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On the Accuracy of the Differential Emission Measure Diagnosticsof Solar Plasmas. Application to AIA / SDO. Part II:

Multithermal plasmas.

C. Guennou1, F. Auchere1, E. Soubrie1 and K. Bocchialini1

Institut d’Astrophysique Spatiale, Batiment 121, CNRS/Universite Paris-Sud, UMR 8617, 91405 Orsay,France

chloe.guennou@ias.u-psud.fr

S. Parenti2

Royal Observatory of Belgium, 3 Avenue Circulaire, B-1180 Bruxelles, Belgium

and

N. Barbey3

SAp/Irfu/DSM/CEA, Centre d’etudes de Saclay, Orme des Merisiers, Batiment 709, 91191 Gif surYvette, France

Accepted for publication in The Astrophysical Journal Supplements 2012 September 5.

ABSTRACT

The Differential Emission Measure (DEM) analysis is one of the most used diagnostic toolsfor solar and stellar coronae. Being an inverse problem, it has limitations due to the presence ofrandom and systematic errors. We present in theses series of papers an analysis of the robustnessof the inversion in the case of AIA/SDO observations. We completely characterize the DEMinversion and its statistical properties, providing all the solutions consistent with the data alongwith their associated probabilities, and a test of the suitability of the assumed DEM model.

While Paper I focused on isothermal conditions, we now consider multi-thermal plasmas andinvestigate both isothermal and multithermal solutions. We demonstrate how the ambiguitybetween noises and multi-thermality fundamentally limits the temperature resolution of the in-version. We show that if the observed plasma is multi-thermal, isothermal solutions tend tocluster on a constant temperature whatever the number of passbands or spectral lines. Themulti-thermal solutions are also found to be biased toward near isothermal solutions around 1MK. This is true even if the residuals support the chosen DEM model, possibly leading to erro-neous conclusions on the observed plasma. We propose tools to identify and quantify the possibledegeneracy of solutions, thus helping the interpretation of DEM inversion.

Subject headings: Sun: corona - Sun: UV radiation

1. Introduction

A convenient approach to study the ther-mal structure of the solar and stellar outer at-mospheres is the Differential Emission Measure(DEM) formalism. The DEM ξ(Te) is a mea-

sure of the amount of emitting plasma along theLine Of Sight (LOS) as a function of the electrontemperature Te. However, the intrinsic difficul-ties involved in this inverse problem lead to manycomplications in its inference, making its inter-pretation ambiguous. The central point of these

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series of papers is to provide new tools to sys-tematically and completely characterize the DEMinversions and assist the DEM interpretation. Us-ing the technique developed for this purpose, ex-haustively described in (Guennou et al. 2012,hereafter Paper I), it is possible to determine andto compare the DEM diagnostic capabilities ofgiven instruments. With only three bands, theprevious generation of imaging telescopes wereshown not to be well-suited to DEM analysis (e.g.Schmelz et al. 2009). The situation has changedwith the availability of new multi-band instru-ments such as the Atmospheric Imaging Assembly(AIA) telescope (Lemen et al. 2012) on boardthe Solar Dynamics Observatory (SDO). Apply-ing our technique to AIA, we show the increasedrobustness of the inversion for isothermal plasmas,but we also found intrinsic biases if the observedplasma is multithermal.

Using the notations of Paper I, the DEM is de-fined as

ξ(Te) = n2e(Te) dp/d(logTe), (1)

where n2e is the square electron density averaged

over the portions dp of the LOS at temperatureTe (Craig & Brown 1976). The observed intensityin the spectral band b of an instrument can beexpressed as a function of the DEM ξ as follows

Ib =1

4π

∫ +∞

0

Rb(ne, Te) ξ(Te) d log Te. (2)

Details and references about the DEM formalismcan be found in Paper I. Given a set of obser-vations in N different bands, the DEM can bein principle inferred by reversing the image ac-quisition process described by Equation 2. How-ever, solving for the DEM proved to be a con-siderable challenge. The complications involvedin its derivation are one of the reasons for con-troversial results regarding the thermal structureof the corona. For example, the DEM is oneof the method used to derive the still debatedphysical properties of plumes (Wilhelm et al.2011). Also, while heating models predict dif-ferent thermal structures for coronal loops de-pending on the processes involved (e.g. Klimchuk2006; Reale et al. 2010), DEM analyses pro-vided ambiguous answers (Schmelz et al. 2009).Several studies (Feldman et al. 1998; Warren

1999; Landi & Feldman 2008) suggest the ubiq-uitous presence of isothermal plasma in the quietcorona. If confirmed, these results would chal-lenge many theoretical models, but the relia-bility of the temperature diagnostics used hasbeen questioned (e.g. Landi et al. 2012). Partof this state of facts is due to technical issues,such as the difficulty to subtract the backgroundemission (Terzo & Reale 2010; Aschwanden et al.2008) or the possible spatial and temporal mis-match of the structures observed in different bandsand/or different instruments. But the fundamen-tal limitations of the DEM inversion clearly playa role.

Despite the many proposed inversion schemes,rigorously estimating levels of confidence in thevarious possible solutions given the uncertaintiesremains a major difficulty. In this work we pro-pose a strategy to explore the whole parameterspace, to detect the presence of secondary solu-tions, and to compute their respective probabil-ities. We illustrate the method by characteriz-ing the inversion of simulated observations of theAIA telescope. This approach allows us to quan-tify the robustness of the inversion by comparingthe inverted DEM to the input of the AIA simula-tions. The main drawback is the limitation to sim-ple DEM forms (i.e. described by a small numberof parameters), but the results provide importantinsights into the properties of more generic DEMinversions. The method can also easily be appliedto any instrument, broad band imagers and spec-trometers alike.

Paper I was dedicated to the analysis of isother-mal plasmas, and showed the existence of multi-ple solutions in the case of a limited number ofbands. Notwithstanding, the statistical methoddeveloped in this work (which provides the respec-tive probabilities of each secondary solution), en-ables to properly interpret the solutions as consis-tent with several plasma temperatures. The use ofthe six AIA coronal bands was shown to increasethe thermal diagnostic capabilities, providing a ro-bust reconstructions of isothermal plasmas. Thedetailed analysis of the squared residuals showedthat the DEM complexity is limited by the redun-dancy of information between the bands.

