Deriving DEMs or their discretized, binned equivalents, the “emission measure dis- tributions” (EMD) in log T from X-ray spectra has been one of the central issues in observational stellar X-ray astronomy. As implied by (7), it is also of considerable im- portance in the context of determining the coronal composition (see Sect. 16). Although a full description of the methodology of spectral inversion is beyond the scope of this review, I will briefly outline the available strategies as well as the current debate on optimizing results. This may serve as an introduction and guide to the more technical literature.
If a spectrum of an isothermal plasma component with unit EM is written in vector form as f(λ, T ), then the observed spectrum is the weighted sum
g(λ)=
f(λ, T )Q(T )dlnT ≡ F · Q. (15)
In discretized form for bins log T , F is a rectangular matrix (in λ and T ). Equation (15) constitutes a Fredholm equation of the first kind for Q. Its inversion aiming at solving for Q is an ill-conditioned problem with no unique solution unless one imposes additional constraints such as positivity, smoothness, or functional form, most of which may not be physically founded. A formal treatment is given in Craig and Brown (1976). The problem is particularly serious due to several sources of unknown and systematic uncertainties,
such as inaccurate atomic physics parameters in the spectral models, uncertainties in the instrument calibration and imprecise flux determinations, line blends (see detailed dis- cussion in van den Oord et al. 1997 and Kashyap and Drake 1998 and references therein) and, in particular, unknown element abundances. The latter need to be determined from the same spectra. They are usually assumed to be constant across the complete DEM although this hypothesis is not supported by solar investigations (Laming et al. 1995; Jordan et al. 1998). The following constrained inversion techniques have turned out to be convenient:3
1. Integral inversion with regularization. A matrix inversion of (15) is used with the additional constraint that the second derivative of the solution Q(log T ) is as smooth as statistically allowed by the data. Oscillations in the data that are due to data noise are thus damped out. This method is appropriate for smooth DEMs, but tends to produce artificial wings in sharply peaked DEMs (Mewe et al. 1995; Schrijver et al. 1995; Cully et al. 1997). A variant using singular value decomposition for a series of measured line fluxes was discussed by Schmitt et al. (1996b) .
2. Multi-temperature component fits. This approach uses a set of elementary DEM build- ing blocks such as Gaussian DEMs centered at various T but is otherwise similar to the traditional multi-component fits applied to low-resolution data (examples were given by Kaastra et al. 1996 and Güdel et al. 1997b).
3. Clean algorithm. This is both a specific iteration scheme and a special case of (2) that uses delta functions as building blocks. The observed spectrum (or part of it) is correlated with predictions from isothermal models. The model spectrum with the highest correlation coefficient indicates the likely dominant T component. A fraction of this spectral component is subtracted from the observation, and the corresponding model EM is saved. This process is iterated until the residual spectrum contains only noise. The summed model EM tends to produce sharp features while positivity is ensured (Kaastra et al. 1996).
4. Polynomial DEMs. The DEM is approximated by the sum of Chebychev polyno- mials Pk. For better convergence, the logarithms of the EM and of T are used: log[Q(T )dlog (T )] = N−1
k=0 akPk(log T ) which ensures positivity. The degree N of the polynomial fit can be adjusted to account for broad and narrow features (Lemen et al. 1989; Kaastra et al. 1996; Schmitt and Ness 2004; Audard et al. 2004). 5. Power-law shaped DEMs of the form Q(T ) ∝ (T /Tmax)α up to a cutoff tempera-
ture Tmaxare motivated by the approximate DEM shape of a single magnetic loop
(Pasquini et al. 1989; Schmitt et al. 1990a).
3 I henceforth avoid expressions such as “global” or “line-based methods” that have often been used in various, ill-defined contexts. Spectral inversion methods should be distinguished by i) the range and type of the data to be fitted, ii) the parameters to be determined (model assumptions), iii) the iteration scheme for the fit (if an iterative technique is applied), iv) the convergence criteria, v) the constraints imposed on the solution (e.g., functional form of DEM, smoothness, positivity, etc), and vi) the atomic database used for the interpretation. Several methods described in the literature vary in some or all of the above characteristics. Most of the methods described here are not inherently tailored to a specific spectral resolving power. What does require attention are the possible biases that the selected iteration scheme and the constraints imposed on the solution may introduce, in particular because the underlying atomic physics tabulations are often inaccurate or incomplete (“missing lines” in the codes).
The ranges and types of data may vary depending on the data in use. Low-resolution spectra are commonly inverted as a whole because individual features cannot be isolated. If a high-resolution spectrum is available, then inversion methods have been applied either to the entire spectrum, to selected features (i.e., mostly bright lines), or to a sample of extracted line fluxes.
As for iteration schemes, standard optimization/minimization techniques are avail- able. Various methods have been developed for fits to samples of line fluxes (e.g., Lemen et al. 1989; Huenemoerder et al. 2001, 2003; Osten et al. 2003; Sanz-Forcada et al. 2003; Telleschi et al. 2004), with similar principles:
1. The DEM shape is iteratively derived from line fluxes of one element only, typically Fe (xvii-xxvi in X-rays, covering T up to≈100 MK), e.g., by making use of one of the above inversion schemes tailored to a sample of line fluxes. Alternatively, one can use T -sensitive but abundance-independent flux ratios between He-like and H-like transitions of various elements to construct the DEM piece-wise across a temperature range of≈ 1–15 MK (Schmitt and Ness 2004).
