Can the Differential Emission Measure constrain the timescale of energy deposition
Chloé Guennou, Frédéric Auchère, James A. Klimchuk, Karine Bocchialini, Susanna Parenti
aa r X i v : . [ a s t r o - ph . S R ] J un Can the Differential Emission Measure constrain the timescale of energydeposition in the corona?
C. Guennou and F. Auch`ere Institut d’Astrophysique Spatiale, Bˆatiment 121, CNRS/Universit´e Paris-Sud, 91405 Orsay,France [email protected] andJ.A. Klimchuk , K. Bocchialini Solar Physics Laboratory, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. andS. Parenti Royal Observatory of Belgium, 3 Avenue Circulaire, B-1180 Bruxelles, Belgium
ABSTRACT
In this paper, the ability of the
Hinode /EIS instrument to detect radiative sig-natures of coronal heating is investigated. Recent observational studies of AR coressuggest that both the low and high frequency heating mechanisms are consistent withobservations. Distinguishing between these possibilities is important for identifying thephysical mechanism(s) of the heating. The Differential Emission Measure (DEM) tool isone diagnostic that allows to make this distinction, through the amplitude of the DEMslope coolward of the coronal peak. It is therefore crucial to understand the uncertain-ties associated with these measurements. Using proper estimations of the uncertaintiesinvolved in the problem of DEM inversion, we derive confidence levels on the observedDEM slope. Results show that the uncertainty in the slope reconstruction stronglydepends on the number of lines constraining the slope. Typical uncertainty is estimatedto be about ± .
0, in the more favorable cases.
Subject headings:
Sun: corona - Sun: UV radiation
1. Motivations
The understanding of how the Sun’s outer atmosphere is heated to very high temperaturesremains one of the central issues of solar physics today. The physical processes that transfer and 2 –dissipate energy into the solar corona remain unidentified and a variety of plausible mechanismshave been proposed (see Parnell & De Moortel 2012; Klimchuk 2006; Walsh & Ireland 2003; Zirker1993, for a review of the various coronal heating models). If the magnetic origin of coronal heat-ing seems to be currently well-accepted (Reale 2010), the details regarding the energy transportfrom the photosphere to the corona or the energy conversion mechanisms are still open issues.Recently, efforts focused on the determination of the timescale of energy deposition in the solarcorona, providing constraint on the properties of the heating mechanisms and allowing for a dis-tinction between steady and impulsive heating scenarios. The nanoflares theory of Parker (1988)for example, is based on the idea that the corona is heated by a series of ubiquitous small andimpulsive reconnection events. However, the term nanoflare is now used in a more general way,referring to any impulsive heating event that occurs on small spatial scale, whatever the nature ofthe mechanism(see Cargill 1994; Cargill & Klimchuk 2004; Klimchuk & Cargill 2001). Even waveheating takes the form of nanoflares by this definition (see Klimchuk 2006).According to the impulsive or steady nature of the heating, coronal loops are predicted topresent different physical properties at a given time. Observations suggest that coronal loops areprobably not spatially resolved. For this reason more often a loop is modeled as a collection ofunresolved magnetic strands, considering a strand as a fundamental flux tube with an isothermalcross-section. Depending on the timescale of the heating mechanisms involved, the plasma withinthe individual strand is allowed or not to cool and drain, via a combination of conductive andradiative cooling (Reale 2010). Therefore, the thermal structure of the whole loops will differ, theproportion of hot to warm material depending on the time delay between heating events.Recently, several authors took a particular interest in one potential diagnostic of the heatingfrequency based on the analysis of the slope of the Differential Emission Measure (DEM) of Ac-tive regions (ARs). Based on both theoretical and observational analysis, earlier analysis reportedthat the coolward part of the DEMs generally follows a power law, up to the emission measurepeak ( ∼ − T ) ∝ T α with α the positive slope index (Jordan 1980; Dere 1982;Brosius et al. 1996). This slope provides indications directly related to the heating timescale: alarge proportion of hot relative to warm material leads to a steep DEM slope, whereas a shallowerslope corresponds to less hot material and more warm material. The former case is consistent withhigh frequency impulsive heating, where the short time delay (lower or equivalent to the coolingtime) between two heating events does not allow the cooling of a large proportion of material. Inthe latter case, the time delay between two heating events (now larger than the cooling time) allowsthe cooling of a significant quantity of the strand material. The limiting case, where the time delaytends to zero, actually corresponds to the steady heating case, where the strand is continuouslyheated. Using different combinations of observations from the Extreme-ultraviolet Imaging Spec-trometer (EIS; Culhane et al. 2007) on board the Japanese mission Hinode (Kosugi et al. 2007),the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) instrument on board the
Solar Dy-namic Observatory (SDO), and the
Hinode soft X-Ray Telescope (XRT; Golub et al. 2007), severalauthors recently carried out new AR observational analysis, estimating slope values ranging from 3 –1.7 to 5.17 for 21 different AR cores (Tripathi et al. 2011; Warren et al. 2011; Winebarger et al.2011; Schmelz & Pathak 2012; Warren et al. 2012).In the present work, we focus on the investigation of the possibilities to derive the DEM fromobservations, and we provide a method to estimate the uncertainties associated with its parameters,especially the slope. We do not refer to any particular physical mechanism, such as magnetic re-connection or dissipation of waves, we only refer to the timescale of the mechanism itself . Technicaldifficulties related to both observational processing and diagnosis complicate the slope derivationand thus the associated physical interpretation. In particular, the DEM inversion problem hasproved to be a real challenge, due to both its intrinsic underconstraint and the presence of randomand systematic errors. Authors were early attentive to examining the fundamental limitations ofthis inversion problem (Craig & Brown 1976; Brown et al. 1991; Judge et al. 1997), and manydifferent inversion algorithms have been proposed (Craig & Brown 1986; Landi & Landini 1997;Kashyap & Drake 1998; McIntosh 2000; Goryaev et al. 2010; Hannah & Kontar 2012). Despiteall these attempts, reliably estimating the DEM and the uncertainties associated with the solutionremain a major obstacle to properly interpret the observations.In this perspective, we developed in recent papers (Guennou et al. 2012a,b, hereafter PaperI and II) a technique, applicable to broadband or spectroscopic instruments, able to completelycharacterize the robustness of the DEM inversion in specific cases. Using a probabilistic approach tointerpret the DEM solution, this technique, briefly recalled in Section 2, is useful for examining theDEM inversion properties and provides new means of interpreting the DEM solutions. Assumingthat the DEM follows a power law, and applying our technique to the
Hinode /EIS instrument, wederive estimates of the errors associated with the reconstructed DEM slopes, described in Section 3.The presence of uncertainties radically changes the conclusions regarding the compatibility betweenobservations and models, as shown by Bradshaw et al. (see 2012) and described in Section 3 and 4,where we also discuss the results in the context of steady vs impulsive coronal heating .