We now generalize the method to DEM modelsable to describe a great variety of plasma condi-tions, from isothermal to broadly multithermal. A

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summary of the technique is given Figure 1 and itsimplementation is exhaustively described in PaperI. The core of our method resides in the proba-bilistic interpretation of the DEM solutions. TheDEM ξI solution of the inversion is the one thatminimizes a criterion C(ξ) defined as the distancebetween the intensities Iobsb observed in Nb bandsand the theoretical ones Ithb . We write

C(ξ) =

Nb∑

b=1

(

Iobsb (ξP )− Ithb (ξ)

σub

)2

ξI =argminξ

C(ξ)

χ2 =minC(ξ)

(3)

where σub is the standard deviation of the un-

certainties and χ2 represents the residuals. Us-ing Monte-Carlo simulations of the systematic andrandom errors, we compute P (ξI |ξP ) the condi-tional probability to obtain a DEM ξI knowing theplasma DEM ξP . Both types of errors are mod-elled as Gaussian random variables with a 25%standard deviation (see Section 2.3.3 of Paper I).Then, using Bayes’ theorem we obtain P (ξP |ξI),the probability that the plasma has a DEM ξP

given the inverted DEM ξI . Using this latterquantity it is possible to identify the range or mul-tiple ranges of solutions consistent with a set ob-servations. Therefore, even if it is not possibleto alleviate the degeneracy of the solutions, it isat least possible to notice and to quantify them.But the computation of the probability P (ξP |ξI)is practical only if the space of solutions is lim-ited. We therefore restrict the possible solutionsto simple forms described by a small number aparameters: Dirac delta function, top hat, andGaussian. We thus do not to propose a genericinversion algorithm, but using this approach, wewere able to completely characterize the behaviorof the inversion in well controlled experiments.

All the results presented in the present paperconcern multithermal plasmas. In Section 2, weanalyse the properties of the isothermal solutionsin order to determine to what extent it is possibleto discriminate between isothermal and multither-mal plasmas. Section 3 is then dedicated to thegeneral case of multithermal inversions. Resultsare summarized and discussed in Section 4.

2. Isothermal solutions

As already mentioned, a recurring question isthat of the isothermality of solar plasmas. TheEM loci technique (e.g. Del Zanna et al. 2002;Del Zanna & Mason 2003, and references therein)was originally proposed as an isothermality test.The EM loci curves are formed by the set of(EM,Te) pairs for which the isothermal theoret-ical intensities exactly match the observations ina given band or spectral line. The isothermal hy-pothesis can then in principle be ruled out if thereis no single crossing point of the loci curves. But afundamental ambiguity arises from the inevitablepresence of measurement errors.

Under the hypothesis of ideal measurements,the observations and the theoretical intensities areboth reduced to I0b . Hence, the isothermal solu-tion for a perfectly isothermal plasma leads to aresidual χ2 exactly equal to 0, and the EM locicurves all intersect at a common point. But, inreality, the presence of errors leads to over orunder-estimations of the theoretical and observedintensities, which conducts to a non-zero χ2. Inthis case, even though the plasma is isothermal,the EM loci curves do not intersect at a singlepoint, each one being randomly shifted from itsoriginal position. But even with perfect measure-ments, the isothermal hypothesis would also yielda residual greater than zero if the observed plasmais multithermal, because in this case the I0b andIthb are intrinsically different. Therefore, measure-ment errors can be incorrectly interpreted as devi-ations from isothermality and vice-versa, a varietyof multithermal plasmas can be statistically con-sistent with the isothermal hypothesis. The ques-tion is then whether or not a perfectly isothermalplasma can be distinguished from a multithermalone and if so, under what conditions.

We thus investigate in this section the isother-mal solutions ξIiso obtained from simulated obser-vations Iobsb of plasmas of different degrees of mul-tithermality. The adopted statistical approach al-lows us to quantify to what extent the robustnessof the inversion is affected as a function of the de-gree of multithermality of the plasma. The plasmais assumed to have a Gaussian DEM defined in

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logTe as

ξPgau(EM,Tc, σ) =EMN (logTe − logTc),

with N (x) =1

σ√2π

exp

(

− x2

2σ2

)

(4)

where EM is the total emission measure, Tc isthe central temperature and σ is the width of theDEM. It represents a plasma predominantly dis-tributed around a central temperature Tc and canrepresent both isothermal and very multithermalplasmas.

The theoretical intensities Ithb are limited to thecase of isothermal solutions, corresponding to aDEM

ξiso(EM,Tc) = EM δ(Te − Tc), (5)

where δ is Dirac’s delta function and EM is thetotal emission measure. The solution ξI of theinversion process then gives the pair (T I

c , EM I)that best explains the observations. The referenceintensities I0b (see the previous Section), used tocompute both Iobsb and Ithb have been tabulatedonce and for all for 500 central temperatures fromlogTc(K) = 5 to 7.5, 200 emission measures fromlogEM(cm−5) = 25 to 33, and for the Gaussianmodel 80 DEM widths from 0 to 0.8, expressed inunit of logTe (see Section 2.3.2 of Paper I).

2.1. Ambiguity between uncertainties and

multithermality

In Figure 2, the criterion C(EM,Tc) (Equa-tion 3) is plotted for different plasma configura-tions. It is represented in gray scale1 and its ab-solute minimum, corresponding to the solution ξI ,is marked by a white plus sign. The criterion is afunction of the two parameters EM and Tc defin-ing the isothermal solutions (Equation (5)). Thewhite curves are the EM loci curves, correspond-ing to the minimum of the contribution of eachband to the total criterion. In all panels, the sim-ulated plasmas have DEMs centered on the typicalcoronal temperature TP

c = 1.5× 106 K, and a to-tal emission measure EMP = 2 × 1029 cm−5, anactive region value guaranteeing signal in all sixbands (see Section 3 and Figure 3 in Paper I). The

1All the criteria presented in this paperare available in color on line at ftp.ias.u-psud.fr/cguennou/DEM AIA inversion/.

top left panel corresponds to an isothermal plasma(σ = 0 which corresponds to the case presented inSection 3.2 of Paper I). Because of systematics,sb, and instrumental noises, nb, and thus the overor under-estimation of the observed and theoret-ical intensities Iobsb and Ithb , the EM loci curvesare shifted up and down along the emission mea-sure axis, with respect to the zero-uncertaintiescase (i.e. nb and sb = 0). Therefore, the curvesnever intersect all six at a single position, thusgiving χ2 > 0. But multithermality has the sameeffect: the criterion presented on the top right cor-responds to a slightly multithermal plasma havinga Gaussian DEM width of σP = 0.1 logTe. Eventhough in this case uncertainties have not beenadded, the loci curves and the absolute minimum,corresponding to the location where the curves arethe closest together, are also shifted relative toeach other along the emission measure axis andthere is no single crossing point. Thus, errorsand multithermality can both produce compara-ble deviations of the observations from the idealisothermal case, hence the fundamental ambiguitybetween the two. The two criteria of the bottompanels have been obtained with the same plasmaGaussian DEM distribution, but now in presenceof random perturbations. These two independentrealizations of the uncertainties illustrate an ex-ample of the resulting dispersion of the solutions.

