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 (), microturbulent velocity (), 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, and , as well as the same absolute value of V but opposite sign, are identical at all . In these cases, it suffices to keep in Eq. (2) unchanged, the dependence upon the sign of V being accounted for by in Eq. (3). Moreover, the low dependence of the populations, and hence of , on V for low velocity values (Fig. 1) implies that we can safely interpolate as a function of V with great precision. A thorough analysis of the dependence of on V has been carried out by Heinzel et al. (1999).
We have adjusted the mesh of the present grid such that any H 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 .
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 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, , , Z and V is rather intricate, a cursory inspection of the composed MALI profiles (background + cloud) for the simple case of a static cloud km s-1 reveals that:
The scheme above illustrating the dependence of the H line profile on the 4 free parameters , T, Z and , 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 is constant inside the cloud. In this way, we can naturally "collapse" Z and 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 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.
The MALI code yields fairly high for moderate and Z. Therefore, we have limited our inversion procedure to an ad hoc range of , which approximately corresponds to cm-5, i.e. the linear part in Fig. 7.
3.4. The inversion strategy
where the index refers to the wavelength positions, and the '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 function, hereafter , 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), denotes the interpolating coefficients. The expression in Eq. (7) is linear along each of the four axes , but it also includes cross products between variables. A multidimensional minimization algorithm is then applied to this polynomial representation to find .
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 is the uncertainty for the k parameter, is the increment in , and is the corresponding diagonal element of , where is the Hessian matrix. For example, if and correspond to parameters T and V respectively, then the element of the Hessian matrix will be:
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.
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999