ON THE ACCURACY OF THE DIFFERENTIAL EMISSION MEASURE DIAGNOSTICS OF SOLAR PLASMAS. APPLICATION TO SDO/AIA. II. MULTITHERMAL PLASMAS

In Figure 2, the criterion C(EM, T_c) (Equation (3)) is plotted for different plasma configurations. It is represented in gray scale⁴ and its absolute minimum, corresponding to the solution ξ^I, is marked by a white plus sign. The criterion is a function of the two parameters EM and T_c defining the isothermal solutions (Equation (5)). The white curves are the EM loci curves, corresponding to the minimum of the contribution of each band to the total criterion. In all panels, the simulated plasmas have DEMs centered on the typical coronal temperature T^P_c = 1.5 × 10⁶ K, and a total emission measure EM^P = 2 × 10²⁹ cm⁻⁵, an active region value guaranteeing signal in all six bands (see Section 3 and Figure 3 in Paper I). The top left panel corresponds to an isothermal plasma (σ = 0, which corresponds to the case presented in Section 3.2 of Paper I). Because of systematics, s_b, and instrumental noises, n_b, and thus the over- or underestimation of the observed and theoretical intensities I^obs_b and I^th_b, the EM loci curves are shifted up and down along the EM axis, with respect to the zero-uncertainty case (i.e., n_b and s_b = 0). Therefore, the curves never intersect all six at a single position, thus giving χ² > 0. However, multithermality has the same effect: the criterion presented in the top right corresponds to a slightly multithermal plasma having a Gaussian DEM width of σ^P = 0.1 log T_e. Even though in this case uncertainties have not been added, the loci curves and the absolute minimum, corresponding to the location where the curves are the closest together, are also shifted relative to each other along the EM axis and there is no single crossing point. Thus, errors and multithermality can both produce comparable deviations of the observations from the ideal isothermal case, hence the fundamental ambiguity between the two. The two criteria of the bottom panels have been obtained with the same plasma Gaussian DEM distribution, but now in the presence of random perturbations. These two independent realizations of the uncertainties illustrate an example of the resulting dispersion of the solutions.

Zoom In Zoom Out Reset image size

In Figure 3, we increased the DEM width to σ^P = 0.3 log T_e (top row) and σ^P = 0.7 log T_e (bottom row) while keeping the same EM and central temperature. For each width, the left and right panels show two realizations of the uncertainties. The corresponding configuration of the loci curves and thus the value and position of the absolute minimum can greatly vary, even though the plasma is identical in both cases. It also appears that privileged temperature intervals exist where the solutions tend to concentrate. This phenomenon is intrinsically due to the shape of the EM loci curves. In the next subsection, we characterize the respective probabilities of occurrence of the various solutions for a wide range of plasma conditions. From the examples presented in Figure 3, we also note that the vertical spread of the loci curves tends to be larger for wider DEMs, which corresponds to larger χ² residuals. In Section 2.3, we will determine up to what DEM width the observations can be considered to be consistent with the isothermal hypothesis.

Zoom In Zoom Out Reset image size

We consider Gaussian DEM plasmas and scan all possible combinations of central temperatures and widths used to pre-compute the reference theoretical intensities. The total EM is kept constant at EM^P = 2 × 10²⁹ cm⁻⁵. For each combination of plasma parameters, 5000 Monte Carlo realizations of the random perturbations n_b and systematics s_b are obtained, leading to 5000 estimates of the observed intensities I^obs_b(ξ^P) in each band b. The isothermal least-square inversion of these 5000 sets of AIA simulated observations provides as many solutions ξ^I, allowing an estimation of P(ξ^I|ξ^P).

