Estimating the properties of hard X-ray solar flares by constraining model parameters

Abstract

We compare four different methods of calculating uncertainty estimates in fitting parameterized models to RHESSI X-ray spectra, considering only statistical sources of error. Three of the four methods are based on estimating the scale-size of the minimum in a hypersurface formed by the weighted sum of the squares of the differences between the model fit and the data as a function of the fit parameters, and are implemented as commonly practiced. The fourth method uses Bayesian data analysis and Markov chain Monte Carlo (MCMC) techniques to calculate an uncertainty estimate. Two flare spectra are modeled: one from the GOES X1.3 class flare of 19 January 2005, and the other from the X4.8 flare of 23 July 2002. The four methods give approximately the same uncertainty estimates for the 19 January 2005 spectral fit parameters, but lead to very different uncertainty estimates for the 23 July 2002 spectral fit. This is because each method implements different analyses of the hypersurface, yielding method-dependent results that differ greatly depending on the shape of the hypersurface. For the 23 July 2002 flare data, there is a 95% probability that the low energy cutoff lies below approximately 40 keV, and a 68% probability that it lies in the range 7-36 keV. The low-energy cutoff for the 19 January 2005 flare is more tightly constrained to 107+/-4 keV (68% probability). Using the Bayesian/MCMC approach, we also estimate for the first time probability density functions (PDFs) for the total number of flare accelerated electrons and the energy they carry. For the 23 July 2002 event, these PDFs are asymmetric with long tails orders of magnitude higher than the most probable value, caused by the poorly constrained value of the low-energy cutoff. For the 19 January 2005 flare spectrum, the PDFs for the total number of flare accelerated electrons and their energy are much more symmetric and narrow.

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• Chi-square and Poissonian Data: Biases Even in the High-Count Regime and How to Avoid them

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