Optimal inversion of hard X-ray bremsstrahlung spectra: I - SVD analysis

The instability properties of bremsstrahlung spectrum inversion to yield source electron spectra in flares are examined in terms of singular value decomposition (SVD) for the cases of thin- and thick-target models for different bremsstrahlung cross-sections. It is shown that, for the Kramers and Bethe-Heitler cross-sections, analytic expressions can be obtained for evaluation of the Gram matrix. These allow quantitative comparison of the instability of inversion for each case and hence assessment of the accuracy and resolution of the electron source function recovery. Based on this analysis we find that: (a) the solution instability is worse for the Bethe-Heitler than for the Kramers approximation; care must therefore be taken in real data analysis to use the most accurate possible cross-section to avoid over- or under-estimating the maximum precision and resolution achievable in the recovered electron spectrum; (b) instability is worse in recovering thick-target injection spectra than thin-target mean source spectra, as expected because of the double deconvolution involved in the former; (c) such SVD analysis should form an important part of future analysis of high resolution flare data such as from the HESSI Mission.