ON THE INVERSION OF THE SCATTERING POLARIZATION AND THE HANLE EFFECT SIGNALS IN THE HYDROGEN Lyα LINE

The chromosphere and the transition region of the Sun lie between the cooler photosphere, where the ratio of gas to magnetic pressure β > 1, and the 10⁶ K corona, where β < 1. It is believed that, in this interface region, the magnetic forces start to dominate over the hydrodynamic forces and that local energy dissipation and energy transport to the upper layers via various fundamental plasma processes are taking place. Recent observations (e.g., Shibata et al. 2007; Katsukawa et al. 2007; De Pontieu et al. 2007; Okamoto et al. 2007; Okamoto & De Pontieu 2011; Vecchio et al. 2009) have revealed ubiquitous dynamical chromospheric activities such as jets, Alfvénic waves, and shocks, which are thought to play a key role in the heating of the chromosphere and corona and in the acceleration of the solar wind. However, we do not have any significant empirical knowledge on the strength and direction of the magnetic field in the upper solar chromosphere and transition region.

The information on the magnetic field of the solar atmosphere is encoded in the polarization that some physical mechanisms introduce in the spectral lines. The familiar Zeeman effect can introduce polarization in the spectral lines that originate in the upper solar chromosphere and the transition region. However, because such lines are broad and the magnetic field there is expected to be rather weak, the induced polarization amplitudes will be very small (except perhaps in sunspots), and the Zeeman effect has limited applicability. Fortunately, the Hanle effect (the magnetic-field-induced modification of the linear polarization caused by scattering processes in a spectral line; Casini & Landi Degl'Innocenti 2008) in some of the allowed UV lines that originate in the upper chromosphere and transition region is expected to be a more suitable diagnostic tool (Trujillo Bueno et al. 2011, 2012; Belluzzi & Trujillo Bueno 2012).

The hydrogen Lyα line (λ = 121.567 nm) is particularly suitable because (1) the line-core polarization originates at the base of the solar transition region, where β ≪ 1 (Trujillo Bueno et al. 2011; Belluzzi et al. 2012; Štěpán et al. 2012), (2) collisional depolarization plays a rather insignificant role (e.g., Štěpán & Trujillo Bueno 2011), and (3) via the Hanle effect, the scattering polarization is sensitive to the magnetic fields expected for the upper chromosphere and transition region (Trujillo Bueno et al. 2011).

The Chromospheric Lyman-Alpha SpectroPolarimeter (CLASP) is a sounding rocket experiment developed by researchers from Japan, the United States, and Europe (Ishikawa et al. 2011; Narukage et al. 2011; Kano et al. 2012; Kobayashi et al. 2012), which is expected to fly in 2015. The first, very important goal of this sounding rocket experiment is the measurement of the linear polarization signals produced by scattering processes in the Lyα line. The second goal is the detection of the Hanle effect action on the core of Q/I and U/I in order to constrain the magnetic field of the transition region from the observed Stokes profiles. CLASP will measure the linear polarization profiles of the Lyα line within a spectral window of at least ±0.05 nm around the line center, where, in addition to the line core polarization itself (where the Hanle effect operates), we expect the largest linear polarization signals produced by the joint action of partial frequency redistribution and J-state interference effects (Belluzzi et al. 2012). Polarization sensitivities of 0.1% and 0.5% are required in the line core (i.e., ±0.02 nm around the line center) and in the line wings (at >  ± 0.05 nm), respectively. In order to achieve these polarization sensitivities, the 400'' spectrograph slit will be fixed at the selected observing target during the CLASP observation time of <5 minutes. Furthermore, after the data recovery, we will add consecutive measurements and perform spatial averaging.

In this paper, we clarify the information we expect to determine with the CLASP experiment, providing a strategy suitable for highlighting the ambiguities of the Hanle effect and the complexity of the ensuing inference problem. To this end, we have used a plane-parallel (one-dimensional) semi-empirical model of the solar atmosphere, and we have created a database of theoretical Stokes Q/I and U/I profiles for all possible strength, inclination, and azimuth values of the magnetic field vector. Then, we investigate the possibility of recovering the magnetic field information using the characteristics of CLASP (noise level, spectral resolution, etc.) and the ambiguities and limitation inherent to the Hanle effect. We also discuss the most suitable observing targets and data analysis strategy for constraining the magnetic field information.

The ambiguities inherent to magnetic field diagnostics can be reduced by exploiting the joint action of the Hanle and Zeeman effect, where both linear and circular polarization signals are used to constrain the magnetic field vector (Landi Degl'Innocenti & Bommier 1993; Landi Degl'Innocenti 1982; Trujillo Bueno et al. 2002; López Ariste & Casini 2003, see also Asensio Ramos et al. 2008; Centeno et al. 2010; Anusha et al. 2011). Unfortunately, while the Lyα line is most advantageous to explore the magnetism of the various regions in the transition region (where β ≪ 1), it is challenging to measure the contribution of the Zeeman effect to the circular polarization in UV lines. Here, we propose alternative ways to alleviate this issue in the subsequent sections.

We assume that the quiet solar atmosphere can be represented by a plane-parallel semi-empirical model atmosphere, even though the upper solar chromosphere is a highly inhomogeneous and dynamic physical system, much more complex than the idealization of the one-dimensional static semi-empirical model used here. Such inhomogeneity and dynamics causes larger amplitudes and spatial variations in the scattering polarization signals (Štěpán et al. 2012, 2014). However, it is of interest to note that Trujillo Bueno et al. (2011) showed that the amplitude and shape of the Q/I profiles calculated in the quiet-Sun plane-parallel semi-empirical model of Fontenla et al. (1993) are qualitatively similar to the temporally averaged profiles obtained from the Stokes I and Q signals computed at each time step of the chromospheric hydro-dynamical model of Carlsson & Stein (1997). Moreover, spatial and temporal averaging of the scattering polarization calculated in three-dimensional (3D) atmospheric models tend to produce Q/I Lyα signals more or less similar to those calculated in plane-parallel semi-empirical model atmospheres (Štěpán et al. 2014). In the case of the CLASP experiment, which will require both spatial (∼10'') and temporal averaging (∼5 minutes) to attain the necessary signal-to-noise ratio, the model atmosphere for a first approximate interpretation of the CLASP data could be a plane-parallel semi-empirical model. We believe that the post-launch data analysis will pave the way for further improvements, for example, via forward modeling calculations of the Lyα scattering polarization signals using increasingly realistic 3D models of the solar chromosphere.

The expected scattering polarization in the core of the hydrogen Lyα line physically originates from population imbalances and quantum coherence between the magnetic sublevels pertaining to the 2p²P_3/2 upper level, both of which are produced from the absorption of anisotropic radiation by the hydrogen atoms of the solar transition region.

Typically, in weakly magnetized stellar atmospheres, the absorption of anisotropic radiation produces atomic level alignment (i.e., population imbalances between magnetic sublevels having different |M| values, M being the magnetic quantum number). On the other hand, atomic level orientation (i.e., population imbalances between sublevels with M > 0 and M < 0) can be neglected in the modeling of the Lyα linear polarization. The anisotropy and symmetry properties of the radiation field that illuminates the atomic system can be conveniently quantified through the so-called radiation field tensor, $J^K_Q$ (see Equation (5.157) of Landi Degl'Innocenti & Landolfi 2004). If the radiation field has cylindrical symmetry along a given direction, and it has no circular polarization, then, if the quantization axis is taken along the symmetry axis, only the components $J^0_0$ and $J^2_0$ are non-zero. The former represents the mean intensity of the radiation field, while the latter quantifies its degree of anisotropy. The explicit expression of the $J^2_0$ component of the frequency-integrated radiation field tensor is given by

$$\begin{equation} J^2_0 = \int {\rm d}x \oint \frac{{\rm d} \Omega }{4 \pi } \frac{\phi _x}{2 \sqrt{2}} [ (3 \mu ^2 - 1) I_{x \Omega } + 3(1 - \mu ^2) Q_{x \Omega} ], \end{equation} \tag{ 1 }$$

where ϕ_x is the absorption profile, x is the normalized frequency distance from line center, and μ = cos θ (where θ is the angle between the radiation beam under consideration and the quantization axis). In the solar atmosphere, the contribution of the Stokes Q_xΩ parameter to $J^2_0$ is very small compared with that caused by the specific intensity I_xΩ. Thus, in analogy with the spherical harmonics of $Y^0_2$, the $J^2_0$ tensor represents whether the local illumination of the atomic system is dominated by predominantly vertical ($J^2_0 > 0$) or predominantly horizontal ($J^2_0 < 0$) illumination. In the case of $J^2_0 > 0$ ($J^2_0 < 0$), photon absorption processes take place predominantly through ΔM = ±1 (ΔM = 0) transitions, which gives rise to population imbalances among the various magnetic sublevels.

In this work, we consider a plane-parallel atmosphere, whose parameters depend only on the height. Taking the quantization axis along the vertical, it can be shown that in the absence of magnetic fields, or in the presence of a vertical magnetic field, any coherence between pairs of magnetic sublevels of the 2p²P_3/2 upper level is zero, while it is non-zero if an inclined magnetic field is present.

Although choosing the quantization axis along the local vertical can be advantageous in some cases, in order to investigate the impact of the magnetic field on the atomic polarization (i.e., the Hanle effect), it is convenient to choose it along the magnetic field direction. In this case, describing atomic polarization through the multipole moments of the density matrix, $\rho ^K_Q$ (e.g., Section 3.7 of Landi Degl'Innocenti & Landolfi 2004), the effect of the magnetic field is described by the equation (see Section 10.3 of Landi Degl'Innocenti & Landolfi 2004)

$$\begin{equation} \rho ^K_Q(J_u) = \frac{1}{1 + {\rm i} Q \Gamma _u} \left[ \rho ^K_Q(J_u) \right]_{B=0} \;, \end{equation} \tag{ 2 }$$

where $\Gamma _u = 8.79 \times 10^6 \, B \, g_{J_u} / A_{u \ell }$ (B is the field strength in gauss, $g_{J_u}$ is the Landé factor, and A_uℓ is the Einstein coefficient for the spontaneous emission process in s⁻¹), and $[\rho ^K_Q(J_u)]_{B=0}$ represents the value of the $\rho ^K_Q(J_u)$ elements for the non-magnetic case. This equation shows that in the magnetic field reference frame, the population imbalances represented by the $\rho ^K_Q(J_u)$ elements with Q = 0 are unaffected by the magnetic field, while the elements with Q ≠ 0, indicating the atomic coherence, are reduced and dephased with respect to the non-magnetic case. The Hanle effect can thus be defined as the modification of the atomic level polarization (in particular, the modification of coherence) and the ensuing observables effects on the emergent Stokes Q and U profiles, caused by the action of an inclined magnetic field. For the hydrogen Lyα line, the magnetic field strength for which Γ_u = 1 (i.e., the critical magnetic field for the onset of the Hanle effect) is B = 53 G.

The Lyα line consists of two blended transitions between the 1p²S_1/2 lower level and the 2p²P_1/2 and 2p²P_3/2 upper levels. To estimate the linear polarization of the Lyα line, we follow the approach of Trujillo Bueno et al. (2011), and we provide a quick overview here (refer to Trujillo Bueno et al. 2011 for details). We consider the quiet Sun semi-empirical model of Fontenla et al. (1993), which is hereafter referred to as the FAL-C model. Thus, our model atmosphere is plane-parallel, and the physical quantities only depend on the coordinate Z. The hydrogen atomic model we have used includes the fine structure of the first two n-levels of the hydrogen, where n is the principal quantum number. The excitation state of each level is quantified by means of the multipolar components of the atomic density matrix, which are self-consistently obtained by solving the statistical equilibrium equations and the Stokes vector radiative transfer equation (see chapter 7 of Landi Degl'Innocenti & Landolfi 2004). Thus, we assume complete frequency redistribution, which is a suitable approximation for the estimation of the scattering polarization at the Lyα line center (Belluzzi et al. 2012). Isotropic collisions with protons and electrons are also taken into account, but these collisions have a negligible depolarizing effect on the scattering polarization of the hydrogen Lyα line at the low plasma densities of the upper chromosphere and the transition region, as discussed by Štěpán & Trujillo Bueno (2011).

By solving the above mentioned non-local thermodynamic equilibrium radiative transfer problem, we created a database of synthetic Stokes profiles (I(λ), Q(λ), and U(λ)) of the hydrogen Lyα line. We consider the presence of a deterministic magnetic field of arbitrary strength B, inclination θ_B, and azimuth χ_B, all of which are assumed to be constant with height (Figure 1). In total, there are 137,751 sets of Q(λ)/I(λ) and U(λ)/I(λ) profiles for different magnetic field strengths (0 ⩽ B ⩽ 250 G in 5 G increments), inclinations (0° ⩽ θ_B ⩽ 180° in 5° increments), and azimuths (0° ⩽ χ_B ⩽ 360° in 5° increments). An example of a synthetic profile is given in Figure 2 (solid lines). For all magnetic parameters, the Stokes I(λ) profiles are virtually identical because only the polarization signals have a measurable sensitivity to the magnetic field. Here, we consider two scattering geometries: disk center (μ = 1.0) and close-to-limb (μ = 0.3).

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Here, we also simulate the CLASP observations. The wavelength resolution of the CLASP optics is ∼0.013 nm, and the Stokes spectra are recorded with a wavelength sampling of 0.005 nm pixel⁻¹. For a spatial area of less than 10'', the polarization sensitivity is 0.1% with respect to the intensity at each wavelength pixel in the line core (121.567 ± 0.02 nm). First, the synthetic profiles I(λ), Q(λ), and U(λ) in the database are convolved with a 0.013 nm FWHM Gaussian and are then sampled with a wavelength step of 0.005 nm. Assuming that the polarization sensitivity will be achieved at a 3σ level, random noise with a standard deviation of σ = 0.033% with respect to the intensity at each wavelength bin is added to the convolved I, Q, and U profiles. Finally, we derive the simulated Q^obs(λ)/I^obs(λ) and U^obs(λ)/I^obs(λ) profiles (squares in Figure 2).

The behavior of the Q/I and U/I signals with respect to the magnetic field parameters (strength, inclination, and azimuth of the magnetic field vector) can be suitably illustrated by a Hanle diagram (see also Trujillo Bueno et al. 2012). In Figure 3, we show Hanle diagrams with a horizontal magnetic field (i.e., the inclination is fixed at θ_B = 90°) for different values of the azimuth angle when the field strength varies between 0 G (black circles) and 250 G (white circles). Figure 3(a) corresponds to a close-to-limb geometry (μ = 0.3), while Figure 3(b) refers to the forward scattering case of a disk center observation (μ = 1.0). The Q/I and U/I signals in this figure refer to the amplitudes of original synthetic profiles Q(λ₀)/I(λ₀) and U(λ₀)/I(λ₀) in the database at the line center, where λ₀ = 121.567 nm. Figure 4 is similar to Figure 3 but the magnetic field is nearly vertical (i.e., the inclination is fixed at θ_B = 20°).

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For the close-to-limb geometry (μ = 0.3), the unmagnetized case (B = 0 G) shown in black circles in Figures 3(a) and 4(a) yields negative Stokes Q/I values, which indicates that the direction of linear polarization caused by the anisotropic radiation field (i.e., the non-magnetic scattering polarization) is perpendicular to the solar limb. As shown by Trujillo Bueno et al. (2011), the anisotropy of the radiation field, $J^2_0$, illuminating the hydrogen atoms in the Lyα line is negative (dominated by horizontal illumination) through the line formation region of the FAL-C model atmosphere. The resulting scattering polarization is perpendicular to the limb. At the disk center (μ = 1.0), where the line of sight (LOS) is parallel to the solar normal (i.e., the symmetry axis of the radiation field), we have Q/I = 0 and U/I = 0 for the non-magnetized case (black circles in Figures 3(b) and 4(b)).

The crossing points seen in the μ = 0.3 Hanle diagrams indicate ambiguity, which occurs when different magnetic field vectors give the same Stokes Q/I and U/I signals. For example, in Figure 3(a), the crossing point at Q/I ∼ −0.3% and U/I ∼ −0.15% corresponds to cases with χ_B = 330° and B = 10 G and with χ_B = 60° and B = 30 G. This ambiguity cannot be solved without using additional information to constrain one of magnetic parameters. At the solar disk center (μ = 1.0), we have 180° ambiguity, where two magnetic field vectors whose azimuths differ by 180° represent the same Q/I and U/I signals (see overlapping solid and dashed lines in Figures 3(b) and 4(b)).

In such Hanle diagrams, the change in linear polarization from 0 G to 50 G is larger than that from 50 G to 250 G, with the exception of nearly vertical fields in the μ = 1.0 forward scattering geometry case (Figure 4(b)). This trend is prominent for the horizontal field case shown in Figure 3, where Q/I and U/I significantly change from 0 to 50 G but show little change from 50 G to 250 G. This indicates that the Hanle effect in the hydrogen Lyα line is sensitive to field strengths where B < 50 G. A field strength of ∼50 G corresponds to the critical field strength for the onset of the Hanle effect in the Lyα line. Above this field strength, the Lyα line approaches the Hanle saturation limit, where the linear polarization is insensitive to the magnetic field strength.

The linear polarization signals produced by the Hanle effect are weaker than 0.1% for a nearly vertical field at the solar disk center (μ = 1.0) (Figure 4(b)) because weak vertical fields do not give rise to a strong symmetry breaking. At the solar disk center, largely inclined and strong fields are more suitable observing targets.

5.1. General Approach for Solving the Inversion Problem

To clarify whether we can retrieve information on the magnetic field from the observed Stokes profiles, we perform a process that mimics the Stokes inversion. Here, we assume that the formation of the Lyα line is modeled with a semi-empirical FAL-C model atmosphere. For this purpose, we introduce the following function:

$$\begin{equation} \chi ^{2}(B,\chi _B,\theta _B)\equiv \sum _{k=1}^{2}\sum _{l=1}^{n} \frac{\left[S_{k}^{{\rm obs}}(\lambda _{l})-S_{k}^{{\rm mod}}(\lambda _{l},B,\chi _B,\theta _B)\right]^{2}}{\sigma ^2}, \end{equation} \tag{ 3 }$$

where S₁ and S₂ indicate Q/I and U/I, respectively. The simulated CLASP observation is $S_{k}^{{\rm obs}}$ (k = 1, 2), i.e., the synthetic Stokes profile taken from the database, convolved with a 0.013 nm FWHM Gaussian, sampled with a wavelength step of 0.005 nm, with added noise (Section 3). Here, $S_{k}^{{\rm mod}}$ represents the noiseless model profile corresponding to the synthetic profile convolved with a 0.013 nm FWHM Gaussian and sampled with a wavelength step of 0.005 nm. In the database, the parameter increments in the synthetic profiles are 5 G increments of field strength and 5° increments of azimuth and inclination. In order to have better accuracy in the inversion, $S_{k}^{{\rm mod}}$ are calculated with 1° steps for azimuth and inclination and with 1 G steps for field strength by linearly interpolating model profiles with adjacent magnetic parameters. We take into account the wavelength range of 121.567 nm ± 0.04 nm, where the linear polarization signals are defined, resulting in n = 17 wavelength points. Finally, σ is the standard deviation of the random noise we assumed in the CLASP observation simulations. We employ σ = 0.033% as the baseline, and in Section 5.4, we will discuss the influence of noise on the inversion results.

We calculate the χ² function using the given observed profile ($S_{k}^{{\rm obs}}$) and all of the model profiles ($S_{k}^{{\rm mod}}$). We find the magnetic parameters with a statistically acceptable χ², defined by $\Delta \chi ^2\equiv \chi ^2-\chi ^2_{\mathrm{min}}\le 3.53$, where $\chi ^2_{\mathrm{min}}$ is the minimum χ² (Press et al. 2007). Because we have three free parameters (B, χ_B, and θ_B), $\Delta \chi ^2\equiv \chi ^2-\chi ^2_{\mathrm{min}}\le 3.53$ is given by the chi-square distribution function for three degrees of freedom with a confidence level of 68.3% (1σ). The parameter region defined by this criteria indicates that there is a 68.3% (1σ) chance for the true field strength, azimuth, and inclination parameters to fall within this region. We call this procedure "inversion" throughout this paper. Furthermore, we use the term "input" to refer to the magnetic parameters of the simulated observation, $S_{k}^{{\rm obs}}$.

Figure 6 shows the dependence of χ² on inclination, azimuth, and field strength for the input parameters of B = 50 G, θ_B = 90°, and χ_B = 120°, which are shown by gray circles. The black region in Figure 6 is defined by $\Delta \chi ^2\equiv \chi ^2-\chi ^2_{\mathrm{min}}\le 3.53$. The model profiles located in this region fit the simulated observations reasonably well, and all magnetic parameters are statistically accepted as results of inversion with a confidence level of 68% (1σ). The model profiles corresponding to the input parameters are shown with dashed lines in Figure 5. The magnetic parameters (chosen as an example in the acceptable χ² region) marked by a gray triangle in Figure 6 correspond to the model profiles shown with solid lines in Figure 5. Within the noise level, these profiles are identical.

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5.2. Properties of the χ² Map

5.2.1. Saturation Regime

Figure 7 represents the χ² maps projected onto two parameter spaces. There are four 1σ confidence level regions (Δχ² ⩽ 3.53) extended along the field strength above 50 G at θ_B = 20°, 30°, 150°, and 160° in Figure 7(b) and at χ_B = 40°, 145°, 195°, and 310° in Figure 7(c). The four dashed circles in Figure 7(a) show the regions on the θ_B-χ_B plane where (θ_B, χ_B) = {(20°, 150°), (30°, 45°), (150°, 195°), (160°, 310°)}, indicating four possible solutions of inclination and azimuth above 50 G. The elongated line, which occurs in only the field strength direction, indicates a large uncertainty where we cannot constrain the field strength. In other words, we do not have the sensitivity to measure the magnetic field strength beyond ∼50 G. Multiple simply connected spaces (1σ confidence level regions) indicate the ambiguity of solutions in which completely different magnetic parameters provide the same Q/I and U/I profiles.

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The elongated regions correspond to the saturation regime, in which the linear polarization weakly depends on the field strength. This is consistent with the Hanle diagrams (Section 4), where the change of linear polarization above 50 G is small compared with that below 50 G. In this saturation regime, the linear polarization signal is only dependent on the inclination and azimuth. Thus, we can determine the azimuth and inclination. However, ambiguity allows four combinations of inclination and azimuth to exist, as shown in Figure 7. The ambiguity corresponds to the Van Vleck ambiguity in the saturation regime, which is inherent to the Hanle effect. These four ambiguous solutions provide magnetic parameters with less inclined (θ_B ∼ 20–30° or θ_B ∼ 150–160°) and relatively strong (B > 50 G) magnetic field vectors.

5.2.2. Non-saturation Regime

Below 50 G, the linear polarization of Q/I and U/I depends on the field strength as well as on the inclination and azimuth (Section 4); this is the non-saturation regime. As shown in Figure 7(a), in the non-saturation regime, there are two isolated 1σ confidence level regions with saturation regimes at both ends (dashed circles, four in total). These two isolated regions are elongated over the wide inclination and azimuth range on the θ_B-χ_B plane. Different magnetic parameters give rise to the same linear polarization signals in these two regions. This can be confirmed by the Hanle diagram in Figure 3(a). If we assume that the magnetic field is horizontal (θ_B = 90°), as shown in the dashed line in Figure 7(a), two solutions are possible: one for χ_B = 120° and B = 50 G (shown in gray circle) and the other for χ_B = 225° and B = 15 G. Indeed, the Hanle diagram with χ_B = 120° and B = 50 G intersects with that with χ_B = 225° and B = 15 G, suggesting the presence of ambiguity.

Below ∼50 G, Figure 7(b) shows that one 1σ confidence level region, which extends from θ_B ∼ 20° to θ_B ∼ 160°, has an apex at B ∼ 15 G and θ_B ∼ 90°. Figure 7(b) further shows that another 1σ confidence level region, which extends in the θ_B = 30° to 150° range, possesses two vertices at θ_B ∼ 60° and θ_B ∼ 120°. The elongated shape over the inclination and field strength shows a strong correlation between these two magnetic parameters; both a weak and inclined field and a stronger and less inclined field provide equally good fitting to the observed spectra. On the χ_B–B plane, the two 1σ confidence level regions have a V-shape with apexes at χ_B ∼ 120° and B ∼ 40 G and at χ_B ∼ 220° and B ∼ 15 G. This also suggests that there is a correlation between the azimuth and field strength and that the wide field strength range is consistent with the data. We note that the azimuth converges to 120° or 220° when the field strength becomes weaker. In general, there is strong correlation among these three magnetic parameters, and it is difficult to uniquely determine a set of these magnetic parameters.

5.2.3. Connection Between Saturation and Non-saturation Regimes

We find four-fold ambiguity in the saturation regime (B > 50 G) and two-fold ambiguity in the non-saturation regime (B < 50 G). As we clearly show in Figure 6, a pair of saturated regimes converges into one of the non-saturated regimes (there are two sets of connections). This connectivity indicates that the Stokes Q/I and U/I profiles for a strong and less inclined magnetic field are similar to those of a weaker and more inclined field. This transition suggests that the degree of the Hanle effect remains the same among strong, less inclined fields and weaker, more inclined fields. In summary, multiple solutions are possible over a broad field strength range. In the saturation regime (B > 50 G), we can only determine the azimuth and inclination, although we have multiple solutions in these parameters. As we enter the non-saturation regime, we have strong correlations among three magnetic parameters in addition to the above ambiguity. These situations make it difficult to uniquely determine the magnetic field vector.