2. Methodology
The approach used in this work is very similar to that used in Paper I and II. The techniqueand the DEM formalism are exhaustively described therein, but a quick summary is given below.
Under the assumption of an optically thin plasma, the observed intensity in a spectral band b can be expressed as I b = 14 π Z + ∞ R b ( T e , n e ) ξ ( T e ) d log T e , (1) 4 –where T e is the electron temperature, ξ ( T e ) = n e ( T e )d p/ d log T e is the DEM that provides ameasure of the amount of emitting plasma as a function of temperature, with n e the square electrondensity averaged over the portions dp of the Line Of Sight (LOS) at temperature T e (Craig & Brown1976). R b ( T e ) is the temperature response function of a given instrument R b ( n e , T e ) = X X,l S b ( λ l ) A X G X,l ( n e , T e )+ Z ∞ S b ( λ ) G c ( n e , T e ) d λ. (2)where the first term refers to the spectral lines l of an atom X of abundance A X , whereas thesecond describes the contribution of the continua. S b ( λ ) is the spectral sensitivity of the spectralband b of the instrument, and G X,l and G c are the contribution functions taking into accountall the physics of the coronal emission processes (Mason & Monsignori Fossi 1994). The totalEmission Measure (EM) is obtained by integrating the DEM over the logarithm of temperature.The inference of the DEM from a set of observations involves the inversion of Equation 1, which ishindered by both the presence of random instrumental perturbations and systematic errors on theinstrument calibration and on the atomic physics. The purpose of our work here is to investigatethe limitations induced by uncertainties in the DEM inversion process concentrating in particularon the determination of the slope of the distribution. Our method is quite general, but we will dealspecifically with observations obtained by the Hinode /EIS spectrometer. Using simulations of the
Hinode /EIS observations I obsb and comparing them to the theoretical expectation I thb , including theperturbations engendered by the uncertainties, it is possible to quantify the reliability of the DEMinversion of the EIS data.In simple terms, our approach is essentially the following. We start with an assumed (called”true” hereafter) DEM with a particular functional form. From this we generate a syntheticspectrum, introducing errors associated with unknown atomic physics, instrumental calibration,and photon counting noise. We then determine the DEM that provides the best fit to the syntheticspectrum, which we take to be the DEM that minimizes the differences in the line intensities. Thisinferred DEM has the same functional form as the true DEM. Only the parameters are different. Themost important parameter is the slope, and by comparing the true and inferred slope, we obtain anerror in the slope measurement for this particular set of atomic physics, calibration, and noise errors.By running many trials, with many different sets of errors chosen from appropriate probabilitydistributions, we finally deduce an estimate of the uncertainty in the slope determination.The core of our method resides in the probabilistic approach of the DEM inversion: let usassume a plasma with a true DEM ξ T ; the DEM solution ξ I is the one that minimizes the criterion C ( ξ ) We choose to define the DEM on a logarithmic scale, but the DEM can also be defined in linear scale as ξ ( T e ) = n e ( T e )d p/ d T e . There is a factor d log T e / d T e = 1 / (ln 10 T e ) between the two conventions. ξ I = arg min ξ C ( ξ ) ,C ( ξ ) = N b X b =1 (cid:18) I obsb ( ξ T ) − I thb ( ξ ) σ ub (cid:19) . (3)The solution ξ I minimizes the distance between the theoretical intensities I thb and the observedones I obsb in N b spectral bands. The normalization σ ub corresponds to the standard deviation ofthe uncertainties. The residuals χ = min C ( ξ ) provide an indication on the goodness of thefit. It is worth noting that as mentioned by Testa et al. (2012); Landi & Klimchuk (2010), andPaper I and II, a low χ does not necessarily imply that the solution be the good one or the onlyone. While our study has broad applicability, we concentrate specifically on observations fromthe EIS spectrometer on Hinode . The criterion is in this case the sum of the contribution of 30components, one per spectral line. We used the set of 30 lines listed in Table 1, identical to the oneused by Bradshaw et al. (2012) and Reep et al. (2013) in order to carry out practical comparisonbetween observations and model predictions (see Section 3), using the uncertainties derived in thiswork. Most of them belong to the more prominent lines in the AR regime (Del Zanna & Mason2005). Some used lines arise from the same ion species, thus we only have 20 different ion formationtemperatures available to constrain the DEM. Column 4 of Table 1 indicates the temperatures wherethe contribution functions peak. However, these additional lines are used in practice as redundantinformation to decrease the uncertainties. Using Monte-Carlo simulations of the instrumentalnoises n b and systematic errors s b (see Section 2.3 for a detailed description of the uncertainties),the conditional probability P ( ξ I | ξ T ) to obtain the inferred DEM ξ I knowing that the true DEMis ξ T can be computed. Then, the inverse conditional probabilities P ( ξ T | ξ I ) giving the probabilitythat the true DEM is ξ T knowing the inferred results can be deduced from Bayes’ theorem. Thislatter quantity contains all the information possible to extract from a set of observations given thelevel of uncertainties.Thus, the range or multiple ranges of solutions able to explain the observations within theuncertainties can be identified. The derivation of P ( ξ T | ξ I ) requires to know P ( ξ I ), and, obviously,because of the uncertainties, a great number of solutions ξ I can be potentially consistent with a setof observations. Therefore, the computation of this probability is practical only if the space of thesolutions is limited, for otherwise it would require the exploration of an infinite number of possibleDEMs. For practical reasons, the number of parameters defining the DEM is limited to four: theslope α , the temperature of the peak T p , the cutoff at high temperature σ and the total EM. 6 – In order to represent in a more realistic way the observed DEMs, we used the following pa-rameterization of the AR DEM model, represented for different sets of parameters on Figure 1 . • A power law for the low temperature wing: T e < T ξ AR ( T e ) = k EM × T αe with k = T − α N . (log T − log T p )and N σ ( x ) = 1 σ √ π exp (cid:18) − x σ (cid:19) (4)where α is the slope of the DEM coolward of the DEM peak, T p is the temperature of theDEM peak and EM is the total emission measure. The normalization constant k is used toensure the continuity and smoothness of the DEM model: the slope must be tangent to thefixed Gaussian connector (see below), at the point T , depending on the slope value. • A Gaussian high temperature wing: T e > T p ξ AR ( T e ) = EM N σ (log T e − log T p ) , (5)where σ is the standard deviation of the Gaussian wing. Thus, beyond the temperature of theDEM peak, the DEM is described by a Gaussian distribution at high temperature, definedby the σ parameter. • A fixed width Gaussian connection: T < T e < T p ξ AR ( T e ) = EM N . (log T e − log T p ) , (6)where T is the point where the slope α is tangent to the fixed Gaussian N . . The connectorhas been added to ensure that the DEM model is continuous and smooth, corresponding toa continuous first derivative.A large range of DEM parameters is explored, computing the reference theoretical intensities I b ,used to deduce I obsb and I thb (see Section 2.3), for electron temperatures T e ranging from log T e = 5 to log T e = 7.5 in steps of 0.005 log T e . The slope α varies from 1.0 to 6.0 in steps of 0.05,and the high temperature wing is explored from σ = 0 .
