  We perform Lyman-$\alpha$ (Ly$\alpha$) Monte-Carlo radiative transfer
calculations on a suite of $2500$ models of multiphase, outflowing media, which
are characterized by $14$ parameters. We focus on the Ly$\alpha$ spectra
emerging from these media, and investigate which properties are dominant in
shaping the emerging Ly$\alpha$ profile. Multiphase models give rise to a wide
variety of emerging spectra, including single, double and triple peaked
spectra. We find that the dominant parameters in shaping the spectra include
(i) the cloud covering factor, $f_c$, in agreement with earlier studies, and
(ii) the temperature and number density of residual HI in the hot ionized
medium. We attempt to reproduce spectra emerging from multiphase models with
`shell models' which are commonly used to fit observed Ly$\alpha$ spectra, and
investigate the connection between shell-model parameters and the physical
parameters of the clumpy media. In shell models, the neutral hydrogen content
of the shell is one of the key parameters controlling Ly$\alpha$ radiative
transfer. Because Ly$\alpha$ spectra emerging from multi-phase media depend
much less on the neutral hydrogen content of the clumps, the shell model
parameters such as HI column density (but also shell velocity and dust content)
are generally not well matched to the associated physical parameters of clumpy
media.

Astrophysics of Galaxies