Likelihood-based reconstruction of muon lateral distribution function using combined integrator and binary detector modes

TL;DR

This work presents a likelihood-based reconstruction of the muon lateral distribution function (MLDF) that simultaneously uses data from dual acquisition modes—binary and ADC—in muon detectors. The authors formulate a joint likelihood L(mu;k,Q) by combining the binary-mode distribution with a Poisson-like term and the ADC-mode log-normal charge distribution, enabling robust muon-count inference even under saturation. They validate the method with simulations for Auger-like underground detectors, showing that the combined approach reduces bias and variance relative to single-mode reconstructions, and demonstrate improved handling of saturated events by fixing the MLDF slope parameter β. The approach enhances muon-density measurements used for cosmic-ray composition analyses and is adaptable to future dual-mode detector designs, with practical guidance on detector dynamic range and saturation treatment.

Abstract

The origin of ultra-high-energy cosmic rays, with energies $E \geq 10^{18}$ eV, remains unknown. Among the key observables used to investigate their nature are the energy spectrum, the arrival direction distribution, and the composition as a function of energy. The composition of the primary cosmic ray is inferred from properties of the extensive air showers they initiate, particularly from parameters sensitive to the primary mass. The most sensitive parameters to the primary mass are the atmospheric depth of the shower maximum, typically measured with fluorescence telescopes, and the muon content of the shower, measured using dedicated muon detectors. A commonly used observable in composition studies is the muon density at a fixed distance from the shower axis, derived by evaluating the reconstructed muon lateral distribution function (MLDF) at a reference distance. A specific type of muon detector features two acquisition modes: binary and integrator (commonly referred to as ADC mode, for Analog-to-Digital Converter). The binary mode allows for direct muon counting, while the ADC mode infers the muon number from the integrated signal of the detector response. Existing methods reconstruct the MLDF using data from either acquisition mode individually, or by combining both, but usually assigning a single mode per detector station in a given event. This work presents a novel method to reconstruct the MLDF based on a likelihood approach that simultaneously incorporates data from both acquisition modes at each detector station. We apply our method to the underground muon detectors of the Pierre Auger Observatory as a case study. However, this general approach can be applied to future detectors with dual acquisition capabilities. Our results demonstrate that the combined method outperforms traditional techniques that rely solely on either binary or ADC mode data.

Figures (12)

• Figure 1: Distributions of $\hat{\mu}$ obtained by maximizing the likelihood functions of the binary, the ADC and the combined one. The parameters used are: $\mu=100$ and $n_\textrm{s}=192$.
• Figure 2: Relative bias (top panel), relative standard deviation (middle panel) and coverage (bottom panel) of $\hat{\mu}$ as a function of $\mu$ obtained for the three methods considered and for $n_\textrm{s}=192$. For the relative bias and relative standard deviation the case of an ideal detector is also included. For the coverage the Gaussian case is also included.
• Figure 3: Standard deviation of $\hat{\mu}$ relative to that of an ideal detector, $\sqrt{\mu}$, as a function of $\mu$ obtained for the three methods considered and for $n_\textrm{s}=192$.
• Figure 4: The negative logarithm of the likelihood ratio as a function of $\mu$ obtained using the three methods considered. Each plot corresponds to a different region of the parameter $\mu$. The number of segments used is $n_\textrm{s}=192$.
• Figure 5: Top panel: relative bias and relative standard deviation of $\hat{\mu}(450)$ as a function of the logarithm of the primary energy for the four reconstruction methods considered. Bottom panel: coverage of $\hat{\mu}(450)$ corresponding to the binary, ADC, and the combined method to reconstruct the MLDF. The dotted line corresponds to the coverage of a Gaussian likelihood. The reconstructed showers are generated by iron primaries of 30$^\circ$ zenith angle.