Figures 4–6 show the resulting temperature probability maps for σ^P = 0.1, 0.3, and 0.7 log T_e, respectively.⁵ The isothermal plasma case (σ^P = 0) is shown in Figure 6 of Paper I. In all figures, the probability P(T^I_c|T_c^P) is obtained by vertically reading panels (a), given the probability of finding a solution T^I_c knowing the plasma central temperature T^P_c. The probability profiles for two specific plasma temperatures, 1.5 × 10⁶ and 7 × 10⁶ K, are shown in panels (b) and (c). Using Baye's theorem, the probability maps P(T^P_c|T_c^I) of panels (e) are obtained by normalizing P(T^I_c|T_c^P) to P(T^I_c), the probability of obtaining T^I_c whatever T^P_c (panels (d)). In the bottom right panels (f) and (g), we show two example horizontal profiles giving the probability that the plasma has a central temperature T^P_c knowing the inverted temperature T^I_c.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In panels (a) and (e) of Figure 4, the most probable solutions are located around the diagonal. However, compared to the case of an isothermal plasma (see Figure 6 of Paper I), the distribution is wider, irregular, and deviations from the diagonal greater than its local width are present. As shown by panel (d) and the nodosities in the map (a), the unconditional probability of obtaining a result T^I_c is nonuniform, meaning that some inverted temperatures are privileged whereas others are unlikely. Compared with Figure 6 of Paper I, profile (b) shows that the probability of secondary solutions at T^P_c = 1.5 × 10⁶ K is increased with respect to the isothermal case. The apparition of these two solutions is illustrated in the bottom row of Figure 2. The bottom right panel corresponds to a realization of the errors yielding a solution close to the diagonal, while the bottom left panel of the same figure illustrates a case where the absolute minimum of the criterion is located at low temperature. Using the map of P(T^P_c|T_c^I), it is, however, possible to correctly interpret the low-temperature solutions as also compatible with 1.5 × 10⁶ K plasma (profile (g)).

In Figure 5, the plasma DEM width is increased to σ^P = 0.3 log T_e. As a result, the above-described perturbations with respect to the isothermal plasma case are amplified. The diagonal structure has almost disappeared, with discontinuities and reinforced and more diffuse nodosities. Multiple solutions of comparable probabilities are present over large ranges of plasma temperatures and consequently, the estimated T^I_c can be very different from the input T^P_c. For example, panel (c) shows that for a 7 × 10⁶ K plasma, the most probable T^I_c is either 1.6 × 10⁵ or 3 × 10⁵ K. The unconditional probability P(T^I_c) of panel (d) is very nonuniform, some ranges of estimated temperatures being totally unlikely (e.g., T^I_c = [1.5 × 10⁶, 4 × 10⁶] K) while others are probable for large intervals of T^P_c (e.g., T^I_c = 3 × 10⁵ K or 10⁶ K). However, despite the jaggedness of P(T^I_c|T_c^P), the map of P(T^P_c|T_c^I) can once again help to properly interpret the result of the inversion. For example, profile (g) shows that for T^I_c = 1.5 × 10⁶ K, the distribution of T^P_c is distributed around T^P_c = 10⁷ K, which is exactly the plasma temperature that can yield an inversion at T^I_c = 1.5 × 10⁶ K (see panel (a)). Panel (f), providing the probability distribution T^P_c knowing that the inversion result is T^I_c = 7 × 10⁶ K, exhibits a broad probability distribution around T^P_c = 1.5 × 10⁷, showing that the plasma temperature thus deduced is very uncertain. This is to be compared to the 0.05 log T^P_c temperature resolution of the isothermal case (see Section 3.2 of Paper I).

As the DEM becomes even larger, the impact on the robustness of the inversion becomes greater. At σ^P = 0.7 log T_e, the probability map P(T^I_c|T_c^P) of Figure 6(a) and the corresponding probability P(T^I_c) clearly show two privileged solutions: T^I_c = 10⁶ and 3 × 10⁵ K. The estimated isothermal temperatures are always the same for any T^P_c, as illustrated by panels (b) and (c). Therefore, the inversion results T^I_c contain no information on the plasma central temperature T^P_c. This is illustrated by the lack of structure in the probability map P(T^P_c|T_c^I) of panel (e). Profile (g) shows that for T^I_c = 1.5 × 10⁶ K, the distribution of T^P_c extends over entire the temperature range.

