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(∊)) is consistent with a purely thermal (locally Maxwellian) distribution of source electrons, and, if so, how to reconstruct the corresponding differential emission measure (ξ(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 ξ(T) from measurements of I(∊) invOlves two successive inverse problems: the first, to recover the mean source-electron flux spectrum ( `(F)]\overline{F} (E)) from I(∊) and the second, to recover ξ(T) from `(F)]\overline{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 δ-like forms of ξ(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)]\overline{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)]\overline{F} (E) derived from a priori known forms of ξ(T) and then applied to hard X-ray spectral data from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).