In Figure 3, we increased the DEM width toσP = 0.3 logTe (top row) and σP = 0.7 logTe

(bottom row) while keeping the same emissionmeasure and central temperature. For each width,the left and right panels show two realizations ofthe uncertainties. The corresponding configura-tion of the loci curves and thus the value and po-sition of the absolute minimum can greatly vary,even though the plasma is identical in both cases.It also appears that it exists privileged tempera-ture intervals where the solutions tend to concen-trate. This phenomenon is intrinsically due to theshape of the EM loci curves. In the next subsec-tion, we characterize the respective probabilitiesof occurrence of the various solutions for a widerange of plasma conditions. From the examplespresented in Figure 3, we also note that the verti-cal spread of the loci curves tends to be larger forwider DEMs, which corresponds to larger χ2 resid-uals. In section 2.3, we will determine up to whatDEM width the observations can be considered to

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be consistent with the isothermal hypothesis.

2.2. Probability maps

We consider Gaussian DEM plasmas and scanall possible combinations of central temperaturesand widths used to pre-compute the reference the-oretical intensities. The total emission measure iskept constant at EMP = 2× 1029 cm−5. For eachcombination of plasma parameters, 5000 Monte-Carlo realizations of the random perturbations nb

and systematics sb are obtained, leading to 5000estimates of the observed intensities Iobsb (ξP ) ineach band b. The isothermal least-square inver-sion of these 5000 sets of AIA simulated observa-tions provides as many solutions ξI , allowing anestimation of P (ξI |ξP ).

Figures 4, 5 and 6 show the resulting temper-ature probability maps for σP = 0.1, 0.3 and0.7 logTe respectively2. The isothermal plasmacase (σP = 0) is shown in Figure 6 of Paper I. Inall figures, the probability P (T I

c |TPc ) is obtained

by reading vertically panels (a), given the prob-ability to find a solution T I

c knowing the plasmacentral temperature TP

c . The probability profilesfor two specific plasma temperatures, 1.5×106 and7× 106 K, are shown in panels (b) and (c). UsingBaye’s theorem, the probability maps P (TP

c |T Ic ) of

panels (e) are obtained by normalizing P (T Ic |TP

c )to P (T I

c ), the probability of obtaining T Ic what-

ever TPc (panels (d)). In the bottom right panels

(f) and (g), we show two example horizontal pro-files giving the probability that the plasma has acentral temperature TP

c knowing the inverted tem-perature T I

c .

In panels (a) and (e) of Figure 4, the most prob-able solutions are located around the diagonal.However, compared to the case of an isothermalplasma (see Figure 6 of Paper I), the distribution iswider, irregular and deviations from the diagonalgreater than its local width are present. As shownby panel (d) and the nodosities in the map (a), theunconditional probability to obtain a result T I

c isnon uniform, meaning that some inverted temper-atures are privileged whereas others are unlikely.Comparing with Figure 6 of Paper I, profile (b)shows that the probability of secondary solutions

2The probability maps for 80 widths from 0 to0.8 log Te are available in color on line at ftp.ias.u-psud.fr/cguennou/DEM AIA inversion/.

at TPc = 1.5 × 106 K is increased with respect to

the isothermal case. The apparition of these twosolutions are illustrated in the bottom row of Fig-ure 2. The bottom right panel corresponds to arealization of the errors yielding a solution closeto the diagonal, while the bottom left panel of thesame Figure illustrates a case where the absoluteminimum of the criterion is located at low temper-ature. Using the map of P (TP

c |T Ic ) it is however

possible to correctly interpret the low tempera-ture solutions as also compatible with 1.5× 106 Kplasma (profile (g)).

In Figure 5, the plasma DEM width is in-creased to σP = 0.3 logTe. As a result, theabove-described perturbations with respect to theisothermal plasma case are amplified. The diag-onal structure has almost disappeared, with dis-continuities and reinforced and more diffuse no-dosities. Multiple solutions of comparable prob-abilities are present over large ranges of plasmatemperatures and consequently, the estimated T I

c

can be very different from the input TPc . For

example, panel (c) shows that for a 7 × 106 Kplasma, the most probable T I

c is either 1.6 × 105

or 3×105 K. The unconditional probability P (T Ic )

of panel (d) is very non uniform, some rangesof estimated temperatures being totally unlikely(e.g. T I

c = [1.5 × 106, 4 × 106] K) while oth-ers are probable for large intervals of TP

c (e.g.T Ic = 3 × 105 K or 106 K). However, despite the

jaggedness of P (T Ic |TP

c ), the map of P (TPc |T I

c ) canonce again help properly interpret the result of theinversion. For example, profile (g) shows that forT Ic = 1.5 × 106 K, the distribution of TP

c is dis-tributed around TP

c = 107 K, which is exactly theplasma temperature that can yield an inversionat T I

c = 1.5 × 106 K (see panel (a)). Panel (f),providing the probability distribution TP

c know-ing that the inversion results is T I

c = 7 × 106 K,exhibits a broad probability distribution aroundTPc = 1.5× 107, showing that the plasma temper-

ature thus deduced is very uncertain. This is tobe compared to the 0.05 logTP

c temperature res-olution of the isothermal case (see section 3.2 ofPaper I).

As the DEM becomes even larger, the im-pact on the robustness of the inversion becomesgreater. At σP = 0.7 logTe, the probability mapP (T I

c |TPc ) of Figure 6 (a) and the correspond-

ing probability P (T Ic ) show clearly two privileged

5

solutions: T Ic = 106 and 3 × 105 K. The es-

timated isothermal temperatures are always thesame for any TP

c , as illustrated by panels (b) and(c). Therefore, the inversion results T I

c containsno information on the plasma central temperatureTPc . This is illustrated by the lack of structure in

the probability map P (TPc |T I

c ) of panel (e). Pro-file (g) shows that for T I

c = 1.5 × 106 K, the dis-tribution of TP

c extends all over the temperaturerange.

