Does H$\alpha$ Stokes~$V$ profiles probe the chromospheric magnetic field? An observational perspective

TL;DR

This study assesses the capability of Hα Stokes V profiles to diagnose the chromospheric magnetic field by comparing weak field approximation results with multi-line inversion methods, revealing partial agreement and limitations.

Contribution

It provides an observational evaluation of Hα Stokes V profiles for magnetic field diagnostics, highlighting their morphological similarities and quantitative differences compared to multi-line inversions.

Findings

Hα Stokes V profiles can trace the magnetic field morphology.

WFA-derived B_LOS is about half of multi-line inversion estimates.

Profiles show consistent magnetic structures at different atmospheric depths.

Abstract

We investigated the diagnostic potential of the Stokes $V$ profile of the H$\alpha$ line to probe the chromospheric line-of-sight (LOS) magnetic field ($B_{\mathrm{LOS}}$) by comparing the $B_{\mathrm{LOS}}$ inferred from the weak field approximation (WFA) with that of inferred from the multi-line inversions of the Ca II 8542 {\AA}, Si I 8536 {\AA} and Fe I 8538 {\AA} lines using the STiC inversion code. Simultaneous spectropolarimetric observations of a pore in the Ca II 8542 {\AA} and H$\alpha$ spectral lines obtained from the SPINOR at the Dunn Solar Telescope on the 4th of December, 2008 are used in this study. The WFA was applied on the Stokes $I$ and $V$ profiles of H$\alpha$ line over three wavelength ranges viz.: around line core ($\Delta\lambda=\pm0.35$ {\AA}), line wings ($\Delta\lambda=[-1.5, -0.6]$ and $[+0.6, +1.5]$ {\AA}) and full spectral range of the line ($\Delta\lambda=\pm1.5$ {\AA}) to derive the $B_{\mathrm{LOS}}$. We found the maximum $B_{\mathrm{LOS}}$ strengths of $\sim+800$ and $\sim+600$ G at $\log\tau_{\mathrm{500}}$ = $-$1 and $-$4.5, respectively in the pore. The morphological map of the $B_{\mathrm{LOS}}$ inferred from the H$\alpha$ line core is similar to the $B_{\mathrm{LOS}}$ map at $\log\tau_{\mathrm{500}}$ = $-$4.5 inferred from multi-line inversions. The $B_{\mathrm{LOS}}$ map inferred from the H$\alpha$ line wings and full spectral range have a similar morphological structure to the $B_{\mathrm{LOS}}$ map inferred at $\log\tau_{\mathrm{500}}$ = $-$1. The $B_{\mathrm{LOS}}$ estimated from H$\alpha$ using WFA is weaker by a factor of $\approx 0.53$ than that of inferred from the multi-line inversions.

Tables

Table 1. Table 1: The first and second column define the spectral blends removed before applying the WFA: λ 0 ± Δ ​ λ plus-or-minus subscript 𝜆 0 Δ 𝜆 \lambda_{\mathrm{0}}\pm\Delta\lambda . The third column indicates the element of the transition in case it is known.

${\lambda }_0$ [Å]	$\Delta {\lambda }_0$ [Å]	Line
6562.44	0.05	V II
6563.51	0.15	Co I
6564.15	0.35	Unknown

Table 2. Table 2: Node positions ( log ⁡ τ 500 subscript 𝜏 500 \log\tau_{\mathrm{500}}   scale) used for inversions of different categories of profiles in T 𝑇 T  , V LOS subscript 𝑉 LOS V_{\mathrm{LOS}}  , V turb subscript 𝑉 turb V_{\mathrm{turb}}  , and B LOS subscript 𝐵 LOS B_{\mathrm{LOS}}

Parameters	Category	Cycle 1	Cycle 2
$T$	Quiescent	$-$5.5, $-$4,5, $-$3.5, $-$2.5, $-$1.5, 0	$-$5.5, $-$4.5, $-$3.5, $-$2.5, $-$1.5, 0
	Emission	$-$4.8, $-$3.8, $-$2.9, $-$1.8, $-$0.9, 0	$-$4.8, $-$3.8, $-$2.9, $-$1.8, $-$0.9, 0
$V_{\mathrm{LOS}}$	Quiescent	$-$4.5, $-$1	$-$4.5, $-$1
	Emission	$-$6, $-$4.5, $-$3, $-$1	$-$6, $-$4.5, $-$3, $-$1
$V_{\mathrm{turb}}$	All	$-$5,$-$4, $-$3, $-$1	$-$5,$-$4, $-$3, $-$1
$B_{\mathrm{LOS}}$	All	–	$-$4.5, $-$1

Equations

$$V(\lambda )=-\Delta {\lambda }_{\mathrm{B}}\bar{g}\cos \theta \frac{\partial I}{\partial \lambda },$$

$$\Delta {\lambda }_{\mathrm{B}}=4.67\times 10^{-13}{\lambda }_0^{2}B,$$

$$B_{\mathrm{LOS}}=-\frac{\Sigma_{\lambda}\frac{\partial I_{\mathrm{\lambda}}}{\partial\lambda}V(\lambda)}{C\Sigma_{\mathrm{\lambda}}(\frac{\partial I_{\mathrm{\lambda}}}{\partial\lambda})^{2}}{\color[rgb]{1,0,0},}$$

$$I_{\mathrm{ref}}^{\mathrm{degraded}}(\lambda )=(1-\nu )I_{\mathrm{ref}}(\lambda )\ast g(\lambda ,\sigma )+\nu \ast I_{\mathrm{ref}}^{\mathrm{c}}(\lambda )$$

$$g_{J_1-l}=1+\gamma (J,1/2,K)+\gamma (J,K,1/2)\gamma (K,J_1,l)\gamma (J_1,S_1,L_1)$$

$$\gamma (A,B,C)=\frac{A(A+1)+B(B+1)-C(C+1)}{2A(A+1)}$$

Full text

Spectropolarimetry, Radiative Transfer, Inversions, Weak Field Approximation

††facilities: SPINOR(DST), STiC††software: SunPy, NumPy, matplotlib, RH, STiC, pyMilne

1 Introduction

Between the bright solar surface and million degrees hot corona, the chromosphere is one of the most dynamic and complex layers of the solar atmosphere. Understanding the magnetic coupling of the chromosphere with the photosphere, transition region and corona may reveal how mass and energy are supplied to the corona and heliosphere. Therefore, simultaneous magnetic field measurements at multiple heights of the solar atmosphere are important.

