  We fix a complex analytic normal singularity germ $(X,o)$ of dimension $\geq
2$ and a (not necessarily irreducible) reduced Weil divisor $(S,o)\subset
(X,o)$. The embedded resolution of the pair determines a multi-index filtration
of the local ring $\mathcal{O}_{X,o}$, which measures the embedded geometry of
the pair. Furthermore, from the (induced) resolution of $(S,o)$ we also
consider a multi-index filtration associated with $(S,o)$. This latter one can
be lifted to a filtration of $\mathcal{O}_{X,o}$ too. The main result proves
that the second filtration of $\mathcal{O}_{X,o}$ can be realized as a `limit'
filtration of the first one (if we blow up certain centers sufficiently many
times).
