Solar energetic particle spectra from the SOHO-EPHIN sensor by application of regularization methods

Abstract

Context. The Electron Proton Helium Instrument (EPHIN) on ESA’s Solar and Heliospheric Observatory (SOHO) measures solar energetic electrons, protons, and alpha particles with a stack of six solid-state detectors forming a telescope. The energy deposit in these detectors must be inverted to derive the original energy of the incident particles, thus leading to the original energy spectrum of solar energetic particles. Normal inversion techniques, such as least-squares methods, rely on fitting a known functional behavior of the spectral dependence (normally a power law) to the measured data with some account taken for the instrument response. Such procedures can fail to retrieve accurate particle spectra, e.g., when count rates are low and unphysical negative counts result from the fitting procedure. Aims. We show how regularization methods can be applied to energetic particle measurements to unambiguously derive the original particle spectrum without any assumptions about its functional behavior, while also satisfying constraints such as non-negative counts. Methods. Such inversion techniques still require knowledge of the instrument response function, however, it is an improvement upon normal least-squares or maximum-likelihood fitting procedures because it does not require any a-priori knowledge of the underlying particle spectra. Given the instrument response function in matrix form (here derived using Monte Carlo techniques), the original Fredholm integral equations reduce to a discrete system of linear algebraic equations that can be solved by ordinary regularization methods such as singular value decomposition (SVD) or the Tikhonov method. This procedure alone may lead to unphysical negative results, requiring the further constraint of non-negative count rates. This technique avoids full deconvolution because it involves the solution of ill-conditioned or singular linear systems. Results. We analyze data from SOHO/EPHIN by full deconvolution of the measured data with the instrument response function. We apply the SVD and Thikonov methods with and without constraints to measured data from SOHO/EPHIN. Conclusions. The derived results agree well with those of other methods that rely on a-priori knowledge of the spectral shape of the particle distribution function, demonstrating the power of the regularization method for more general cases.