  Our previous analysis indicates that small-scale fluctuations in the
intracluster medium (ICM) from cosmological hydrodynamic simulations follow the
lognormal distribution. In order to test the lognormal nature of the ICM
directly against X-ray observations of galaxy clusters, we develop a method of
extracting statistical information about the three-dimensional properties of
the fluctuations from the two-dimensional X-ray surface brightness.
  We first create a set of synthetic clusters with lognormal fluctuations.
Performing mock observations of these synthetic clusters, we find that the
resulting X-ray surface brightness fluctuations also follow the lognormal
distribution fairly well. Systematic analysis of the synthetic clusters
provides an empirical relation between the density fluctuations and the X-ray
surface brightness. We analyze \chandra observations of the galaxy cluster
Abell 3667, and find that its X-ray surface brightness fluctuations follow the
lognormal distribution. While the lognormal model was originally motivated by
cosmological hydrodynamic simulations, this is the first observational
confirmation of the lognormal signature in a real cluster. Finally we check the
synthetic cluster results against clusters from cosmological hydrodynamic
simulations. As a result of the complex structure exhibited by simulated
clusters, the empirical relation shows large scatter. Nevertheless we are able
to reproduce the true value of the fluctuation amplitude of simulated clusters
within a factor of two from their X-ray surface brightness alone.
  Our current methodology combined with existing observational data is useful
in describing and inferring the statistical properties of the three dimensional
inhomogeneity in galaxy clusters.