We have shown that as the width of the plasma DEM increases, multiple solutions to the isothermal inversion appear. This phenomenon has been already mentioned by Patsourakos & Klimchuk (2007), using triple-filter TRACE data. After a proper treatment of the uncertainties, the authors found that their observations of coronal loops were consistent with both a high (≈1.5 × 10⁶ K) and a low (≈5 × 10⁵ K) isothermal plasma temperature. They correctly concluded that without a priori knowledge of the physical conditions in these loops, they could not rule out the cool plasma solutions. Even though we used six bands, multiple solutions appear anyway with increasing plasma width. In addition, as we have seen in Paper I, multiple solutions can exist even with an isothermal plasma if only a limited number of bands is available. This is another illustration of the similar effects of errors and multithermality.

The isothermal temperature solutions become progressively decorrelated from the plasma central temperature as the width of the DEM increases. For very large DEMs (Figure 6), the inversion process yields exclusively either 3 × 10⁵ K or 10⁶ K whatever the plasma T^P_c. These two temperatures correspond to the preferential locations of the minima shown in the criteria of Figure 3. This is a generalization of the phenomenon analyzed by Weber et al. (2005) in the simpler case of the TRACE 19.5 over 17.3 nm filter ratio. The authors showed that in the limit of an infinitely broad DEM, the band ratio tends to a unique value equal to the ratio of the integrals of the temperature response functions. Furthermore, they showed that as the width of the DEM increases, the temperature obtained from the band ratio becomes decorrelated from the DEM central temperature. We have found a similar behavior in the more complex situation of six bands. This is not, however, an intrinsic limitation of AIA. We can predict that the same phenomenon will occur with any number of bands or spectral lines. Indeed, for an infinitely broad DEM, since the observed intensities are equal to the product of the total EM by the integral of the response functions (Equation (2)), they are independent from the plasma temperature. Therefore, the inversion will yield identical results for any plasma temperature T^P_c, whatever the number of bands or spectral lines.

2.3.1. Defining Isothermality

As already noted in Section 2.1 and in Figure 2, larger DEM widths correspond to larger squared residuals. From Paper I, the distribution of residuals to be expected for an isothermal plasma is known. Examining, then, the residuals for the solutions given in the probability maps of Section 2.2, the solutions may not all be statistically consistent with the isothermal hypothesis. We will thus analyze the distribution of residuals to define rigorously a test of the adequacy of the isothermal model used to interpret the data.

Because both the random and systematic errors n_b and s_b have been modeled by a Gaussian random variable, if the DEM model used to interpret the data can represent the plasma DEM, the residuals are equal to the sum of the square of six normal random variables (see Equation (3)). Since we adjust the two parameters EM and T_c, the residuals should thus behave as a four-degree χ² distribution.

Figure 7 shows the distribution of the squared residuals for all of the 80 DEM widths considered in the simulations. The shades of gray in the top panel correspond to the probability of obtaining a given χ² value (abscissa) as a function of the width of the plasma DEM (ordinate). In the bottom panel, four profiles give the distribution of squared residuals obtained for an isothermal plasma (thin dotted line) and for the three DEM widths discussed in Section 2.2: σ^P = 0.1 (thick dotted line), 0.3 (thick dashed curve), and 0.7 log T_e (thick dash-dotted curve), corresponding to the white horizontal lines on the top panel. The theoretical χ² distributions of three (thin solid curve) and four degrees (dashed curve) are also plotted.

Zoom In Zoom Out Reset image size

If the plasma is isothermal, the distribution of the residuals is slightly shifted toward a three-degree χ² instead of the expected four-degree one. In Paper I, we interpreted this as a correlation between the six AIA coronal bands. The distributions of residuals progressively depart from the isothermal case as the DEM width of the simulated plasma increases. The distributions become broader and their peaks are shifted toward higher values, forming the diagonal structure in the top panel of Figure 7. This behavior stops around σ^P = 0.4 log T_e. Above, the peaks of the distributions are shifted back toward smaller values and remain constant. As the DEM becomes wider, the simulated observations become independent from the plasma parameters, and all inversions tend to give the same solution and the same residuals.