2.3. Interpretation

We have shown that as the width of the plasmaDEM increases, multiple solutions to the isother-mal inversion appear. This phenomenon has beenalready mentioned by Patsourakos & Klimchuk(2007) using triple-filter TRACE data. After aproper treatment of the uncertainties, the authorsfound that their observations of coronal loops wereconsistent with both a high (≈ 1.5 × 106 K) anda low (≈ 5 × 105 K) isothermal plasma tempera-ture. They correctly concluded that without a pri-ori knowledge of the physical conditions in theseloops, they could not rule out the cool plasma so-lutions. Even though we used six bands, multiplesolutions appear anyway with increasing plasmawidth. In addition, as we have seen in Paper I,multiple solutions can exist even with an isother-mal plasma if only a limited number of bands isavailable. This is another illustration of the simi-lar effects of errors and multithermality.

The isothermal temperature solutions becomeprogressively decorrelated from the plasma cen-tral temperature as the width of the DEM in-creases. For very large DEMs (Figure 6), the in-version process yields exclusively either 3× 105 Kor 106 K whatever the plasma TP

c . These twotemperatures correspond to the preferential loca-tions of the minima shown in the criteria of Fig-ure 3. This is a generalization of the phenomenonanalyzed by Weber et al. (2005) in the simplercase of the TRACE 19.5 over 17.3 nm filter ra-tio. The authors showed that in the limit of aninfinitely broad DEM, the band ratio tends to aunique value equal to the ratio of the integrals ofthe temperature response functions. Furthermore,they showed that as the width of the DEM in-creases, the temperature obtained from the bandratio becomes decorrelated from the DEM centraltemperature. We have found a similar behavior in

the more complex situation of six bands. This isnot however an intrinsic limitations of AIA. Wecan predict that the same will occur with anynumber of bands or spectral lines. Indeed, for aninfinitely broad DEM, since the observed inten-sities are equal to the product of the total emis-sion measure by the integral of the response func-tions (Equation 2), they are independent from theplasma temperature. Therefore, the inversion willyield identical results for any plasma temperatureTPc , whatever the number of bands or spectral

lines.

2.3.1. Defining isothermality

As already noted in section 2.1 and in Figure 2,to larger DEM widths correspond larger squaredresiduals. From Paper I the distribution of resid-uals to be expected for an isothermal plasma isknown. Examining then the residuals for the so-lutions given in the probability maps of section 2.2,all the solutions may not all be statistically con-sistent with the isothermal hypothesis. We willthus analyze the distribution of residuals to definerigorously a test of the adequacy of the isothermalmodel used to interpret the data.

Because both the random and systematic er-rors nb and sb have been modeled by a Gaussianrandom variable, if the DEM model used to in-terpret the data can represent the plasma DEM,the residuals are equal to the sum of the squareof six normal random variables (see Equation 3).Since we adjust the two parameters EM and Tc,the residuals should thus behave as a degree fourχ2 distribution.

Figure 7 shows the distribution of the squaredresiduals for all the 80 DEM widths considered inthe simulations. The shades of grey in the toppanel correspond to the probability to obtain agiven χ2 value (abscissa) as a function of the widthof the plasma DEM (ordinate). In the bottompanel, four profiles give the distribution of squaredresiduals obtained for an isothermal plasma (thindotted line) and for the three DEM widths dis-cussed in section 2.2: σP = 0.1 (thick dottedline), 0.3 (thick dashed curve) and 0.7 logTe (thickdash-dotted curve), corresponding to the whitehorizontal lines on the top panel. The theoreticalχ2 distributions of degree three (thin solid curve)and four (dashed curve) are also plotted.

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If the plasma is isothermal, the distribution ofthe residuals is slightly shifted toward a degree 3χ2 instead of the expected degree four. In paper I,we interpreted this as a correlation between the sixAIA coronal bands. The distributions of residualsprogressively depart from the isothermal case asthe DEM width of the simulated plasma increases.The distributions become broader and their peaksare shifted toward higher values, forming the diag-onal structure in the top panel of Figure 7. Thisbehavior stops around σP = 0.4 logTe. Above,the peaks of the distributions are shifted back to-ward smaller values and remains constant. As theDEM becomes wider, the simulated observationsbecome independent from the plasma parameters,and all inversions tend to give the same solutionand the same residuals.

These distributions of residuals provide a ref-erence against which to test the pertinence of theisothermal model. The isothermal hypothesis canfor example be invalidated for the solutions bi-ased towards T I

c ≈ 1 MK and corresponding tovery multithermal plasmas (e.g. σP = 0.7, seesection 2.2 and Figure 6). Indeed, for a giveninversion and its corresponding residual, the toppanel of Figure 7 gives the most probable widthof the plasma, assuming it has a Gaussian DEM.Let us assume that an isothermal inversion re-turns a residual equal to 5. Analyzing the his-togram corresponding to the bottom row of thetop panel of Figure 7 (σP = 0), we can show thatan isothermal plasma has 68.2% chance to yieldχ2 ≤ 5. This residual can therefore be consid-ered consistent with an isothermal plasma. Butreading the plot vertically, we see that the proba-bility P (χ2 = 5) is greater for multithermal plas-mas and peaks around σP = 0.12, which is thusin this case the most probable Gaussian width.For larger residuals the situation is more complexbecause several plasma widths can have equallyhigh probabilities. Past χ2 = 3.5 the plasma hasa higher probability to be Gaussian than isother-mal and a Gaussian inversion is required to prop-erly determine its most probable width and centraltemperature.

Figure 7 is a generalization of the results ob-tained by Landi & Klimchuk (2010). It is equiv-alent to their Figure 2, identifying our χ2 to theirFmin and their solid and dashed lines to respec-tively the peak and the contours at half maximum

of our χ2 distribution as a function of σP . TheirFigure 2 was computed using 13 individual spec-tral lines for a 1 MK plasma and extends only upto σP ≈ 0.2, while our Figure 7 was computedfor the 6 AIA bands over a wide range of centraltemperatures and for widths up to σP = 0.8. De-spite these differences, the two figures exhibit thesame global behavior. Indeed, for any number ofspectral lines or bands, the residuals of the isother-mal inversion tend to increase as the width of theplasma DEM increases.

