An improved method of estimating the uncertainty of air-shower size at ultra-high energies

The collection of a statistically significant number detected of cosmic rays with energy above $10^{17}$ to $10^{18}$ eV requires widely-spaced particle detectors at the ground level to detect the extensive air showers induced in the atmosphere. The air-shower sizes, proxies of the primary energies, are then estimated by fitting the observed signals to a functional form for expectations so as to interpolate the signal at a reference distance. The functional form describes the rapid falloff of the expected signal with the distance from the shower core, using typically two logarithmic slopes to account for the short-range and long-range decreases of signals. The uncertainties associated to the air-shower sizes are determined under the assumption of a quadratic dependence of the log-likelihood on the fitted parameters around the minimum, so that a meaningful variance–covariance matrix is provided. In this paper, we show that for an event topology where one signal is much larger than the others, the quadratic dependence of the fitted function around the minimum is a poor approximation that leads to an inaccurate estimate of the uncertainties. To restore a quadratic shape, we propose to use the polar coordinates around the detector recording the largest signal, projected onto the plane of the shower front, to define the likelihood function in terms of logarithmic polar distances, polar angles and logarithmic shower sizes as free parameters. We show that a meaningful variance–covariance matrix is then recovered in the new coordinate system, as the dependence of the fitted function on the modified parameters is properly approximated by a quadratic function. The use of the uncertainties in the new coordinate system for subsequent high-level analyses is illustrated.