These distributions of residuals provide a reference against which to test the pertinence of the isothermal model. The isothermal hypothesis can, for example, be invalidated for the solutions biased toward T^I_c ≈ 1 MK and corresponding to very multithermal plasmas (e.g., σ^P = 0.7; see Section 2.2 and Figure 6). Indeed, for a given inversion and its corresponding residual, the top panel of Figure 7 gives the most probable width of the plasma, assuming it has a Gaussian DEM. Let us assume that an isothermal inversion returns a residual equal to 5. Analyzing the histogram corresponding to the bottom row of the top panel of Figure 7 (σ^P = 0), we can show that an isothermal plasma has a 68.2% chance of yielding χ² ⩽ 5. This residual can therefore be considered consistent with an isothermal plasma. But reading the plot vertically, we see that the probability P(χ² = 5) is greater for multithermal plasmas and peaks around σ^P = 0.12, which is thus in this case the most probable Gaussian width. For larger residuals, the situation is more complex because several plasma widths can have equally high probabilities. Past χ² = 3.5 the plasma has a higher probability to be Gaussian than isothermal and a Gaussian inversion is required to properly determine its most probable width and central temperature.

Figure 7 is a generalization of the results obtained by Landi & Klimchuk (2010). It is equivalent to their Figure 2, identifying our χ² with their criterion F_min and their solid line to the maximum of our χ² distribution as a function of σ^P. Their dashed lines are equivalent to our values of the half-peak, as a function of σ^P, given by the resolution of χ²(σ^P) = χ²_max(σ^P)/2. Their Figure 2 was computed using 13 individual spectral lines for a 1 MK plasma and extends only up to σ^P ≈ 0.2, while our Figure 7 was computed for the six AIA bands over a wide range of central temperatures and for widths up to σ^P = 0.8. Despite these differences, the two figures exhibit the same global behavior. Indeed, for any number of spectral lines or bands, the residuals of the isothermal inversion tend to increase as the width of the plasma DEM increases.

Investigating multithermal Gaussian solutions now, the least-square criterion given by Equation (3) thus has three dimensions C(ξ^gau) = C(EM, T_c, σ). This three-dimensional parameter space is systematically scanned to locate the theoretical intensities best describing the simulated observations. Figure 8 shows this criterion for three cases illustrating different degrees of multithermality. In each row, from top to bottom, the simulated plasma has a DEM width σ^P of 0.1, 0.3, and 0.7 log T_e, respectively, centered on the temperature T_c = 1.5 × 10⁶ K. On the left panels, the background image represents C(EM, T_c, σ^I), the cut across the criterion in the plane perpendicular to the DEM width axis at the width σ^I corresponding to the absolute minimum (white plus sign). The curves are the equivalent of the loci EM curves in a multithermal regime: for each band b and for a given DEM width σ^P, they represent the loci of the pairs (EM, T_c) for which the theoretical intensities I^th_b are equal to the observations I^obs_b. As σ varies, they thus describe a loci surface in the three-dimensional criterion. The difference with the isothermal loci curves is that the theoretical intensities I^th_b have been computed for the multithermal case (i.e., considering a Gaussian DEM), and thus the parameter σ must be now considered. The right panels of Figure 8 display C(EM^I, T_c, σ), the cuts across the criterion in the plane perpendicular to the EM axis at the EM^I corresponding to the absolute minimum of the criterion (also represented by a white plus sign).