01 to 0 .
05 log T e in steps of 0.01. Thetotal EM varies between 3 × and 3 × cm − with a resolution of 0.1 in logarithmic scale,and the temperature of the peak T p varies between log T p = 5 . T p = 6 . I b as a function of the four parameters α, σ, T p , EM, for each of the thirty For color version of all plots presented in this paper, see the online color version. cm − while the others parameters α, σ and T p are allowed to vary. The five curves on the left are alldrawn for the same peak temperature T p = 10 K and a fixed Gaussian high temperature wing of σ = 0 . T e , whereas the slopes varies between 1 and 5. The last five curves on the right displaythe variation of the high temperature wing: the central temperature T p and the slope α are nowfixed to respectively T p = 10 . K and α = 5 whereas the σ parameter varies between 0.05 and 0.49. Following the initial reasoning of Paper I, the theoretical intensities I thb and I obsb can be ex-pressed as I thb = I b + s b and I obsb = I b + n b , where I b are called the reference theoretical intensities, n b are the random perturbations and s b are systematic errors. The reference theoretical intensitiesare equal to I obsb and I thb in case of a hypothetically perfect knowledge of the atomic physics andobservations. They have been computed via Equation 1 and Equation 2 and using the given ARDEM model ξ AR (see Section 2.2). We used the CHIANTI 7.1 atomic database (Dere et al. 1997;Landi et al. 2013), and for each of the spectral lines b listed in Table 1, the EIS reference theoreticalintensities have been calculated using the function eis eff area (Mariska 2010) of the InteractiveDate Language Solar Software (SSW) package.The different nature of the random and systematic uncertainties n b and s b affects the observa-tions in distinct ways (Taylor 1997). The random errors affect the data in an unpredictable way, i.e.they could be revealed by a hypothetically large number of experiments, the error made on eachmeasurements differing for each attempts. A set of Hinode /EIS observations is randomly perturbedby various factors: the Poisson photon shot noise and the detection noises, such as thermal or readnoise, often assumed to be Gaussian. These phenomenona are well-known and can be realisticallysimulated: Poisson perturbations P λ and σ ccd = 6 e − rms (McFee 2003) of Gaussian CCD readnoise are added, before conversion to digital numbers (DNs), using the conversion gains of the EISspectrometer.In contrast, the systematic uncertainties can not be revealed by the repetition of the sameexperience, always pushing the results in the same direction and thus leading to a systematic and unknown over or under-estimation. Besides, it is difficult to estimate the probability distribution ofthe systematics. In the following, the probability distribution of such kind of uncertainties will beconsidered to be Gaussian, as it generally assumed. The observational intensities I obsb are affectedby the uncertainty associated with the calibration of the instrument, estimated by Culhane et al.(2007) to be around σ cal = 25% for the two different CCDs cameras of the EIS instrument. Thisuncertainty refers to the absolute calibration. We used two independant Gaussian variables tomodel it, one for each camera. All the lines falling on one camera are perturbated by the same 8 –amount for each random realization of the uncertainties. The difference between the two camerascan be as large as 40%. In the second set of uncertainties described in Section 3, this difference isreduced to 20%. Besides, the degradation of the instrument response over time can also include anadditional systematic uncertainty, biasing the results in a given direction.The theoretical expectations I thb are impacted by a complex chain of uncertainties of differentnature. Thus, the estimation of the errors on the contribution functions G c and G X,l (see Equa-tion 2) is a more challenging task. In particular, recasting the expression of the observed intensitiesinto Equation 1 is possible only via several implicit physical assumptions (Judge et al. 1997): theplasma is considered as an optically thin gas, in statistical and ionization equilibrium. The elec-tron velocity distributions function are generally considered to be Maxwellian, as in the CHIANTIdatabase, and the abundance of each element must be constant over the LOS. A discrepancy ofthe observed coronal plasma with one of these assumptions potentially affects the interpretationof the data. For example, the observed enhancement of the low first ionization potential (FIP)elements (Young 2005) in the solar corona possibly induces a non-uniformity of the abundancesalong the LOS.Incompleteness in the atomic databases, such as missing transitions, or inaccuracy in somephysical parameters such as ion-electron collision cross sections, de-excitation rates, etc..., also re-sults in systematic uncertainties. For example, the recent release from version 7.0 to version 7.1of the CHIANTI spectral code (Landi et al. 2013), including important improvements in the softX-ray data, clearly shows that the version 7.0 of the CHIANTI database was incomplete in the50-170 ˚ A wavelength range, leading to strong inaccuracy in the emissivity calculations of some Feions from Fe VIII to Fe XIV. These updates particularly affect the temperature response functionof 94 and 335˚ A channels of the SDO/AIA instrument. Atomic structure computations are basedon two different types of electron scattering calculations: the distorted wave (see Crothers 2010,for details) or the close coupling approximation (see McCarthy & Stelbovics 1983, for details), thelatter being generally more accurate. Ionization balance implies equilibrium between the ionizationand recombination processes, but if the plasma is out-of-equilibrium or in a dynamic phase, theCHIANTI calculations of line intensities are not consistent with the observations. For example inthe case of low frequency heating, the plasma can be out of ionization equilibrium, so that the DEMdetermination will be incorrect (Sturrock et al. 1990). In that case, temperature-sensitive line ra-tios of individual ions may be a better way to constrain the models (Raymond 1990). However,these effects should not be important except for very hot plasmas produced by impulsive heat-ing (Bradshaw & Klimchuk 2011; Reale & Orlando 2008). Within the used temperature range, theevolution is slow enough and the density is high enough that ionization equilibrium is generally agood approximation. Impacts of a deviation of the electron velocity distributions from a Maxwellianon the ionization equilibrium and on the electron excitation rates have been studied by Dzifˇc´akov´a(1992) and Dzifˇc´akov´a (2000), showing that the intensities of spectral lines can be significantlyaltered. The effects of radiative losses inaccuracy have also been investigated by Reale & Landi(2012), demonstrating that changes in the radiative losses have important impacts on the plasma 9 –cooling time, which itself impacts the conclusions of the impulsive heating models. Some studieshave been recently carried out to evaluate the impact of using inconsistent atomic physics datain the DEM inversion process (Landi & Klimchuk 2010; Landi et al. 2012; Testa et al. 2012) andfound that the DEM robustness can be significantly altered, leading to important uncertainties onthe reconstruction accuracy.To take into account all these effects, we include the uncertainties in our Monte Carlo sim-ulations using normally distributed random variables. For each realization (each simulation), wechoose a number randomly from a Gaussian distribution with a halfwidth σ i , considering the fourfollowing separate classes: • Class 1 : the first uncertainty class σ at involves errors that are different for each and everyspectral line, thus we used 30 independent Gaussian random variables to model it (i.e. adifferent random number for each line). These include errors in the radiative and excitationrates, atomic structure calculations, etc. • Class 2 : the second class σ ion involves errors that are the same for every line of a given ion,but different for different ions. We used the same random number for multiple lines of thesame ion (e.g., Fe XIV 264, 270, and 274 ˚ A ), but different random numbers for different ions,resulting thus in 20 independent Gaussian random variables (3 different Mg ions, 3 Si ions,8 Fe ions, 2 S ions and 4 Ca ions). This class corresponds to errors in the ionization andrecombination rates. • Class 3 : the third class σ abu involves errors that are the same for every line of a givenelement, but different for different elements, thus we used 5 different Gaussian variables (oneper element). These are errors in the elemental abundances that are unrelated to the firstionization potential (FIP) effect. • Class 4 : finally, the fourth class σ fip involves the additional errors that are the same forevery low-FIP elements corresponding to errors on the coronal abundance of such elements.In order to simulate this effect, we adopted a mean FIP bias of 2.5, adding then an uncertaintyof σ fip on this enhancement factor itself, through an identical Gaussian variable. All our setsof spectral lines, except the two Sulfur lines are finally perturbed in the same way. • In addition to these atomic physics uncertainties, a generic uncertainty of σ ble = 15% is addedon the blended lines, to account for the added technical difficulties to extract a single lineintensity from the data. Blended lines are underlined by a b in the EIS spectral lines list inTable 1.Each theoretical line intensity I thb , is then modified by the sum of the four random numbersrepresenting the four uncertainty classes (plus a fifth random number in case of blended lines),leading to I thb = [(1 + R )(1 + R )(1 + R )(1 + R , if low FIP)(1 + R , if blended)] I b . Note that 10 –the R i are equally likely to be positive or negative, and the amplitude of the random number isvery likely to be less than the Gaussian halfwidth, but will occasionally be larger and on rareoccasion will be much larger. All the random numbers are reset for each new realization. Theresulting uncertainty of each spectral line is reported in column 4, where the σ unc is obtained byquadratically summing all the sources of uncertainty, as is appropriate if the errors are independent: σ unc = σ at + σ ion + σ abu + σ cal (+ σ fip + σ ble if applicable).In order to determine appropriate amplitudes for the four classes of uncertainty related toatomic physics, we polled a group of well-known solar spectroscopists (G. Del Zanna, G. Doschek,M. Laming, E. Landi, H. Mason, J. Schmelz, P. Young). There was a good consensus that thegeneric amplitudes are approximately σ at = 20% for the class 1 and σ ion = σ abu = σ fip = 30%for each of the other three classes. It was noted, however, that the errors could be substantiallylarger or smaller for specific spectral lines. Adding these uncertainties in quadrature leads to atotal atomic physics uncertainty ranging between 46 . . σ cal = 20%. The results corresponding to both these sets of uncertaintiesare presented in Section 3. Ultimately, a customized set of uncertainties should be developed forthe specific line lists that have been used in published studies. This is beyond the scope of ourpresent investigation, but is something we plan for the future. Until such customized uncertaintiesare available, it is our opinion that the primary set of uncertainties (20%, 30%, 30%, 30%) are mostappropriate for estimating the uncertainties in the DEM slope. Atomic physics uncertainties aredifficult to determine, but the associated systematic errors decreased in the last decades, thanks tomore sophisticated computation facilities, and more accurate atomic physic experiments.Even though we have tried to simulate the systematic errors in a realistic way, some additionalsophistications could also be added in our model. Our treatment of the class 1 and 2 uncertaintiesas intensity modifications is an approximation. In reality, errors in excitation, ionization, andrecombination rates are manifested as modifications in the G X,l and G c contribution functions ofthe lines (see Equation 2). These functions change shape and central position as well as amplitude.A given modification in G X,l or G c will therefore produce an intensity change that depends on theDEM. Treating this properly could be done in the future but is beyond the scope of this initialwork. Future studies might also account for the correlation between various uncertainties. For 11 –example, if the class 2 error for Fe XIV is positive, the class 2 error for Fe XIII and Fe XV is likelyto be negative.
