction shows the predictions made, and the evolution of these predictions as observations arrive.

This allows to have quite good prevision for short and middle term studies. but it also shows, that somtimes the model without correction can be more precise than with the given corrections. The importance given to each observed parameter should be very carefully studied, as well as the influence of different parameters on eachothers.

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