5.3. Additional Information for Constraining Magnetic Parameters

5.4. Inversion with Different Noise Levels

Here, we investigate the influence of the noise level on the inversion results. Figure 8 represents the dependence of the χ² values on inclination, azimuth, and field strength for noise levels of 1σ = 0.1%, 1σ = 0.03%, and 1σ = 0.01%. The input parameters for all cases are B = 50 G, θ_B = 90°, and χ_B = 120°. Note that Figure 8(b) is the same as Figure 6. The χ² distributions are similar for all noise levels. Above 50 G, four isolated 1σ confidence regions with Δχ² ⩽ 3.53 are extended only in the direction of the field strength, representing the saturation regime. Around 50 G, these regions are connected to the two isolated regions with largely inclined, weak fields. Even though we increase the signal-to-noise ratio, the ambiguity remains intact, and we find similar correlation between the magnetic parameters. Thus, we require additional observables to constrain the solution, even in low noise situations. The difference caused by different noise levels is equivalent to the thickness of the 1σ confidence level region. These regions are thinner for lower noise levels, indicating lower uncertainty in the determination of the magnetic parameters. For example, in the saturation regime, the thicknesses correspond to 40° for 1σ = 0.1%, ∼10° for 1σ = 0.033%, and ∼5° for 1σ = 0.01% (see Figures 8(d), (e), and (f)).

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5.5. Inversion for Different Observing Regions and Input Parameters

To see whether the properties of the χ² maps (i.e., result of inversion) identified in Section 5.2 depend on the choice of the input parameters, we perform inversions for different sets of input parameters and different observing locations on the solar disk. We study input parameters with weak field (B = 10 G), marginal field (B = 50 G), and strong field (B = 250 G) with both horizontal (θ_B = 90°) and almost vertical (θ_B = 20°) configurations. Note that all azimuths are fixed at χ_B = 120°. With this set of input parameters, we consider both the close-to-limb case (μ = 0.3) and the disk center case (μ = 1.0) and execute 12 total inversions.

Figure 9 shows χ² maps for the case of a horizontal magnetic field. With the exception of the disk center case (μ = 1.0) with B = 50 G (Figures 9(h) and (k)) and B = 250 G (Figures 9(i) and (l)), the χ² distributions are similar to those in Section 5.2, and any field strength, from weak to strong, is consistent with the data. For the disk center, with horizontal magnetic fields of 50 and 250 G, the 1σ confidence level region appears only in the saturation regime. Thus, in this case, there is no possibility to have a wrong solution with weak magnetic fields. This is a distinct advantage; however, it is impossible to determine the field strength in this case. The number of ambiguous regions is different depending on the input parameters.

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Figure 10 shows χ² maps for the case of vertical magnetic fields (θ_B = 20°). Again, we find the same properties of χ² distributions, indicating that any field strength can be possible as a solution. For the disk center case with B = 10 G and B = 50 G (Figures 10(g) and (h)), the 1σ confidence regions with Δχ² ⩽ 3.53 spread out on the plane with vertical fields (θ_B = 0° or θ_B = 180°) and on the plane with zero magnetic fields (see Figures 10(j) and (k)), indicating that the inversion does not work. This result is reasonable because the magnitude of linear polarization is quite small and below the noise level when the magnetic fields are almost vertical, as shown in Figure 4(b). If we employ a smaller noise level, properties of the χ² distribution, similar to those in Section 5.2, will appear. For CLASP observations, an observing target with a largely inclined and/or strong magnetic field strength is appropriate for the disk center observation.

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6.1. Required Additional Information

6.2. Observing Target

6.3. Atmospheric Model

This work was done during R.I.'s visit to IAC, which was supported by the Grant-in-Aid for "Institutional Program for Young Researcher Overseas Visits" from the Japan Society for the Promotion of Science (JSPS). R.I. was also supported by JSPS KAKENHI grant No. 25887051. J.Š. recognizes support from the Grant Agency of the Czech Republic through the grant P209/12/P741 and the project RVO:67985815. Financial support by the Spanish Ministry of Economy and Competitiveness through projects AYA2010–18029 (Solar Magnetism and Astrophysical Spectropolarimetry) and Consolider-Ingenio CSD2009-00038 (Molecular Astrophysics: The Herschel and Alma Era) are gratefully acknowledged. A.A.R. also acknowledges financial support through the Ramón y Cajal fellowships. The CLASP experiment has been financially supported by a Grant-in-Aid for Scientific Research (S).