Evaluation of algorithms for reconstructing electron spectra from their bremsstrahlung hard X-ray spectra

The Ramaty High Energy Solar Spectroscopic Imager (RHESSI) has yielded solar flare hard X-ray spectra with unprecedented resolution, enabling reconstruction of mean source electron energy spectra F(E) by deconvolution of photon energy spectra I(epsilon). While various algorithms have been proposed, the strengths and weaknesses of each have yet to be explored in a systematic fashion. For real data F(E) is unknown, so these various algorithms must instead be tested on simulated data for which the "true'' F(E) is known. Accordingly, we devised several forms of F(E) with "interesting'' features, generated the corresponding (noise-added) I (epsilon), and recovered F(E) using a variety of algorithms, including zero- and first-order Tikhonov regularizations, triangular matrix row elimination, and forward fitting using a parametric form consisting of a double power law with low/high cutoffs plus an isothermal component. All inversion methods reconstructed the general magnitude and form of F(E) well, suffering only from (1) blurring of sharp features and (2) poor recovery at low electron energies E in cases in which F'(E) was positive and large. Addition of a steep thermal component at low E did not prevent recovery of features at higher values of E. Forward fitting did recover large-scale forms and features well but, inevitably, failed to recover local features not expressible within the parametric used. This confirms that inversions are the most dependable way to discover such features. However, examination of the pattern of I(epsilon) residuals can suggest feature locations and so help refine the parametric form used. Since quite smooth F(E) forms do reproduce the observed I(epsilon) form with relatively small residuals, it appears that sharp features may be uncommon in actual flares.