Zoom In Zoom Out Reset image size

The impact of multithermality on the criterion topology is clearly visible in the loci curves, inducing a smoothing of the curves as the multithermality degree increases. Indeed, the intensities I⁰_b as a function of the central temperature can be expressed as the convolution product between the instrument response functions and the DEM (see Section 2.3.2 of Paper I). Therefore, the reference intensities I⁰_b(EM, T_c, σ) computed for multithermal plasmas are equal to the isothermal ones smoothed along the T_c axis. As a result, the criterion exhibits a smoother topology and the minimum areas become broader and smoother as the multithermality degree increases, introducing more indetermination in the location of the absolute minimum. For each row, the realization of the uncertainties n_b and s_b yields a solution that is close to the simulation input. The cuts are therefore made at similar locations in the criterion in order to best illustrate the modification of its topology. We will now analyze to what extent the distortion of the criterion affects the robustness of the inversion.

In the case of DEM models defined by three parameters, and since the plasma EM^P is fixed, the probability matrices P(EM^I, T^I_c, σ^I|EM^P, T_c^P, σ^P) resulting from the Monte Carlo simulations have five dimensions. In order to illustrate the main properties of these large matrices, we rely on combinations of fixed parameter values and summation over axes.

The associated probability maps are displayed in Figure 9 for simulated plasmas characterized by σ^P = 0.1 (top panel) and σ^P = 0.7 log T_e (bottom panel).⁶ The probability maps are represented whatever the EM and DEM width obtained by inversion, i.e., the probabilities P(EM^I, T^I_c, σ^I|EM^P, T_c^P, σ^P) are integrated over EM^I and σ^I. In case of a narrow DEM distribution (σ^P = 0.1, top panels), the solutions are mainly distributed along the diagonal, with some secondary solutions at low probabilities. The accuracy of the determination of the central temperature T^P_c is improved compared to the isothermal inversion of the same plasma as shown in Figure 4: the diagonal is more regular and P(T^I_c) is more uniform. However, increasing the width of the DEM of the simulated plasma reduces the robustness of the inversion process. In the bottom panels (σ^P = 0.7), we observe a distortion and an important spread of the diagonal. In particular, the horizontal structure in panel (a) and the consequent peak of P(T^I_c) show that the estimated temperature is biased toward T^I_c ∼ 1 MK, especially for plasma temperatures T^P_c < 2 MK. This is to be compared with the privileged isothermal solutions in Figure 6, and the same reasoning applies. For very broad DEMs, because of the smoothing of the I⁰_b(EM, T_c, σ) along the T_c axis, the observed intensities are only weakly dependent on the central temperature, and thus all inversions tend to yield identical results. As a consequence, the probability map P(T^P_c|T_c^I) shows that over the whole temperature range considered there is a large uncertainty in the determination of T^P_c. The smoothing of the I⁰_b(EM, T_c, σ) is due to the width of the plasma DEM and not to the properties of the response functions. It is therefore to be expected that the uncertainty in the determination of T^P_c persists even if using individual spectral lines.

Zoom In Zoom Out Reset image size

In the same way, the conditional probabilities P(σ^I|σ^P) and P(σ^P|σ^I) of the DEM width are displayed on Figure 10. The probabilities are represented whatever the estimated temperature T^I_c and emission measure EM^I, for a simulated plasma having a DEM centered on T^P_c = 1 MK. The conditional probability P(σ^I|σ^P) on panel (a) exhibits a diagonal that becomes very wide for large σ^P and from which another broad branch bifurcates at σ^I = 0.15 log T_e. This branch is the analog of those observed on the temperature axis (top panels of Figure 9). For plasmas having a DEM width greater than σ^P = 0.3 log T_e, the estimated width σ^I is decorrelated from the input σ^P. Profiles (b) and (c) provide the conditional probabilities of σ^I for plasma DEM widths σ^P = 0.2 and 0.6 log T_e, respectively. As shown by panel (d), the unconditional probability P(σ^I) of obtaining σ^I whatever T^I_c for this 10⁶ K plasma is biased toward σ^I = 0.12 log T_e. Exploring the full probability matrix P(EM^I, T^I_c, σ^I|EM^P, T_c^P, σ^P), we discovered that if the plasma DEM is broad, the small width solutions (≈0.12 log T_e) are the ones that were also biased toward a central temperature of 10⁶ K in Figure 6. This is caused by the shape of the temperature response functions R_b(T_e). A minimum is formed in the criterion C(T_c, σ) around (T_c = 1 MK, σ = 0.12 log T_e) that tends to be deeper than the other local minima. Since for a very broad DEM plasma the simulated observed intensities I^obs_b are almost always similar, most of the random realizations of the perturbation s_b and n_b lead to an absolute minimum located in this deeper area of the criterion and thus, to the formation of the narrow solutions branch in P(σ^I|σ^P) panel (a). Therefore, very multithermal plasmas tend to be systematically measured as near isothermal and centered on 10⁶ K.

