4. Inversion of filament data: an example
4.1. Data acquisition and processing
To illustrate the use of the inversion procedure we have applied the algorithm to an MSDP H data set including a solar filament (Fig. 8), which was acquired on September , 1996 with the MSDP instrument of the VTT telescope at the Observatorio del Teide on the Canary Islands (Mein 1991). Nine two-dimensional images were recorded simultaneously at 9 wavelengths along the H line on a pixel CCD camera. By successive steps across the solar disk, large areas were thus scanned in relatively short time intervals. Moreover, the H line profile was computed at every pixel in the final image. For our filament study, only the 5 central wavelength positions could be used, due to the lack of contrast at the remaining far wing wavelengths.
To calculate the mean H background profile we examined the full-disk H Meudon spectroheliogram corresponding to the same date. The filament was not far from a facular region, part of which can be seen in Fig. 8 on the lower right side of the filament. No sign of any remarkable chromospheric activity could be found, on the contrary, on the upper left side of the filament, and consequently the background profile was calculated with pixels from that area.
To perform a correct comparison between observed and theoretical profiles, we have simulated on the latter the effect of observing with the MSDP. The theoretical profiles from the grid have been integrated numerically in intervals of 0.015 nm centered at wavelengths (referred to H line center) -0.0528 nm, -0.0264 nm, 0 nm, nm and nm. To carry out the calibration between observed and grid intensities we compare the respective background profiles, and , by first adjusting line center wavelengths, and then calculating a proporcionality factor K such that:
K is easily computed via least-squares:
The remaining, although small, disagreement between the two profiles in Fig. 9 may be due either to instrumental and observational reasons (e.g. poor correction from stray-light, variable seeing), or to actual differences between the quiet Sun areas, or both.
4.2. Data inversion
A total of 7543 H profiles selected from the filament region of Fig. 8 has been inverted with the inversion code described in the previous sections. The mean CPU time invested during each inversion was s on a 400 MHz-type workstation. Figs. 10 and 11 show the distributions of temperature, emission measure, microturbulence and bulk velocity inside the filament body. Notice that "gaps" in the distribution pinpoint places where the inversion was not successful, i.e. no could be found within the grid's limits.
From a close inspection of both figures we can conclude the following:
4.3. Uniqueness of the profile determinations
Having a grid of models at our disposal, it becomes possible to readily explore the whole space of parameters and search for other local solutions. For one of the observed filament profiles from Fig. 8, we have plotted in Fig. 13 the and the planes through the corresponding 4-dimensional distribution function, for (see e.g. Press et al. 1988for a definition of ). In order to give an estimate of the 's in Eq. (6), we have adopted a constant for all i's, and equal to 1% of the intensity of the nearby H continuum.
This two examples show clearly the interdependence between variables that has already been pointed out in Sect. 3.3. It becomes evident that there is only one possible solution within the range of the grid (no other local minima are found). Such a simple example illustrates the power implicit in the present inversion method, i.e. the global is always found (for the range of parameter values which have been assumed).
Although the example depicted here, in which only 4 free parameters are used to describe the grid, is relatively simple, the strategy is readily applicable to inversions with a larger number of parameters, since the ability to explore the whole grid of models enables to discriminate (and even classify) the different minima. In this latter case, on the other hand, gradient-based minimization methods can be confused by the complicated geometry of the function in the space of parameters.
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999