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

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3. Description of the inversion procedure

3.1. Constructing a grid of models

A first step in the implementation of our grid-based inversion procedure is the construction of the grid itself. Building on our previous experience with smaller grids, we have constructed a grid of 83160 MALI models. As free parameters we have chosen kinetic temperature (T), electronic density ([FORMULA]), microturbulent velocity ([FORMULA]), cloud velocity (V) and geometrical thickness (Z).

In Table 1 the range of values taken by all the parameters is shown. They closely follow the so-called Hvar Reference Atmosphere of a Quiescent Prominence (Engvold et al. 1990; Jensen & Wiik 1990) which was put forward to summarize our present knowledge about the principal physical parameters in quiescent prominences. Notice that, due to the axial symmetry of the incident radiation, atomic populations calculated for clouds which have the same T, Z, [FORMULA] and [FORMULA], as well as the same absolute value of V but opposite sign, are identical at all [FORMULA]. In these cases, it suffices to keep [FORMULA] in Eq. (2) unchanged, the dependence upon the sign of V being accounted for by [FORMULA] in Eq. (3). Moreover, the low dependence of the populations, and hence of [FORMULA], on V for low velocity values (Fig. 1) implies that we can safely interpolate [FORMULA] as a function of V with great precision. A thorough analysis of the dependence of [FORMULA] on V has been carried out by Heinzel et al. (1999).

[FIGURE] Fig. 1. [FORMULA] vs. [FORMULA], for [FORMULA] km s-1 (solid line), 2.5 km s-1 (dotted line) and 5 km s-1 (dashed line). In this example, [FORMULA] K, [FORMULA] km s-1, [FORMULA] km, and [FORMULA] cm-3. Hereafter, and unless specified otherwise, intensities and [FORMULA] values are given relative to the intensity at Sun center of the nearby H[FORMULA] continuum.

We have adjusted the mesh of the present grid such that any H[FORMULA] profile can be linearly interpolated at any position in the 5-D "hyperspace" of parameters, with a precision (i.e. the difference between it and the actual profile computed with the MALI code with a similar set of parameters) of approximately [FORMULA].

Calculating such a large set of models is a time demanding task which requires a careful design of the computational strategy. The MALI radiative transfer algorithm is such that it can use the output atomic populations from previous model computations as new input estimates, which helps saving CPU time. In all, the 83160 MALI models have required [FORMULA] days of computer time on a 400 MHz-type workstation.

3.2. Dependence of the profiles on the 5 grid parameters

Although the line profile's dependence on the parameters T, [FORMULA], [FORMULA], Z and V is rather intricate, a cursory inspection of the composed MALI profiles (background + cloud) for the simple case of a static cloud [FORMULA] km s-1 reveals that:

  • Higher T produces brighter H[FORMULA] intensities near line core, where the cloud is more opaque, as should be expected since the rate of collisions increases with T. Such effect can readily be observed in Fig. 2. In the wings, on the other hand, the strong dependence of W on T through Eq. (4) becomes more important, causing a decrease in line wing intensities at wavelengths farther than [FORMULA] nm from line center.

  • The effect of [FORMULA] on the H[FORMULA] line profile is similar to that of T (Fig. 3). Larger [FORMULA] produce brighter line core intensities as well as broader profiles in the wings (and therefore, darker intensities at fixed wavelength positions).

  • There is a noticeable dependence of line intensity on [FORMULA]. As we move from small to large [FORMULA], the H[FORMULA] line center intensity goes through a minimum, and then increases again, giving rise to the "elbow" seen in Fig. 4. One sees that the elbow is conspicuous at all temperatures. How all this affects the whole line profile is shown in Fig. 5. The contrast reversal occurs at all wavelengths as one moves towards larger [FORMULA]. This feature introduces an ambiguity when interpreting observed profiles, i.e. a given line center intensity may be matched by two very different [FORMULA] values. For very high [FORMULA], T, Z and [FORMULA], on the other hand, the intensity values go into emission (relative to the background chromosphere). Similar results have been presented and examined by Giovanelli (1967) and Heinzel & Schmieder (1994), and will not be further discussed here.

  • Larger Z increases the optical thickness of the cloud, which is then less transparent to the underneath radiation. Furthermore, large Z's have an immediate effect on the radiative transfer problem of line formation inside the cloud. As shown in Fig. 6, the final effect is similar to that of increasing [FORMULA] for low to medium [FORMULA].

[FIGURE] Fig. 2. H[FORMULA] line profile in relative units, for [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] km and [FORMULA] cm-3. The solid profile corresponds to the H[FORMULA] chromospheric background. The dotted line corresponds to [FORMULA] K, dashed is [FORMULA] K and dotted-dashed is [FORMULA] K.

[FIGURE] Fig. 3. H[FORMULA] line profile in relative units, for [FORMULA] km s-1, [FORMULA] K, [FORMULA] km and [FORMULA] cm-3. The solid profile corresponds to the H[FORMULA] chromospheric background. The dotted line corresponds to [FORMULA] km s-1, dashed to [FORMULA] km s-1, dotted-dashed to [FORMULA] km s-1 and three-dotted-dashed to [FORMULA] km s-1.

