  Extended X-ray emission can be studied either spatially (through its surface
brightness profile) or spectrally (by analyzing the spectrum at various
locations in the field). Both techniques have advantages and disadvantages, and
when the emission becomes particularly faint and/or extended, the two methods
can disagree. We argue that an ideal approach would be to model the events file
directly, and therefore to use both the spectral and spatial information which
are simultaneously available for each event. In this work we propose a first
step in this direction, introducing a method for spatial analysis which can be
extended to leverage spectral information simultaneously. We construct a model
for the entire X-ray image in a given energy band, and generate a likelihood
function to compare the model to the data. A critical goal of this modeling is
disentangling vignetted and unvignetted backgrounds through their different
spatial distributions. Employing either maximum likelihood or Markov Chain
Monte Carlo, we can derive probability distribution functions for the source
and background parameters together, or we can fit and subtract the background,
leaving the description of the source non-parametric. We calibrate and
demonstrate this method against a variety of simulated images, and then apply
it to Chandra observations of the hot gaseous halo around the elliptical galaxy
NGC 720. We are able to follow the X-ray emission below a tenth of the
background, and to infer a hot gas mass within 35 kpc of 4-5x10^9 Msun, with
some indication that the profile continues to at least 50 kpc and that it
steepens as the radius increases. We derive much stronger constraints on the
surface brightness profile than previous studies, which employed the spectral
method, and we show that the density profiles inferred from these studies are
in conflict with the observed surface brightness profile. (abridged)
