Chapter 3 Coronal seismology with kink oscillations in the era of
3.4 Coronal Seismology using Bayesian inference
The seismological study presented in the last section was updated to employ the Bayesian inference and model comparison approaches discussed in Sections 1.7.3 and 1.7.4. This allowed additional parameters to be included in the fitted model, and their relevance in interpreting the data to be tested. This approach is also used in the next section as well as the next chapter.
The seismological analysis was updated to include additional physical ef- fects. In particular the analysis was modified to describe a time-dependent period of oscillation, additional longitudinal harmonics of the kink mode, and the decayless regime of kink oscillations, which as described in the introduction, can be detected in many coronal loops. The procedure for describing the background trend is also updated. The method is based on spline interpolation and is better at describing the dynamical background behaviour exhibited in these observations. The new method is built directly into the model function, as opposed to the detrending made prior to the fitting in Section 3.3. Bayesian analysis and MCMC sampling are used to investigate the dependence of results on model parameters and perform quantitative model comparison.
In Fig. 3.8 the oscillation for loop 1 is shown, with the different panels showing different aspects of the analysis. It is described in detail in the figure caption. In the top left panel the data and most credible model and its confidence interval are plotted. It can be seen that the model describes the observational data well. The wavelet plot in the top right panel and the detrended time series in the middle left show that the contribution of the additional longitudinal harmonics (up to the third) predicted by the model is not significant for this particular loop, however in the corresponding paper [Pascoe et al.,2017a] cases with more significant amplitude in the higher harmonics are presented. The major result of this work is presented in the middle right panel. These two density structure parameters are uniquely determined for the first time seismologically. The red error bars represent the 95% credible intervals determined from the histograms below.
The Bayes factor used in this work compares how well a particular model describes the data considering the whole explored parameter space, whereas a good- ness of fit test, for exampleχ2, compares only the best fits. This was a limitation of the work presented in Section 3.2. Morton & Mooroogen[2016] apply an alternative approach to loop oscillation model comparison using the Kolmogorov–Smirnoff test, for some of the same oscillation events.
For loop 1, the favoured model was one including the additional longitudinal harmonics but without dispersion (i.e the period ratios are fixed to integer values). This gave estimates of = 1.15`0´0..7235 and ρ0{ρe = 1.71`0´0..2219, in agreement with the
Figure 3.5: Left: least-squares fits of the time series of the fitted loop centre po- sition. In blue is the fit with a purely Gaussian envelope, in red with a purely exponential envelope and in green the generalised damping profile which includes both regimes. The dashed line corresponds to the fitted background trend and the dashed-dotted line marks the start of the oscillation. Right: the extrema of the oscillations detrended using the fitted background trend. The colour scheme for the fits is the same as for the left panel. The dashed lines denotes the fitted value ofts,
Figure 3.6: Left: seismologically determined values of and ρ0{ρe and their un-
certainties (red points). The solid line corresponds to the inversion curve obtained by using the exponential damping time alone, and the dashed lines correspond to the error bars. Right: density profiles for the transverse density structure obtained seismologically. The corresponding LOS intensity for that density profile is shown by the dashed curves.
Figure 3.7: The seismologically determined transverse structure of the of the coronal loops forward modelled to the corresponding EUV emission (blue). The observed transverse intensity profile is given by the crosses. The green dashed curves corre- spond to a Gaussian fit to the intensity profile.
was favoured, giving = 0.70`0´0..2115 and ρ0{ρe = 1.93`0´0..2418. For loop 3 the model
with a decayless component and the effect of stratification on the period ratios was favoured. From this values of = 0.42`0´0..1816 and ρ0{ρe = 3.49`6´0..6190 were obtained,
which are significantly different from those found in the previous section. These differences in the obtained transverse density structure of the three loops analysed could be due to differences in their respective formation process within the active regions, or differences in their evolution over time prior to being observed.
The improvements to the physical model combined with the use of Bayesian inference and MCMC produces improved estimates of model parameters and their uncertainties. By allowing the period of oscillation to vary in time the time series used for analysis in Section 3.3 can be extended. The consideration of additional longitudinal harmonics aimed to account for the non-harmonic shapes of some of the oscillations. It can also provide additional seismological information from the ratios these periods, as described in Section 1.4.3. Here the effects of density stratification and loop expansion are also considered, which were not included in the previous analysis. The results of the model comparison show a lack of evidence for interpret- ing these period ratios in terms of either of these effects however. The exception is for Loop #3 (not shown here), for which there was very strong evidence for the stratified model, or any other model which describesP1{nPnă1.