# 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 $$\mathcal{H}$$.

Think of $$\mathbf{H}$$ as a matrix operator which allows you to extract the part of the modelled state vector, which is observable. Just to give you a trivial example - assuming your state vector consists of 2D location and two velocity components and what you observe is just 2D location (e.g. obtained from GPS): $$\mathbf{y_o} = \left( \begin{matrix} x_o(t)\\ y_o(t)\\ \end{matrix} \right)$$ $$\mathbf{x_b} = \left( \begin{matrix} x(t)\\ y(t)\\ v_x(t)\\ v_y(t) \end{matrix} \right)$$ $$\mathbf{H}=\left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{matrix} \right)$$ To check how $$\mathbf{H}$$ works, calculate on a sheet of paper the following products: $$\mathbf{H x_b}$$ and $$\mathbf{H^T y_o}$$. To construct a proper $$\mathbf{H}$$ operator you need to understand both what are you observing and what are you modelling.