Introduction to data assimilation 

The Baye's rule is based on conditional probabilities.

By definition,

and this is the Baye's rule.

What des it means for our job?

Classical notations for data assimilation have been introduced by [ICGL97], we try to follow them as possible as we can. We denote by

the digital representation of the system (atmosphere, ocean,..) at time

the observations of the system at time With these notations, the Baye's rule states that

or more simply  where the normalization term is forgotten. This is called the analysis step.

If we combine the forecast step and the analysis step, we obtain the following evolution of information:

where

• denotes the uncertainty on the state at time knowing observations until time ,

• the one for at time conditionned by the knowledge of the new observations at time , this is the analysis step

• denotes the uncertainty on the state at time knowing observations until time , this is the forecast step

Keep in mind that here, we have three processes:

1. The real process , that is the realization of the system (the weather we are seeing day after days).

2. The analysis process , that is our knowledge of the real process  knowing all observations until time .

3. The forecast process  that is the time evolution thanks to the dynamics of the analysis process, that is (without model error) .

An optimal state is a state that minimizes the variance of error.

It can be shown that

minimize the variance of error that is, if  (we assume this random value is centered, that is  reaches its minimum for .

is the analysis error and the matrix

is the analysis covariance matrix.

Similarly,

is also an optimal estimator at time .

The error  is the forecast error and the matrix  is the forecas error covariance matrix.