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Astron. Astrophys. 345, 618-628 (1999)

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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[FORMULA] data set including a solar filament (Fig. 8), which was acquired on September [FORMULA], 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[FORMULA] line on a [FORMULA] pixel CCD camera. By successive steps across the solar disk, large areas were thus scanned in relatively short time intervals. Moreover, the H[FORMULA] 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.

[FIGURE] Fig. 8. MSDP H[FORMULA] line center image of the solar filament and neighbouring chromospheric regions. The bright patch at the lower right part of the figure signals the presence of an active region. To compute the background H[FORMULA] profile, the chromospheric area at the upper left part of the filament has been used.

To calculate the mean H[FORMULA] background profile we examined the full-disk H[FORMULA] 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[FORMULA] line center) -0.0528 nm, -0.0264 nm, 0 nm, [FORMULA] nm and [FORMULA] nm. To carry out the calibration between observed and grid intensities we compare the respective background profiles, [FORMULA] and [FORMULA], by first adjusting line center wavelengths, and then calculating a proporcionality factor K such that:

[EQUATION]

K is easily computed via least-squares:

[EQUATION]

Fig. 9 compares the observed background H[FORMULA] line profile, with that of David (1961) after these corrections. Both profiles are in good agreement, their difference being always [FORMULA].

[FIGURE] Fig. 9. Relative intensity of the H[FORMULA] background profiles from David's observations (stars and solid line) and from the present study (diamonds and dashed line), in arbitrary units. David's H[FORMULA] profile has been corrected both for a proportionality factor and for a net Dopplershift relative to the MSDP profiles (see text). Wavelengths are given relative to line center (656.2808 nm).

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[FORMULA] 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 [FORMULA] 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 [FORMULA] could be found within the grid's limits.

[FIGURE] Fig. 10. Results from the inversion of filament H[FORMULA] profiles. Maps correspond to distribution of temperature (T) and emission measure (Q) inside the filament's body.

[FIGURE] Fig. 11. Same as Fig. 10, for microturbulence [FORMULA] and velocity V distributions. Negative and positive V's correspond to motions toward and away from the observer, respectively.

From a close inspection of both figures we can conclude the following:

  1. In the darker regions, T appears to be lower at the center of the filament than near its border. Such results agree well with previous findings (Hirayama 1971). The fact that the border of some of the gaps in the map are contoured by colder ([FORMULA] K) pixels suggests that the very dark cores in the filament body may in fact harbour regions of [FORMULA] K.

  2. Q, which correlates well with [FORMULA] (not shown), dramatically increases from the border towards the center of the filament. What one witnesses here is obviously the combined effect of Z and [FORMULA] through Eq. (5).

  3. It is difficult to distinguish any clear trend in the [FORMULA] distribution inside the filament. Although there are cases where higher [FORMULA] values seem to be arranged at the filament's border, the situation is less convincing than for T.

  4. One sees apparent velocity structures inside the filament, which agree well with those calculated by applying the method described by Mein N. et al. (1996), who assumed a parabolic shape for the source function inside the filament, to our data set. In Figs. 12a and b we observe the excellent agreement between the results from both methods, especially in the case of V, which shows a striking linear relationship. In the case of the optical thickness the correspondence is less good for relatively thicker slabs, which may stem from the inability of the simple parabolically-shaped [FORMULA] approach adopted by Mein N. et al. (1996) to explain the exact dependence of [FORMULA] upon [FORMULA].

[FIGURE] Fig. 12. a  Comparison between Doppler velocities computed with the method presented in this work ([FORMULA]) and those calculated with the method of Mein et al. cited in the text ([FORMULA]). b  Same as a , but for the optical thickness.

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 [FORMULA] and the [FORMULA] planes through the corresponding 4-dimensional [FORMULA] distribution function, for [FORMULA] (see e.g. Press et al. 1988for a definition of [FORMULA]). In order to give an estimate of the [FORMULA]'s in Eq. (6), we have adopted a [FORMULA] constant for all i's, and equal to 1% of the intensity of the nearby H[FORMULA] continuum.

[FIGURE] Fig. 13. a  Example of an observed H[FORMULA] MSDP profile (diamonds) from the filament of Fig. 8, and corresponding MALI grid profile (solid line) which minimizes the [FORMULA] merit function of Eq. (6). In this example, the parameters take the values [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] K and [FORMULA] cm-5. Uncertainty estimates are given as calculated with Eq. (8). b  [FORMULA] cut through the [FORMULA] distribution for the profile in a . c  [FORMULA] cut through the [FORMULA] distribution for the profile in a . In b and c , darker tones reveal lower [FORMULA] values, which are also denoted by iso-contour lines.

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 [FORMULA] 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 [FORMULA] function in the space of parameters.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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