  Absorption spectroscopy is traditionally used to determine the average gas
temperature and species concentration along the laser line-of-sight by
measuring the magnitude of two or more absorption transitions with different
temperature dependence. Previous work has shown that the nonlinear temperature
dependence of the absorption strength of each transition, set by the
lower-state energy, E", can be used to infer temperature variations along the
laser line-of-sight. In principle, measuring more absorption transitions with
broader bandwidth light sources improves the ability to resolve temperature
variations. Here, we introduce a singular value decomposition framework in
order to explore the theoretical limits to resolving temperature distributions
with single-beam line-of-sight absorption measurements. We show that in the
absence of measurement noise or error, only the first ~14 well-selected
absorption features improve the temperature resolution, and a Tikhonov
regularization method improves the accuracy of the temperature inversion,
particularly for recovery of the maximum gas temperature along the laser beam.
We use inversion simulations to demonstrate that one can resolve a selection of
temperature distributions along a laser beam line-of-sight to within 3% for the
sample cases analyzed. In part II of this work, we explore the influence of
measurement noise and error, and experimentally demonstrate the technique to
show that there is benefit to measuring additional absorption transitions under
real conditions.
