At ultra-high energies, the flux of cosmic rays is too low for direct measurements to be meaningful.
When a cosmic ray enters the atmosphere, it initiates an extensive air shower, producing a cascade of secondary particles that propagate toward the ground.
Large arrays of surface detectors are used to measure these secondary particles upon arrival.

The signal detected at a specific reference distance from the shower core serves as a proxy for the shower size and, consequently, as a reliable estimator of the energy of primary cosmic ray.
However, shower development is influenced by attenuation effects: measured signals at the ground depend on the amount of traversed atmospheric density (column density) through which the shower evolves.
Since the column density varies with the inclination of the shower, it is important to account for these attenuation effects to ensure accurate energy estimation.

In this study, we derive physics-and-geometry-based functional forms to describe attenuation and propose appropriate expansion terms using simple one-dimensional shower-development models, incorporating one or two main particle-cascade components.
We then evaluate the applicability and effectiveness of these functional forms using a Monte-Carlo dataset that includes various primary cosmic-ray particles.
By directly calibrating the the shower size derived from ground signals to the Monte-Carlo energy, we characterize attenuation behavior across different primary particles, assess the energy dependence of attenuation, and quantify systematic uncertainties introduced by different functional forms.