3. Multithermal solutions

We now focus on multithermal solutions. Theability of AIA to reconstruct the DEM given theuncertainties is evaluated, and the probabilitymaps associated to all parameters are computed,allowing to take into account all the GaussianDEMs consistent with the simulated observations.Both cases of consistent and inconsistent DEMsmodels between the simulations and the inversionassumptions are examined. After the previous sec-tion on isothermal solutions, this generalizes thestudy of the impact of a wrong assumption on theDEM shape.

As in the previous section, the simulated ob-servations remain Gaussian, but we now considerGaussian solutions, i.e. theoretical intensities Ithbtabulated for the Gaussian DEM model. Themodel can thus in principle perfectly represent theplasma conditions.

3.1. Three-dimensional criterion

Investigating now multithermal Gaussian solu-tions, the least-square criterion given by Equa-tion (3) has thus three dimensions C(ξgau) =C(EM,Tc, σ). This three dimensional parameterspace is systematically scanned to locate the the-oretical intensities describing the best the simu-lated observations. Figure 8 shows this criterionfor three cases illustrating different degrees of mul-tithermality. In each row, from top to bottom, thesimulated plasma has a DEM width σP of 0.1, 0.3and 0.7 logTe respectively, centered on the tem-perature Tc = 1.5× 106 K. On the left panels, thebackground image represents C(EM,Tc, σ

I), thecut across the criterion in the plane perpendicu-lar to the DEM width axis at the width σI cor-responding to the absolute minimum (white plus

7

sign). The curves are the equivalent of the lociemission measure curves in a multithermal regime:for each band b and for a given DEM width σP ,they represent the loci of the pairs (EM,Tc) forwhich the theoretical intensities Ithb are equal tothe observations Iobsb . As σ varies, they thus de-scribe a loci surface in the three-dimensional cri-terion. The difference with the isothermal locicurves is that the theoretical intensities Ithb havebeen computed for the multithermal case (i.e. con-sidering a Gaussian DEM), and thus the param-eter σ must be now considered. The right pan-els of Figure 8 display C(EM I , Tc, σ), the cutsacross the criterion in the plane perpendicular tothe emission measure axis at the EM I correspond-ing to the absolute minimum of the criterion (alsorepresented by a white plus sign).

The impact of multithermality on the criteriontopology is clearly visible on the loci curves, in-ducing a smoothing of the curves as the multi-thermality degree increases. Indeed, the intensi-ties I0b as a function of the central temperaturecan be expressed as the convolution product be-tween the instrument response functions and theDEM (see Section 2.3.2 of Paper I). Therefore,the reference intensities I0b (EM,Tc, σ) computedfor multithermal plasmas are equal to the isother-mal ones smoothed along the Tc axis. As a result,the criterion exhibits a smoother topology and theminimum areas become broader and smoother asthe multithermality degree increases, introducingmore indetermination in the location of the ab-solute minimum. For each row, the realizationof the uncertainties nb and sb yields a solutionthat is close to the simulation input. The cuts aretherefore made at similar locations in the criterionin order to best illustrate the modification of itstopology. We will now analyse to what extent thedistortion of the criterion affects the robustness ofthe inversion.

3.2. Probability maps

In the case of DEM models defined by three pa-rameters, and since the plasma EMP is fixed, theprobability matrices P (EM I , T I

c , σI |EMP , TP

c , σP )resulting from the Monte-Carlo simulations havefive dimensions. In order to illustrate the mainproperties of these large matrices, we rely on com-binations of fixed parameter values and summa-tion over axes.

The associated probability maps are displayedin Figure 9 for simulated plasmas characterizedby σP = 0.1 (top panel) and σP = 0.7 logTe

(bottom panel)3. The probability maps are repre-sented whatever the emission measure and DEMwidth obtained by inversion, i.e. the probabili-ties P (EM I , T I

c , σI |EMP , TP

c , σP ) are integratedover EM I and σI . In case of a narrow DEM dis-tribution (σP = 0.1, top panels) the solutions aremainly distributed along the diagonal, with somesecondary solutions at low probabilities. The ac-curacy of the determination of the central tem-perature TP

c is improved compared to the isother-mal inversion of the same plasma as shown in Fig-ure 4: the diagonal is more regular and P (T I

c ) ismore uniform. However, increasing the width ofthe DEM of the simulated plasma reduces the ro-bustness of the inversion process. In the bottompanels (σP = 0.7), we observe a distortion and animportant spread of the diagonal. In particular,the horizontal structure in panel (a) and the con-sequent peak of P (T I

c ) tell that the estimated tem-perature is biased toward T I

c ∼ 1 MK, especiallyfor plasma temperatures TP

c < 2 MK. This is tobe compared with the privileged isothermal solu-tions in Figure 6, and the same reasoning applies.For very broad DEMs, because of the smoothing ofthe I0b (EM,Tc, σ) along the Tc axis, the observedintensities are only weakly dependent on the cen-tral temperature, and thus all inversions tend toyield identical results. As a consequence, the prob-ability map P (TP

c |T Ic ) shows that over the whole

temperature range considered there is a large un-certainty in the determination of TP

c . The smooth-ing of the I0b (EM,Tc, σ) is due to the width of theplasma DEM and not to the properties of the re-sponse functions. It is therefore to be expectedthat the uncertainty in the determination of TP

c

persists even if using individual spectral lines.

In the same way, the conditional probabilitiesP (σI |σP ) and P (σP |σI) of the DEM width aredisplayed on Figure 10. The probabilities arerepresented whatever the estimated temperatureT Ic and emission measure EM I , for a simulated

plasma having a DEM centered on TPc = 1 MK.

The conditional probability P (σI |σP ) on panel (a)exhibits a diagonal that becomes very wide for

3The probability maps for 80 widths from 0 to0.8 log Te are available in color on line at ftp.ias.u-psud.fr/cguennou/DEM AIA inversion/.

