Solar Physics (2006) 237: 61–83 DOI: 10.1007/s11207-006-0029-1 C  Springer 2006 REGULARIZED RECONSTRUCTION OF THE DIFFERENTIAL EMISSION MEASURE FROM SOLAR FLARE HARD X-RAY SPECTRA M. PRATO Dipartimento di Matematica, Universit` a di Genova, via Dodecaneso 35, I-16146 Genova, Italy M. PIANA Dipartimento di Informatica, Universit` a di Verona, Ca’ Vignal 2, Strada le Grazie 15, 37134 Verona, Italy (e-mail: michele.piana@univr.it) J.C. BROWN Department of Physics and Astronomy, University of Glasgow, The Kelvin Building, G12 8QQ, U.K. A.G. EMSLIE Department of Physics, Oklahoma State University, Stillwater, OK 74078, U.S.A. E.P. KONTAR Department of Physics and Astronomy, University of Glasgow, The Kelvin Building, G12 8QQ, U.K. and A.M. MASSONE CNR-INFM LAMIA, via Dodecaneso 33, I-16146 Genova, Italy (Received 8 July 2005; accepted 24 April 2006; Published online 11 July 2006) Abstract. We address the problem of how to test whether an observed solar hard X-ray bremsstrahlung spectrum ( I (ǫ )) is consistent with a purely thermal (locally Maxwellian) distribution of source elec- trons, 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 (ǫ ) invOlves two successive inverse problems: the first, to recover the mean source-electron flux spectrum ( ¯ F ( E )) from I (ǫ ) and the second, to recover ξ (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 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 ( 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 ξ (T ) and then applied to hard X-ray spectral data from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI).