SPECTRAL INVERSION OF THE Hα LINE FOR A PLASMA FEATURE IN THE UPPER CHROMOSPHERE OF THE QUIET SUN

We here review the formulation of the BCM because it is the basis of the ECM we propose. The absorption profile is assumed to be uniform inside the feature and to be subject to Doppler broadening only. Then the optical thickness profile is given by

$$\begin{equation} \tau _{\lambda } = \tau _{0} \, \exp \left[ - \left({\lambda - \lambda _1 \over W } \right)^2 \right], \end{equation} \tag{ 2 }$$

where τ₀ is the peak absorption optical thickness, λ₁ is the peak absorption wavelength of the profile related to the line-of-sight velocity v and the rest wavelength λ₀ via

$$\begin{equation} v ={\lambda _1- \lambda _0 \over \lambda _0}c \end{equation} \tag{ 3 }$$

and W is the Doppler width

$$\begin{equation} W={\lambda _0 \over c} \sqrt{ {2kT \over M} + \xi ^2} \, \end{equation} \tag{ 4 }$$

that is determined by the temperature T, the speed of non-thermal motion or microturbulence ξ, and the atomic mass of the line-emitting element M. The source function S is assumed to be constant over wavelength and position inside the feature. The constancy of S over wavelength is a general property of lines formed under the complete frequency redistribution. The observed intensity profile of the line emergent out of the feature I_{λ, obs} is then determined by the intensity of light incident on the feature from the opposite side I_{λ, in} as well as S and τ_λ

$$\begin{equation} I_{\lambda, \rm obs} = I_{\lambda, \rm in} \exp (-\tau _\lambda) + S [ 1-\exp (-\tau _\lambda)]. \end{equation} \tag{ 5 }$$

With this solution, the contrast profile defined by

$$\begin{equation} C_\lambda \equiv {I_{\lambda, \rm obs} - I_{\lambda, \rm in} \over I_{\lambda, \rm in}} \end{equation} \tag{ 6 }$$

is modeled like

$$\begin{equation} C_{\lambda }= \left({ S \over I_{\lambda, \rm in}} -1 \right) [ 1- \exp (- \tau _\lambda)]. \end{equation} \tag{ 7 }$$

This BCM has four free parameters: S, τ₀, λ₁, and W.

The remaining question is how to reasonably specify I_{λ, in}. As mentioned above, the BCM usually adopts the assumption

$$\begin{equation} I_{\lambda, \rm in} = R_{\lambda, \rm obs} \, \end{equation} \tag{ 8 }$$

as illustrated in Figure 2, which leads to the equality $C_{\lambda, \rm obs} = D_{\lambda, \rm obs}$. Even though not explicitly made, this assumption is a crucial component of the model, being applicable to features lying high above the chromospheric forest. The success of the cloud model often depends on the plausibility of this assumption.

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Figure 3 shows the fitting of the BCM to the Hα spectral data of the four features, respectively. Note that the specific method of model fitting is described in Section 3.3. This model satisfactorily fits the Hα profile of feature 1 that appear in absorption at all the wavelengths against the reference, but fails to fit those of the other features that appear in emission at some wavelengths. The model profile looks similar to the observed profile of feature 2 but is not satisfactory at all. The fitting of features 3 and 4 are even worse.

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This failure originates from the fact that the BCM rules out emission at wavelengths away from the center (Grossmann-Doerth & von Uexküll 1971). This can be easily understood with the help of Equation (7). The contrast C_λ becomes positive when S exceeds R_{λ, in}. In the upper chromosphere, however, S is usually lower than the minimum value of I_{λ, in} at the line center. Therefore, C_λ turns out to be usually negative. S can be larger than the minimum value of I_{λ, in} in some cases but still remains lower than the values of I_{λ, in} at wavelengths off the line center. This means that if there is any emission, it is likely to appear near the line center only, and not at off-center wavelengths.

We have realized that the failure of the BCM should be attributed to the inadequacy of the assumption Equation (8). As already noted, this assumption easily breaks down when the feature of interest does not lie high above the reference features, especially when the feature of interest is embedded in the chromospheric forest (see Figure 2). Hence we propose to replace the assumption by a new one which holds in more general,

$$\begin{equation} I_{\lambda, \rm in} = R_{\lambda, \rm in}, \end{equation} \tag{ 9 }$$

where R_{λ, in} is the average intensity of light at the same atmospheric height as I_{λ, in}. Note that I_{λ, obs} and R_{λ, obs} are observable, but I_{λ, in} and R_{λ, in} are not.

To relate the non-observable R_{λ, in} to the observed reference profile R_{λ, obs}, we formulate the radiative transfer for the ensemble average light in the following way. We first consider the radiative transfer along each ray, indexed by k, in the reference ensemble as expressed in

$$\begin{equation} I_{\lambda, \rm obs}^k = I_{\lambda, \rm in}^k \exp \left(-\tau _\lambda ^k\right) + S^k \left[ 1-\exp \left(-\tau _\lambda ^k\right) \right] \end{equation} \tag{ 10 }$$

and

$$\begin{equation} \tau _{\lambda }^k = \tau _{0}^k \, \exp \left(- \left({\lambda - \lambda _1^k \over W^k } \right)^2 \right) \, \end{equation} \tag{ 11 }$$

that have the same form as Equations (5) and (2). Introducing the ensemble-average quantities:

$$\begin{eqnarray} &&R_{\lambda, \rm obs} \equiv \left\langle I_{\lambda, \rm obs}^k \right\rangle \,, \quad R_{\lambda, \rm in} \equiv \left\langle I_{\lambda, \rm in}^k \right\rangle \,, \quad s \equiv \langle S^k \rangle \, \end{eqnarray} \tag{ 12 }$$

and defining t_λ as in

$$\begin{equation} \exp (-t_\lambda) \equiv \frac{ \left\langle \left(I_{\lambda, \rm in}^k - S^k\right) \exp \left(-\tau _\lambda ^k\right) \right\rangle }{ \left\langle I_{\lambda, \rm in}^k - S^k \right\rangle }, \end{equation} \tag{ 13 }$$

we can reduce the ensemble average of Equation (10) to the form

$$\begin{equation} R_{\lambda, \rm obs} = R_{\lambda, \rm in} \exp (- {t}_{\lambda }) + {s} [ 1-\exp (-{t}_{\lambda }) ]. \end{equation} \tag{ 14 }$$

For the tractability of the model, we adopt the assumption that t_λ depends on λ in the same form as Equation (2)

$$\begin{equation} t_{\lambda } = t_{0} \, \exp \left[- \left({\lambda - \lambda _2 \over w} \right)^2 \right] \, \end{equation} \tag{ 15 }$$

with the three new parameters: t₀, λ₂, and w.

Note that the observed contrast profile D_λ defined in Equation (1) is now distinct from C_λ defined in Equation (6) because $I_{\lambda,\rm in} \ne R_{\lambda, \rm obs}$ any longer. Combining Equations (1), (5), (9), and (14), we obtain the ECM equation of the observed contrast profile

$$\begin{eqnarray} &&D_{\lambda }= \left({s \over R_{\lambda, \rm obs}} - 1 \right) \left[ 1- \exp (-\Delta \tau _\lambda) \right] + \left[ 1 - \exp (-\tau _\lambda) \right]{ \Delta S \over R_{\lambda, \rm obs}}\nonumber \\ \end{eqnarray} \tag{ 16 }$$

with the abbreviations Δτ_λ ≡ τ_λ − t_λ and ΔS ≡ S − s. This equation indicates that the non-zero contrast of a spectral feature is attributed to either Δτ_λ or ΔS. Note that positive Δτ_λ leads to negative contrast while positive value ΔS, to positive one. Note that in the limit t_λ → 0, the ECM is reduced to the BCM.

To determine the parameters, we use the constrained fitting of the model (Chae et al. 1998) that minimizes the functional

$$\begin{equation} H= \chi ^2 + \Pi, \end{equation} \tag{ 17 }$$

where the "reduced" chi-square χ² defined as

$$\begin{equation} \chi ^2 \equiv {1 \over N } \sum _{i=1}^N \left({D_i^{\rm obs} - D_i^{\rm mod} \over \sigma _D } \right)^2 \end{equation} \tag{ 18 }$$

is to make the model $D_i^{\rm mod}$ as get close to the data $D_i^{\rm obs}$ as possible,—where N is the number of wavelength points and σ_D is the standard error of each data value—and the constraint Π defined as

$$\begin{equation} \Pi \equiv \sum _{j=1}^M \left({p_j - p_{j,d} \over \sigma (p_j) } \right) ^2 \end{equation} \tag{ 19 }$$

is to regularize each parameter p_j by constraining it to its default value p_{j, d} within the extent defined by σ(p_j), the standard deviation of p_j from the default value. Note that in the limit σ(p_j) → ∞ the parameter p_j behaves as an unconstrained free parameter independent of the default value, and in the other limit σ(p_j) → 0, as a fixed parameter equal to the default value. By adopting an appropriate value of σ(p_j) in the intermediate range, we can also constrain p_j to the default value either loosely or tightly.

The ECM has eight parameters: τ₀, λ₁, W, S, t₀, λ₂, w, and s. Among these, we regard the four parameters τ₀, λ₁, W, S only as unconstrained free parameters, and t₀ as a free parameter loosely constrained to 0 with σ(t₀) = 1. We fix the other parameters λ₂, w, and s to the default values: −1.8 pm, 35 pm, and 0.12I_c, which we obtained from the analysis of root-mean-square (rms) contrast profiles of intensity in the reference ensemble as described in Appendix A.

Our model fitting consists of three stages. First, we construct a number of model profiles $D_i^{\rm mod}$ for every set of parameters in the parameter space. In this space, each of [τ₀, W, S, t₀] has five different values, and λ₁ have 11 different values, the other parameters are fixed to the default values, so the total number of synthetic profiles in the archive is 6875. Next, the observed contrast profile is compared with all of these profiles in the archive, and the closest matching profile is selected, and the parameter values of this selected profile is used as the initial guess for the nonlinear least-square fitting. Finally, the iteration of the parameters is done with the Marquardt's gradient-expansion algorithm that is implemented in the standard procedure curvefit" of the Interactive Data Language.

The value of σ_D needs to be specified for the calculation of χ². The smaller σ_D is, the bigger χ² is. After a few trials, we have adopted a value for σ_D ensuring that the median of χ² values in the observed quiet region is close to 1. Then, with this choice of σ_D, χ² = 1 represents a typical goodness of fitting in the quiet region.

The top panels of Figure 4 show the result of the ECM fitting. We find that the new model fitting is no better than the BCM in feature 1. However, it is much better in feature 2 and is excellent in features 3 and 4. Note that the contrast profiles of features 3 and 4 cannot be fitted by the BCM at all (see Figure 3).

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The strength of the new model originates from the newly introduced parameter t₀. The uniqueness and role of this parameter in the model can be inferred from the plots of χ² as a function of t₀ presented in the middle panels of Figure 4. Note that with each given value of t₀, the other parameters have been chosen to ensure the minimum of χ², and the special case of t₀ = 0 corresponds to the BCM.

There exist different patterns in the variation of χ². In the case of feature 1, χ² is fairly small for t₀ = 0, slowly decreases with t₀, and displays a shallow minimum at t₀ > 1. The profiles of this type can be adequately fitted by the BCM as well, and t₀ of the ECM remains ill-determined unless the constraint is used. The role of the imposed constraint Π is thus crucial in this type. It forces the minimum of H to occur at t₀ as small as possible, making the solution be close enough to the BCM. The second type is well represented by the other features. In this type, χ² at t₀ = 0 is very big, implying the inadequacy of the BCM. The minimum occurs at t₀ ∼ 1. If the minimum is deep (e.g., features 2 and 4), the effect of the constraint on the solution is minor. If not (e.g., feature 3), the constraint becomes significant, forcing t₀ to be as small as possible.

The bottom panels of Figure 4 show the inferred R_{λ, in} in comparison with R_{λ, obs} and I_{λ, obs}. The bigger t₀ is, the more R_{λ, in} deviates from R_{λ, obs}. Note that R_{λ, in} is higher than I_{λ, obs} at every wavelength in all the features. It implies that with the choice of $I_{\lambda, \rm in}=R_{\lambda, \rm in}$, C_λ is negative at all the wavelengths and can be well fitted by the BCM. We also note that a big value of t₀ is not preferred. If t₀ is too big, then R_{λ, in} deviates too much from R_{λ, obs} and might deviate much from the real as well.

Table 1 presents the determined values of the parameters with their probable standard errors estimated using the bootstrap method. With these values, we can understand the emission/absorption characteristic seen in each contrast profile. The absorption at all the wavelengths in feature 1 is mostly due to Δτ_λ > 0. The emission in the red wing in feature 2 is the combined effect of ΔS > 0 and the large redshift of τ_λ that leads to Δτ_λ < 0 (hence emission) in the blue wing. The absorption/emission characteristic of feature 3 originates from the blueshift of τ_λ (upward moving plasma) causing Δτ_λ > 0 (hence absorption) in the blue wing and Δτ_λ < 0 (hence emission) in the red wing. Finally, the physical origin of emission characteristic of feature 4 is a little hard to understand. It appears that the emission characteristic is mainly due to W < w which leads to Δτ_λ < 0 at all the wavelengths. The redshift of τ_λ is in charge of the blue-red asymmetry; it causes the significant negative value of Δτ_λ (hence strong emission) in the blue wing, and the insignificant negative value of Δτ_λ (hence week emission) in the red wing.

The ECM is superior to the BCM in that it can produce the complete parameter maps of the observed region on the quiet Sun as is demonstrated in Figure 5. Thanks to the flexibility associated with a new free parameter t₀, the ECM can be applied not only to network features, but also to internetwork features, and hence to all the quiet region features. The χ² map indicates that the ECM fit is excellent in the internetwork region. In the network region, χ² is bigger, but the goodness of fit is still satisfactory at most points as illustrated in the fits of features 1 and 2 (see Figure 4). These fits are no worse than any of cloud model fits published so far, e.g., Figure 4 of Grossmann-Doerth & von Uexküll (1971) and Figure 2 of Tsiropoula et al. (1999).

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In contrast, recalling that the ECM in the limit t₀ → 0 is reduced to the BCM, we find from Figure 5 that the applicability of the BCM is limited. The white-colored (t₀ = 0) area of t₀-map where the BCM is applicable excludes most of the internetwork region and a significant portion of the area near the center of the rosette structure. This limited applicability of the BCM explains why the maps of parameters in a rosette previously obtained using the BCM (e.g., Tsiropoula et al. 1993) were left unfilled in many places.

Figure 5 shows a few spatial patterns of the parameters. First, the central part of the rosette is redshifted whereas its outer part is blueshifted. Even inside an individual fibril (say the one including feature 1, the inner (or lower) part is redshifted, and the outer (upper) part, blueshifted, which confirms the earlier studies (e.g., Tsiropoula et al. 1993). Second, bright mottles and dark mottles are distinguished by the source function, quite in line with the previous study of Bray (1973) and Heinzel & Schmieder (1994). In addition, the different appearance of dark mottle contrast in the red wing and blue wing is due to the Doppler shift. Finally, W in the network region is systematically larger than in the internetwork region, confirming the result of Cauzzi et al. (2009). In fact, we find from Figure 6 that the number distribution of W in the quiet region is bimodal. The peak around 30 pm represents the internetwork region, and the one around 46 pm the network region. If ξ does not differ much between these regions, the difference in W reflects difference in T. With the choice of, for example, ξ = 8 km s⁻¹, these Doppler widths yield the estimates T = 7500 K and 23,000 K, respectively.

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Figure 7 compares the inferred line-of-sight velocities between the lambdameter method and the ECM method. The lambdameter method fits a horizontal chord (lambdameter) of width 2Δλ into the observed line profile. The wavelength corresponding to the center of the lambdameter is adopted as the measured wavelength of the line, and hence a measure of Doppler shift. Since this method works without failure as long as the Hα line appears in absorption, it has been often used to produce the complete Doppler shift map of the observed region (e.g., Chae et al. 2013b). As expected, the figure shows that the ECM velocities are strongly correlated with the lambdameter velocities. The Peterson coefficient is found to be 0.87. Note, however, that there is a big difference in velocity magnitude between the two methods: the standard deviation of the lambdameter velocities is 1.3 km s⁻¹ and that of the ECM velocities, 4.5 km s⁻¹. Supposing that the ECM velocities are close to the real, we find that the lambdameter velocities are seriously underestimated, on average, by a factor of 3 to 4, which was well-known from previous studies (e.g., Alissandrakis et al. 1990; Tsiropoula et al. 1993). More serious is that despite the strong correlation, the one-to-one correspondence between lambdameter velocities and the ECM velocities is far from unique. For example, the lambdameter velocity of 2 km s⁻¹ corresponds not to a single value of the ECM velocity but to its broad range from 3 km s⁻¹ to 15 km s⁻¹.

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We have introduced the ECM for the spectral inversion of the Hα line for a plasma feature in the upper layer of the chromosphere. Like the Beckes' cloud model (BCM), the ECM assumes that the absorption profile (characterized by Doppler width W and line-of-sight velocity v) and the source function S are uniform inside the feature of optical thickness τ₀. The ECM, however, differs from the BCM in the way of adopting I_{λ, in}. As illustrated in Figure 2, I_{λ, in} is identified with R_{λ, obs} in the BCM, but it is identified with R_{λ, in} in the ECM. We have proposed to relate between R_{λ, obs} and R_{λ, in} via the radiative transfer of ensemble-average light across the average chromospheric layer, which is parameterized by optical thickness t₀, source function s and Doppler width w. The ECM has practically one additional free parameter t₀ with the other additional parameters being either fixed or tightly constrained to the default values. With the flexibility associated with t₀, the ECM can fit all the intensity profiles of the observed quiet region and produce the complete maps of line-of-sight velocity, Doppler width, optical thickness, source function, and so on.

In determining line-of-sight velocity the ECM is superior either to the lambdameter method or to the BCM in two aspects. The lambdameter method can give us some qualitative information such as the spatial distribution of redshift and blueshift, but cannot produce reasonable velocities for a quantitative analysis. For this purpose, the BCM is preferred to the lambdameter method. The BCM, however, is limited in its applicability; there are many regions where the BCM cannot be applied. The ECM has no such limitation; it can produce not only the complete velocity map of the observed region, but also it can produce reasonable velocity values suited for quantitative studies.

The critical parameter in the ECM, t₀, has a clear physical meaning. It refers to the atmospheric level where the bottom of the feature is located. If t₀ is much smaller than 1, the feature lies high above the chromospheric forest, like a cloud. If t₀ becomes comparable to or larger than 1, the bottom of the feature is embedded inside the chromospheric forest. Thus, t₀ provides us with some information on the geometric height. Roughly speaking, the feature with bigger t₀ extends down to a deeper level than the one with smaller t₀. By constraining t₀ to be as small as possible, we aim to infer the physical parameters of features in the outer layer of the chromosphere.

Our ECM can be used to chromospheric features in other environments as well if the reference ensemble is well established. The reference ensemble we used in the present study consists of the spectral profiles extracted from the internetwork region. It provided us with the observed reference profile R_{λ, obs} and the default parameters of λ₂, w, and s. With these data, the spectral profiles of the internetwork features were excellently fitted by the model. Those of the network features were fitted by the model a little worse than the internetwork features. We do not know the main reason for the degraded goodness of fitting in the network features, but we suspect that the poorer fitting may be partly because the reference ensemble is not from the network environment itself, but from the internetwork environment. Nevertheless, we think the radiation field may not be much different between the two environments because the model fitting of network features is still good enough. Each environment of active regions such as sunspot umbrae, sunspot penumbrae, and plages may be quite different from that of the quiet Sun. Therefore, it would not be reasonable to use R_{λ, obs}, λ₂, w, and s obtained here for the study of features in the above environments, and it is necessary to establish a new set of these data for each environment.

The ECM may be compared with other variants of the cloud model introduced for the same purpose: to overcome the difficulty of finding the suitable I_{λ, in}. Assuming I_{λ, in} is symmetric, Liu & Ding (2001) applied the fitting to the profile of intensity difference between the blue side and the red one where I_{λ, in} disappears. This method is good when the observed line profile is significantly asymmetric so that it highly deviates from the supposed I_{λ, in}. Some asymmetric line profiles in the quiet region may be fitted with this method, but there are many features in the quiet Sun displaying line profiles that are little asymmetric, where the method is not effective. Another interesting approach is to adopt the theoretical reference profile constructed by using the non-LTE profile synthesis. Bostanci & Al Erdoğan (2010) found that the synthetic profile computed from the Kurucz radiative-equilibrium model is the best choice as I_{λ, in} for the cloud modeling of Hα features in the quiet Sun. With this choice almost all the profiles were fitted by the BCM, and the determined cloud model parameters were in good agreement with the VAL-C atmospheric model. This approach of Bostanci & Al Erdoğan (2010) is similar to the ECM in that it admits $I_{\lambda, \rm in} \ne R_{\lambda, \rm obs}$, but it differs from the ECM in that it uses one synthetic profile as an input I_{λ, in} for all the features while the ECM uses R_{λ, obs} as an input for all the features and yields I_{λ, in} as an output for each individual feature.

5.2.1. Can the Source Function be Assumed to be Constant Inside Each Embedded Cloud?

Like the BCM, the ECM assumes the constancy of the source function S inside the feature and in the ensemble average chromosphere, which is the basis of Equations (5) and (14). The assumption of constant S holds most satisfactorily in an optically thin homogeneous cloud lying high over the chromosphere where the BCM can be successfully applied. Generally speaking, however, S is likely to vary with position.

Since S of the Hα line is mostly determined by the angle-averaged mean intensity in the upper atmosphere, and the contribution of lateral illumination and hence the mean intensity increase with depth, it is likely that S increases with optical depth both inside the feature and in the average chromosphere. In fact, Mein et al. (1996)'s non-LTE models of a horizontal slab irradiated from below indicates that the source function increases with optical depth inside the slab in all the models, even for optically thin slab models. Mein et al. (1996) also suggested a variant of the cloud model which makes use of this result of model computations. The variation of S over optical depth is approximated by a quadratic function of optical depth the coefficients of which are fully specified by the optical thickness of the slab.

The variation of the source function in a stratified mean atmosphere can be inferred from the values of hydrogen population given in the Table 17 of Vernazza et al. (1981). Using Equation (A9), we have calculated s/I_c as a function of height in the VAL Model C, and found that at height ⩽2000 km, s/I_c monotonically increases with depth. A noticeable feature of the model is that s/I_c varies too slowly from 0.073 to 0.085 with depth at heights between 1400 and 2000 km, representing the upper chromosphere, so that it can be assumed constant (∼0.08) in this range.