Magnetic field measurements in the photosphere are routinely made, and significant progress has been made in the techniques that allow for chromospheric magnetic field measurements. The most widely used lines to probe the magnetic field of the solar chromosphere are the Ca II 8542 Å  and He I 10830 Å  lines (Lagg et al., 2017). While using these lines to probe the chromospheric magnetic field has its advantages like the relatively well-understood line formation, and they can be interpreted utilizing non-local thermodynamic equilibrium (non-LTE) inversion codes (e.g. Socas-Navarro et al., 2000a; Asensio Ramos et al., 2008; de la Cruz Rodríguez et al., 2016, 2019; Ruiz Cobo et al., 2022), they also have some limitations. The He I 10830 Å  line forms in a narrow range of heights in the upper chromosphere, and line formation depends on coronal and transition region EUV radiation (Andretta & Jones, 1997; Leenaarts et al., 2016). The Ca II 8542 Å  line forms from the upper photosphere to the mid-chromosphere, however, the Ca II ion gets ionized to the Ca III ion in flaring regions, and hence the Ca II 8542 Å  line itself may sample deeper layers of the solar atmosphere (Kerr et al., 2016; Kuridze et al., 2018; Bjørgen et al., 2019).

On the other hand, the H $\alpha$  line seems to probe a relatively wider temperature range. Carlsson & Stein (2002) have shown that the H $\alpha$  opacity in the upper chromosphere is determined by the ionization degree and radiation field. Using 3-D radiative transfer calculations, Leenaarts et al. (2012a) have shown that ionization degree and the radiation field are insensitive to local temperature variations over time but determined by mass density. More recently, Bjørgen et al. (2019) have also confirmed that the H $\alpha$  line retains opacity even in flaring active regions by synthesizing spectra using a 3D rMHD simulation. Therefore, the H $\alpha$  line always retains opacity in the chromosphere.

In spite of the diagnostic capabilities of the H $\alpha$  line, very few spectropolarimetric observations have been reported in the literature. For example, Abdussamatov (1971) estimated vertical magnetic field gradient using simultaneous measurements of the Fe I 6302.5 Å  and H $\alpha$  lines. Simultaneous spectropolarimetric observations from the H $\alpha$  line and lines of Fe I atom have been reported in the literature to compare the photospheric and chromospheric magnetic fields in sunspots (Balasubramaniam et al., 2004; Hanaoka, 2005; Nagaraju et al., 2008). Radial variation of the line-of-sight magnetic field in the chromosphere and photosphere of a sunspot were discussed by Nagaraju et al. (2020a). Magnetic fields in prominences were diagnosed using spectropolarimetric observations of the H $\alpha$  line (López Ariste et al., 2005). Jaume Bestard et al. (2022) studied linear and circular polarization signals near the North and South Solar Limb and inferred the LOS magnetic field using the weak field approximation (WFA).

The reason for so few spectropolarimetric observations of the H $\alpha$  line have been reported and why this has not been a preferred line for the chromospheric magnetometry is because it is challenging to model the H $\alpha$  line. Though the Stokes $V$ signal is dominated by the Zeeman effect, both the intensity and polarization profiles of this line are sensitive to the 3D radiation field. Furthermore, in case of weakly magnetized atmospheres, the Stokes $Q$ & $U$ signals are sensitive to atomic polarization (Štěpán & Trujillo Bueno, 2010, 2011), making this line difficult to model using the currently available inversion codes which adopt 1.5D plane-parallel geometry. It is also important to note that in the weak field regime (when Zeeman Splitting, $\Delta\lambda_{\mathrm{B}}$ , is much smaller than the Doppler width, $\Delta\lambda_{\mathrm{D}}$ ), the amplitude of circular polarization is proportional to the ratio of $\Delta\lambda_{\mathrm{B}}$  to $\Delta\lambda_{\mathrm{D}}$ , and the linear polarization is proportional to the square of that ratio (for more details see page 405 of Landi Degl’Innocenti & Landolfi, 2004). This ratio, owing to the light weight of the Hydrogen atom-thus correspondingly large Doppler width, is typically small compared to that in the case of heavier atoms like Calcium.

In a study by Socas-Navarro & Uitenbroek (2004) who calculated 1-D response functions of Stokes parameters of the H $\alpha$  line and showed that it exhibits significant sensitivity to the photospheric magnetic fields in addition to the chromospheric magnetic fields. However, it has been shown by Leenaarts et al. (2012a) that 1-D radiative transfer is not a good approximation to model the H $\alpha$  line, treatment of radiative transfer in 3-D is necessary. When the radiative transfer is treated in 3-D, the H $\alpha$  line traces the chromospheric magnetic features like fibrils, and it has been shown that it is a good chromospheric diagnostic. This finding is further supported by recent work by Bjørgen et al. (2019).

