Regularization methods used in error analysis of solar particle spectra measured on SOHO/EPHIN

Context. The telescope EPHIN (Electron, Proton, Helium INstrument) on the SOHO (SOlar and Heliospheric Observatory) spacecraft measures the energy deposit of solar particles passing through the detector system. The original energy spectrum of solar particles is obtained by regularization methods from EPHIN measurements. It is important not only to obtain the solution of this inverse problem but also to estimate errors or uncertainties of the solution. Aims. The focus of this paper is to evaluate the influence of errors or noise in the instrument response function (IRF) and in the measurements when calculating energy spectra in space-based observations by regularization methods. Methods. The basis of solar particle spectra calculation is the Fredholm integral equation with the instrument response function as the kernel that is obtained by the Monte Carlo technique in matrix form. The original integral equation reduces to a singular system of linear algebraic equations. The nonnegative solution is obtained by optimization with constraints. For the starting value we use the solution of the algebraic problem that is calculated by regularization methods such as the singular value decomposition (SVD) or the Tikhonov methods. We estimate the local errors from special algebraic and statistical equations that are considered as direct or inverse problems. Inverse problems for the evaluation of errors are solved by regularization methods. Results. This inverse approach with error analysis is applied to data from the solar particle event observed by SOHO/EPHIN on day 1996/191. We find that the various methods have different strengths and weaknesses in the treatment of statistical and systematic