## 4 Reconstruction of the Energy and Mass of the Primary Particle  
The crucial point in air shower observations with EAS arrays is the reconstruction of the primary particle properties (especially energy and mass number) from the measured quantities. In fact, analysis of shower data consists in the disentanglement of a threefold problem involving primary energy, primary mass and modelling of hadronic interactions (for a discussion about hadronic interactions in CR physics see, as an example, refs. [47,48] and references therein). An intrinsic ambiguity affects the interpretation of data. Different combinations of the two following elements can produce similar showers. As an example, a _\"short\"_ shower can be produced by a large cross-section, high inelasticity or heavy primary mass. On the contrary, a _\"long\"_ shower, penetrating deeper in the atmosphere, can be produced by small cross-section, low inelasticity or light mass. 1. shower development, mainly governed by the inelasticity and by the inelastic cross section 2. elemental composition of the primary flux, that we don't know and want to measure Strictly speaking, when operating with shower arrays there are no observables directly related to the mass of the primary particle, and its measurement is very indirect. We note, however, that the Cherenkov light emitted by a primary heavy nucleus high up in the atmosphere (the so-called _\"direct Cherenkov light\"_) is directly related to the charge (and therefore to the mass) of the primary particle [49; 50]. Since this light is proportional to \(Z^{2}\), heavy nuclei are more suited for detection. Charge resolution is about 10% for \(Z>10\). The main limitation is that it can only be used over a small energy range for each atomic charge Z. The majority of experiments with shower arrays can therefore apply only'statistical' methods according to a classical scheme: 1. From the experimental data, via some phenomenological functions determined by Monte Carlo simulations for the particular array, the measured observables (\(N_{e},N_{hadr}\), \(N_{\mu}\), \(X_{max}\),...) are reconstructed. 2. The distributions of such quantities are compared with those extracted from a detailed simulation of the EAS development in the atmosphere in which a trial CR spectrum is used. 3. The input spectrum is varied in order to optimize the agreement between the reconstructed and calculated distributions of measured observables. Therefore, a typical data analysis consists in finding a combination of primary energy spectrum, elemental composition and hadronic interaction characteristics to obtain a consistent description of the experimental results. Clearly, this is not a measurement, but only a consistency check of some trial models. In case of discrepancy, it is difficult to identify the origin; in case of agreement, is the parameter combination unique? Due to the reduced resolution in the measurement of the primary mass (see Section 3), the majority of shower arrays displayed the results only as a function of the total energy per particle with the so-called _\"all-particle\"_ energy spectrum, that is, as a function of the total energy per nucleus, and not per nucleon. Any tentative to infer informations about elemental composition are limited, at most, to study the evolution of the _\"light\"_ (\"proton-like\") or _\"heavy\"_ (\"iron-like\") components as a function of the energy, with results which critically depend on Monte Carlo predictions. In the last two decades, a number of multi-component experiments have started to measure, with high statistics, at the same time, different shower observables, on an event-by-event basis. This fact allowed to exploit sophisticated analysis techniques to infer the characteristics of the primary particle by measuring the correlation between different components (for a review see, for example, the Refs. [39; 51] and references therein). In a nutshell, * _How to obtain the energy spectrum in shower arrays?_ This is the first step in the analysis of CR data.