Note that the ECM deals with not the whole chromosphere of large range of optical depth but with the embedded cloud and the outermost layer of the average chromosphere whose optical thickness in the Hα center is order of unity. We expect the variation of the source function within this range to be small enough to be well described by low order polynomials of optical depth. The approach we adopt in the present study is to keep only the lowest order term, that is, the constant term. If this choice of approximation leads to a successful reproduction of the observed spectral profiles, we can be satisfied with it; if the model too much deviates from the data, then we may have to make the model more sophisticated by including the first order term or even the second order term. In Appendix B, we have presented the generalization of the ECM that takes into account the first order term of the source function δS as well.

As a matter of fact, we have fitted the contrast profiles of the four features with this generalized ECM and compared the results with the ECM with a constant source function. In features 3 and 4, δS is practically zero, and there is no significant difference in the value of the parameters between the two models. In feature 1, the ECM with variable source function yields S/I_c = 0.14 and δS/I_c = 0.14, which are significantly different from the ECM with constant source function S/I_c = 0.13 and δS/I_c = 0. There is little difference, however, in the values of other parameters and χ² between the two models. The biggest difference between the two models occurs in feature 2. As shown in Figure 8, the model with the variable source function (S/I_c = 0.28 and δS/I_c = 0.51) better fits the observed profile than the one with the constant source function S/I_c = 0.18. We find that the inclusion of the δS term in the model affected the value t₀ the most, changing it from 1.6 to 0.90, and then the value of τ₀, changing it from 1.9 to 2.1. The other parameters λ₁ and W changed little. From these comparisons, we conclude that the ECM with the variable source function can better fit the observed line profiles at the network regions where the ECM fitting with constant source function is poorer.

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5.2.2. Can the Incident Radiation be the Same at Every Point in the Lower Parts of the Features?

One assumption of the ECM is that I_{λ, in} is the same at every point in the horizontal plane t_c = t₀ as expressed in Figure 2 and Equation (9) or, practically speaking, I_{λ, in} varies much less in the horizontal plane than I_{λ, obs} does. This assumption is reasonable in a couple of ways. We first note that the specific form of I_{λ, in} originates from the low atmosphere t_c > t₀ while that of I_{λ, obs} is determined by the whole atmosphere t_c > 0 that includes the above low atmosphere. As a consequence, the horizontal variation of I_{λ, in} reflects the inhomogeneity of the low atmosphere only, but that of I_{λ, obs} does not only this inhomogeneity but also the inhomogeneity of the outermost layer 0 < t_c < t₀. Moreover, the inhomogeneity of plasma increases with height in the solar atmosphere because the dynamic and thermal interaction of plasma across field lines is more and more inhibited as the plasma β decreases with height, and velocity fluctuation usually increases with height. Thus, the inhomogeneity of the outermost layer may be much stronger than that of the low atmosphere. As a matter of fact, observations indicate that the rms contrast of a quiet region is much bigger in the core—that mostly originates from the outermost layer of the atmosphere, rather than in the wings, which is what it does from the lower atmosphere (see Figure 9). For these reasons, we neglect the spatial variation of I_{λ, in}, and we set it equal to R_{λ, in}, which makes the model simple and tractable.

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There is a situation, however, when the assumption $I_{\lambda, \rm in}=R_{\lambda, \rm in}$ is not valid. For example, when more than one chromospheric feature lies along the line of sight, I_{λ, in} incident on the outermost feature originates not only from the low atmosphere but also from the other lower-lying chromospheric features, so that it differs much from R_{λ, in} inferred from R_{λ, out} of a large area of the internetwork region. In this case, the BCM appears to be a better choice since the average of the rays over a small local area nearby containing similar chromospheric features can be reasonably used as the background profile for the BCM.

5.2.3. How do We have to Interpret the Maps of Parameters?

5.2.4. How can the ECM Inversion be Compared with the Forward MHD and Non-LTE Modeling?