In this study, we explore the diagnostic potential of the H $\alpha$  line to probe the chromospheric magnetic field using spectropolarimetric observations simultaneously recorded in the H $\alpha$  and Ca II 8542 Å  lines. We compare the magnetic field inferred using the WFA method on the Stokes $I$ and $V$ profiles of the H $\alpha$  spectral line with the stratification (from the photosphere to the chromosphere) of line-of-sight (LOS) magnetic field inferred using the inversions of the Ca II 8542 Å  line using a non-LTE inversion code.

2 Observations

The observations were made with the Spectro-Polarimeter for Infrared and Optical Regions (SPINOR: Socas-Navarro et al., 2006) instrument at the Dunn Solar Telescope (Dunn, 1969) at the Sacramento Peak Observatory in the H $\alpha$  and Ca II 8542 Å  lines simultaneously. The Fe I 6569 Å  line was recorded in the H $\alpha$  spectrum and the Si I 8536 Å  and Fe I 8538 Å  lines were recorded in the Ca II 8542 Å  spectrum. The spectral sampling of the H $\alpha$  and Ca II 8542 Å  lines data are 22 mÅ  and 33 mÅ , respectively. The pixel scale corresponds to $\sim 0\farcs 38$ on the solar surface along the slit. The observed field of view (FOV) consists of a pore centered at North 28o.3 and East 16o.7 in Stonyhurst Heliographic coordinates system on December 4, 2008, starting at 15:35 UT with a viewing angle $\cos\theta=\mu=0.8$. Here, $\theta$ is the angle between the LOS direction and the local surface normal. Twenty spectropolarimetric raster scans of 20 slit positions with a step size of $0\farcs 375$ were recorded; however, only four scans had good seeing conditions, and only the first scan is used in this study. Adaptive optics (AO; Rimmele, 2000) were used during the observations. The data were corrected for dark and flat field variations and instrumental polarization. The details are given in appendix A. In spite of strictly simultaneous observations, there was a spatial shift, due to atmospheric refraction, between the images at these two wavelengths, which were taken care of by co-aligning the H $\alpha$  data with the Ca II 8542 Å  data by cross-correlating the far wing images of the H $\alpha$  and Ca II 8542 Å  lines. The signal-to-noise ratio (SNR) in the H $\alpha$  data is higher ($5\times 10^{3}$) than in the Ca II 8542 Å  data ($10^{3}$).

An overview of observations is shown in Fig. 1. A pore almost centrally located in the FOV can be seen in the far wing images of the Ca II 8542 Å  and H $\alpha$  lines (see panels (a) and (b) of Fig. 1). The pore morphologically has a different shape in the Ca II 8542 Å  and H $\alpha$  lines and is brighter in the H $\alpha$  line core image (see panels (c) and (d) of Fig. 1). Panels (e) and (f) show the LOS magnetic field ($B_{\mathrm{LOS}}$ ) inferred from the Ca II 8542 Å  and Fe I 6569 Å  lines, respectively. More details on the methods to infer the $B_{\mathrm{LOS}}$  are discussed in section 2. There is an opposite polarity region near the pore visible in the photosphere that is absent in the WFA $B_{\mathrm{LOS}}$  map of the chromosphere.

The quiet-Sun profiles for the H $\alpha$  and the Ca II 8542 Å  lines were calculated by averaging profiles of a few pixels in a region away from the pore with negligible signal in the Stokes $V$ profiles. As explained in appendix A, the wavelength calibration was done by comparing the quiet-Sun profile with the BASS 2000 atlas profile. The Ca II 8542 Å  and H $\alpha$  data were also corrected for spectral veil.

In the following paragraphs, we discuss a few selected profiles which are chosen such that they show distinct features from one another corresponding to various dynamics observed in the FOV. In the top two panels of Fig. 2 we present sample spectral images of Stokes $I$ and $V$ corresponding to a slit position marked by a brown line in Fig. 1. In the bottom two panels of Fig. 2, we show a few profiles from selected spatial locations marked using colored horizontal lines in the top two panels. The left and right columns of panels correspond to the Ca II 8542 Å  and H $\alpha$  spectrum, respectively.

There is a hint of enhancement in the intensity in the red wing of the Ca II 8542 Å  line profile at 8542.38 Å  ($\Delta v=+10.31$ km s*-1*), at slit position near $\sim 4\farcs 2$ with the line core position (8542.04 Å) slight blue-shifted ($\Delta v=-$1.5 km s*-1*) compared to the quiet-Sun profile (see violet-colored profile). The corresponding H $\alpha$  profile shows nominal absorption. The violet-colored profile lies in the opposite polarity region. The Stokes $V$ profiles of the Ca II 8542 Å  and H $\alpha$  lines have weak signal with sign opposite to that of the Stokes $V$ profile of the Fe I 6569 Å . There is hardly any signal in the Stokes $V$ profile of the Si I 8536 Å  and Fe I 8538 Å  lines that is above the noise level. This is because, as noted above, the SNR in Ca II 8542 Å  spectrum is about 5 times lower compared to that of H $\alpha$  spectrum.

The intensity profile of the Ca II 8542 Å  line at slit position near $7\farcs 8$ (blue-colored profile) shows emission in the blue wing at 8541.9 Å  ($\Delta v=-$6.56 km s*-1*) and line core (8542.12 Å) redshifted ($\Delta v=+$1.4 km s*-1*), signature of surge flow (more details are discussed in Nagaraju et al. under prep.). The corresponding H $\alpha$  and the wings of Ca II 8542 Å  intensity profiles also show red excursion compared to the quiet-Sun profile. The Stokes $V$ profile of the Ca II 8542 Å  line shows a sign reversal compared to the Stokes $V$ profile of the Fe I 6569 Å  which is not due to change in the polarity of the magnetic field but due to emission in the blue wing (Sanchez Almeida, 1997). Such emission feature can only be caused by a change in the gradient of the source function, which also affects the sign of the Stokes $V$ signal.

