INVESTIGATING THE RELIABILITY OF CORONAL EMISSION MEASURE DISTRIBUTION DIAGNOSTICS USING THREE-DIMENSIONAL RADIATIVE MAGNETOHYDRODYNAMIC SIMULATIONS

The heating mechanism that is responsible for the million degree solar corona remains unknown, though several candidates exist. It is one of the most important open issues in astrophysics. Constraining the properties of this heating mechanism is a difficult task, but usually performed through spectral and imaging observations of the solar corona, which provide diagnostics of the plasma temperature distribution. The latter has important implications for the energy balance of the corona (see, e.g., Klimchuk 2006; Reale 2010 and references therein).

The thermal distribution of the plasma—or emission measure distribution, EMD (in Section 2 we define the relationship between EMD and the differential emission measure, DEM, which is also often used to describe the plasma thermal distribution)—is crucial in testing heating models. For instance, the EMD in coronal loops strongly depends on the spatial and temporal properties of the energy release (e.g., Klimchuk & Cargill 2001; Cargill & Klimchuk 2004; Testa et al. 2005). The presence of a high-temperature (T ≳ 5 MK) component, e.g., in the EMD of an active region, is a good tracer of the properties of the heating (e.g., Patsourakos & Klimchuk 2006), and it has recently been addressed by several studies (e.g., Patsourakos & Klimchuk 2006; Reale et al. 2009a, 2009b; Ko et al. 2009; Schmelz et al. 2009a; Testa et al. 2011; Testa & Reale 2012). Also, the thermal distribution in the cross-field direction can help discern between a monolithic, single strand loop (isothermal at a given location along the loop; e.g., Aschwanden et al. 2000; Aschwanden & Nightingale 2005; Aschwanden & Boerner 2011; Del Zanna 2003; Landi et al. 2002, 2006; Landi & Feldman 2008) and a loop structure composed of several strands (multi-thermal plasma; e.g., Schmelz et al. 2001, 2005; Warren & Brooks 2009; Brooks et al. 2009).

Ample effort has been devoted to the accurate determination of the thermal structuring of coronal plasma to derive robust observational constraints on the coronal heating mechanism(s). The plasma temperature distribution of the quiet corona and of active regions has been investigated through imaging data and spectroscopic observations (e.g., Brosius et al. 1996; Landi & Landini 1998; Aschwanden et al. 2000; Testa et al. 2002; Del Zanna & Mason 2003; Reale et al. 2007; Landi et al. 2009; Shestov et al. 2010; Sylwester et al. 2010). Several recent studies have focused on EUV spectra obtained with the Hinode Extreme Ultraviolet Imaging Spectrometer (EIS; Culhane et al. 2007) which provides good temperature diagnostic capability, together with higher spatial resolution and temporal cadence than previously available (e.g., Watanabe et al. 2007; Warren et al. 2008; Patsourakos & Klimchuk 2009; Brooks et al. 2009; Warren & Brooks 2009; Testa et al. 2011; Tripathi et al. 2011). Imaging observations obtained by the Solar Dynamics Observatory (SDO) Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) in narrow EUV bands have also been used for studying the plasma temperature distribution, especially in active region loops (e.g., Schmelz et al. 2010; Aschwanden & Boerner 2011; Aschwanden et al. 2011; Schmelz et al. 2011), also in conjunction with EIS data (e.g., Brooks et al. 2011; Warren et al. 2011; Testa & Reale 2012).

The optically thin nature of the coronal EUV and X-ray emission implies that the emission observed in the resolution element (instrument pixel) is generally produced by several independent structures along the line of sight (LOS). Most plasma diagnostics rely on some homogeneity assumptions (e.g., constant plasma density along the LOS) whereas these overlapping structures can in principle have very different plasma parameters. This poses significant challenges for interpreting the meaning of the derived plasma parameters (which are by necessity weighted averages of the distributions along the LOS) in terms of the actual physical conditions of the plasma. Further assumptions typically made for specific diagnostics (e.g., on the functional form and smoothness of the EMD) also have a potentially significant impact on the accuracy of the diagnostics. Several efforts have been carried out to test the accuracy of the plasma temperature diagnostics. For instance, the notorious challenges in determining the EMD and its confidence limits have been addressed in several studies (e.g., Craig & Brown 1976; Judge et al. 1997; McIntosh 2000; Judge 2010; Landi & Klimchuk 2010; Landi et al. 2012; Hannah & Kontar 2012). The main limitation of these previous efforts lies in the necessarily simplified test cases adopted.

In this paper we use advanced three-dimensional (3D) radiative MHD (rMHD) simulations of the solar atmosphere with the Bifrost code (Gudiksen et al. 2011) to carry out detailed tests of coronal plasma temperature diagnostics. We focus on the plasma temperature diagnostics using current spectral (Hinode/EIS) and imaging (SDO/AIA) data. We synthesize EIS and AIA data from 3D rMHD simulations and analyze them like real data, and use the comparison of the thermal distributions inferred from the synthetic data with the "true" distributions, which in the case of the simulations are known, to carefully assess the accuracy of the diagnostics. These 3D simulations provide us with the opportunity to improve upon previous work by exploring more realistic configurations with significant superposition of different structures along the LOS, allowing a statistical approach to determine the accuracy and limitations of the plasma diagnostics, for a variety of spatial and thermal structuring of the plasma. The 3D nature of the simulations also allows us to explore a variety of realistic viewing angles, reproducing typical distributions of structuring ranging from on-disk to limb observations.

In Section 2, we describe the analysis methods, include a short description of the Bifrost code, and discuss the characteristics of the 3D rMHD models and of the corresponding synthetic observables used in this work. The analysis of the synthetic data and the results of the determination of the plasma temperature distribution are presented and discussed in Section 3. We summarize our findings and draw our conclusions in Section 4.