2. The Fe abundance (and thus the DEM normalization) is found by requiring that the continuum (formed mainly by H and He) agrees with the observations.
3. The abundances of other elements are found by comparing their DEM-predicted line fluxes (e.g., assuming solar abundances), with the observations.
The advantage of such schemes is that they treat the DEM inversion and the abun- dance determination sequentially and independently. Huenemoerder et al. (2001) and Huenemoerder et al. (2003) used an iteration scheme that fits DEM and abundances si- multaneously based on a list of line fluxes plus a continuum. Kashyap and Drake (1998) further introduced an iteration scheme based on Markov-chain Monte Carlo methods for a list of line fluxes. This approach was applied to stellar data by Drake et al. (2001). Genetic algorithms have also, albeit rarely, been used as iteration schemes (Kaastra et al. 1996 for low-resolution spectra).
There has been a lively debate in the stellar community on the “preferred” spectral inversion approach. Some of the pros and cons for various strategies are: Methods based on full, tabulated spectral models or on a large number of individual line fluxes may be compromised by inclusion of transitions with poor atomic data such as emissivities or wavelengths. On the other hand, a large line sample may smooth out the effect of such uncertainties. Consideration of all tabulated lines further leads to a treatment of line blends that is self-consistent within the limits given by the atomic physics uncertainties. A most serious problem arises from weak lines that are not tabulated in the spectral codes while they contribute to the spectrum in two ways: either in the form of excess flux that may be misinterpreted as a continuum, thus modifying the DEM; or in the form of unrecognized line blends, thus modifying individual line fluxes and the pedestal flux on which individual lines are superimposed. A careful selection of spectral regions and lines for the inversion is thus required (see discussion and examples in Lepson et al. 2002 for the EUV range).
If DEMs and abundances are iterated simultaneously, numerical cross-talk between abundance and DEM calculation may be problematic, in particular if multiple solutions exist. Nevertheless, each ensemble of line flux ratios of one element determines the same DEM and thus simultaneously enforces agreement. If a list of selected line fluxes
is used, e.g., for one element at a time, DEM-abundance cross talk can be avoided, and the influence of the atomic physics uncertainties can be traced throughout the reconstruction process. But there may be a strong dependence of the reconstruction on the atomic physics uncertainties and the flux measurements of a few lines. The lack of a priori knowledge on line blends affecting the extracted line fluxes will introduce systematic uncertainties as well. This can, however, be improved if tabulated potential line blends are iteratively included.
The presence of systematic uncertainties also requires a careful and conservative choice of convergence criteria or smoothness parameters to avoid introduction of spu- rious features in the DEM. The result is a range of solutions that acceptably describe the data based on a goodness-of-fit criterion, in so far as the data can be considered to be represented by the spectral database in use. Within this allowed range, “correctness” cannot be judged on by purely statistical arguments. The spectral inversion is non-unique because the mathematical problem is ill-posed – the atomic data deficiencies cannot be overcome by statistical methodology but require external information.
Direct comparisons of various methods, applied to the same data, are needed. Mewe et al. (2001) presented an EMD for Capella based on selected Fe lines that compares very favorably with their EMD derived from a multi-T approach for the complete spec- tra, and these results also seem to agree satisfactorily with previously published EMDs from various methods and various data sets. EMDs of the active HR 1099 found from spectra of XMM-Newton RGS (Audard et al. 2001a) and from Chandra HETGS (Drake et al. 2001) agree in their principal features, notwithstanding the very different recon- struction methods applied and some discrepancies in the abundance determinations. Telleschi et al. (2004) determined EMDs and abundances of a series of solar analogs at different activity levels from polynomial-DEM fits to selected spectral regions and from an iterative reconstruction by use of extracted line-flux lists. The resulting EMDs and the derived abundances of various elements are in good agreement. Schmitt and Ness (2004) compared two approaches within their polynomial DEM reconstruction method, again concluding that the major discrepancies result from the uncertainties in the atomic physics rather than from the reconstruction approach, in particular when results from EUV lines are compared with those from X-ray lines. A somewhat different conclusion was reported by Sanz-Forcada et al. (2003) from an iterative analysis of Fe-line fluxes of AB Dor; nevertheless, their abundance distribution is in fact quite similar to results reported by Güdel et al. (2001b) who fitted a complete spectrum.
The evidence hitherto reported clearly locates the major obstacle not in the inversion method but in the incompleteness of, and the inaccuracies in, the atomic physics tabu- lations. Brickhouse et al. (1995) gave a critical assessment of the current status of Fe line emissivities and their discrepancies in the EUV range, together with an analysis of solar and stellar spectra. The effects of missing atomic transitions in the spectral codes were demonstrated by Brickhouse et al. (2000) who particularly discussed the case of Fe transitions from high n quantum numbers, i.e., of transitions that have only recently been considered.