A sample profile over the pore is shown in cyan color. The Ca II 8542 Å  and H $\alpha$  lines show asymmetric Stokes $V$ profiles with blue and red lobe amplitudes of ($\sim$9%, $\sim$6%) and ($\sim$1.3%, $\sim$0.9%), respectively. In contrast, the Si I 8536 Å , Fe I 8538 Å  and Fe I 6569 Å  lines show relatively symmetric Stokes $V$ profiles of amplitudes $\sim$1.6%, $\sim$1% and $\sim$3.5% respectively.

There is a hint of enhancement in the red wing of the Ca II 8542 Å  line profile at the edge of the pore (yellow colored profile). The Stokes $V$ profiles of the Ca II 8542 Å  and H $\alpha$  lines (similar to cyan-colored profile) show amplitude asymmetry. This could be because of the presence of multiple Stokes components within one resolution element (Socas-Navarro et al., 2000b) or gradients in LOS velocity and magnetic field (Sankarasubramanian & Rimmele, 2002).

The brown-colored profile is an example of a quiescent profile away from the pore region. The Stokes $V$ amplitudes of the Ca II 8542 Å  and H $\alpha$  line profiles are similar (0.2%).

Fig. 3 shows spectropolarimetric images and spectral profiles for the slit position shown in green color in Fig. 1 which passes over a location with stronger field within the negative polarity region and the pore edge. The profiles over the negative polarity region are shown in green and purple colors. The Stokes $V$ amplitudes of the Si I 8536 Å  and Ca II 8542 Å  lines are $\sim$0.7%. The green and purple colored intensity profiles of the Ca II 8542 Å  line are blueshifted by about $\Delta\lambda=-0.04$ and $-$0.02 Å  ($\Delta v=-1.4$ and 0.7 km s*-1*), respectively, whereas there is no Doppler shift seen in the Si I 8536 Å  and Fe I 8538 Å  lines. The sign of the Stokes $V$ profiles of the Si I 8536 Å , Fe I 8538 Å  and Fe I 6569 Å  lines is opposite to that of the sign of the Stokes $V$ profiles of the Ca II 8542 Å  and H $\alpha$  lines which suggest that the Stokes $V$ profile of the H $\alpha$  line probes the chromospheric magnetic field. This most probably is because the H $\alpha$  and Ca II 8542 Å  line cores are sampling the canopy fields extending from a nearby region and overlying the opposite polarity region. To further demonstrate the existence of such field configuration, we show in Fig. 4 sample spectral images and profiles of the same region but from a raster scan recorded at a later time (16:08 UT). As clearly seen in this figure that the sign of the Stokes $V$ profiles corresponding to the core of Ca II 8542 Å  and H $\alpha$  lines are opposite to that of Stokes $V$ profiles of the photospheric lines viz., Si I 8536 Å , Fe I 8538 Å , and Fe I 6569 Å (see spectral images from 2$\arcsec$ to 6$\arcsec$). On the other hand, the Stokes profiles in the wings of the Ca II 8542 Å  and H $\alpha$  lines show the polarity the same as that of the photospheric lines. This is consistent with the canopy field scenario as explained above since the line wings of H $\alpha$  and Ca II 8542 Å  lines form deeper in the solar atmosphere, and hence the Stokes $V$ profiles have the same sign as that of the photospheric lines. However, as noted before, under weak field conditions, the Stokes $V$ profiles change sign when the spectral features change from absorption to emission or vice-versa. In order to make sure that the change in sign of Stokes $V$ profiles from core to wing of the chromospheric lines is actually because of the change in polarity of the magnetic field but not due to change in spectral features, we have over-plotted the first derivative of Stokes $I$ ($\frac{dI}{d\lambda}$: the dotted green curves in the bottom panel of Fig. 4 with the sign changed) over the Stokes $V$ profiles. The reason for doing this is that under weak field conditions, Stokes $V$ profile resembles $\frac{dI}{d\lambda}$ profile (see section 3.1). The comparison between the Stokes $V$ and $\frac{dI}{d\lambda}$ profiles clearly demonstrate that the Stokes $V$ sign change from core to wings of the chromospheric lines is due to the change in the polarity of the magnetic field but not due to change in the emission or absorption features of Stokes $I$.

The profile shown in maroon color (Fig. 3) is another example of a profile in pore region. There is a red excursion in the Ca II 8542 Å  line wings and H $\alpha$  line core profile corresponding to a surge flow. The Stokes $V$ profiles of the Ca II 8542 Å , H $\alpha$ , and Fe I 6569 Å  lines show amplitude asymmetry, and the blue lobe show a positive sign suggesting positive polarity of the $B_{\mathrm{LOS}}$ .

The pink and khaki-colored profiles show profiles away from the pore region. The intensity profiles of the Ca II 8542 Å  and H $\alpha$  lines are similar to that of the quiet-Sun profile. Amplitude asymmetry is present in the Stokes $V$ profiles of the Ca II 8542 Å  and H $\alpha$  lines.

3 Methods

3.1 Weak field approximation

The magnetic field from the H $\alpha$  spectral line was inferred using the WFA. Under WFA the Stokes $V$ is linearly related to $B_{\mathrm{LOS}}$  and ($\partial I/\partial\lambda$) through (Landi Degl’Innocenti & Landolfi, 2004)

[TABLE]

and

[TABLE]

where $\Delta\lambda_{\mathrm{B}}$  is expressed in Å , $B$ in Gauss, $\bar{g}$ is effective Landé factor, $\theta$ is the inclination of $B$ with respect to the LOS, and $\lambda_{0}$ is the central wavelength of the spectral line (expressed in Å).

The $B_{\mathrm{LOS}}$  can be derived from Eq.1 using the linear regression formula (e.g., Martínez González & Bellot Rubio, 2009),

[TABLE]

where $C=4.66\times 10^{-13}\;\bar{g}\;\lambda_{\mathrm{0}}^{2}$.

We have used $\bar{g}=1.048$ following the investigation done by Casini & Landi Degl’Innocenti (1994). We derived three values of $B_{\mathrm{LOS}}$ , one from the line core (H $\alpha$ $\pm$0.35 Å), the line wings ($[-1.5,-0.6]$ and $[+0.6,+1.5]$ Å ), and over the full H $\alpha$  spectral line (H $\alpha$ $\pm$1.5 Å). The spectral blends listed in Table 1 were excluded while calculating $B_{\mathrm{LOS}}$  using the WFA, as have done by Jaume Bestard et al. (2022) and Nagaraju et al. (2020a). The WFA is applied on the Ca II 8542 Å  line within the wavelength range $\lambda_{0}\pm 0.25$ Å  and the inferred $B_{\mathrm{LOS}}$  map is shown in panel (e) of Fig. 1. The uncertainties in the values of $B_{\mathrm{LOS}}$  inferred from applying the WFA to the Ca II 8542 Å  and H $\alpha$  data were estimated to be 23 and 18 G, respectively.

3.2 Milne-Eddington inversion

We performed Milne-Eddington (ME) inversions (see chapter 11 of del Toro Iniesta, 2007) of the Fe I 6569 Å  data to infer the $B_{\mathrm{LOS}}$  (panel (f) of Fig. 1) utilizing pyMilne code, a parallel C++/Python implementation222https://github.com/jaimedelacruz/pyMilne (de la Cruz Rodríguez, 2019). The $B_{\mathrm{LOS}}$  from the H $\alpha$  line is inferred from the WFA as explained in section 3.1. The stratification of the $B_{\mathrm{LOS}}$  from the photosphere to the chromosphere is inferred using simultaneous multi-line non-LTE inversions explained in section 3.3.

3.3 Non-LTE inversion

The MPI-parallel STockholm inversion Code (de la Cruz Rodríguez et al., 2019, 2016) is used to retrieve the stratification of atmospheric parameters. STiC is based on a modified version of the RH radiative transfer code (Uitenbroek, 2001) and solves the polarised radiative transfer equation using cubic Bezier solvers (de la Cruz Rodríguez & Piskunov, 2013). In non-LTE, assuming statistical equilibrium, it can fit multiple spectral lines simultaneously. It employs the fast approximation to account for partial re-distribution effects (PRD) (for more details Leenaarts et al., 2012b). STiC assumes plane-parallel geometry to fit the intensity in each pixel (also called the 1.5D approximation). STiC uses an LTE equation-of-state obtained from the library functions in the Spectroscopy Made Easy (SME) package code (Piskunov & Valenti, 2017). The optical depth scale at 5000 Å  (500 nm), abbreviated $\log\tau_{\mathrm{500}}$ , is used to stratify atmospheric parameters.

We have inverted the Stokes $I$ and $V$ profiles of the Ca II 8542 Å , Si I 8536 Å  and Fe I 8538 Å  lines simultaneously to infer the stratification of temperature ($T$ ), LOS velocity ($V_{\mathrm{LOS}}$ ), microturbulence ($V_{\mathrm{turb}}$ ) and LOS magnetic field ($B_{\mathrm{LOS}}$ ). We used a 6-level Ca II atom. The Ca II H&K profiles were modeled in PRD (Milkey & Mihalas, 1974; Uitenbroek, 1989), and Ca II IR lines were synthesized in complete re-distribution (CRD) approximation. The atomic parameters of the Si I 8536 Å  and Fe I 8538 Å  lines were obtained from the Vienna Atomic Line Database (VALD3) (Ryabchikova et al., 2015), and Kurucz’s line lists (Kurucz, 2011), respectively and synthesized under LTE approximation. The upper level of the Si I 8536 Å  transition was treated with $J_{1}$-$l$ ($JK$) coupling scheme as described in appendix B. The latest version of the STiC inversion code has been upgraded to allow for the treatment of atomic levels of Kurucz’s lines in $JK$ coupling. An empirical $\log\mathrm{gf}$ value of $-1.4$ was used for the Fe I 8538 Å  transition (Socas-Navarro, 2007).

We used the $k$-means clustering to group the Stokes $I$ profiles in different clusters such that similar-shaped profiles were grouped in one cluster. We then inverted the mean profile of each of those clusters to derive the stratification of $T$ , $V_{\mathrm{LOS}}$ , and $V_{\mathrm{turb}}$ . Finally, the inferred stratification was used as the initial guess atmosphere to infer the stratification of the atmospheric parameters of the FOV, similar to the approach used by Nóbrega-Siverio et al. (2021) and Mathur et al. (2022). In the second cycle, we used the values of $B_{\mathrm{LOS}}$  derived from the ME inversions from the Fe I 6569 Å  line and WFA of the Ca II 8542 Å  line as the guess values of the $B_{\mathrm{LOS}}$  at $\log\tau_{\mathrm{500}}$  = $-$1 and $-$4.5, respectively. Earlier studies have found that the Stokes $V$ profiles of the Ca II 8542 Å  line have maximum response to the perturbations in the $B_{\mathrm{LOS}}$  between $\log\tau_{\mathrm{500}}$  = $-$4 and $-$5 (Quintero Noda et al., 2016a; Joshi & de la Cruz Rodríguez, 2018; Murabito et al., 2019). Table 2 describes the node positions used for different categories of profiles. Quiescent profiles are nominal absorption profiles (910 profiles), and emission profiles are profiles that have an emission (or a hint of emission) in either blue or red or both the wings of the Ca II 8542 Å  line (110 profiles). The quality of inversion fits are discussed in appendix C.