In this work we investigate how accurately the thermal distribution of the plasma can be inferred from coronal observations that are currently available. The observed intensities of a set of spectral emission lines constrain the plasma temperature distribution, as they depend on the abundance A_Z of the emitting element, the plasma emissivity of the spectral feature G_λ(T, n_e) as a function of temperature T and electron density n_e, and the differential emission measure distribution DEM(T):

$$\begin{equation} I_{\lambda } = A_Z \int _{T} G_{\lambda }(T,n_e) {\rm DEM}(T) \,dT, \end{equation} \tag{ 1 }$$

where DEM(T) = n²_e dV/dT (cm⁻³ K⁻¹). Analogously, in the case of imaging observations in broad/narrow passbands, the observed intensity in a channel will depend on the temperature response function, R_chan(T):

$$\begin{equation} I_{\rm chan} =\int _{T} R_{\rm chan}(T) {\rm DEM}(T) \,dT. \end{equation} \tag{ 2 }$$

Throughout the paper, instead of the DEM(T) we will discuss the emission measure distribution EM(T) which is obtained by integrating the differential emission measure distribution DEM(T) in each temperature bin; here we use a temperature grid with constant binning in logarithmic scale (Δlog T = 0.05).

Several methods have been developed to reconstruct EMDs from a set of observed intensities in lines, or passbands in the case of imaging observations (see, e.g., review by Phillips et al. 2008, and the discussion and references in Hannah & Kontar 2012). Here we test the Monte Carlo Markov chain (hereafter MCMC) forward modeling method (Kashyap & Drake 1998), which is widely used and considered to provide robust results (see, e.g., Landi et al. 2012; Hannah & Kontar 2012; see also http://www.lmsal.com/~boerner/demtest/ for a recent comparative analysis of results from different methods, applied to AIA). With respect to several other methods, the MCMC method has the advantages of not imposing a pre-determined functional form for the solution, and, most importantly, of estimating the uncertainties associated with the resulting EMD (see, e.g., Kashyap & Drake 1998; Testa et al. 2011 for additional details). We use the Package for Interactive Analysis of Line Emission (PINTofALE; Kashyap & Drake 2000) which is available as part of SolarSoft. Though no pre-determined functional form is imposed, the MCMC method does apply some smoothness criteria which are locally variable and based on the properties of the temperature responses/emissivities for the used data, instead of being arbitrarily determined a priori.

For a given simulation snapshot, which provides the electron density n_e and temperature values for each grid point (voxel) in a 3D box (see Section 2.1 for a description of the characteristics of the simulations we analyzed), and a selected LOS we proceed as follows.

• 1.  
  Using the n_e and T values, and the emissivities from CHIANTI (Dere et al. 1997, 2009), we synthesize intensities of a set of EIS lines and in the six AIA coronal channels, using the optically thin approximation and statistical ionization equilibrium.
• 2.  
  By integrating through the box along the LOS and degrading the spatial resolution to the instrument resolution, we obtain synthetic coronal images in the different lines/passbands.
• 3.  
  We consider two cases: with or without photon (Poisson) noise; i.e., in the latter case we randomize the intensities according to the photon counting statistics. The noise level and the uncertainties associated with the intensities are calculated by assuming signal-to-noise ratios (S/Ns) typical of actual observations.
• 4.  
  We use the intensities and uncertainties, in each pixel, as input for the MCMC routine to calculate the EM(T) solution, pixel by pixel.
• 5.  
  We analyze the results, using several parameters to assess the ability of the method to reproduce the input intensities and the "true" EM(T), which in these test cases are known.

In Section 2.1, we describe in detail the choices and assumptions made, and the characteristics of the selected simulations and of the resulting synthetic data.

2.1. 3D Models and Synthetic Observables

The model considered spans from the upper layer of the convection zone up to the low corona, and self-consistently produces a chromosphere and hot corona, through Joule dissipation of electrical currents. In order to investigate the robustness of the EM(T) reconstruction method for a wide range of plasma conditions, we selected snapshots from two simulations that are characterized by significantly different parameters (strength and spatial distribution of the seed magnetic field, and dimensions of the box), which leads to significantly different distributions of temperature and density throughout the box.

We analyze one of the snapshots from the simulation that was previously used to investigate the relative contribution of different lines to the emission in the AIA passbands (Martínez-Sykora et al. 2011). This simulation covers a volume (x, y, z, where z is the vertical direction) of 16 × 8 × 16 Mm³ (512 × 256 × 365 grid points; with Δx = Δy ∼ 32 km and Δz non-uniform and smaller where gradients are large with values of Δz ≈ 28 km up to a height of 4 Mm and increasing to ≈150 km at the top of the computational box), and is highly dynamic, yielding coronal temperatures that increase with time. We used an intermediate snapshot (t = 1200 s), where the distribution of temperature peaks around log (T[K]) ∼ 6.25 and log T_max ∼ 6.4. As discussed in Martínez-Sykora et al. (2011), the plasma conditions of this simulation represent a good comparison for quiet Sun/coronal hole conditions with some small emerging flux regions, e.g., small bright points. In the following we use the label "C" to refer to this cooler and smaller snapshot.

The other snapshot we consider (hereafter H) is from a simulation modeling a larger region (24 × 24 × 16 Mm³; 768 × 768 × 768 grid points; Δx = Δy ∼ 31 km, Δz ≈ 14 km up to a height of 4 Mm and increasing to ≈80 km at the top of the computational box; see Figure 1) with a magnetic field configuration consisting of two small regions of opposite polarities similar to a small active region (M. Carlsson et al., in preparation). The distribution of coronal temperatures is also similar to typical active region values with a peak around log (T[K]) ∼ 6.6 and log T_max ∼ 6.7.