Zoom In Zoom Out Reset image size

Reading horizontally in panel (e) the inverse conditional probability P(σ^P|σ^I), a large range of DEM widths σ^P is consistent with the estimation σ^I. Both plots (f) and (g), representing two cuts at σ^I = 0.2 and 0.6 log T_c, can be used to deduce the most probable plasma DEM with σ^P. However, at σ^I = 0.6 log T_e, almost all plasma widths have significant probabilities, considerably restricting the possibility of inferring relevant DEM properties. The situation improves for smaller widths, as shown in panel (g), even though the probability distribution is still broad. For narrow DEM distributions (σ^P < 0.2), the width of the distribution decreases to about 0.15 log T_e, which thus represents the width resolution limit.

The analysis of the probability maps demonstrates that the robustness of the inversion is substantially affected by the degree of multithermality of the observed plasma. Furthermore, as we already noted, the simulations presented have been made in a favorable configuration where a significant signal is present in all six bands. For narrow plasma DEMs (σ^P < 0.15 log T_e), the six AIA coronal bands enable an unambiguous reconstruction of the DEM parameters within the uncertainties. The precision of the temperature and DEM width reconstruction is then given by the widths of the diagonals in Figures 9 and 10. For the temperature, that width is consistent with the resolution of [0.1, 0.2] log T_e given by Judge (2010). However, both the accuracy and the precision of the inversion decrease as the multithermality degree of the simulated plasma increases, a wider range of solutions becoming consistent with the observations. In the case of a very a large DEM plasma, the solutions are skewed toward narrow 1 MK Gaussians. The isothermal solutions biased toward 1 MK (Section 2.3) could be invalidated on the basis of their correspondingly high residuals (Section 2.3.1). But the biased Gaussian solutions are by definition fully consistent with a Gaussian plasma. This result generalizes that of Weber et al. (2005), and since our method ensures that the absolute minimum of the criterion is found, it is a fundamental limitation and not an artifact of the minimization scheme. Instead of being evidence for underlying physical processes, the recurrence of common plasma properties derived from DEM analyses may be due to biases in the inversion processes. However, using the type of statistical analysis presented here, it is possible to identify theses biases and correct for them.

The distribution of the sum of squared residuals is displayed in the top panel of Figure 11 as a function of the plasma DEM width, σ^P, of the simulated plasma. The three white horizontal lines represent the locations of the cuts at σ^P = 0.1 (dotted line), 0.3 (dashed line), and 0.7 log T_e (dotted dashed line) shown in the bottom panel. Unlike for the isothermal solutions (Figure 3), the distributions are similar for all plasma widths and resemble a three-degree χ² distribution (thin solid line). The most probable value is about 1 and 95% of the residuals are between 0 and 10. Any DEM inversion yielding a χ² smaller than 10 can thus be considered consistent with the working hypothesis of a Gaussian DEM plasma. It does not mean, however, that a Gaussian is the only possible model, but that it is a model consistent with the observations. Since we perform a least-square fit of the six values I^obs_b − I_b^th by three parameters (EM, T_c, σ), this behavior is expected. The small correlation between the residuals observed in the isothermal case disappears with increasing DEM width. This can be explained by the smoothing of the criterion (see Figure 8) that reduces the directionality of its minima.