We set the average velocity in the pore region in the photosphere ($\log\tau_{\mathrm{500}}$ range of [$-1$, [math]]) to rest for the absolute velocity calibration. With respect to the average profile in the pore, the quiet-Sun profile is blue-shifted, that is, after velocity calibration, the $V_{\mathrm{LOS}}$  stratification inferred from the quiet-Sun profile shows an upflow of about $-$3 km s*-1*.

4 Results and discussion

4.1 Results from the WFA and ME inversions

The $B_{\mathrm{LOS}}$  map inferred from the ME inversion of Stokes $I$ and  $V$ profiles of Fe I 6569 Å  line is shown in panel (f) of Fig. 1. The maximum $B_{\mathrm{LOS}}$  strength found is +800 G. The panel (e) of Fig. 1 shows the $B_{\mathrm{LOS}}$  map inferred from WFA applied on the Ca II 8542 Å  within the wavelength range of $\lambda_{0}\pm 0.25$ Å  and the maximum $B_{\mathrm{LOS}}$  field strength found is +600 G. The region in the photosphere with opposite polarity of the $B_{\mathrm{LOS}}$  with respect to the pore has a field strength of about $-$200 G which is absent in the chromospheric $B_{\mathrm{LOS}}$  map.

4.2 Non-LTE inversion results

In this section we discuss about the results from the inversion of Stokes $I$ and  $V$ profiles of the Ca II 8542 Å , Si I 8536 Å  and Fe I 8538 Å  lines using the STiC inversion code.

In Fig. 5 we show inversion results of profiles averaged (to increase SNR) from a 3$\times$3 box about the selected regions of the FOV, viz., pore, surge flow, and region with opposite polarity of the magnetic field. The average profile for the pore is calculated by averaging the pixels about the darkest pixel in the pore, for the surge flow profile by averaging about the blue-colored profile in Fig. 2 and for the opposite polarity profile by averaging about the green colored profile in Fig. 3.

The $T$  stratification inferred from the pore profile has lower value of $T$  at the photospheric layers ($\log\tau_{\mathrm{500}}$ $\sim-$1) and higher values at the chromospheric layers ($\log\tau_{\mathrm{500}}$ $\leq-$4) compared to that inferred from the inversion of the quiet-Sun profile. The $V_{\mathrm{LOS}}$  at photospheric layers ($\log\tau_{\mathrm{500}}$ $\sim-$1) is zero (calibrated) and at the chromospheric layers ($\log\tau_{\mathrm{500}}$  = $-$4.5) show upflow of about $-$2 km s*-1*. The $V_{\mathrm{turb}}$  is non-zero between $\log\tau_{\mathrm{500}}$  = $-$1 and $-$4. The value of $B_{\mathrm{LOS}}$  at the photospheric and chromospheric layers is about $+$800 G and $+$400 G, respectively.

The reversal of the sign of the Stokes $V$ profile of the Ca II 8542 Å  line shown in Fig. 5 corresponding to surge flow region is a result of emission feature in the blue wing of the Ca II 8542 Å  line and is not an indication of any change in polarity of the $B_{\mathrm{LOS}}$ . Accordingly the $T$  show enhancement of about 600 K compared to $T$  inferred from the quiet-Sun profile at $\log\tau_{\mathrm{500}}$  = $-$4.5 simultaneous with an upflow of about $-$4 km s*-1*. For the pixels having spectral profiles similar to the surge flow, i.e., an emission feature in the blue wing and a red excursion in the Stokes $I$, the $V_{\mathrm{turb}}$  is small at all depth positions of $\log\tau_{\mathrm{500}}$ . It may be possible that such low values of $V_{\mathrm{turb}}$  may have some contribution in the higher temperature values inferred for the pixels in the surge flow region. Díaz Baso et al. (2022) have shown that there is a degeneracy between the $T$  and the $V_{\mathrm{turb}}$ . When the spectral profiles are in emission, the $T$  and $V_{\mathrm{turb}}$  are anti-correlated; that is, an increase in $T$  broadens the spectral line, and thus to maintain the same width, the $V_{\mathrm{turb}}$  must be decreased. However, the enhancement in $T$  and minimal values of $V_{\mathrm{turb}}$  are necessary to achieve a satisfactory fit of the emission feature (for more discussion see Mathur et al., 2022). The value of $B_{\mathrm{LOS}}$  at the photospheric and chromospheric layers is about $+$50 G and $+$200 G, respectively. The reason for the increased value of the magnetic field in the chromosphere compared to the photosphere is because the line core of the Ca II 8542 Å  line is sampling the canopy fields overlying the pore and the nearby opposite polarity region.

The $T$ , $V_{\mathrm{LOS}}$  and $V_{\mathrm{turb}}$  stratification inferred from the opposite polarity profile is similar to that inferred from the quiet-Sun profile. The value of $B_{\mathrm{LOS}}$  at the photospheric and chromospheric layers is about $-$220 G and $+$250 G, respectively, again suggesting a canopy structure.

As described in section 2, above the positive polarity region, the sign of the Stokes $V$ profile in the wings of the Ca II 8542 Å  and H $\alpha$  lines is opposite to that of in the core. This is probably because the line cores of the H $\alpha$  and Ca II 8542 Å  lines are sampling the canopy fields overlying the opposite polarity region. To verify that the wings and the core of the Ca II 8542 Å  line are indeed sampling different layers of the atmosphere, in Fig. 6 we show the response to perturbations of the $B_{\mathrm{LOS}}$  at one wavelength position in the wing ($\Delta\lambda$ = $-$1.03 Å) and another one in the core ($\Delta\lambda$ = $-$0.11 Å) of the line where the sign of the Stokes $V$ is opposite. Beckers & Milkey (1975) define the response function (RF) for a physical parameter $X$ as $RF_{X}(\tau,\lambda)\;=\;\delta I(\lambda)/\delta X(\tau)$. Response functions contain information on how the Stokes parameters at different wavelength positions are sensitive to perturbations of a physical parameter at different $\log\tau_{\mathrm{500}}$ . The response function in the line wing of the Ca II 8542 Å  line shows a dominant contribution of the photospheric fields ($\log\tau_{\mathrm{500}}$ $\simeq-$1.5), in contrast, the response function near the line core shows a dominant contribution of the chromospheric fields ($\log\tau_{\mathrm{500}}$  $\leq$ $-3$).