8

large σP and from which another broad branchbifurcates at σI = 0.15 logTe. This branch is theanalog of those observed on the temperature axis(top panels of Figure 9). For plasmas having aDEM width greater than σP = 0.3 logTe, the esti-mated width σI is decorrelated from the input σP .Profiles (b) and (c) provide the conditional prob-abilities of σI for plasma DEM widths σP = 0.2and 0.6 logTe respectively. As shown by panel (d),the unconditional probability P (σI) to obtain σI

whatever T Ic for this 106 K plasma is biased to-

wards σI = 0.12 logTe. Exploring the full prob-ability matrix P (EM I , T I

c , σI |EMP , TP

c , σP ), wediscovered that if the plasma DEM is broad, thesmall width solutions (≈ 0.12 logTe) are the onesthat were also biased towards a central temper-ature of 106 K in Figure 6. This is caused bythe shape of the temperature response functionsRb(Te). A minimum is formed in the criterionC(Tc, σ) around (Tc = 1 MK, σ = 0.12 log Te)that tends to be deeper than the other local min-ima. Since for a very broad DEM plasma the sim-ulated observed intensities Iobsb are almost alwayssimilar, most of the random realizations conductto this deeper minimum and to the formation ofthe narrow solutions branch in P (σI |σP ) panel(a). Therefore, very multithermal plasmas tendto be systematically measured as near-isothermaland centered on 106 K.

Reading horizontally in panel (e) the inverseconditional probability P (σP |σI), a large range ofDEM widths σP is consistent with the estimationσI . Both plots (f) and (g), representing two cutsat σI = 0.2 and 0.6 logTc, can be used to deducethe most probable plasma DEM with σP . How-ever, at σI = 0.6 logTe, almost all plasma widthshave significant probabilities, restricting consid-erably the possibility of inferring relevant DEMproperties. The situation improves for smallerwidths, as shown in panel (g), even though theprobability distribution is still broad. For narrowDEM distributions (σP < 0.2), the width of thedistribution decreases to about 0.15 logTe, whichthus represents the width resolution limit.

The analysis of the probability maps demon-strates that the robustness of the inversion is sub-stantially affected by the degree of multithermalityof the observed plasma. Furthermore, as we al-ready noted, the simulations presented have beenmade in a favorable configuration where signifi-

cant signal is present in all six bands. For narrowplasma DEMs (σP < 0.15 logTe), the six AIAcoronal bands enable an unambiguous reconstruc-tion of the DEM parameters within the uncertain-ties. The precision of the temperature and DEMwidth reconstruction is then given by the widthsof the diagonals in Figures 9 and 10. For the tem-perature, that width is consistent with the reso-lution of [0.1, 0.2] logTe given by Judge (2010).However, both the accuracy and the precision ofthe inversion decrease as the multithermality de-gree of the simulated plasma increases, a widerrange of solutions becoming consistent with theobservations. In the case of a very a large DEMplasma, the solutions are skewed toward narrow1 MK Gaussians. The isothermal solutions bi-ased towards 1 MK (section 2.3) could be inval-idated on the basis of their correspondingly highresiduals (section 2.3.1). But the biased Gaussiansolutions are by definition fully consistent witha Gaussian plasma. This result generalizes thatof Weber et al. (2005), and since our method en-sures that the absolute minimum of the criterionis found, it is a fundamental limitation and not anartefact of the minimization scheme. Instead ofbeing evidence for underlying physical processes,the recurrence of common plasma properties de-rived from DEM analyses may be due to biases inthe inversion processes. However, using the typeof statistical analysis presented here, it is possibleto identify theses biases and correct for them.

3.3. Residuals and model testing

The distribution of the sum of squared residualsis displayed in the top panel of Figure 11 as a func-tion of the plasma DEM width, σP , of the simu-lated plasma. The three white horizontal lines rep-resent the locations of the cuts at σP = 0.1 (dot-ted line), 0.3 (dashed line) and 0.7 logTe (dotteddashed line) shown in the bottom panel. Unlikefor the isothermal solutions (Figure 3), the distri-butions are similar for all plasma widths and re-semble a degree 3 χ2 distribution (thin solid line).The most probable value is about 1 and 95% of theresiduals are between 0 and 10. Any DEM inver-sion yielding a χ2 smaller than 10 can thus be con-sidered consistent with the working hypothesis ofa Gaussian DEM plasma. It does not mean how-ever that a Gaussian is the only possible model,but that it is a model consistent with the observa-

9

tions. Since we perform a least square fit of the sixvalues Iobsb −Ithb by three parameters (EM , Tc, σ),this behavior is expected. The small correlationbetween the residuals observed in the isothermalcase disappears with increasing DEM width. Thiscan be explained by the smoothing of the criterion(see Figure 8) that reduces the directionality of itsminima.

For model testing, a reduced chi-squared χ2red =

χ2/n, where n is the number of degree of freedom,is sometimes preferred over a regular χ2 test, forit has the advantage of being normalized to themodel complexity. Usually, n is the number ofobservations minus the number of fitted param-eters. This assumes that the measurements areindependent which is what we wanted to test. Inpractice, a small correlation between the residualsis found in the isothermal case, but the effect issmall and a reduced χ2 can be used. It shouldbe noted that the residuals have a non negligibleprobability to be greater than the peak of the cor-responding χ2 distribution. For plasmas havingGaussian DEMs for example, the residuals haveabout a 43% chance to be greater than 3 (Fig-ure 11). This implies that seemingly large resid-uals are not necessarily a proof of the inadequacyof the chosen DEM model. The working hypothe-sis can only be invalidated if it can be shown thatanother model has a greater probability to explainthe obtained residuals. This was for example thecase in Section 2.3.1 where we have shown thatif an isothermal inversion gives a squared residualgreater than 3, the plasma DEM is more proba-bly Gaussian than isothermal. We now explore asimilar situation for Gaussian inversions.

Indeed, in reality the DEM shape is not knownand it is interesting to test if a Gaussian is a per-tinent model or if AIA has the capability to dis-criminate between different models. For this pur-pose, still considering Gaussian DEM solutions,the simulations of the observations are now per-formed using a top hat DEM distribution definedas

ξhat(Te) =EM ΠTc,σ(Te),

with ΠTc,σ(Te) =

{

1

σif | log(Te)− log(Tc)| < σ

2

0 else

(6)

Like a Gaussian, this parametrization can repre-

sent narrow and wide thermal structures.

Figure 12 gives the associated temperatureprobability maps. The top panels correspond tosimulated plasmas with a top hat distribution ofwidth σP = 0.1. Most of the solutions are con-centrated around the diagonal, even though therobustness is somewhat affected for temperaturesin the range 5 × 105 < TP

c < 106 K, where lowprobability secondary solutions exist. These twoplots are very similar to the top panels of Figure 9that give the probability of the Gaussian solutionsto a Gaussian plasma. Therefore, even though theDEM assumed for the inversion is different fromthat of the plasma, the central temperature of thetop-hat is determined unambiguously.

