  We show that the structure factor $S(q)$ of water can be obtained from X-ray
synchrotron experiments at grazing angle of incidence (in reflection mode) by
using a liquid surface diffractometer. The corrections used to obtain $S(q)$
self-consistently are described. Applying these corrections to scans at
different incident beam angles (above the critical angle) collapses the
measured intensities into a single master curve, without fitting parameters,
which within a scale factor yields $S(q)$. Performing the measurements below
the critical angle for total reflectivity yields the structure factor of the
top most layers of the water/vapor interface. Our results indicate water
restructuring at the vapor/water interface. We also introduce a new approach to
extract $g(r)$, the pair distribution function (PDF), by expressing the PDF as
a linear sum of Error functions whose parameters are refined by applying a
non-linear least square fit method. This approach enables a straightforward
determination of the inherent uncertainties in the PDF. Implications of our
results to previously measured and theoretical predictions of the PDF are also
discussed.