The maps of $T$ , $V_{\mathrm{LOS}}$  and $V_{\mathrm{turb}}$  inferred from the inversions of the FOV at $\log\tau_{\mathrm{500}}$  = $-$4.5, $-$3 and $-$1 are presented in Fig. 7. The morphological structure of the $T$  map at $\log\tau_{\mathrm{500}}$  = $-$1 is similar to the far wing image of the Ca II 8542 Å  line (see panel (a) of Fig. 1) with a decrease in $T$  of about $300$ K in the pore compared to surrounding background. There is a weak downflowing ($+$2 km s*-1*) region near ($x$, $y$) = ($3\farcs 5$, $8\farcs 8$) while the rest of the pore does not show any signature of plasma flows. The value of $V_{\mathrm{turb}}$  is about 0–1 km s*-1* in the $V_{\mathrm{turb}}$  maps at $\log\tau_{\mathrm{500}}$  = $-$1 except at a few regions where the $V_{\mathrm{turb}}$  is about 2–3 km s*-1* such as the pore boundary (6$\arcsec$, $10\farcs 8$).

The $T$  at $\log\tau_{\mathrm{500}}$  = $-$3 ranges from 4–5.5 kK with a region of higher temperature ($T$ $\sim$5.4 kK) seen near the pore boundary (6$\arcsec$, $10\farcs 8$). The temperature near the negative polarity region (in ME $B_{\mathrm{LOS}}$  map) is about 4.2 kK. The dark fibril-like lanes starting at (2$\arcsec$, 8$\farcs$8), indicated by the ellipse-shaped contour in Fig. 7, can also be seen in the H $\alpha$  line core image (see panel (d) of Fig. 1). The majority of the pore region show upflows of up to $-3$ km s*-1*. The $V_{\mathrm{turb}}$  inside the pore is about 0–3 km s*-1* and outside the pore is about 3–5 km s*-1*.

The morphological structure of the $T$  map at $\log\tau_{\mathrm{500}}$  = $-$4.5 looks similar to the Ca II 8542 Å  line core image (see panel (c) of Fig. 1). The brightest and darkest pixels in panel(c) of Fig. 1 correspond to $T$  of about $5.7$ kK and $3.8$ kK, respectively. In general, the FOV show upflows of up to $-$3 km s*-1* with two small regions showing downflows of about $+$2 km s*-1*. The strong upflows up to $-$5 km s*-1* are located in regions of higher temperature. There is almost zero $V_{\mathrm{turb}}$  in the $V_{\mathrm{turb}}$  map in all regions of the FOV.

The panels (a) and (b) of Fig. 8 show maps of the magnetic field at $\log\tau_{\mathrm{500}}$  = $-$1 and $-$4.5. The change in the stratification of |$$B_{\mathrm{LOS}} $|$ inferred from inversions is shown in the middle panel of the bottom row of Fig. 8. The morphological structure of the $B_{\mathrm{LOS}}$  map at the photospheric layers (at $\log\tau_{\mathrm{500}}$  = $-$1, panel (a) of Fig. 8) is similar to panel (e) of Fig. 1 with strong positive polarity in the pore ($\sim+$800 G) and nearby regions ($\sim+$350 G). An opposite polarity with a magnitude of $\sim-$200 G is seen in the $B_{\mathrm{LOS}}$  map. The field at $\log\tau_{\mathrm{500}}$  = $-$4.5 has positive polarity with a maximum $B_{\mathrm{LOS}}$  of $\sim+600$ G in the pore and about $\sim+$300 G in the opposite polarity region. In general, the structure of the $B_{\mathrm{LOS}}$  at $\log\tau_{\mathrm{500}}$  = $-$4.5 is more spread out compared to that of at $\log\tau_{\mathrm{500}}$  = $-$1 (see panel (b) of Fig. 8). In addition, in the regions outside of the pore the $B_{\mathrm{LOS}}$  strength at $\log\tau_{\mathrm{500}}$  = $-$4.5 has increased compared to that at $\log\tau_{\mathrm{500}}$  = $-$1, suggesting a magnetic canopy-like structure (see middle panel in the bottom row of Fig. 8). Using multiple spectral lines, the magnetic canopies around pores have also been reported by many authors in the recent literature (for eg. Stauffer et al., 2022; Tapia & Bellot Rubio, 2022; Buehler et al., 2019; Murabito et al., 2016; Shimizu et al., 2012; Keppens & Martinez Pillet, 1996). The magnetic field strengths inferred at $\log\tau_{\mathrm{500}}$  = $-$1 (the photosphere) and $\log\tau_{\mathrm{500}}$  = $-$4.5 (the chromosphere) are comparable with the values reported by many authors in recent literature, who studied the stratification of $B_{\mathrm{LOS}}$  in pore using spectral lines of the Fe I atom, and Ca II 8542 Å  and He I 10830 Å  lines (Sowmya et al., 2022; Nagaraju et al., 2020b; Yadav et al., 2019; Quintero Noda et al., 2016b; Jurčák et al., 2015; Sobotka et al., 2013; Criscuoli et al., 2012).

