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 ($F(E)$) from $I(\epsilon)$ and the second, to recover $\xi(T)$ from $F(E)$. 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 $F(E)$,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 $F(E)$ 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).