Key assumptions: (see also: astro-ph/0506588 ) All the observed quantities dF/du^* in the high energy Extensive Air Shower (EAS) physics are obtained via convolutions of the energy spectra dI_A/dE₀ of primary nuclei (A=H,He,... at least up to Ni ) over the atmosphere with studied spectra dW_A(E₀)/du of  EAS parameters {u=N_e,N_m,s,...} at the observation level and corresponding response functions of a given experiment - dR(u,A,E₀)/du^*.
   In general, it looks like:

where dD is an element of the multidimensional phase space (D) of EAS detection, W_A(E₀)/du - are the corresponding spectra of EAS parameter (u) at a given energy E₀ and a given kind of primary nucleus (A), and, in general case, depending on interaction model. (see also [S.V.Ter-Antonyan and L.S.Haroyan hep-ex/0003006 (2000).// S.V.Ter-Antonyan & P.L.Biermann astro-ph/0106091 (2001)].
   The multidimensional integral above rather to calculate by Monte-Carlo simulation especially since, the spectra dW_A(E₀)/du one may compute more or less precisely only by simulation.

Simulation Strategy: We computed the shower spectra dW_A(E₀)/du  {u=N_e,N_m,s ...} at the observation level of GAMMA experiment (700 g/cm²) using the CORSIKA6030(NKG,EGS) EAS simulation code [D.Heck et al., FZKA, 6019, 1998] with QGSJET01 and SIBYLL2.1 interaction models for 4 groups of primary nuclei: A=H, He, O, Fe at power law spectra with spectral index (-1.5). EGS mode of CORSIKA was used for calculation of the response functions of GAMMA experiment taking into account the EAS gamma-quanta contributions and to select the corresponding parameters of an adequate NKG mode.
   Each secondary particle (e,μ,h,γ) obtained from CORSIKA at the observation level is online passed through the GAMMA scintillator and further the transformation of energy deposit to the corresponding ADC code and final decoding into a number of detected particles is simulated. All EAS muons with energy E_m>4 GeV at the GAMMA observation level are passed through the 2300 g/cm² rock taking into account fluctuations of the ionization losses and electron (positron) accompaniment due to the muon bremsstrahlung, direct pair production, knock on and photo-nuclear interactions.

The direct computations of expected spectra {dF/du^*}_expected are performed using expression (1) and simulated values of spectra dW_A(E₀)/du and dR(u,A,E₀)/du^* taking into account the isotropic angular distribution of primary nuclei. Primary energy spectra were used according to P.Biermann`s multicomponent model predictions with rigidity-dependent steepening parameterizations from [Samvel Ter-Antonyan and Peter Biermann, "Primary Cosmic Ray Spectra in the Knee Region." Proceedings 28th ICRC, Tsukuba, Japan, p.235, (2003).  ] which agree well with KASCADE data. Universal approximation of primary energy spectra for energy range 10¹⁵-10¹⁷eV taking into account the sharpness of knee for each of nuclear species (A=H, He, C, O, Si, Fe ) was reported in ANI98 Workshop (Hor-Amberd, Armenia, 1998) by S.Ter-Antonyan [hep-ex/0003006 (2000)]

for computation of expected spectra at observation level according to expression (1).
The results of direct computation of expected EAS size spectra and EAS muon spectra in comparison with corresponding GAMMA data have been reported in GAMMA workshop and partly presented in Welcome and Underground Muon Hall pages.

EAS Inverse Problem that is the reconstruction of primary energy spectra dI_A/dE₀ by the measured spectra {dF/du^*}_measured from GAMMA experiment are carried out both by the solution of the expression (1) as a set of integral equations at the known object functions dW_A(E₀)/du^* (see refs. S.Ter-Antonyan (2007, 2003, 2000) in Publications page) and direct reconstruction method i.e. event-by-event estimation of the primary energy by the observable quantities u^*=N_e,N_m, EAS age (s) and zenith angle.
   Obtained by GAMMA Team the best fit for the last method (E₀ ~ f(u^*)) has a 6-parametric form at the accuracy less that 15%. This results was reported in GAMMA 2004 workshop and presented in Ground-based Array page (see Menu). The preliminary results (so called α-parametric method) presented in  ref. [A.P.Garyaka, L.W.Jones, R.M.Martirosov,J.Procureur (2003)] (see Publication page).
   The all-particle primary energy spectra obtained by ourselves using all above methods are presented below (Current Results 2007-2009) and at the beginning of Welcome page.

Below the comparisons of detected single particle spectra, CORSIKA NKG-EGS adaptation data,  calibration of the measurement errors and EAS particle density spectra obtained by the GAMMA experiment are presented along with the corresponding expected spectra obtained by the simulation strategy described above and the direct calculations using (1) at the parametrization of primary energy spectra according to the approximations of balloon and satellite data [B.Wiebe-Soothl and P.Biermann,  Roma (1999)].

Single-particle spectra from GAMMA detectors (dotted lines) and corresponding simulated expected spectra (red symbols). Dashed lines are the expected spectra without measurement errors (presented in workshop GAMMA2004).

Comparison EGS and adapted NKG mode of CORSIKA for the SIBYLL and QGSJET interaction model. The gamma-quanta contribution and geomagnetic field have also been taken into account for the CORSIKA EGS mode. The simulations were performed for the Gamma EAS array at the 3 kind of primary nuclei in energy range 0.5-50 PeV.

Measurement errors of GAMMA detectors (callibration). Circle symbols are the combined measured (red) and simulated (black) discrepancies {n_i,k-(n_1,k+n_2,k+n_3,k)/3}_i=1,2,3, of each k=1,...28 registration station of GAMMA array. The star symbols show the resulting measurement errors of a single detector versus number of charged particles (Workshop GAMMA2004).

GAMMA EAS charged particle and muon density spectra along with corresponding simulated spectra at the SIBYLL and QGSJET interaction models. Primary spectra were taken from [S.Ter-Antonyan and P.Biermann, astro-ph/0106076, (2001)]. (Workshop GAMMA2004)