3. Results
In order to quantify the influence of both random and systematic errors, we performed severalMonte-Carlo simulations with the uncertainties described in Section 2.3 and the AR DEM modeldescribed in Section 2.2. The thirty lines described in Table 1 have been used. The simulatedobservations I obsb and the theoretical intensities I thb have been calculated with the same AR DEMmodel. In this way, the model can perfectly represent the simulated EIS data. Since the solutionscorrespond by definition to the absolute minimum of the least-square criterion (Equation 3), all so-lutions are fully consistent with the simulated data. Thus, comparison between the input simulateddata and the inversions reveal limitations associated with the presence of uncertainties, and notby the inversion scheme itself . We argue that this is actually an optimistic case, since a practicalanalysis of real observations generally uses blind inversion. The different existing DEM solvingalgorithms, whether they are based on forward or inverse methods include additional assumptionsto ensure uniqueness, such as the smoothness of the solution. Thus, the mathematical difficultiesinherent to solving the inverse problem generally introduce additional ambiguity on the results,while our method allows to separate the sources of error and to study the impact of uncertaintiesonly.In the following, the four parameters defining the simulated observations with a true ARDEM are denoted EM T , T Tp , σ T and α T respectively, whereas the associated inferred parametersresulting from the least-square minimization are noted EM I , T Ip , σ I and α I . It is useful to thinkof the coronal plasma parameters as the ”true” values, while the inverted one as the ”observed”values. To reduce the number of dimensions and for the sake of clarity, we choose to fix the EMof the simulated observations I obsb to a constant value EM TAR = 10 cm − , typical of ARs. Sincewe focus our attention in the ability to reconstruct the slope coolward of the peak of the DEM ( α parameter), we also fix the width of the high temperature wing σ in both our simulated observations I obsb and theoretical expectations I thb : only the EM, α and T p are solved for here. The width σ isfixed to the arbitrary constant value σ T = σ I = 0 . T e . We verified that the value of σ doesnot affect results on the slope. Thus, the probability matrices P (EM I , T Ip , σ I = 0 . , α I | EM T =EM TAR , T Tp , σ T = 0 . , α T ) are finally reduced to five dimensions. To illustrate the main propertiesof these large matrices, we display them by different combinations of fixed parameter values andsummation over axes.The probability maps resulting from such a simulation are displayed on Figure 2 for DEMscharacterized by a peak temperature of T Tp = 10 . K. The probabilities are presented whatever theEM I and the peak temperature T Ip by integrating them over EM I and T Ip , even though EM I and T Ip are of course solved for. This allows us to plot two-dimensional probability maps. Panel (a) of 12 –Figure 2 displays the conditional probability P ( α I | α T ) of finding a solution α I knowing the slope α T . Vertically cuts through panel (a) give probability profiles are shown in panels (b) and (c) forthe two specific values of α T = 3 and α T = 5 respectively.The main diagonal structure indicates that the solutions α I are linearly correlated with theinput α T . In P ( α I | α T ) in panel (a) of Figure 2, the spreading of the solutions around the diagonalimplies that a range of inferred results α I is consistent with the same true slope parameter α T ,given the level of uncertainties involved in this problem. We can also note that for steep slopes,the spreading of the solutions is greater. This is due to the fact that the emission is, in thesecases, dominated by higher temperatures, leading to a loss of low temperature lines, which furtherreduces the temperature range available to constrain the slope. Panels (b) and (c) show ranges ofpossible inferred solutions for the same true input parameter: considering α T = 3 (panel (b)), thedistribution of the solutions α I is peaked around 3, with more probable values in the 2 . − α I consistent with the input true slope α T = 5may be in the 2-6 interval with a quasi uniform distribution. If no additional independent a priori information is available, the results of inversion is thus highly uncertain.However, the computed probability map P ( α I | α T ) is not usable in a practical way, i.e. withDEM inversion of true observations. Indeed, since the systematics are in reality identical for allmeasurements, the output α I will be always biased in the same way. Ignoring to what extentthe theoretical intensities are over or under-estimated, we must take into account all the potentialinferred solutions. Therefore, in order to deduce the probability distribution of the true parameters α T consistent with a given inferred result α I we computed the inverse probability map P ( α T | α I )using Bayes’s theorem (see Section 2.2 of Paper I for more details). This quantity is therefore therelevant one for interpreting a given inferred result α I . Thus, using Bayes’ theorem as described inSection 2 and the total probability P ( α I ) displayed in panel (d), the inverse conditional probability P ( α T | α I ) shown in panel (e) can be computed. A horizontal cut through panel (e) give theprobability distribution of the true slope α T for a given observed slope α I . Panels (f) and (g) showexamples for α I = 5 and α I = 3. The lack of structure in the first case indicates that a large rangeof true slopes is consistent with the inferred results: 3 < α T <
6. In the second case, the mostlikely value of the true slope is similar to the observed slope of 3, but there is again a wide rangeof true slopes that are consistent with this observed slope.The probability distribution of panel (e) is very useful to assist the DEM inversion interpreta-tion: from this we can compute descriptive statistic quantities such as the standard deviation andthe mean of the probability distribution for a given α I , which give a quantitative representation ofthe reconstruction quality and uncertainty. From panel (f), we derived a mean value of α P = 4 . α I = 5. The standard deviation, evaluated to 0.87 in this case, characterizingthe dispersion of the results, is an estimation of the confidence level on the slope reconstruction.From this, a proper interpretation of the DEM inversion result can be derived, providing a final Defined as the probability for the solutions to be between α and α + ∆ α .
13 –result of α T = 4 . ± .
87, for a given inferred result of α I = 5. In panel (g), the mean valueis estimated to be α P = 3 .
39, whereas the inferred slope was α I = 3. The associated standarddeviation is 1.07, leading to a final result of α T = 3 . ± . T Tp = 10 . (top) and T Tp = 10 K (bottom). Compared to the previous case, theprobability distributions are clearly wider and less regular. Whatever the inferred result α I , theprobability distribution of the possible true solutions α T extends over the entire possible range.For T Tp = 10 . K, we found a typical standard deviation of 1.3-1.4, similar to the one computed forthe extreme low temperature peak of 10 K. For completeness, the probability maps for 63 peaktemperatures T Tp , from 10 . to 10 . K, and an animation showing the whole amplification of theperturbations are available on-line at ftp: // ftp.ias.u-psud.fr / cguennou / DEM EIS inversion / slope / slope / . This deterioration can be explained by the cumulative effects of the decreasing ofnumber of EIS lines and the smaller temperature range available to constrain the slope part of theDEM. In Figure 2, the DEM temperature peak is T Tp = 10 . K and thus, all 30 lines constrain theslope and the temperature range in which the slope is allow to vary covers 1 .