Figure 2 shows the distributions of electron density and temperature throughout the simulation box for the two selected snapshots, including only plasma with log T higher than 5.3. These distributions show that simulation "C," which models the smaller region with emerging flux, is characterized by lower coronal temperature than the larger simulation "H," and it has a broader distribution of densities.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Intensities of a selection of EIS lines and AIA passbands were computed for each voxel of the 3D simulation box V_xyz using the plasma electron density and temperature values and using the density- and temperature-dependent contribution functions G(T, n_e) from CHIANTI (Dere et al. 1997, 2009), assuming coronal abundances (Feldman 1992), and the ionization fractions of chianti.ioneq. For AIA we compute synthetic intensities in the coronal passbands (94 Å, 131 Å, 171 Å, 193 Å, 211 Å, 335 Å) by calculating the spectra (line by line, taking into account the temperature and density dependence of the contribution function for each line, and adding the continuum emission) and folding them through the effective area of each channel (Boerner et al. 2012). In the case of EIS we selected a set of lines that provide good coverage of the temperature range log T ∼ 5.6–6.7, but using a relatively limited number of lines, representative of a typical EIS observation: We selected 14 lines, which are listed in Table 1.

Table 1. EIS Lines Synthesized from 3D Models and Used for Reconstructing the Emission Measure Distribution

λ(Å)a	Ion	log (Tmax[K])	Notesb
268.991	Mg vi	5.65	sb
185.213	Fe viii	5.70
278.404	Mg vii	5.80
275.361	Si vii	5.80
188.497	Fe ix	5.90
184.537	Fe x	6.05
188.216	Fe xi	6.15
195.119	Fe xii	6.20	sb
274.204	Fe xiv	6.30
284.163	Fe xv	6.35
262.976	Fe xvi	6.45
208.604	Ca xvi	6.70
192.853	Ca xvii	6.75	bl
254.347	Fe xvii	6.75

Notes. ^aThe wavelengths are from CHIANTI (in case of self-blend we list the wavelength of the strongest line). ^bThe label "sb" indicates that the spectral feature is a self-blend of lines from the same ion, all included in synthesizing the data. The label "bl" for the Ca xvii line indicates that at the EIS spectral resolution this line is blended with the Fe xi and O v lines (Ko et al. 2009). For this paper however we only synthesize the Ca xvii intensity, i.e., we do not calculate the blending lines.

Download table as:  ASCIITypeset image

Images are then derived for each snapshot, and for two different LOS, by integrating the emission through the simulated box along the LOS: along the horizontal direction y for the "side view," and along the vertical direction z in the case of the "top view," analogously to the case presented in Martínez-Sykora et al. (2011). Figures 3 and 4 show examples of such images, at the intrinsic resolution of the simulations. In addition, we degrade the spatial resolution to the instrumental spatial resolution: 0.6 arcsec pixel⁻¹ for the AIA synthetic data and 1 arcsec pixel⁻¹ for EIS.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In order to compute the uncertainties of the simulated intensities we assume an S/N typical of well-exposed active region observations (e.g., Reale et al. 2011). For the bright channels of AIA (171 Å, 193 Å, 211 Å), we use the DN pixel⁻¹ assuming typical exposure times in a single image (t_exp ∼ 2 s). For the other channels (94 Å, 131 Å, 335 Å) which typically have significantly fainter emission we use the S/N calculated assuming the exposure of 10 summed images (corresponding to t_exp ∼ 30 s). In the case of EIS, we calculate the S/N by scaling the images so that the Fe xii 195 Å line, which is usually one of the brightest emission lines in EIS observations, has 500 counts, as in typical observations, and computed the uncertainties from photon counting statistics. We first considered the case without Poisson noise, and then included the Poisson noise by randomizing the intensity values assuming the photon counting statistics of typical AIA and EIS observations as described above.

In Appendix A we present a comparison of the synthetic intensities derived from the simulations, as described above, with measured intensities from recent SDO/AIA observations. This comparison shows that the simulations considered here yield emission values of similar order of magnitude to real observations, and therefore provide a sensible test case for the coronal diagnostics.

Following the procedure described in detail in Section 2, we analyzed the synthetic data to diagnose the plasma temperature distributions, pixel by pixel, using the MCMC reconstruction method. We will first discuss the results obtained using the EIS synthetic data, and then describe the results obtained with the AIA intensities.

When analyzing real data, the only measure of the goodness of the inferred EMD consists in the agreement between the measured intensities and the emissions predicted using the best-fit EMD. Therefore, as a first step, we show the maps of χ²₀ (=Σ_j[(I_{j, pred} − I_{j, obs})/σ_j]²/df, where I_{j, obs} and σ_j are the synthetic intensity and associated error for the line/channel j, I_{j, pred} is the intensity predicted by the best-fit EMD, and df are the degrees of freedom⁴), for both snapshots and both LOS, using the EIS synthetic intensities (snapshots C and H in Figures 5 and 6, respectively). We show both the case without Poisson noise (left) and including Poisson noise (right). In both cases the errors on the intensities, σ_j, are calculated as the Poisson error on the counts values. For both snapshots the χ²₀ maps indicate that the inferred EMD on average reproduce the "measured" fluxes adequately. The case without noise is characterized by χ²₀ lower than 1 over large areas, while areas with large χ²₀ present clear structuring, which is still present in the case including the effect of noise.

For the top view case, the poorer results in those areas appear to be related to the plasma electron density. To investigate this effect, we use the diagnostic based on the Fe xii line ratio 186.88 Å/195.12 Å, which is one of the most useful density diagnostics accessible with EIS spectra (e.g., Young et al. 2009), and included in many EIS observing programs.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In Figure 7 we show the top view map of electron density obtained from the ratio of the synthetic Fe xii intensities, computed from the two snapshots. By comparing these density maps to the χ²₀ plots (top row, Figures 5 and 6), it is clear that the high-density regions, such as for instance the emerging flux region of snapshot C or footpoint ("moss") regions of snapshot H, generally correspond to high χ²₀ areas.

Zoom In Zoom Out Reset image size

As an example, in Figure 8 we show the electron density distribution from the simulations within two pixels with significantly different χ²₀ (with pixel A showing lower χ²₀ than B), in the top view case for each of the two snapshots. The selected pixels are marked in Figures 5 and 6 (top row panels). We find that the plasma in pixel B (χ²₀ ≳ 2) has a density distribution along the LOS that is much broader than the plasma in pixel A. This leads to a worse determination of the emission measure and worse match of the observed fluxes: Even if the selected lines, which are similar to typical analyses of real data, are mostly insensitive to electron density, their small density dependency significantly affects the diagnostics of EMDs when there is significant mixing of plasmas with different electron densities within the LOS. The same density mixing effect is causing the large χ²₀ regions at the loop footpoints in the side views of both snapshots (bottom panels of Figures 5 and 6).

