We define the {\em level} of a subset $X$ of Euclidean space to be the
dimension of the smallest subspace such that the distance between each
element of $X$ and the subspace is bounded. We prove that the number
of faces in the $n$-dimensional extended Shi arrangement
${\widehat{\mathcal A}}_{n}^{r}$ having codimension $k$ and level $m$
is given by
$m \cdot {n \choose k} \cdot
\Delta_{r}^{k} \Delta^{m-1} x^{n-1}
\left| {_{x = r n - 1}} \right.$
where $\Delta$ is the difference operator and $\Delta_{r}$ is the
difference operator of step $r$, that is,
$\Delta_{r} p(x) = p(x) - p(x-r)$.
This generalizes a result of Athanasiadis which counts the number of
faces of different dimensions from the Shi
arrangement~${\widehat{\mathcal A}}_{n}$. The proof relies on
extending Athanasiadis' result to ${\widehat{\mathcal A}}_{n}^{r}$ and
applying a multi-variated Abel identity.