35 decades. Consideringthe case displayed on the top of Figure 3, where T Tp = 10 . K, this number of lines decreases to 26,whereas the temperature range decreases to about 1 decade. In the extreme case of T Tp = 10 K,only 8 lines constrain the DEM slope, while the temperature range is reduced to only 0.35 decades.The potential discrepancy between the true DEM ξ T and the inferred one ξ I is illustratedin Figure 4 and 5, by showing three different realizations of uncertainties (Fig. 4 -bottom- andFig. 5), as well as the perfect case (Fig. 4 -top-). The EM loci curves, formed by the set of (EM, T e ) pairs for which the isothermal theoretical intensities exactly match the observations for a givenspectral line (see Del Zanna & Mason 2003, for more details), are represented for each case asa function of both the element, given by the line type, and the relative intensity, given by thecolor from pale yellow (faintest) to dark red (strongest). In the case 1 of Figure 4, the loci curvesare perfectly aligned, and thus the estimated DEM ξ T perfectly match the initial true DEM ξ T .The case 2 (Figure 4) shows a realization of the perturbation n b and s b , each loci curves beingrandomly shifted from its original position. This corresponds to a deviation of the solution ξ I , theestimated temperature peak being underestimated from T Tp =4 MK to T Ip =2.8 MK and the slopeincreased to the steeper value of α I =3.4 while the initial true slope was α T = 2 .
0. Note that therelative intensity of each line plays a key role in the reconstruction: the more intense lines havemore important weight in the inversion process, even though we normalize the χ by the differentuncertainties sources, including the photon noise (see Equation 3). The cases 3 and 4 in Figure 5show another different realizations of errors leading in the case 3 to an overestimation of the totalEM, and in the case 4 to a significant deviation of the peak temperature T p .The reconstruction of the temperature peak is much better constrained than the slope. Figure 6displays the probability maps associated to the T p parameter, for a true shallow slope α T = 1 . TAR . Probabilities are now represented whatever the EM I
14 –and α I by integrating them over the EM I and α I axes. Results are very similar whatever thechosen input α T , and the probability maps presented here are typical . Most of the solutions arecondensed around the diagonal. The use of the thirty lines provides an unambiguous determinationof the peak temperature. However, the confidence interval remains quite large: we found a typicalstandard deviation between 0.7 and 0.85 MK associated to the spread of the solutions aroundthe diagonal for the different tested plasma slopes, with extreme values varying between 0.1 and1.3 MK.These results can finally be summarized in the two graphs of Figure 7. The first one, on theright, displays the mean slope value of the initial true α T , knowing the inferred result α I . On thetop, the map shows the slope mean value, represented as a function of both the peak tempera-ture T Tp and the inferred results α I . The quantity α T has been computed from the probabilitydistribution P ( α T | α I ), in the same way than described previously. The three different horizontalprofiles displayed on the bottom and denoted by the horizontal white lines on the top, correspondto the three different probability maps displayed on Figure 2 and Figure 3. Using these curves,it is possible to correctly interpret the results of the inferred α I , providing thus the slope meanvalue computed from the probability distribution of all true slopes consistent with a given inferredresults. The diagonal (black solid line) correspond to a perfect agreement between α T and α I .The bias of α T strongly affects the results for the low temperature profiles T Tp = 1 MK (red solidline) and T Tp = 3 . T Tp = 6 . P ( α I | α T ) previously presented, and taken into accountby computing the inverse probability maps P ( α I | α T ). For low temperature peaks, the correspond-ing probability distributions are very wide, almost covering the whole space of the solutions(seeFigure 3). Consequently, the slope mean value approaches a roughly constant value of α T = 3 . σ α T = 1 . − . T Tp = 10 . K, as expected in light of the above. For the hightemperature peak T Tp = 10 . K, confidence level extends between 0.3 and 1.15, depending on thevalue of the inferred slope α I .The summarized results regarding the second set of uncertainties used in this work and de-scribed in Section 2.3 is displayed in Figure 8. In this case, the atomic physics uncertainties aregreatly reduced from 20% to 10% for class 1 and from 30% to 10% for classes 2 through 4, whilethe calibration uncertainties are reduced from 25% to 20%. The resulting total uncertainty variesbetween 25-30% depending on the line. As expected, the reduced uncertainties lead to an improvedcorrelation between the estmated slope α I and the true one α T , particularly for medium temper-ature peak around 10 . K. As a result, the standard deviation is decreased, ranging now between The probability maps of the peak temperature for 101 values of α T ranging from 1 to 6 are available on-line atftp: // ftp.ias.u-psud.fr / cguennou / DEM EIS inversion / slope / temperature /
15 –0.2 and 0.8 for T Tp = 10 . K, 0.3 and 1.2 for T Tp = 10 . K, and approaching the same constantvalue as before, around σ α T = 1 .
4. Maps like these in Figures 7 and 8 are useful for interpretingthe DEM inversions from true observations: given the slope and the temperature of the peak, bothmean value and confidence level can be derived.The confidence levels derived in the present work can be used to evaluate the agreement betweentheoretical model predictions and DEM measurements. In the recent paper of Bradshaw et al.(2012), the authors carried out a series of low-frequency nanoflare simulations. They investigateda large number of heating and coronal loop properties, such as the magnitude and duration of thenanoflares and the length of the loop. They concluded that the low frequency heating mechanismcannot explain DEM slopes α ≥ .
6, similar to the findings of Mulu-Moore et al. (2011). Comparingtheir results to the current observations of AR cores (see Section 1 for corresponding references),they found that 36% of observed AR cores are consistent with low-frequency nanoflare heating ifuncertainties in the slope measurements are ignored. Using then the slope uncertainties estimatedaround ∆ α ± . ≤ α ≤ . α ±
1, they concluded that 86% to 100% of current AR coreobservations are consistent with such trains.The determination of the uncertainties associated with the atomic physic processes is no sim-ple matter, as discussed in Section 2.3, that is why we have tested two different sets of uncer-tainties. However, the most important issue here, considering the temperature peaks currentlyderived in observational analysis, is that whatever the set of uncertainties used to determine theconfidence level on the reconstructed slope, their typical values remain important relative to whatis necessary to strongly constrain the timescale of the coronal heating. Warren et al. (2012) andWinebarger et al. (2012) for example, derived temperature peak generally around log T e = 6 . T e = 6 . T e = 6 .
