  An I-surface $X$ is a surface of general type with $K_X^2 =1$ and $p_g(X)
=2$. This paper studies the asymptotic behavior of the period map for
I-surfaces acquiring simple elliptic singularities. First we describe the
relationship between the deformation theory of such surfaces and their
$d$-semistable models. Next we analyze the mixed Hodge structures on the
$d$-semistable models, the corresponding limiting mixed Hodge structures, and
the monodromy. There are $6$ possible boundary strata for which the relevant
limiting mixed Hodge structures satisfy: $\dim W_1 = 4$, and hence $W_2/W_1$ is
of pure type $(1,1)$. We show that, in each case, the nilpotent orbit of
limiting mixed Hodge structures determines the boundary stratum and prove a
global Torelli theorem for one such stratum.
