  We perform a systematic study of how the inhomogeneities in the
Inter-Galactic Medium (IGM) affect the observability of Lyman Alpha Emitters
(LAEs) around the Epoch of Reionization. We focus on the IGM close to the
galaxies as the detailed ionization distribution and velocity fields of this
region could significantly influence the scattering of Ly-alpha photons off
neutral H atoms as they traverse the IGM after escaping from the galaxy. We
simulate the surface brightness (SB) maps and spectra of more than 100 LAEs at
z=7.7 as seen by an observer at z=0. To achieve this, we extract the source
properties of galaxies and their surrounding IGM from cosmological simulations
of box sizes 5-30 Mpc/h and follow the coupled radiative transfer of ionizing
and Ly-alpha radiation through the IGM using CRASH-alpha. We find that the
simulated SB profiles are extended and their detailed structure is affected by
inhomogeneities in the IGM, especially at high neutral fractions. The
detectability of LAEs and the fraction of the flux observed depend heavily on
the shape of the SB profile and the SB threshold (SB_th) of the observational
campaign. Only ultradeep observations (e.g. SB_th ~ 10^-23
ergs/s/cm^2/arcsec^2) would be able to obtain the true underlying
mass-luminosity relation and luminosity functions of LAEs. The details of our
results depend on whether Ly-alpha photons are significantly shifted in the
galaxy to longer wavelengths, the mean ionization fraction in the IGM and the
clustering of ionizing sources. These effects can lead to an easier escape of
Ly-alpha photons with less scattering in the IGM and a concentrated SB profile
similar to the one of a point source. Finally, we show that the SB profiles are
steeper at high ionization fraction for the same LAE sample which can
potentially be observed from the stacked profile of a large number of LAEs.
