  The investigation of cosmic rays holds significant importance in the realm of
particle physics, enabling us to expand our understanding beyond atomic
confines. However, the origin and characteristics of ultra-high-energy cosmic
rays remain elusive, making them a crucial topic of exploration in the field of
astroparticle physics. Currently, our examination of these cosmic rays relies
on studying the extensive air showers (EAS) generated as they interact with
atmospheric nuclei during their passage through Earth's atmosphere. Accurate
comprehension of cosmic ray composition is vital in determining their source.
Notably, the muon content of EAS and the atmospheric depth of the shower
maximum serve as the most significant indicators of primary mass composition.
In this study, we present two novel methods for reconstructing particle
densities based on muon counts obtained from underground muon detectors (UMDs)
at varying distances to the shower axis. Our methods were analyzed using Monte
Carlo air shower simulations. To demonstrate these techniques, we utilized the
muon content measurements from the UMD of the Pierre Auger cosmic ray
Observatory, an array of detectors dedicated to measuring extensive air
showers. Our newly developed reconstruction methods, employed with two distinct
UMD data acquisition modes, showcased minimal bias and standard deviation.
Furthermore, we conducted a comparative analysis of our approaches against
previously established methodologies documented in existing literature.
