  This paper describes an adaptation of the Optimal Localized Averaging (OLA)
inversion technique, originally developed for geo- and helioseismological
applications, to the interpretation of solar spectroscopic data. It focuses on
inverting the thermodynamical properties of the solar atmosphere assuming that
the atmosphere and radiation field are in Local Thermodynamic Equilibrium
(LTE). We leave inversions for magnetic field and non-LTE inversions for future
work. The advantage with the OLA method is that it computes solutions that are
optimally resolved (in depth) with minimal cross-talk error between variables.
Additionally, the method allows for direct assessment of the vertical
resolution of the inverted solutions. The primary challenges faced when
adapting the method to spectroscopic inversions originate with the possible
large amplitude differences between the atmospheric model used to initiate the
inversion and the underlying atmosphere it aims to recover, necessitating the
development of an iterative scheme. Here we describe the iterative OLA method
we have developed for both single and multi-variable inversions and demonstrate
its performance on simulated data and synthesized spectra. We note that when
carrying out multi-variable inversions, employing response function
amplification factors can address the inherent spectral-sensitivity bias that
makes it hard to invert for less spectrally-sensitive variables. The OLA method
can, in most cases, reliably invert as well as or better than the frequently
employed Stokes Inversion based on Response functions (SIR) scheme, however
some difficulties remain. In particular, the method struggles to recover
large-scale offsets in the atmospheric stratification. We propose future
strategies to improve this aspect.