Zoom In Zoom Out Reset image size

For the side view, we note that for snapshot H, in the higher atmosphere, where the temperature distribution in each pixel is essentially isothermal at high temperatures (see also Appendix B), the case including the noise tends to have slightly lower χ²₀ values (Figure 6). We interpret this effect as due to the extremely narrow (isothermal) distribution: the MCMC method applies some physically based, locally variable smoothness criteria based on the properties of the temperature responses/emissivities for the data used (see also discussion in Kashyap & Drake 1998; Testa et al. 2011), and therefore it does not perfectly recover purely isothermal distributions; the effect of the noise mimics slight departures from isothermal distributions and therefore the MCMC methods find better matches to the intensities. A similar effect is also observed for small regions of snapshot C (e.g., the high χ²₀ strip at z = 19 pixels and x from pixel 7 to 14).

We now consider the χ²₀ maps obtained for the analysis of the AIA synthetic intensities; Figures 9 and 10 show the maps for snapshot C and H, respectively (analogous to the EIS results of Figures 5 and 6). For snapshot C, the top view shows a few pixels with higher χ²₀ in the region corresponding to the dense emerging flux region, similar to the EIS case. For snapshot H, there is no clear correspondence between the χ²₀ value and the regions with high-density mixing. This can be explained by looking at the density sensitivity of the AIA bands, which we have already discussed in Martínez-Sykora et al. (2011) and summarized in Figure 11. The plots in Figure 11 show that most channels have limited density sensitivity, which becomes negligible at high temperature. Therefore we expect less of a density effect for snapshot H where high-temperature plasma (log T > 6.5; see also Appendix B) largely contributes to the observed intensities. In the side view case, the intensities are reproduced quite well in the cases that do not include Poisson noise, whereas the cases including noise show large areas of high χ²₀ values, especially at large heights. The larger effect of the noise for AIA is due to the fact that for the side view, the AIA intensities are rather low (see also Figure 26) in some channels, especially in 131 Å for both snapshots and in 94 Å for snapshot C. These channels are more sensitive to the cool plasma (though, for snapshot H, the 94 Å intensity is largely due to the Fe xviii emission from the high-temperature plasma), which has a very small or zero contribution higher up in the atmosphere, where the plasma is close to isothermal (as we will show in detail in Section 3.1). The low statistics in some channels makes the fit sensitive to the noise. As a result, the fits become worse when including the effect of Poisson noise. We have investigated the results for significantly larger S/N for the side view case of snapshot H. We re-run the case including noise, but assuming exposure times that correspond to 50 images for all AIA channels. We find that for very large S/N values the χ²₀ values for the intensities tend to get large because of the very small errors associated with the intensities (down to the level of ∼0.1% for the brightest channels). We find that in order to get reasonable χ²₀ values, the MCMC procedure needs to be run with a significantly larger number of simulations to reproduce the intensities to that level of accuracy. However, as we will discuss later in Section 3.2, results show that the effect of noise is not the main factor that determines the ability of the MCMC method to reconstruct EMDs from AIA data.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For snapshot H, for both LOSs, for the case without noise, the χ²₀ values are generally worse than the corresponding C snapshot (Figure 10). We interpret this to be an effect of the broader temperature distributions (especially in the top view) extending to rather high temperatures (log T[K] ≳ 6.6; see also Figures 16 and 17, Appendix B, and Section 3.1). In this temperature regime AIA provides less reliable temperature diagnostics than at lower temperatures (log T[K] ∼ 6–6.3) where several AIA channels have high sensitivity and their relative shapes in temperature response provide better constraints to the plasma temperature distributions.

We note that the uncertainties we assume here are generally significantly smaller than the errors typically associated with intensity measurements in real data for the EMD reconstruction. Those errors are typically larger than the uncertainties associated with photon counting statistics because they also take into account uncertainties in atomic data which are difficult to quantify. In the test presented in this paper, there are no uncertainties to be associated with the atomic data, since we use the same atomic data to synthesize the intensities and to infer the EMD from them. Nevertheless, the small errors we assume can explain in part the sometimes large values of χ²₀ we obtained here.

The results discussed so far provide us with a measure of how well the temperature distributions inferred using the MCMC method are able to reproduce the "measured" intensities. We now discuss in detail the comparison of the derived EMD with the true EMDs. We looked at different parameters to assess the robustness and limits of the reconstruction method in recovering the plasma temperature distributions: (1) the temperature at which the EMD peaks (T(EMD_max)) within the temperature range log T(K) = [5.5–6.7], (2) the EM values integrated in broad temperature bins, and (3) the full EMD (i.e., at the fine temperature resolution). The temperature of the EMD peak is not necessarily a good indicator to evaluate the goodness of the MCMC solution, especially for broader and more structured EMDs. We therefore present the analysis of T(EMD_max) in Appendix A, summarizing here only the main results, while in the main text we will focus on the other two comparisons (Sections 3.1 and 3.2).

The results presented in Appendix A show that the MCMC method performs well in recovering the T(EMD_max) when the EMD is close to isothermal. For broad EMDs, the method tends to generally underestimate T(EMD_max) due to several factors, including poor constraints beyond the log T(K) = [5.5–6.7] range and limited capability of the method to recover sharp features.

3.1. Emission Measure Maps

One of the approaches we use to determine the accuracy of the EMD diagnostics is by looking at the maps of emission measure in different temperature ranges.

