  We consider the orbit type filtration on a manifold $X$ with locally standard
action of a compact torus and the corresponding homological spectral sequence
$(E_X)^r_{*,*}$. If all proper faces of the orbit space $Q=X/T$ are acyclic,
and the free part of the action is trivial, this spectral sequence can be
described in full. The ranks of diagonal terms are equal to the $h'$-numbers of
the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend
only on the orbit space $Q$ but not on the characteristic function. If $X$ is a
slightly different object, namely the model space $X=(P\times T^n)/\sim$ where
$P$ is a cone over Buchsbaum simplicial poset $S$, we prove that $\dim
(E_X)^{\infty}_{p,p} = h''_p(S)$. This gives a topological evidence for the
fact that $h''$-numbers of Buchsbaum simplicial posets are nonnegative.