[FIGURE] Fig. 4. Intensity at H[FORMULA] line center as a function of electronic density, for a static ([FORMULA] km s-1) case. In this example, [FORMULA] km s-1 and [FORMULA] km. The horizontal solid line corresponds to H[FORMULA] line center intensity from David (1961). The solid curve corresponds to [FORMULA] K, dotted to [FORMULA] K, dashed to [FORMULA] K, dotted-dashed to [FORMULA] K, three-dotted-dashed to [FORMULA] K and long-dashed to [FORMULA] K.

[FIGURE] Fig. 5. H[FORMULA] line profile in relative units, for [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] K and [FORMULA] km. The solid profile corresponds to the H[FORMULA] chromospheric background. The dotted line corresponds to [FORMULA] cm-3, dashed to [FORMULA] cm-3, dotted-dashed to [FORMULA] cm-3, and three-dotted-dashed to [FORMULA] cm-3. Also shown in long-dashed lines (in emission relative to the chromospheric background) is the profile for [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] K, [FORMULA] km and [FORMULA] cm-1.

[FIGURE] Fig. 6. H[FORMULA] line profile in relative units, for [FORMULA] km s-1, [FORMULA] km s-1, [FORMULA] K, and [FORMULA] cm-3. The solid profile corresponds to the H[FORMULA] chromospheric background. The dotted line corresponds to [FORMULA] km, dashed to [FORMULA] km dotted-dashed to [FORMULA] km, three-dotted-dashed to [FORMULA] km and long-dashed to [FORMULA] km.

The scheme above illustrating the dependence of the H[FORMULA] line profile on the 4 free parameters [FORMULA], T, Z and [FORMULA], makes clear that ambiguities may arise during the inversion of observed profiles if the uncertainties in the observed profiles are large.

3.3. Dependence on the emission measure

In order to reduce the number of variables in our problem, we have employed the so-called emission measure, Q, which can be defined as follows (see e.g. Tandberg-Hanssen 1995):


since [FORMULA] is constant inside the cloud. In this way, we can naturally "collapse" Z and [FORMULA] into a new parameter, reducing the number of dimensions of the grid from 5 to 4. We hereby confirm the finding by Heinzel et al. (1994): there is a clear relationship between the integrated H[FORMULA] intensity (i.e. the second term on the right-hand-side of Eq. 2) and Q (Fig. 7). It is worthwhile to point out that Heinzel et al. modelled prominences as vertical slabs standing above the chromosphere, and being illuminated from both sides, whereas our MALI code models solar filaments as horizontal slabs illuminated only from below.

[FIGURE] Fig. 7. Integrated intensity E(H[FORMULA]) emitted by the slab itself, (second term on the right-hand-side of Eq. (2)), vs. Q. E(H[FORMULA]) is given in units of erg s-1 cm- 2 sr-1, and Q is in units of cm-5. Data points for all [FORMULA] are shown.

The MALI code yields fairly high [FORMULA] for moderate [FORMULA] and Z. Therefore, we have limited our inversion procedure to an ad hoc range of [FORMULA], which approximately corresponds to [FORMULA] cm-5, i.e. the linear part in Fig. 7.

3.4. The inversion strategy

We compare the observed spectral line profile, [FORMULA], with each of the grid's model profiles, [FORMULA], and then find the best match between the two by minimizing the [FORMULA] function given by:


where the index [FORMULA] refers to the wavelength positions, and the [FORMULA]'s represent the measurement errors. Once the minimum of Eq. (6) has been found at the resolution given by the grid's mesh, the search for a minimum of the [FORMULA] function, hereafter [FORMULA], is extended further by interpolating the calculated profiles with a 4-dimensional polynomial of the form:


which requires 16 points, located at the vertices of each grid cell, for each wavelength position. In Eq. (7), [FORMULA] denotes the interpolating coefficients. The expression in Eq. (7) is linear along each of the four axes [FORMULA], but it also includes cross products between variables. A multidimensional minimization algorithm is then applied to this polynomial representation to find [FORMULA].

An important aspect of our inversion strategy is the ability to estimate the uncertainty in the parameters. Assuming that our minimization algorithm has taken us to the true minimum, it is easy to see that (e.g. Bevington 1969):


where [FORMULA] is the uncertainty for the k parameter, [FORMULA] is the increment in [FORMULA], and [FORMULA] is the corresponding diagonal element of [FORMULA], where [FORMULA] is the Hessian matrix. For example, if [FORMULA] and [FORMULA] correspond to parameters T and V respectively, then the element [FORMULA] of the Hessian matrix will be:


The elements of [FORMULA] can be approximated in terms of the first partial derivatives only (Bevington 1969; Press et al. 1988), which saves computing time.

It should be pointed out that Eq. (8) is strictly valid if and only if measurement errors are normally distributed. On the other hand, it is evident that differences between observed and MALI model profiles may actually reflect the fact that solar filaments are not 1-D isothermal slabs; rather, they possess a fairly complicated filamentary structure which is not accounted for by our model. These differences may in turn appear as systematic errors, which could however be explained satisfactorily by a more complete filament model.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999