In Figures 12 and 13, we show, for top and side view of snapshot C, respectively, the maps of the true emission measure (left column) in six relatively broad temperature bins spanning the temperature range log T(K) = 5–7. Figures 16 and 17 show analogous plots for snapshot H. Figures 14 and 15 (for snapshot C) and Figures 18 and 19 (for snapshot H) also show the EM maps derived at the AIA spatial resolution for direct comparison with the corresponding maps derived from the analysis of the synthetic data. These plots show, for the top view, how the plasma is characterized by rather broad temperature distributions in most locations, especially in locations associated with the emerging flux region for snapshot C, and with the loop footpoints ("moss") for snapshot H (see Figures 3 and 4). The simulated corona in snapshot C lacks significant plasma volumes at temperatures above log T(K) = 6.4, while for snapshot H the coronal plasma reaches even higher temperatures than log T(K) = 6.7 in a hot loop-like feature, which emits brightly in, e.g., Ca xvii (Figure 4). The side view shows loop-like features in the lower corona of both snapshots, with a rather broad temperature distribution. Higher up in the corona the plasma is characterized by narrow temperature distributions peaking around log T(K) ∼ 6.2 in snapshot C, and log T(K) ∼ 6.6 in snapshot H.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The corresponding maps for the EMD derived from the analysis of EIS and AIA synthetic data are shown next to the true maps for all the cases, without or including noise (in the center and right columns, respectively, of Figures 12–15 and 16–19).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For EIS, the general features of the EMD in the central temperature bins (i.e., between log T[K] ∼ 5.5 and ∼6.4 for snapshot C and in the range log T[K] ∼ 5.5–6.7 for snapshot H) are recovered quite well, also when including the effect of the noise. In the top view case, for both snapshots, in the temperature bin at the low end of the constrained range (log T[K] ∼ 5.5–5.8), the EM is overestimated in several pixels because of the same effects discussed in Appendix B—the lack of constraints at lower temperatures. The noisy results at the low- and high-temperature end are expected considering the poor constraints provided by the selected EIS lines at those temperatures.

With AIA, though some structures such as the side view loops are still apparent in the reconstructed EM maps to a lesser extent, the EM maps derived from the analysis of the synthetic data are not reproducing the true maps very accurately, especially for the top view case. For instance, for the top view case of snapshot C (14) the maps derived from AIA present significant EM, especially in the emerging flux region, in the bin temperature log T[K] ∼ 6.4–6.7, where the real EM is zero almost everywhere.

In Figure 20, we show, for all the cases including the effect of the noise, the scatter plots of the derived versus true EM for the four central temperature bins, i.e., log T[K]: 5.5–5.8, 5.8–6.1, 6.1–6.4, 6.4–6.7. These plots show that when integrating in broad temperature ranges (Δlog T = 0.3), the EM derived from EIS reproduces the true EM rather well, while the EM derived from AIA has significantly larger scatter for the whole range of EM values. We note that these bins are significantly wider than typically used by observers when deriving EMDs.

Zoom In Zoom Out Reset image size

3.2. Full Emission Measure Distributions

In the previous section we have addressed a comparison of the general properties of the EMDs derived from synthetic EIS and AIA data with the input EMDs from 3D rMHD simulations. In this section we will assess how well the derived EMDs reproduce the true EMDs in their details on the finest temperature scale, and explore to what level of detail the MCMC EMD reconstruction method is reliable.

In Figure 29 we have already shown the full EMDs for a selection of pixels chosen to explain the systematic discrepancies in T(EMD_max). That set of plots already shows that, while as discussed above some general features of the EMD is recovered by the MCMC method, when comparing the EMDs on the fine temperature scale (i.e., here we used Δlog T(K) = 0.05 which is the typical intrinsic resolution of atomic data, and also the typical resolution used by observers), the derived EMD can significantly depart from the true distributions.

In Figure 21 we show the comparison of the true and derived EMD for a pixel selected for each case (for every snapshot and LOS, for the cases including noise), to show examples where, even if the intensities are reproduced within the errors (χ² ≲ 1), the derived and true EMD present significant discrepancies.

Zoom In Zoom Out Reset image size

First we note that in most of the examples of Figures 29 and 21, there are several temperature bins within the log T = [5.5–6.7] range where the true EMD is not compatible with the derived EMD within the uncertainties. In general we find that the EMDs resulting from the analysis of EIS synthetic data reproduce the general properties of the EMDs—such as the bulk of the emission and the general shape—significantly better than the corresponding AIA EMDs, as also shown in Section 3.1. However, even in the temperature range log T(K) ∼ [5.5–6.7], where the EMD should be well constrained by the selected spectral lines, the best-fit EMD can be very noisy and show peaks and valleys that are not present in the true EMD.

We note that, as discussed by Kashyap & Drake (1998) and Testa et al. (2011), the uncertainties associated with the EMD solution are correlated in the different temperature bins, and therefore what are shown as error bars in the single temperature bins cannot strictly be interpreted as error bars for the EMD value in that bin; instead, they describe the range containing 68% of the sets of solutions. Nevertheless, since they are typically used as error bars we also investigate here the limitations of this assumption.

In order to assess the ability of the MCMC method to recover the full EMD for all the different cases studied, we use two different ways to parameterize the "goodness of the fit." First, we use the full information of the set of EMD solutions produced by the MCMC routine for each pixel. For a given pixel and a given temperature bin (we use here the full temperature resolution, i.e., Δlog T(K) = 0.05), we can calculate the fraction of solutions included in the range between the true EMD value and the best-fit value. This can be interpreted to be a probability of the EMD value to be part of the distribution of Monte Carlo solutions. Since it is not trivial to combine the results of the different temperature bins, we compute an average of these fractions over all temperature bins between log T = 5.5 and 6.7. If the average is close to 1, this implies that for most temperature bins the true EMD is far from the bulk of the distribution. We show the maps of the values obtained from the EIS EMDs in this fashion in the left panels of Figure 22 (for snapshot C) and 23 (for snapshot H). The corresponding maps for the AIA EMDs are not shown but are qualitatively very similar to the EIS cases shown here. For the top view cases the maps present similar structuring to the χ²₀ maps shown in Figures 5 and 6, with the worst match of the EMD found in the regions with large superpositions of plasma structures with different densities and temperatures. For the side view, while lower in the atmosphere the values are similar to the top view case, they become ∼1 everywhere at larger heights. The reason for this is that at larger heights the true EMD is close to isothermal, dropping to zero everywhere outside the peak region. For all the temperature bins where the true EMD is zero, the derived EMDs are non-zero, causing large discrepancies in most of the temperature bins.