Zoom In Zoom Out Reset image size

For model testing, a reduced chi-squared χ²_red = χ²/n, where n is the number of degree of freedom, is sometimes preferred over a regular χ² test because it has the advantage of being normalized to the model complexity. Usually, n is the number of observations minus the number of fitted parameters. This assumes that the measurements are independent, which is what we wanted to test. In practice, a small correlation between the residuals is found in the isothermal case, but the effect is small and a reduced χ² can be used. It should be noted that the residuals have a nonnegligible probability to be greater than the peak of the corresponding χ² distribution. For plasmas having Gaussian DEMs, for example, the residuals have about a 43% chance of being greater than 3 (Figure 11). This implies that seemingly large residuals are not necessarily proof of the inadequacy of the chosen DEM model. The working hypothesis can only be invalidated if it can be shown that another model has a greater probability of explaining the obtained residuals. This was, for example, the case in Section 2.3.1 where we have shown that if an isothermal inversion gives a squared residual greater than 3, the plasma DEM is more probably Gaussian than isothermal. We now explore a similar situation for Gaussian inversions.

Indeed, in reality the DEM shape is not known and it is interesting to test if a Gaussian is a pertinent model or if AIA has the capability to discriminate between different models. For this purpose, still considering Gaussian DEM solutions, the simulations of the observations are now performed using a top-hat DEM distribution defined as

$$\begin{eqnarray} \xi _{{\rm hat}}(T_e) = & {\rm EM}\ \Pi _{T_c, \sigma }(T_e),\nonumber\\ \mbox{with } \Pi _{T_c, \sigma }(T_e) = &\left\lbrace \begin{array}{@{}l@{\quad }l@{}}\frac{1}{\sigma } & {\rm if }\ |\log (T_e)-\log (T_c)| < \frac{\sigma }{2}\\ \ms 0 & {\rm else} \end{array}\right.. \end{eqnarray} \tag{ 6 }$$

Like a Gaussian, this parameterization can represent narrow and wide thermal structures.

Figure 12 gives the associated temperature probability maps. The top panels correspond to simulated plasmas with a top-hat distribution of width σ^P = 0.1. Most of the solutions are concentrated around the diagonal, even though the robustness is somewhat affected for temperatures in the range 5 × 10⁵ < T^P_c < 10⁶ K, where low probability secondary solutions exist. These two plots are very similar to the top panels of Figure 9, which give the probability of the Gaussian solutions for a Gaussian plasma. Therefore, even though the DEM assumed for the inversion is different from that of the plasma, the central temperature of the top hat is determined unambiguously.

Zoom In Zoom Out Reset image size

The picture is different for wide plasma DEM distributions. In the bottom panels of Figure 12, with σ^P = 0.7, the diagonal has become very wide, structured and does not cross the origin any more. Some plateaus appear, meaning that for some ranges of plasma temperature T^P_c, the temperature T^I_c provided by the Gaussian solutions will be invariably the same. For 5 × 10⁵ < T^P_c < 3 × 10⁶ K, for example, a constant solution T^I_c = 1 MK appears, as shown also by P(T^I_c) (bottom panel (b)). As a result, the inverse conditional probability map P(T^P_c|T_c^I) (bottom panel (c)) indicates that for an inversion output of 1 MK there is a large indetermination on the central temperature. The behavior of the solutions is thus globally equivalent to that described in Section 3.2 for the inversion of Gaussian DEM plasmas.

The observed distribution of the sum of the squared residual is very similar to those obtained with consistent Gaussian DEM models and is close to a three-degree χ² (see Figure 8). If the working hypothesis were a top-hat DEM, thus consistent with the plasma, the residuals would also be close to a three-degree χ², as for any DEM model described by three parameters (see discussion above). The solar plasma has no reason to actually have a top-hat DEM. However, this numerical experiments shows that, using AIA data only and the χ² test, the discrimination between two very different multithermal DEMs is practically impossible.