  In the framework of the hierarchical model the intra-cluster medium
properties of galaxy clusters are tightly linked to structure formation, which
makes X-ray surveys well suited for cosmological studies. To constrain
cosmological parameters accurately by use of galaxy clusters in X-ray surveys,
a better understanding of selection effects related to the detection method is
needed. We aim at a better understanding of galaxy cluster morphologies to
include corrections between the different core types and covariances with X-ray
luminosities in selection functions. We stress the morphological deviations
between a newly described surface brightness (SB) profile characterization and
a commonly used single $\beta$-model. We investigate a novel approach to
describe SB profiles, where the excess cool-core emission in the galaxy cluster
centres is modelled using wavelet decomposition. Morphological parameters and
the residuals are compared to classical single $\beta$-models. Using single
$\beta$-models to describe the ensemble of overall SB profiles leads on average
to a non-zero bias ($0.032 \pm 0.003$) in the outer part of the clusters, i.e.
a $\sim 3\%$ systematic difference in the SB at large radii. In addition
$\beta$-models show a general trend towards underestimating the flux in the
outskirts for smaller core radii. Fixing the $\beta$ parameter to $2/3$ doubles
the bias and increases the residuals from a single $\beta$-model up to more
than $40\%$. Modelling the core region in the fitting procedure reduces the
impact of these two effects significantly. We find a positive scaling between
shape parameters and temperature, as well as a negative correlation
($\sim-0.4$) between extent and luminosity. Our non-parametric analysis of the
self-similar scaled emission measure profiles indicates no systematic core-type
differences of median profiles in the galaxy clusters outskirts.
