The background state \(\underline x^b\) can be:
The observation vector
No, the observation vector is denoted by \(\underline y^o\).
An a priori knowledge of the initial state of an evolution model.
Yes, this is the case when the analysis \(\underline x^a\) is looked for as an initial state.
An a priori knowledge of model parameters.
Yes, this is the case when the analysis \(\underline x^a\) is looked for a vector of parameters for the model.