7. For these typical values, the slope uncertainties varies between ∆ α = ± . α > ± .
4. Summary and conclusions
The slope of the DEM distribution coolward of the coronal peak can potentially be used todiagnose the timescale of energy deposition in the solar corona. Indeed the DEM slope providesimportant information on the proportion of hot to warm material, which is useful to determine the 16 –heating timescale. Recent observational studies of AR cores suggest that some active region coresare consistent with low frequency heating mechanisms, where the plasma cools completely beforebeing reheated, while other show consistency with high frequency energy deposition, where rapidreheating causes the temperature to fluctuate about a particular value. Distinguishing betweenthese possibilities is important for identifying the physical mechanism of the heating. It is thereforecrucial to understand the uncertainties in measurements of observed DEM slopes.In this work, we presented an application of our recently developed technique in the specificcase of typical AR DEMs, in order to properly estimate confidence level of the observed DEM slopesand assist the DEM interpretation. Using a probabilistic approach and Monte-Carlo simulationsof uncertainties to interpret the DEM inversion, our method is useful for examining the robust-ness of the DEM inversion, and to analyze the DEM inversion properties. Comparing simulatedobservations of the
Hinode /EIS spectrometer with inferred results, the range or multiple ranges ofsolutions consistent with a given set of measurement can be estimated, along with their associatedprobabilities. From such probability distributions, statistical quantities can be derived, such as thestandard deviation, providing rigorous confidence levels on the DEM solutions.In this way, we carefully assess the errors in the DEM slopes determined from
Hinode /EISdata. Both random and systematic errors have been taken into account. We paid particularattention to the description of the systematic errors related to the atomic physics process andabundances. Uncertainties associated with ionization fractions, elemental abundances, FIP effectand a combination of uncertainties in the radiative and excitation rates have been simulated.Additional systematic errors have been added on the blended lines, to take into account the technicaldifficulties in isolating a single line intensity. We argue that our work actually provides an optimisticestimation of the slope confidence levels: the mathematical difficulties intrinsic to solving an inverseproblem introduce additional ambiguity, while our method allows to focus only on the impact ofintrinsic uncertainties. The fact that our inverted DEMs have the same functional form than thetrue ones, known a priori , means that our slope uncertainties are lower limits. In reality, the formof the true DEM is unknown, and this introduces additional uncertainty, through the use of blindinversion.In Section 3, we demonstrated how the slope reconstruction is affected by the uncertainties.The analysis of the probability maps provides the range of slopes consistent with the observedDEM slopes. These maps show that in most cases, a large range of solutions is consistent with themeasurements. The presence of uncertainties degrades the quality of the inversion, leading to typicalconfidence levels around 0.9-1.0. However, the inversion robustness, and thus the confidence level,largely depends on the number of lines constraining the slope. For DEMs with high temperaturepeaks [5-6 MK], about 20 lines contain suitable information, while low temperature peaks [1-3 MK]reduce this number to less than 10. For these latter cases, the effect of uncertainties leads to largerconfidence levels, about 1.3 and more in some cases.The slope confidence levels derived in the present work are useful for quantifying the degree 17 –of agreement between theoretical models and observations. Current slope reconstructions can thusbe properly compared to theoretical expectations. However, the typical derived confidence levelsremain significant, comparing to the majority of observed slopes values concentrated between 1.5and 5. The sizable confidence levels make it difficult to draw definitive conclusions about thesuitability of a given heating model, implying in one hand, that a model might be consistent withthe majority of observations or, in the other hand, with none at all (see Bradshaw et al. 2012, for apractical application of these confidence levels). When relaxing the constraint on the DEM slopesas Reep et al. (2013), the slope DEM diagnostic does not allow to distinguish between differentscenarios, because observations can thus be explained by a variety of different heating models.Our generic approach can be improved for specific datasets, and additional sophistication incan be incorporated (see Section 2.3). We could, for example, use a customized set of uncertaintiesfor a given set of lines. However, the main important point of our work, is that, even for uncer-tainties that would seem to be on the low end of what is feasible (our second set of uncertainties),the corresponding uncertainty in the measured slope may be too large to definitively exclude orcorroborate a given heating scenario in many cases. The methodology presented here can also beused to establish the optimal set of lines required to obtain the smallest possible confidence levels.Such kind of preliminary investigations can be very helpful to optimize the future instruments,whether it be spectrometer or broad band imagers, in order to maximize their DEM diagnosticcapabilities.S.P. acknowledges the support from the Belgian Federal Science Policy Office through theinternational cooperation programmes and the ESA-PRODEX programme and the support of theInstitut d’Astrophysique Spatiale (IAS). F.A. acknowledges the support of the Royal Observatoryof Belgium. The work of J.A.K. was supported by the NASA Supporting Research and TechnologyProgram. The authors would like to thank G. Del Zanna, H. Warren, G. Doschek, M. Laming,E. Landi, H. Mason, J. Schmelz and P. Young for fruitful discussions and comments about atomicphysic uncertainties. Discussions with H. Mason, H. Warren, and P. Testa at the second meetingof the Bradshaw/Mason International Space Science Institute Team were also very helpful.
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This preprint was prepared with the AAS L A TEX macros v5.2.