Zoom In Zoom Out Reset image size

Another way to calculate a measure of how well the derived EMDs match the true EMDs is to calculate a χ²₀(EMD) for the EMD, which we define as =Σ_j[(EMD_{j, MCMC} − EMD_{j, true})/σ(EMD)_j]²/df, where EMD_{j, true} and EMD_{j, MCMC} are respectively the values of the true and derived EMD for the temperature bin j, σ(EMD)_j is the uncertainty of the EMD in the T bin j defined by the range including 68% of the solutions, as described above, and df stands for the degrees of freedom (as defined at the beginning of Section 3). We find that when calculating the so-defined χ²₀(EMD) using the fine temperature grid (Δlog T = 0.05) the χ²₀(EMD) values are very large (≫10) for both the EIS and AIA cases. Therefore, we decrease the temperature resolution until the χ²₀(EMD) values become on average more reasonable. We find that this threshold is Δlog T ∼ 0.2 for EIS, and we show the corresponding maps of χ²₀(EMD) in the middle panels of Figures 22 and 23. These maps show the usual pattern of the features with large superposition of different structures, where the agreement is worse, but for the other regions the integration over large temperature bins yields a better match of the derived and true EMDs, by smoothing out the large variability on the small temperature scale shown in Figures 29 and 21. We note that for the side view cases, high in the corona, the uncertainties of these EMDs on the coarser temperature grid even appear overestimated, since the χ²₀(EMD) drops to very low values. This is consistent with the findings of Landi et al. (2012) who explored the ability of the MCMC methods to diagnose isothermal EMD. Landi et al. (2012) find indeed that for isothermal plasma a bin width of Δlog T ∼ 0.05 is sufficient to diagnose the temperature distribution from spectral data. For AIA, even with Δlog T ∼ 0.3, the χ²₀(EMD) values (maps are shown in the right panels of Figures 22 and 23) are large for the top view cases, in agreement with the results discussed in Section 3.1. The side view cases show that with AIA, the temperature bin size Δlog T ∼ 0.3 is adequate to diagnose isothermal plasmas, as shown by the low χ²₀(EMD) values at high z values, but is still insufficient to guarantee a good match of the true EMD where the distributions are significantly multi-thermal. The EMDs derived from AIA are much less accurate reproductions of the true EMDs compared to EIS. This points to a significant problem in relying on AIA data exclusively to diagnose the plasma temperature distribution. This can be explained by considering two main factors: (1) the smaller number of constraints—AIA data provide a considerably more limited number of constraints (at best 6) compared to EIS and (2) imaging data have significantly more limited temperature diagnostics than spectral data because of their broader temperature sensitivity (even for the narrow AIA passbands) compared to the resolved spectral lines. In this respect, though outside the scope of this paper, we note that narrowband coronal imagers also have the additional disadvantage with respect to broadband instruments of being critically sensitive to the uncertainties in the atomic data (see, e.g., Testa et al. 2012). We note that the EMDs derived from AIA present significant discrepancies with the true EMDs even in the case that does not include noise, or in the test case with very high S/N discussed at the beginning of this section when we discussed the fit to the observed intensities. We therefore conclude that the limitations in diagnosing EMDs with AIA data are mainly due to the intrinsic characteristics of the (broad) AIA temperature responses, and that the noise is not the main cause of the discrepancies between true and derived EMDs.

While our results imply that in general the MCMC method does not reproduce the true EMDs at the highest temperature resolution, it is preferable for several reasons to run the method at high resolution (Δlog T ∼ 0.05) and then rebin the solution a posteriori. As discussed above (and shown in Figures 22 and 23), the high resolution allows one to find nearly isothermal solutions. Also, we have run the MCMC for a few pixels on the EIS synthetic data using different values of Δlog T and found that adopting too large a bin size leads to less stable solutions due to the rebinning of the temperature response function and consequent loss of temperature information. For instance, if the temperature bins become significantly large compared to the peaks in the temperature response functions, the rebinned responses often end up with peaks that are much less prominent and significantly displaced in temperature.

Zoom In Zoom Out Reset image size

We have presented the results of a test of the limitations of plasma temperature diagnostics that are currently available from spectral and imaging observations of the solar corona. Determining the plasma EMD is fraught with difficulties, both because of the intrinsic nature of the mathematical inversion problem, which is ill posed, and because of the limitations of the available data. While imaging instruments provide only moderate temperature diagnostics (compared with spectrographs that can resolve spectral lines), they typically provide significantly better spatial coverage and resolution, and temporal cadence.

Here we used advanced 3D rMHD simulations of the solar atmosphere (Hansteen et al. 2007; Gudiksen et al. 2011) as realistic test cases, from which we produced synthetic observables, and applied an MCMC forward modeling technique to derive the EMDs. We then compared the results of EMD reconstruction from imaging (AIA) and spectral (EIS) synthetic data (based on the coronal properties in the models) with the input "true" EMD to establish the limitations of the temperature diagnostic power of the available instruments, and how these results depend on the characteristics of the underlying thermal distributions. We also investigated the effect of the photon counting noise by running the analysis with or without randomization, accounting for Poisson noise.

These 3D simulations provide us with the opportunity to improve upon previous work by exploring more realistic configurations with significant superposition of different structures along the LOS, allowing a statistical approach to determine the accuracy and limitations of the plasma diagnostics, for a variety of spatial and thermal structuring of the plasma. The 3D nature of the simulations also allows us to explore a variety of realistic viewing angles, reproducing typical distributions of structuring ranging from on-disk to limb observations.

We assess the robustness of the EMD reconstruction method by using several parameters. We explored the ability of the method in: (1) reproducing the "measured" intensities, (2) determining the temperature of the peak of the EMD, (3) deriving the emission measure values in broad temperature bins, (4) reproducing the true emission distribution in its details.

