## 4 Reconstruction of the Energy and Mass of the Primary Particle
We measure the spectrum in one observable and make a conversion to the energy spectrum. The observable typically used is the shower size because, as discussed in Section 3.2, the number of electrons at shower maximum is nearly independent on the primary mass: \(N_{e|_{max}}^{A}\approx N_{e|_{max}}^{p}\). However, surface detectors are usually located deep in the atmosphere and do not measure the number of electrons at shower maximum. Beyond the maximum, the number of electrons is a mass-sensitive parameter, with a larger electron number for air showers initiated by light primaries, according to a relation of the type \[N_{e}(E,A)=\alpha(A)\cdot E^{\beta(A)}\] (26) where the parameters \(\alpha\) and \(\beta\) depend on the primary mass \(A\). This implies a degeneracy in the reconstruction procedure because to recover the primary energy from the size spectrum we must assume a given elemental composition to be measured. If the composition changes in the investigated energy range, the relationship between the measured electron size and inferred energy will also vary. The number of electrons in the core region has been used in some experiment, as well as the particle density at a suitable given distance from the shower axis, in some large arrays (see, for example, the Refs. [11; 52]). In both cases this densities, according to Monte Carlo simulations, are nearly independent of the primary mass. * _How do we measure elemental composition at ground?_ The inelastic cross-section \(\sigma_{inel}^{Fe-Air}\) of iron at 1 PeV is about six times larger than for protons of equal energy. Hence, nuclei develop showers higher in atmosphere (smaller \(X_{max}\)) than protons, dissipating their energy much faster. Due to the shorter interaction length and the smaller energy per nucleon and because of the reduced attenuation of the muon component, nuclei-induced showers contain less particles in the e.m. component deep in the atmosphere, but they carry more muons than a proton shower of the same energy. This is the basis of the electron-muon correlation method. Therefore, as discussed in Section 3.4, the measurement of electron and muon contents simultaneously (with their fluctuations) has become the first and most commonly employed technique to infer the CR elemental composition with arrays. However, intrinsic shower to shower fluctuations limit mass resolution to a few mass groups (see Section 3.4) and electron and muon numbers are not independent. In addition, the muon component is heavy dependent on the details of the hadronic interactions and the results strongly depend on the particular model used to interpret the data. The other common technique, below \(10^{18}\) eV, involves the observation of the Cherenkov light and the study of its shape. In fact, the characteristics of the photon distribution depend on the depth of the shower maximum, therefore on the mass of the primary particle. The overall Cherenkov intensity provides a calorimetric measurement of the CR energy. Cherenkov light has been measured, for instance, in hybrid experiments by ARGO-YBJ and Tunka apparatus. The KASCADE multi-component array was the first experiment that claimed the measurement of the energy spectra of 5 different mass groups (p, He, CNO, MgSi, Fe) through a complex unfolding of the \(N_{e}/N_{\mu}\) diagram [22; 23; 24]. In the last two decades, other multi-component experiments measured a number of observables that, in principle, are mass-sensitive: steepness of the lateral distribution, characteristics of shower core region, distribution of the relative arrival times and angles of incidence of the muon component, characteristics of the lateral distribution of high energy muons (the so-called _\"muon bundles\"_) measured underground, pulse shape and lateral distribution of the air Cherenkov light, depth X\({}_{max}\) of the shower maximum (see, for example, ref. [15; 17]). But the study of the muon component has remained the most used technique.