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>I</i>(∊)) is consistent with a purely thermal (locally Maxwellian) distribution of source electrons, and, if so, how to reconstruct the corresponding differential emission measure (ξ(<i>T</i>)). 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 ξ(<i>T</i>) from measurements of <i>I</i>(∊) involves two successive inverse problems: the first, to recover the mean source-electron flux spectrum (<i><div style="text-decoration: overline;">F</div></i>(<i>E</i>)) from <i>I</i>(∊) and the second, to recover ξ(T) from <i><div style="text-decoration: overline;">F</div></i>(<i>E</i>). 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 δ-like forms of ξ(<i>T</i>) 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 <i><div style="text-decoration: overline;">F</div></i>(<i>E</i>), 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 <i><div style="text-decoration: overline;">F</div></i>(<i>E</i>) derived from a priori known forms of ξ(<i>T</i>) and then applied to hard X-ray spectral data from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).<i><div style="text-decoration: overline;">F</div></i&gt