21 –Ions Wavelength (˚A) log( T [ K ]) Total uncertainty σ unc Mg V 276.579 5.45 61.03 %Mg VI 268.991 5.65 61.03 %Mg VI 270.391 5.65 61.03 %Mg VII b b b b Hinode /EIS spectral lines used in our simulations. Lines are sorted by elementsas a function of the peak temperature of the contribution functions. The blended lines are specifiedwith the index b . The fourth column indicate the percentage of total uncertainty applied to eachspectral lines, resulting of both systematic and random errors. 22 –
23 24 25 26 27 2810 D i f e r e n t i a l E m i ss i o n m e a s u r e [ c m - ] T e [K] •log T p = 6; σ = 0.1 •log T p = 6.8; α = 5 α = 1.0 α = 2.0 α = 3.0 α = 4.0 α = 5.0 σ = 0.05 σ = 0.10 σ = 0.20 σ = 0.30 σ = 0.40 Fig. 1.— Some examples of the parameterization of the AR DEM model (see Section 2.2). Thetotal emission measure is adjusted to the typical AR value of EM AR = 10 cm − . The left groupillustrates the slope variations, whereas the right group depicts variety of high temperature wingparameterizations. In the first case, the temperature of the coronal peak and the width of the hightemperature part are fixed to respectively T p = 10 K and σ = 0 . T e , while the slope of the fivedistinct parameterizations varies between 1 and 5. On the right, the peak temperature is increasedto T p = 10 . K and the slope is fixed to α = 5, while the σ parameter varies between 0.05 and 0.4. 23 – P( α I | α T ) (c)(b) (f)(g)(d)
1 2 3 4 5 6 α T α I
0 0.05 (a) T pT = 10 K P( α I | 5)
0 0.02 P( α I | 3)
0 0.02 P( α I )
1 2 3 4 5 6 P( α T | 5)
1 2 3 4 5 6 P( α T | 3) P( α T | α I )
1 2 3 4 5 6 α T α I
0 0.05 0.1 0.15 (e) T pT = 10 K Fig. 2.— Maps of probability for the DEM slope, considering an Active Region (AR) DEM (seeFigure 1), and achieved by 1000 Monte-Carlo realizations of the random and systematic errors n b and s b . In this case, the true DEM is characterized by constant emission measure EM TAR =10 cm − , a fixed high temperature wing of σ T = 0 . T e and a peak temperature of T Tp =10 . K; only the α T parameter is investigated here. (a) : Probability map P ( α I | α T ), verticallyreading. (b) and (c) : Probability profiles of α I for true parameter α T = 3 and 5 corresponding tovertical lines in panel(a). (d) : Total probability P ( α I ) to obtain α I whatever α T . (e) Vice-versa ,probability map P ( α T | α I ), horizontally reading, inferred by means of Bayes’ theorem. (f ) and (g) : Probability profiles of α T knowing that the inversion results are, from top to bottom, 5 and3. From these probability distributions, the slope mean and confidence level are estimated to be α T = 4 . ± .
87 for panel (f) and α T = 3 . ± .
07 for panel (g) (see text in Section 3 for details). 24 – P( α I | α T ) (b)
1 2 3 4 5 6 α T α I
0 0.05 (a) T pT = 10 K
0 0.02 P( α I ) P( α T | α I )
1 2 3 4 5 6 α T α I
0 0.05 0.1 (c) T pT = 10 K P( α I | α T ) (b)
1 2 3 4 5 6 α T α I
0 0.05 0.1 (a) T pT = 10 K
0 0.02 P( α I ) P( α T | α I )
1 2 3 4 5 6 α T α I
0 0.05 0.1 (c) T pT = 10 K Fig. 3.— Same as Figure 2, but with a true DEM characterized by peak temperatures of respectively T Tp = 10 . and T Tp = 10 K, from top to bottom. The decrease of number of lines constrainingassociated with the uncertainties clearly deteriorate the quality of the inversion, increasing theconfidence level to typical value of 1.3 (see also Figure 7). 25 – l o g E M [ c m - ] log T [K] EM loci curves ➫ ξ I = ξ T ❶ FeCaSSiMg ξ T (T e ) d log T e × Relative Intensity -4 -3 -2 -1 l o g E M [ c m - ] log T [K] EM loci curves ➫ ξ I ≠ ξ T ❷ α T = 2 α I = 3.4 FeCaSSiMg ξ T (T e ) d log T e × ξ I (T e ) d log T e × Relative Intensity -4 -3 -2 -1 Fig. 4.— Illustration of the potential discrepancy between the true DEM ξ T (blue solid line) and theestimated one ξ I (green solid line), due to the presence of both random and systematic errors. TheEM loci curves are represented as a function of the elements, sorted by line type, and as a functionof their relative intensity, sorted by color, from pale yellow (faintest) to dark red (strongest). Top: no uncertainty in this first case, thus the inferred DEM ξ I is equal to the initial one ξ T . Bottom:
Agiven realization of systematic and random errors, leading to discrepancy between true and inferredDEMs (see Fig 4). 26 – l o g E M [ c m - ] log T [K] EM loci curves ➫ ξ I ≠ ξ T ❸ α T = 2 α I = 2.15 FeCaSSiMg ξ T (T e ) d log T e × ξ I (T e ) d log T e × Relative Intensity -4 -3 -2 -1 l o g E M [ c m - ] log T [K] EM loci curves ➫ ξ I ≠ ξ T ❹ α T = 2 α I = 3.85 FeCaSSiMg ξ T (T e ) d log T e × ξ I (T e ) d log T e × Relative Intensity -4 -3 -2 -1 Fig. 5.— Same as Figure 4 but for two different realizations of systematic and random errors. Thebottom case illustrates an extreme case, leading to strong discrepancy between input and inferredDEMs. 27 –
P(T pI |T pT ) (b) T pT T pI
0 0.05 0.1 0.15 0.2 (a)
P(T pI ) P(T pT |T pI ) T pT T pI
0 0.05 0.1 0.15 0.2 (c)
Fig. 6.— Maps of probability for the peak temperature, represented for a simulated observationwith a true DEM slope of α T = 1 .
5. Results originate from same simulations framework thanFigure 2, showing that if the slope is strongly impacted by the presence of uncertainties, the peaktemperature is still well constrained, providing confidence levels between 0.7 and 0.85 MK. 28 –
Mean α T
1 2 3 4 5 T pT Mean α T α I T pT = 10 K T pT = 10 KT pT = 10 K Standard deviation σ α T
1 2 3 4 5 σ α T α I T pT = 10 K T pT = 10 KT pT = 10 K Fig. 7.— Mean and standard deviation of the true slopes α T consistent with a given inversionresult α I . Top : Mean (left) and standard deviation (right) maps represented as a function of thepeak temperature and the inversion result α I . Bottom : Cut across the mean (left) and standarddeviation (right), corresponding to the white horizontal lines. The peak temperature are fixed torespectively T Tp = 10 K (solid lines), T Tp = 10 . K (bold dashed lines) and T Tp = 10 . K (dashedlines), corresponding to the probability maps displayed in Figures 2 and 3. 29 –
Mean α T
1 2 3 4 5 T pT Mean α T α I T pT = 10 K T pT = 10 KT pT = 10 K Standard deviation σ α T