The picture is different for wide plasma DEMdistributions. In the bottom panels of Figure 12,with σP = 0.7, the diagonal has become very wide,structured and does not cross the origin any more.Some plateaus appears, meaning that for someranges of plasma temperature TP

c , the tempera-ture T I

c provided by the Gaussian solutions will beinvariably the same. For 5×105 < TP

c < 3×106 Kfor example, a constant solution T I

c = 1 MK ap-pears, as shown also by P (T I

c ) (bottom panel (b)).As a result, the inverse conditional probabilitymap P (TP

c |T Ic ) (bottom panel (c)) indicates that

for an inversion output of 1 MK there is a largeindetermination on the central temperature. Thebehavior of the solutions is thus globally equiva-lent to that described in Section 3.2 for the inver-sion of Gaussian DEM plasmas.

The observed distribution of the sum of thesquared residual is very similar to those obtainedwith consistent Gaussian DEMmodels and is closeto 3 χ2 (see Figure 8). If the working hypothe-sis were a top hat DEM, thus consistent with theplasma, the residuals would also be close to a de-gree 3 χ2, as for any DEM model described bythree parameters (see discussion above). The so-lar plasma has no reason to really have a top hatDEM. However, this numerical experiments showsthat, using AIA data only and the χ2 test, thediscrimination between two very different multi-thermal DEMs is practically impossible.

4. Summary and discussion

In this work we described a complete charac-terization of the statistical properties of the DEM

10

inversion, rigorously treating both systematic andrandom errors. The developed methodology hasbeen illustrated in the specific case of the AIAtelescope, but the technique is generic and canbe applied to any other instrument, spectrome-ters as well as imaging telescope. By restrictingourselves to parametric DEMs, we could analyzein details what occurs during the inversion pro-cess, and could therefore point out the fundamen-tal difficulties involved in the DEM reconstruc-tion. Even though only a few simple DEM distri-butions have been studied, important and genericconclusions regarding the robustness of the inver-sion problem have been reached. The method canbe applied to other forms of DEMs as long as theycan be defined by a small number of parameters.

Our technique provides new tools to facilitatethe interpretation of the DEM inversion. By com-puting the P (ξI |ξP ) probability matrices the ro-bustness can be evaluated, secondary solutionscan be detected and their probabilities quantified.Since we do not know whether the systematic er-rors are over or under-estimated, their random-ization in the computation of the inverse condi-tional probability P (ξP |ξI) ensures that all possi-ble DEMs consistent with the solution ξI given theuncertainties are taken into account. The specificlocation of secondary solutions in the space of thepossible DEMs ξ can be interpreted by analyzingthe topography of the criterion C(ξ), which is in-trinsically linked to the shapes of the temperatureresponse functions. The simulated distributions ofthe sum of the squared residuals provide referencesagainst which to asses the pertinence of the DEMmodel used to perform the inversion. Applyingthe technique on real data, and applying the toolspresented here on the results, the interpretation ofDEM inversions can be facilitated.

By investigating the isothermal solutions, wehave shown that the six AIA coronal bands greatlyenhance the robustness of the DEM reconstructioncompared to previous EUV imagers, even thoughthe temperature resolution varies between 0.03and 0.11 logTc. This resolution depends on thelevel of uncertainties assumed (see Section 3.2 inPaper I). Because of the ambiguity between mea-surement errors and deviation from isothermality,a variety of plasma conditions is consistent withthe isothermal hypothesis. As the degree of mul-tithermality of the observed plasma increases, the

isothermal solutions become progressively decorre-lated from the input. We have demonstrated that,in the limit of very broad multithermal plasmas,the isothermal solutions are biased toward twopreferred solutions: 3×105 or 106 K depending onthe systematics. But the analysis of the behaviorof the residuals allows to measure the adequacy ofthe isothermal hypothesis. Past a critical value ofthe residuals, a multithermal DEM is more proba-ble than an isothermal plasma. That critical valuehas to be estimated from Monte-Carlo simulationsfor the considered set of bands or spectral lines.Our procedure provides a rigorous quantificationof the classical EM loci isothermal test.

Investigation of the multithermal Gaussian so-lutions shows that AIA has the ability to recon-struct simple DEM forms. However, the greaterthe width of the plasma DEM, the lower the ac-curacy and precision on the determination of theDEM parameters. We argued that this would bethe case also if using individual spectral lines in-stead of narrow bands. In the extreme case of verybroad DEMs distributions the quality of the inver-sion is strongly affected, because of the combinedeffects of a smoothed criterion and simulated ob-servations weakly dependent on the plasma pa-rameters. The AIA Gaussian solutions are bi-asS.P. acknowledges the support from the BelgianFederal Science Policy Office through the inter-national cooperation programmes and the ESA-PRODEX programme and the support of the In-stitut d’Astrophysique Spatiale (IAS). F.A. ac-knowledges the support of the Royal Observatoryof Belgium. The authors would like to thank J.Klimchuk for fruitful discussions and comments.edtowards a near isothermal plasma (σI = 0.15) cen-tered on Tc = 1 MK. Furthermore, the analysis ofthe distribution of the residuals has revealed thatit is difficult to discriminate between different mul-tithermal DEMs models using AIA alone.

These results can have far reaching implicationsfor the determination of the thermal structure ofthe solar corona. Indeed, the thermal structure ofcoronal loops for example is still under debate, dif-ferent proposed heating scenarios leading to oppo-site DEM properties. For example, the nanoflarestheory (e.g. Parker 1983; Cargill 1994; Klimchuk2006), suggesting that the stored energy is thenreleased impulsively over small scales, predicts awide thermal structure at a given time, in case of

11

sparse events inside a single strand. Conversely,the steady heating processes should produce loopswith narrow DEMs. Our analysis revealed thateven if the residuals support the pertinence ofthe solution, the DEM analysis of AIA observa-tions can lead to the erroneous conclusion thatthe plasma is nearly isothermal while it is in factbroadly multithermal. Different biases can be ex-pected for different sets of bands or spectral lines,different instruments or atomic physics data.

The methodology developed in this series of pa-pers can also be used to establish the optimal setof bands needed to maximize the robustness of theDEM inversion, for broad band instrument as wellas spectrometers. Performing this type of prepara-tory simulations can thus be very useful to opti-mize the design of future solar instruments if theyare to provide reliable temperature diagnostics.

