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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 $\mathcal{H}$.

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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.

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