5 Conclusions

Our SVD and condition number analysis of the flare bremsstrahlung spectrum inversion problem has established the following important results:

• analytic expressions can be obtained for computing the singular functions and values for thick- and thin-target problems with both the Kramers and the Bethe-Heitler cross-sections. These make it straightforward to assess quantitatively the accuracy and resolution achievable in the electron spectrum solution in each case;
• the condition number (error magnification) as usual increases with the number of recovery points. It is considerably larger for the Bethe-Heitler than for the Kramers cross-section. Insofar as the Bethe-Heitler expression is physically more accurate, this means that if we used the Kramers cross section for numerical inversions we would overestimate the accuracy of the recovered solution - i.e. undersmooth it (alternatively we can say that in finding a regularized solution we will need to smooth the Kramers case less than the Bethe-Heitler one). In fact, however, there are aspects of the Kramers expression which are physically more accurate than the Bethe-Heitler. Therefore, given that our results show the ill-posedness to be sensitive to the cross-section, it will be worthwhile to pursue further our present analysis by numerical analysis of the inversion properties of more precise cross-sections;
• the ill-conditioning of the thick-target inversion is considerably worse than for the thin-target, regardless of cross-section, so that the accuracy with which we can recover the injected flux spectrum from prescribed data is a great deal lower than that of the mean bremsstrahlung source (thin-target) spectrum. This is as expected because of the double convolution (cross-section and electron path) involved between the thick-target injection spectrum and the HXR data. It is also consistent with the second derivative analytic solution given by Brown & Emslie (1989) and the heuristic numerical solution by Johns & Lin (1992);
• in the case of the thermal model the inversion of the photon spectrum to determine the differential emission measure is much more ill-conditioned than the inverse problems considered in the two non-thermal models.

Acknowledgements

We gratefully acknowledge financial support of this work by a NATO Collaborative Research Grant, a UK PPARC Grant and Visitor Grant and by the Italian INFM.