Herschel observations of EXtra-Ordinary Sources: Analysis of the HIFI
  1.2 THz Wide Spectral Survey Toward Orion KL I. Methods
  We present a comprehensive analysis of a broad band spectral line survey of
the Orion Kleinmann-Low nebula (Orion KL), one of the most chemically rich
regions in the Galaxy, using the HIFI instrument on board the Herschel Space
Observatory. This survey spans a frequency range from 480 to 1907 GHz at a
resolution of 1.1 MHz. These observations thus encompass the largest spectral
coverage ever obtained toward this high-mass star-forming region in the sub-mm
with high spectral resolution, and include frequencies $>$ 1 THz where the
Earth's atmosphere prevents observations from the ground. In all, we detect
emission from 39 molecules (79 isotopologues). Combining this dataset with
ground based mm spectroscopy obtained with the IRAM 30 m telescope, we model
the molecular emission from the mm to the far-IR using the XCLASS program which
assumes local thermodynamic equilibrium (LTE). Several molecules are also
modeled with the MADEX non-LTE code. Because of the wide frequency coverage,
our models are constrained by transitions over an unprecedented range in
excitation energy. A reduced $\chi^{2}$ analysis indicates that models for most
species reproduce the observed emission well. In particular, most complex
organics are well fit by LTE implying gas densities are high ($>$10$^6$
cm$^{-3}$) and excitation temperatures and column densities are well
constrained. Molecular abundances are computed using H$_{2}$ column densities
also derived from the HIFI survey. The distribution of rotation temperatures,
$T_{\rm rot}$, for molecules detected toward the hot core is significantly
wider than the compact ridge, plateau, and extended ridge $T_{\rm rot}$
distributions, indicating the hot core has the most complex thermal structure.