The analysis of the spectral EIS synthetic data shows that the measured intensities are reproduced generally well by the inferred EMD, even when including the effect of noise, with the exception of regions along the LOS that have significantly different densities mixing. The temperature of the peak of the EMD is reproduced reasonably well for the side view in larger areas where the EMD is close to isothermal. For the top view, where the EMD is broad and in particular where there are large amounts of dense material and large superposition of different structures (emerging flux region, moss), the temperature of the peak EMD appears to be systematically underestimated. The maps of the emission measure, when integrated in broad temperature bins, are similar to the true distributions in the well-constrained range (log (T[K]) ∼ [5.6–6.4]); the low- and high-temperature ends of the considered range are rather noisy and present spurious components. However, when considering the direct comparison of the derived EMD curves with the true distributions, at the full temperature resolution, we find that the detailed properties of the true EMD on the smallest temperature scale are not accurately reproduced. The best-fit EMD can present peaks which are not displayed by the true EMD, or vice versa, narrow peaks might be missed by the reconstruction method. The worst cases show how the uncertainties do not generally account for these discrepancies, and are therefore likely underestimated by the method. We explore the temperature resolution at which the discrepancies between the true and derived EMD become smaller, and we find that for EIS, Δlog T needs to be ≳ 0.2. We therefore conclude that the temperature diagnostic provided by spectral data with enough observational constraints and good temperature coverage are on average reliable to derive the general characteristics of the EMD, such as the emission measure in broad temperature bins, but that the results are not robust enough as far as the fine details of the EMD, such as the presence of narrow peaks, are concerned, and that features on scales below ≳ 0.2 in log T cannot be trusted.

The results obtained from the analysis of the AIA synthetic data are more unsatisfactory. The synthetic intensities for the side view LOS are low in some channels, making the results more sensitive to the effect of the noise. The maps of the temperatures at which the EMDs peak show that, as for the EIS case, the temperatures are not very well reproduced and appear systematically underestimated, especially in the top view case. Also, the maps of the emission measure integrated in broad temperature bins are a poor match for the true distributions, especially for the top view case. Finally, direct comparison of the derived and true EMD curves at the full temperature resolution indicates discrepancies well beyond the errors, implying that the EMD is not at all well constrained by the AIA data, often not even in their general characteristics. Even when degrading the temperature resolution to Δlog T = 0.3, the derived EMDs do not accurately match the true EMDs, especially where the temperature distributions are broad. We therefore conclude that the limited number of constraints and the broad temperature sensitivity of the AIA passbands, which often present multiple peaks, critically hamper the capability of diagnosing the plasma temperature distribution on the basis of AIA imaging observations exclusively. The addition of simultaneous Hinode/XRT data, which provide complementary imaging observations in X-ray broad bands, might considerably improve our ability to reconstruct EMDs from AIA data by providing additional constraints, especially at the high-temperature end (log T ∼ 6.5–7), where AIA provides limited information. We plan to explore this issue in a follow-up paper. In this paper we have not addressed the use of EIS and AIA together because, as discussed here and in other papers (see, e.g., Warren et al. 2011; Testa et al. 2011), the spectral data, when available, provide much better constraints to the temperature distribution of the plasma compared with imaging data, and the imaging data only provide limited additional constraints. However, imaging data provide much better temporal and spatial coverage of the coronal plasma, so we focused on investigating the usefulness of imaging data for thermal diagnostics when spectral data are not available, as in the large majority of solar observations.

The approach adopted here, using 3D simulations, has provided us with the opportunity to improve upon previous work by exploring more realistic configurations with significant superposition of different structures along the LOS, allowing a statistical approach to determine the accuracy and limitations of the plasma diagnostics, for a variety of spatial and thermal structurings of the plasma. These results provide a stringent and accurate assessment of the limitations of the temperature diagnostics, indicating the extent of the sensitivity of EMD measurements to the structuring along the LOS, and to characteristics inherent to the instrumentation. However, we emphasize that the exercise we carried out focuses on the use of the MCMC reconstruction method, and therefore it only assesses the robustness of this particular method, which however is largely adopted and trusted for studies of EMDs of solar and stellar coronal plasmas. We note that these results actually present an optimistic view because they do not account for several effects that further complicate real observations, such as for instance uncertainties in both instrument calibration and atomic data, influence of blends in fitting the spectral data, and distribution of element abundances. Finally, the 3D simulations adopted here model-limited coronal volumes, therefore likely underestimating the effects of superposition along the LOS.

We thank Vinay Kashyap, Karel Schrijver, Fabio Reale, Ed DeLuca, Paul Boerner, and Enrico Landi for useful discussions and comments that have helped improve the paper. P.T. was supported by contract SP02H1701R from Lockheed-Martin, NASA contract NNM07AB07C to the Smithsonian Astrophysical Observatory, and NASA grants NNX10AF29G and NNX11AC20G. B.D.P. gratefully acknowledges support by NASA grants NNX08AH45G, NNX08BA99G, and NNX11AN98G. The 3D simulations have been run on clusters from the Notur project, and the Pleiades cluster through computing grants SMD-07-0434, SMD-08-0743, SMD-09-1128, SMD-09-1336, SMD-10-1622, SMD-10-1869, SMD-11-2312, and SMD-11-2752 from the High End Computing (HEC) division of NASA. We thankfully acknowledge the computer and supercomputer resources of the Research Council of Norway through grant 170935/V30 and through grants of computing time from the Programme for Supercomputing. This work has benefited from discussions at the International Space Science Institute (ISSI) meeting on "Coronal Heating—Using Observables (flows and emission measure) to Settle the Question of Steady versus Impulsive Heating" 2012 February 27–March 1, where many topics relevant to this work were discussed with other colleagues. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement n° 291058.

In this appendix we present a comparison of the synthetic intensities derived from the simulations, as described in Section 2.1, with measured intensities from recent SDO/AIA observations. We considered AIA observations taken on 2012 January 6, around 23UT (the 193 Å full disk image is shown in Figure 24), and selected a few small regions (100 pixels × 100 pixels, corresponding to ∼60 arcsec × 60 arcsec) sampling a variety of coronal features, ranging from quiet Sun to active regions, and including areas both on-disk and above limb.

