How to Infer the Primary Energy Spectrum from Eas Observations Demonstrated with Kascade Data

The energy spectrum of primary cosmic rays in the energy range of 1014 1016 eV is infered from the information about the electron/photon and muon components from the KASCADE array. We combine the EAS simulation program CORSIKA and a detector simulation program, based on the GEANT package, with a multivariate statistical analysis program for an event-by-event analysis of real data. Approaches of neural network classification and Bayesian decision making are used to get a reliable information about the energy of the primaries. We demonstrate the power of the procedures by first results from measured data. INTRODUCTION The knowledge of the energy spectra of various components of primary cosmic rays around the knee region is important for testing alternative hypotheses of the cosmic ray (CR) origin and acceleration. In previous experiments registering extensive air showers (EAS) the mass resolution has been rather poor, and only general trends have been obtained from comparison of various oneor twodimensional distributions of observables with Monte Carlo (MC) simulations. Taking advantage of the larger degree of sampling of muon and soft components by the KASCADE experiment (Klages et al., 1997), proceeding with detailed simulations of the EAS development in the atmosphere with the CORSIKA code (version 5.2: VENUS model) (Capdevielle et al., 1992), and calculating the full detector response function with GEANT and reconstruction programs, we present first results of more detailed analysis of EAS. The problem of event-by-event analysis of CR data (Chilingarian et al., 1987; Chilingarian, 1989) has been addressed by developing the unified methodology of nonparametric multivariate statistical inference providing the tools for an estimation of the primary particle type and energy. ENERGY ESTIMATORS Following previous attempts of comparing the alternative energy estimation methods (A.Chilingarian et al., 1997) we present results obtained with three different nonparametric regression methods. The algorithms ”training” were performed with 6000 events, simultated with the CORSIKA code, for five primaries with energies uniformly distributed in the range 1014 1016 eV. The following parameters were used: so called truncated numbers of muons (Nμ(truncated)) and electrons (Ne(truncated)) along with the age parameter s30 associated with a Moliere radius value of 30 m instead of the commonly used 79 m (Weber et al.,1997). As indicated in Table 1, the newly introduced parameters show the same correlation features as the “classical” shower parameters EAS size Ne and total number of muons Nμ. But they are more suitable for reconstruction and comparisons . Figures 1 a-b) indicate a further advantage of the new variables: improved linearity of energy dependence for all nuclei. corresponding author: e-mail: [email protected] †A full list of authors and institutions of the KASCADE Collaboration is given at the end of this volume. ; ρE0;Ne(truncated) ρE0;Ne ρE0;Nμ(truncated) ρE0;Nμ ρE0;s30 ρE0;s79 p 0.94 0.94 0.96 0.95 -0.39 -0.37 He 0.96 0.95 0.97 0.97 -0.46 -0.44 O 0.98 0.95 0.97 0.97 -0.55 -0.53 Si 0.98 0.94 0.98 0.98 -0.61 -0.60 Fe 0.98 0.92 0.98 0.98 -0.70 -0.69 Table 2: The Mean Square Deviation (MSD) of Energy Estimation Primary p He O Si Fe MSD (Ne(truncated)) 0.61 0.60 0.40 0.34 0.34 MSD (Ne(truncated), Nμ(truncated)) 0.45 0.44 0.32 0.30 0.28 For energy estimation we use the well known K Nearest Neighbours (KNN) nonparametric regression method. The incorporation of the muon information (Table 2) improves the accuracy of the energy estimation. As expected the energy of the iron primary is measured much more accurately as compared with protons. Along with the KNN estimator other methods were used. The Parzenian window nonparametric regression: Ê(v) = MT S ∑ i CiEi with MT S ∑ i Ci = 1 (1) where Ci 1 2πd=2hd detRK e r2 i j=h2Wj: (2) Here, d is the feature space dimensionality, MTS is the number of events in the Training Sample (TS) , ri j is the distance from the observable vi to the j-th point u j of the TS in the Mahalanobis metric: ri j = (ui v j)T RK 1(ui v j) (3) RK is a sampling covariance matrix of the class to which the training event ui with energy Ei belongs, Wj are the event weights, h is the width of the kernel. The estimate (Eqs. 1-3) is calculated for different prechosen values of kernel widths h. The median of the estimates sequence is used as final estimate (Parzen,1962). The neural regression with modified quality (objective) function: Though the presently used CORSIKA simulations include only few high energy events, we use a special procedure to weigh them, to avoid a too large bias to smaller reconstructed energies in the high energy region. The following expression was used in neural network training: Q j = MT S ∑ i (Êi Ei)s1W s2 i ; (4) where Q j and fÊig are the current values of the quality function and the energy estimates of training sample events respectively, and the parameters s1 and s2 influence the bias and it’s energy dependence. Figure 2a) demonstrates the lack of bias of the energy estimates for the whole investigated energy log(EMC) (GeV) ln( N a) log(EMC) (GeV) ln( N e (tr un ca ted ) b) log(EMC) (GeV) log (E Es t. ) (G eV ) c)