Inverse problem or Spectral unfolding

Key assumptions: (see also: astro-ph/0506588 ) All observed spectra dF/dû in the high energy Extensive Air Shower (EAS) physics have been obtained via the convolutions of the energy spectra dℑ/dE_A of primary nuclei (A ≡ H, He, ... at least up to Ni) over the atmosphere with the expected spectra ∂W_A(E_A, u)/∂u of EAS vector parameter u={N_e , N_μ , s, θ, ...} at the observation level taking into account the corresponding response function of a given experiment, ∂ℜ_A(u, E_A, x, y, θ)/∂û, which itself is the probability to detect and reconstruct a shower with vector parameter û instead of true vector parameter u. For a horizontal observation level and the shower core coordinates x, y, θ, φ, the mentioned convolution looks like:

dF/dû = ∑_A∫dℑ/dE_A⋅K_A(E_A,û)dE_A  ,                         (1)

where K_A(E_A,û) = _∫∫(∂W_A/∂u)⋅(∂ℜ_A/∂û)dudD is the kernel function defined in the multivariate shower detection space with  dudD=du⋅2πcosθdxdydcosθ  space element. In general, the shower spectra W_A(E_A,u) from (1) depend on the interaction model (see also [S.V.Ter-Antonyan and L.S.Haroyan, hep-ex/0003006 (2000), or S.V.Ter-Antonyan & P.L.Biermann, astro-ph/0106091 (2001)]. The multidimensional integral (1) is better to calculate by the Monte-Carlo simulation since the shower spectra W_A(u) and corresponding response functions ℜ_A(u,û) one can estimate more or less precisely only by the simulations.
The integral equation (1) is not a Fredholm equation (or a set of equations) of the first kind due to a set of object (unknown) functions (dℑ/dE_A), and the theory of integral equations is not applicable for the equation (1), [S. Ter-Antonyan, arXiv:0706.4087, arXiv:1405.5472].

We computed the kernel functions K_A(E_A,û) of integral equations (1) for û ≡ {N̂_{ch(e,γ,μ,h)}, N̂_μ , ŝ, θ ̂} 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 the 4 groups of primary nuclei: A ≡ H, He, O-like, Fe-like at the power law primary energy spectra
f(E_A) ∝ E^−1.5,  . Each secondary shower particle (e, μ, h, γ) obtained at the observation level from CORSIKA shower simulation code is passed online through the GAMMA shower array (scintillators). Furthermore, the transformation of computed energy deposit to the corresponding ADC code and final decoding into a number of detected particles simulates also the digitization errors of readout system.

Simulation of shower particle passage through the GAMMA detectors

All EAS muons with energy E_μ > (4 GeV)/cosθ at the GAMMA observation level are passed through the 2500 g/cm² rock taking into account the fluctuations of ionization losses and electron (positron) accompaniment due to muon bremsstrahlung, direct pair production, knock-on and photo-nuclear interactions.
The reconstructions of shower parameters at the observation level (N̂_ch, N̂_μ , ŝ) on the basis of the simulated (detected) number of particles in the scintillators are carried out using NKG and Exponential lateral distribution functions for ground-based charged particle and high energy underground muon shower components respectively.

    The computations of expected spectra {dF/dû} are performed using integration (1) for the simulated kernel function K_A(E_A,û) and universal parameterization of primary nuclei energy spectra reported in ANI98 Workshop (Hor-Amberd, Armenia, 1998 [hep-ex/0003006 (2000)]) by S. Ter-Antonyan:

dℑ_A/dE = Φ_A⋅E ^−γ_₁ [ 1 + ( E /E_{knee, A} ) ^ε ] ^{−Δγ /ε}  ,                 (2)

where the parameter ε describes the sharpness of spectral knee and Δγ=γ_₂−γ_₁ for the asymptotic energy spectral power indices below (γ_₁≃2.7 ± 0.02) and above (γ_₂≃3.15 ± 0.05) the spectral rigidity-dependent knee energy, E_{knee, A} ≃ (3 PV)⋅Z_A for the Z_A charge of primary nucleus, A. The parameterization (2) has been used as the solutions of inverse problem in the energy range 10¹⁵-10¹⁷eV (see arxiv.org/abs/1405.5472).