Zoom In Zoom Out Reset image size

In Figure 25 we show the distribution of the observed intensities (in units of DN s⁻¹ pixel⁻¹) in the six selected AIA EUV bands for the different regions shown and labeled in Figure 24. These plots clearly show that: (1) the average absolute intensities change by more than an order of magnitude from region to region; (2) the distribution in each channel in each of the regions is rather narrow; (3) the distributions of intensities in the 94 Å, 131 Å, and 335 Å channels typically peak at values about two orders of magnitude lower than the intensities in the other three bright channels; and (4) the intensities in the 211 Å and 335 Å channels typically have relative lower values in cooler regions (quiet Sun and cool fan loops; regions QS and CL).

Zoom In Zoom Out Reset image size

In Figure 26 we show the distribution of the synthetic intensities in the AIA passbands for the two lines of sight for each of the two snapshots. The smaller, cooler snapshot (C) is characterized by intensities similar to the values observed in cool fan loops (region CL) and quiet Sun (region QS), whereas the larger and hotter snapshot (H) produces intensities more similar to the observed values of active region plasma (regions AR1 and AR2). We note that, as expected, the top view ("xy") provides a more realistic comparison for real coronal observations, while for the side view ("xz") the distributions have tails at low intensity values, which are not observed in the AIA-selected data. This is because the small box size does not accurately capture the enormous LOS at the limb in the optically thin corona. However, for the side view of snapshot C the inversion of the peaks of the 335 Å and 131 Å channels with respect to the top view (i.e., for the side view the 131 Å distribution peaks at lower intensity values than the 335 Å) reproduces what is observed in actual data at the limb (region LM) compared with quiet on-disk regions (QS, CL). We also note that the distributions for snapshot H appear generally broader than the observed distributions. For snapshot C, the distribution of each passband peaks around a different value, whereas the observed distributions typically show a common peak for the weaker channels and a separate common peak for the stronger channels. Finally, the intensities in the 94 Å channel are systematically lower than the observed values, but this is expected on the basis of recent studies arguing for incompleteness of atomic databases, especially in the 94 Å passband (e.g., Testa et al. 2012). By and large, these qualitative comparisons indicate that the simulations produce emission values of order of magnitude similar to real observations, and therefore represent a reasonable test case for the coronal diagnostics.

Zoom In Zoom Out Reset image size

In this appendix we discuss the capability of the MCMC method to recover the temperature at which the EMD peaks (T(EMD_max)) within the temperature range log T(K) = [5.5–6.7]. The EIS lines we use (like typical EIS studies aimed at temperature diagnostics) provide good temperature coverage of these temperatures, and the EMDs are expected to be well constrained in this range. For AIA, the determination of the temperature range where the derived EMDs are expected to be well constrained is less straightforward because of the broader temperature responses due to the contribution of several lines. Nevertheless, this is a reasonable range to assume for temperature diagnostics in, e.g., active region plasma.

In Figure 27 we compare for the two snapshots the maps of the temperature at which the true EMD peaks (left panels), with the corresponding maps for the EMD derived using EIS synthetic data, with or without including the effect of noise (center and right panels, respectively). The corresponding temperature maps for the EMD derived from AIA data are shown in Figure 28.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The results are significantly different for the top and side view. For the side view, for all cases, the true temperature maps are recovered reasonably well, at least at heights above a few Mm from the photosphere, where the plasma temperature distributions are narrow, as shown in more detail in Section 3.1. For the lower atmospheric layers, where the EMDs are broader and there is significant mixing of structures with plasma at different densities, the derived temperature maps show some more significant departures from the true values, similar to the top view case. For the top view case for both EIS and AIA, we note that, while some general features are roughly recovered, there are significant discrepancies between the derived and true temperature; in particular, the temperature appears to be systematically underestimated.

In Figure 29 we show some examples of EMDs derived in selected pixels of the top view cases (including the noise) to illustrate the different causes explaining these results. The pixels where the derived T(EMD_max) is at the lower end of the selected range, i.e., close to log T = 5.5 (dark blue in Figures 27 and 28), while the true T(EMD_max) is much larger (log T ∼ 6.3 for snapshot C and log T ∼ 6.7 for snapshot H), generally have an EMD with a prominent peak above log T = 6, with a sharp drop on the high T side, while going toward low T their EMDs have a slower decrease reaching a minimum and then increasing again at lower T. Example of these EMDs for each case is shown in panels (b) (snapshot C, EIS), (d) (C, AIA), (f) (H, EIS), and (h) (H, AIA) of Figure 29. Since, as discussed above, the EIS and AIA data poorly constrain the EMDs at temperatures below log T ∼ 5.5, for these EMDs, the MCMC method tends to put more emission measure at the low-temperature end of the range because of the lack of constraints from the data (similar to the "toothpaste" effect discussed by Schmelz et al. 2009b). In panels (a), (c), and (e) we show some examples of pixels where the EMD does not show the rise below log T ∼ 5.5, and the derived T(EMD_max) is close to the true T(EMD_max). Besides the cases discussed above, where the temperature is largely underestimated and pushed to the low end of the temperature range, there are several pixels where the T(EMD_max) is closer to the true temperature, and yet systematically underestimated, which is clear especially for snapshot H from the large number of "yellow" pixels (log T ∼ 6.4–6.5) corresponding to "red" areas (log T ∼ 6.6–6.7) in the maps of true T(EMD_max) (bottom two rows of Figures 27 and 28). Panels (e) and (g) allow us to investigate this effect: When the EMD is fairly structured at high temperature, i.e., it does not present a single peak, the derived EMD, even when reproducing the general shape of the true EMD, might peak at a lower temperature, as shown in (e), where the two peaks at high temperatures are not perfectly reproduced. As discussed above, the method does not always capture very sharp drops in the EMD, which in these simulations are common at the high-temperature end of the EMD, as in many of these examples, and this may skew the high-temperature peak toward slightly lower temperatures. Finally, we note that these results are not significantly affected by the noise.

Zoom In Zoom Out Reset image size