Observation Operator - Data Assimilation

In data assimilation, one assumes the existence of a observation operator $\mathcal{H}$ that maps the model-state vector $\bf{x_b}$ to $ \bf{y_b}$ (the model-equivalent of the observations $\bf{y_o}$) according to a reference I'm using to develop a preliminary understanding of DA.

Can someone please elaborate on the precise meaning of:

model-equivalent of the observations $\bf{y_o}$

and the methods one can use to estimate the operator $\mathcal{H}$.