EAS Inverse Problem that is the reconstruction of primary energy spectra dℑ/dE_A (object functions) has been carried out by the solution of integral equations (1) as a set of the parameterized equations at the known kernel functions K_A(E_A,û), parameterizations (2) [see refs. S.Ter-Antonyan (2014, 2008, 2003, 2000) in Publications page] for rigidity-dependent steepening primary nuclei energy spectra in the knee region and the detected shower spectra {dF/dû} from GAMMA experiment.

The all-particle energy spectrum, dℑ/dE₀ = ∑_Adℑ/dE_A , has also been obtained by the event-by-event primary energy E₀ estimation method regardless of primary nucleus (A). The primary energy estimator, lnE_₀ = ƒ(û), was obtained from the best fit of shower simulated dataset {(E, A, θ) ⇒ u ⇒ û}, where û = {N̂_ch , N̂_μ , ŝ} is the observable shower vector parameter taking into account the measurement errors and uncertainties of GAMMA experiment. The derived 6-parametric best fit for the energy estimator of GAMMA shower array is the following:

lnE_₀ =a_₁ lnN_ch+a_{₂ √ s /} cosθ +a_₃ cosθ +a_₄ +a_₅ /( lnN_ch+a_₆ lnN_μ) ,        (3)

where a_₁ , a_₂ , ..., a_₆ are approximation parameters. The observable (reconstructed and measured) shower parameters in the expression (3) are: N_ch - the EAS size, N_μ - a truncated (R_μ<50 m) number of EAS muons with energies E_μ>5 GeV, s - the EAS age parameter, and θ - the zenith angle of shower core. The expression (3) provides the accuracy of energy estimator ΔE_₀/E_₀≲15% for the primary energies E_₀≲200 PeV.

The method was first reported in GAMMA 2004 workshop and presented in ref. [Ter-Antonyan et al., astro-ph/0506588 (2005)] and [arXiv:1405.5472 [astro-ph.HE] (2014), Physical Review D 89, 123003 (2014)] (see also Publications in menu).

RESULTS for 2007-2009

Romen M. Martirosov, Samvel V. Ter-Antonyan, Anatoly D. Erlykin, Alexandr P. Garyaka, Natalya M. Nikolskaya, Yves A. Gallant and Lawrence W. Jones, "Galactic diffuse gamma-ray flux at the energy about 175 TeV", arXiv:0905.3593 , 31^th International Cosmic Ray Conference, Lodz, Poland (2009).

A.P. Garyaka, R.M. Martirosov, S.V. Ter-Antonyan, A.D. Erlykin, N. Nikolskaya, Y.A. Gallant, L. Jones, and J. Procureur: "All-particle primary energy spectrum in the 3-200 PeV energy range", Journal of Physics G: Nuclear and Particle Physics, 35 (2008) 115201. http://stacks.iop.org/0954-3899/35/115201 or arXiv:0808.1421.

A.P. Garyaka, R.M. Martirosov, S.V. Ter-Antonyan, N. Nikolskaya, Y.A. Gallant, L. Jones, J. Procureur. "Rigidity-dependent cosmic ray energy spectra in the knee region obtained with the GAMMA experiment". Astroparticle Physics, 28, 2 (2007) p. 169, arXiv:0704.3200.

PRELIMINARY DATA (2005)

S.V. Ter-Antonyan, R.M. Martirosov, A.P. Garyaka, V. Eganov, N. Nikolskaya, T. Episkoposyan, J. Procureur, Y. Gallant, L. Jones: "Primary Energy Spectra and Elemental Composition. GAMMA Experiment". astro-ph/0506588 (2005).

S.V. Ter-Antonyan, Y.A. Gallant, A.P. Garyaka, L.W. Jones, R.M.Martirosov, N.M. Nikolskaya, J. Procureur, "All-particle primary energy spectrum in the knee region", 29^th ICRC, Pune, India, (2005)  .

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, hep-ph/0106076, (2001)]. (Workshop GAMMA2004)