Regularized reconstruction of the differential emission measure from solar flare hard X-ray spectra

We address the problem of how to test whether an observed solar hard X-ray bremsstrahlung spectrum ( I(epsilon)) is consistent with a purely thermal ( locally Maxwellian) distribution of source electrons, and, if so, how to reconstruct the corresponding differential emission measure (xi(T)). Unlike previous analysis based on the Kramers and Bethe-Heitler approximations to the bremsstrahlung cross-section, here we use an exact (solid-angle-averaged) cross-section. We show that the problem of determining xi(T) from measurements of I (epsilon) invOlves two successive inverse problems: the first, to recover the mean source-electron flux spectrum ((F) over bar (E)) from I (epsilon) and the second, to recover. ( T) from (F) over bar (E). We discuss the highly pathological numerical properties of this second problem within the framework of the regularization theory for linear inverse problems. In particular, we show that an iterative scheme with a positivity constraint is effective in recovering delta-like forms of xi(T) while first-order Tikhonov regularization with boundary conditions works well in the case of power-law-like forms. Therefore, we introduce a restoration approach whereby the low-energy part of (F) over bar (E), dominated by the thermal component, is inverted by using the iterative algorithm with positivity, while the high-energy part, dominated by the power-law component, is inverted by using first-order regularization. This approach is first tested by using simulated (F) over bar (E) derived from a priori known forms of xi(T) and then applied to hard X-ray spectral data from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).