With 20 spectral lines () and 8 solar regions (), our inversion problem has unknowns ( and for each line, and for each region). If all 20 lines were recorded in all of the 8 regions, there would be observables, but since the observations did not have this degree of completeness, the actual number of "observables" ( values) was 99.
Several of the 20 lines were unidentified or blended and therefore of questionable usefulness for Hanle diagnostics. As such lines may contribute to making the inversion problem severely ill posed, it is necessary to first condition the problem by exploring the degree of consistency in the behavior of the different spectral lines, to sort out before the inversion those lines that do not exhibit any consistent pattern with respect to the Hanle effect. This sorting was done as follows.
For a given spectral line the extracted values of should decrease with increasing value of for the solar regions. If one ranks the different solar regions in terms of their values, this ranking should be approximately the same for all the spectral lines if one ignores the difference in height of formation and the variation of with height. In practice the rankings are not identical for all the lines, e.g. because of measurement errors, an invalid procedure for determining , or different formation heights of the lines. Still there turns out to be a high degree of consistency between the different rankings, which is encouraging and indicates that our procedure is at least partially valid.
To explore the consistency behavior of the lines we first determine for each line and region the normalized line polarization , where is the maximum value of among the 8 solar regions for a given spectral line. The normalization is done to avoid giving undue weight to lines with high polarization amplitudes. Then we determine for each solar region the average of over all the spectral lines. This average is then used to rank the solar regions in a sequence of monotonically increasing (the numerical values of which are so far entirely unknown). Next we compare this average ranking with the rankings of the regions that are obtained if we use the values for each spectral line separately. The quantified difference between the individual and average rankings defines a "goodness index" for each line. To understand the meaning of this difference we express it in units of the difference that would be expected if the region ranking would be random for a given spectral line. Ideally, for a "perfect" line, the "goodness index" would be zero, while for a bad line (that gives random values) it would be around unity. For 12 of the 20 lines the goodness index is smaller than 0.5. We have selected these 12 lines for use in the inversion while rejecting the other lines. This leaves us with 59 observables and 32 unknowns.
The whole scale of the field strengths and is free floating unless we fix the value of for one of the lines before we start the inversion (thus leaving us with 31 unknowns). We have done this for the Ba II 4934 Å line, which has one of the best "goodness"rankings of all our lines, and for which is found to be 16.3 G from the known life time (obtained from the tables of Wiese & Martin 1980) and Landé factor of its upper state. Since it is a strong line it is formed fairly high in the solar atmosphere, where it can be estimated that the collisional depolarization rate should be negligible in comparison with the natural decay rate, such that may be assumed to be unity. We thus fix the value of for the Ba II 4934 Å line to 16.3 G. As is always , a value of that is different from unity would lead to an increase of all the derived values by the factor .
The inversion is done with a standard, non-linear, iterative least-squares fitting technique (cf. Marquardt 1963; Bevington 1969), by minimizing the standard deviation between the model and the data. The problem is thus to find the main minimum of the -hypersurface in the 31-dimensional parameter space. If the problem is ill posed there are many separate hyperspace minima to which the iteration may converge, such that the solution becomes dependent on the starting values that are chosen for the iteration.
The situation is not much helped by finding the global minimum, since there is no assurance why this global minimum should represent the "correct" solution. For any set of minima there will of course always be one that is the deepest, but this by no means implies that it has anything to do with reality. From our experience with inversion techniques, the best way to establish confidence in the validity of a solution is to do many inversion runs with widely separated starting values, both small and large. If all the inversions converge to the same solution, then one may have confidence in it. If there is a dependence on the initial values, then the solution should be discarded.
When running our Hanle inversion problem with the 31 unknowns and 59 observables, we find that all the resulting values of as well as the values of and for 8 of the 12 spectral lines are independent of all the widely chosen starting values, while the 4 remaining lines are sensitive to the choice of initial values. In spite of the mixed results for the spectral lines the solution for the turbulent field strengths remains stable, which is the main objective of the inversion. Although this stable convergence is no guarantee that our model and reduction procedure are physically correct, it suggests that we are on the right track in our initial attempts to apply the differential Hanle effect for diagnostics of magnetic fields on the solar disk. The derived values of the turbulent field strengths vary between 4 and 40 G for the 8 solar regions that we have studied.
We here refrain from presenting a tabulation of the numerical values of the solutions for the parameters and , since these values are not at all to be regarded as definite determinations of intrinsic line parameters. They depend on a large number of idealizations, which at the present time were necessary to reach a formal solution. In particular the values depend on our formal and physically questionable procedure of extracting a line polarization, with our crude estimates for the zero point of the polarization scale and the parameter in Eq. (8). In future more quantitative work it will be unavoidable to use detailed radiative transfer calculations, since the continuum and line polarizations are so entangled that the formal extraction procedure of Sect. 3.2in general cannot be trusted. Our main aim with the inversion exercise is rather to illustrate how the differential Hanle effect may in principle be used for diagnostic purposes.
The 20 spectral lines that we started with to explore the differential Hanle effect are all the lines in Figs. 1 and 2 that had an apparently significant polarization signal in solar regions with minimal Hanle depolarization. Most of the lines that did not make it among the final 8 (because they either did not satisfy the "goodness" criterion, or they had unstable convergence) are found in the 5682-5687 Å range of Fig. 2. We do not understand why this is so, but it could be that the application of identical formal extraction procedures for the line polarization to widely separated wavelength regions is incorrect and may lead to internal inconsistencies.
Let us finally note that the line polarizations and the derived values are strongly affected by instrumental broadening in the spectrograph and by macroturbulence on the Sun. Our use of the differential polarization effects (line ratios) between different spectral lines and solar regions helps to minimize the possible effects of such broadening on the determined turbulent field strengths.
© European Southern Observatory (ESO) 1998
Online publication: November 24, 1997