S.P. acknowledges the support from the BelgianFederal Science Policy Office through the inter-national cooperation programmes and the ESA-PRODEX programme and the support of the In-stitut d’Astrophysique Spatiale (IAS). F.A. ac-knowledges the support of the Royal Observatoryof Belgium. The authors would like to thank J.Klimchuk for fruitful discussions and comments.

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This 2-column preprint was prepared with the AAS LATEXmacros v5.2.

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Fig. 1.— Summary of the technique used in this work. The AIA temperature response functions arecomputed using CHIANTI 7.0. The simulated observations Iobsb are calculated assuming a simple DEMmodelξP (isothermal, Gaussian or top hat), and adding the instrumental random perturbations and systematicerrors. The theoretical intensities Ithb are computed assuming a DEMmodel ξ, for a large range of parameters.The inverted DEM ξI is evaluated by least-square minimization of the distance between the theoreticalintensities and the simulated observations. Using a Monte-Carlo scheme, N realizations of the random andsystematic errors nb and sb are computed. For a given set of parameters (EMP , TP

c , σP ) of the plasma DEMξP , the quantity P (ξI |ξP ) is then evaluated. The probability P (ξP |ξI) is then derived using Bayes’ theorem.

13

Fig. 2.— Equivalence between uncertainties and multi-thermality. The criterion C(EM,Tc) given by Equa-tion 3, represented in gray scale and its absolute minimum, marked by a white plus sign. The EM loci curvesfor each of the six AIA coronal bands are also represented (white lines), superimposed to the criterion. Thelocation of the absolute minimum provides the isothermal solution ξI for a plasma having DEMs ξP centeredon TP

c = 1.5 MK, and a total emission measure of 2×1029 cm−5. Top left : isothermal plasma having a DEMξP = ξiso in presence of random and systematic errors. Top right : Multithermal plasma having a GaussianDEM ξP = ξgauss with a width σP = 0.1 logTe, without uncertainties. Uncertainties and multithermalityboth produce comparable deviations of the EM loci curves. Bottom: Same as top right, but for two differentrealizations of the random and systematic errors nb and sb.

14

Fig. 3.— Isothermal solutions for different thermal widths. Same as Figure 2 for larger DEMs plasmawidths. In the top row the plasma has a Gaussian DEM σP = 0.3 logTe, and σP = 0.7 logTe in the bottomrow. For each thermal width, two different realizations of the random and systematic errors are presented,showing that the location of the absolute minimum can greatly vary.

15

Fig. 4.— Maps of probabilities considering a Gaussian DEM plasma ξP = ξgau having a narrow thermaldistribution of σP = 0.1 logTe, obtained by 5000 Monte-Carlo realizations of the random and systematicserrors nb and sb, and investigating the isothermal solutions. (a): Probability map P (T I

c |TPc ), vertically

reading. The central temperature T Ic resulting from the inversion is presented whatever the total emission

measure EM I . (b) and (c): Probability profiles of T Ic for plasma central temperatures TP

c = 1.5× 106 and7 × 106 K (vertical lines in panel(a)). (d): Total probability to obtain T I

c whatever TPc (see Section 1 and

Section 2.1 of paper I). (e): Vice-versa, probability map P (TPc |T I

c ), horizontally reading, inferred by meansof Bayes’ theorem using P (T I

c |TPc ) and P (T I

c ). (f) and (g): Probability profiles of TPc knowing that the

inversion result is, from top to bottom, 7× 106 and 1.5× 106 K.

16

Fig. 5.— Same as Figure 4, but with a plasma DEM width is increased to σP = 0.3 logTe. Many pertur-bations, already visible in the Figure 4 are amplified: the distribution is wider, irregular and the diagonalstructure disappeared (see panel (a)). The presence of multiples solutions of comparable probabilities isincreased for a large range of plasma temperatures TP

c , leading to very different estimated T Ic from the

input TPc . The probability map P (T I

c |TPc ) can help us to properly interpret the inversion result, taking into

account the secondary solutions and providing their respective probability.

17

Fig. 6.— Same as Figure 4, but in case of the plasma DEM width is increased to σP = 0.7 logTe. Theimpact on the robustness of the inversion is clearly increased in this case, showing two privileged isothermalsolutions T I

c , totally decorrelated from the input TPc . As a result, no information regarding the central

temperature of the DEM can be extracted from the inversion.

18

Fig. 7.— In the top panel, the distributions of the sum of the squared residuals corresponding to isothermalinversions of Gaussian DEM plasmas, as a function of the DEM width σP . Higher probabilities correspondto lighter shades of grey. The bottom panel shows cuts at σP = 0 (isothermal), 0.1, 0.3 and 0.7 log Te alongwith theoretical χ2 distributions of degree 3 and 4.

19

Fig. 8.— Criterion in case of Gaussian multithermal DEM inversions. The three-dimensional criterion isa function of EMP , TP

c , and σP . The simulated plasma has a Gaussian DEM centered on 1.5 MK and adifferent width for each row. From top to bottom: σP = 0.1, 0.3 and 0.7 logTe. The superimposed curveson the left panels represent the equivalent of the EM loci emission measure curves in a multithermal regime(see the detailed description in Section 3.1).

20

Fig. 9.— Same as figure 4 but investigating now the multithermal solutions. The simulated observationsare computed for DEM widths of σP = 0.1 (top row) and 0.7 logTe (bottom row). The probabilitiesare represented here whatever the emission measure EM I and the Gaussian width σI returned by theminimization scheme. In the case of low degree of multithermality plasmas, the solutions remain distributedaround the diagonal. However, for broad DEM distributions, the solutions appear to be biased toward∼1 MK, especially for plasmas exhibiting central temperature lower than 2 MK.

21

Fig. 10.— Maps of probabilities for the DEM width for a simulated plasma having a Gaussian DEMcentered on TP

c = 1 MK. Probabilities are represented whatever the emission measure EM I and the centraltemperature T I

c obtained. The solutions appear to be biased toward σI ∼ 0.12 logTe whatever the plasmawidths, even if the situation improves for smaller widths.

22

-

Fig. 11.— Same as Figure 7 for Gaussian solutions.

23

Fig. 12.— Same as Figure 9 but for a plasma having a top hat DEM. The simulated observations arecomputed for DEM widths of σP = 0.1 (top row) and 0.7 logTe (bottom row).

24