4.3 Comparison of $B_{\mathrm{LOS}}$  inferred from the Ca II 8542 Å  and H $\alpha$  lines

The panels (c), (d) and (e) of Fig. 8 show maps of the magnetic field inferred from the WFA method applied to the Stokes $I$ and $V$ profiles of the H $\alpha$  line. The difference in amplitude of the |$$B_{\mathrm{LOS}} $|$ inferred from the H $\alpha$  line with that of inferred from the inversions is shown in the two rightmost panels of the bottom row of Fig. 8. Figure 9 shows the scatter plots between the |$$B_{\mathrm{LOS}} $|$ inferred from inversions and the WFA.

The field strengths inferred from inversions of the Ca II 8542 Å  line at $\log\tau_{\mathrm{500}}$  = $-$4.5 are comparable to those of inferred from the WFA of the Ca II 8542 Å  line (see panel (a) of Fig. 9), suggesting consistency between the two methods in inferring the $B_{\mathrm{LOS}}$ . The magnetic field map inferred from the WFA of the H $\alpha$  line core (H $\alpha$ $\pm$0.35 Å) has a similar morphological structure to the map inferred from inversions of Ca II 8542 Å  line at $\log\tau_{\mathrm{500}}$  = $-$4.5 (see panels (b) and (c) of Fig. 8) suggesting the H $\alpha$  line core probes the chromospheric magnetic field. The $B_{\mathrm{LOS}}$  field strength inferred from WFA of H $\alpha$  line core is almost half (0.53 times) of that of inferred from inversions at $\log\tau_{\mathrm{500}}$  = $-$4.5 and the WFA of the Ca II 8542 Å  line (see the rightmost panel in the bottom row of Fig. 8 and panel (b) and (c) of Fig. 9). This could be due to sensitivity of the core of the H $\alpha$  line to the magnetic field in the higher atmospheric layers than that of Ca II 8542 Å  line. Or due to systematic underestimation of $B_{\mathrm{LOS}}$  using the WFA from the H $\alpha$  line. A detailed analysis of the H $\alpha$  Stokes $I$ and $V$ profiles synthesized using current state-of-the-art model atmospheres of active regions and quiet Sun, taking into account the fine-structure sub-levels ($nlj$) of the atomic levels ($n$), 3D radiative transfer and the Zeeman effect, is required.

The maps of $B_{\mathrm{LOS}}$  inferred from the WFA method on the wings and the full spectral range of the H $\alpha$  line show a morphological structure similar to that of $\log\tau_{\mathrm{500}}$  = $-$1 (see panel (d) and (e) of Fig. 8), suggesting a significant contribution from the photospheric fields. However, the negative polarity region is not clearly seen. This is because, as explained in section 2, there are very few spectral pixels with a clear signal of opposite sign in Stokes $V$ profile in the line core and wings of the H $\alpha$  line. The negative polarity region is very well reproduced through inversions because of good signal in Stokes $V$ profiles of the Si I 8536 Å  and Fe I 8538 Å  lines. The |$$B_{\mathrm{LOS}} $|$ inferred from the full spectral range of the H $\alpha$  line is weaker by a factor of 0.42 than that of $B_{\mathrm{LOS}}$  at $\log\tau_{\mathrm{500}}$  = $-$1, which is consistent with previous studies (see panel (d) of the Fig. 9) (Abdussamatov, 1971; Balasubramaniam et al., 2004; Hanaoka, 2005; Nagaraju et al., 2008).

The above comparison of $B_{\mathrm{LOS}}$  inferred from the H $\alpha$  and the Ca II 8542 Å  lines suggest that the line core of the H $\alpha$  line is sensitive to the chromospheric magnetic fields while the wings and the full H $\alpha$  line exhibit significant sensitivity to the photospheric magnetic fields.

5 Conclusions

In this paper, we presented an analysis of spectropolarimetric observations recorded simultaneously in the H $\alpha$  and Ca II 8542 Å  lines of a pore with positive magnetic polarity, as well as the surrounding region that covers also a negative polarity magnetic field region. This is towards the goal of exploring the diagnostic potential of the H $\alpha$  line to probe the chromospheric magnetic field. To estimate $B_{\mathrm{LOS}}$  field from the H $\alpha$  line we have used the WFA, and from Ca II 8542 Å  line we have used both the WFA and inversion methods. The similarity between the magnetic field morphology inferred by applying the WFA on the core of the H $\alpha$  and Ca II 8542 Å  lines and inferred from inversions at $\log\tau_{\mathrm{500}}$ =$-$4.5 is a clear evidence that the H $\alpha$  line core probes the chromospheric magnetic field. This evidence is even more striking in the region above the negative polarity region in which the H $\alpha$  and Ca II 8542 Å  line cores exhibit positive polarity where as the line wings exhibit negative polarity. This is because of the canopy fields from the dominant positive polarity region extended at the chromospheric heights overlying above the negative polarity region, which is mostly confined to photospheric heights. It is found from the quantitative comparison that $B_{\mathrm{LOS}}$  fields estimated from the H $\alpha$  line core are about $\approx 0.53$ times that of estimated from the Ca II 8542 Å  line core. This may suggest that the magnetic sensitivity of the H $\alpha$  line core is located in the higher layers in the solar atmosphere than that of the Ca II 8542 Å  line core. However, there is a possibility that $B_{\mathrm{LOS}}$  values are systematically underestimated from the H $\alpha$  line under WFA. Further investigation through multi-line spectropolarimetric observations and 3D radiative transfer calculations are required to fully understand the diagnostic potential of the H $\alpha$  line to probe the chromospheric magnetic field.

We are grateful to the anonymous referee for their valuable feedback, which helped to enhance the quality of this manuscript. This research has made use of the High-Performance Computing (HPC) resources (NOVA cluster) made available by the Computer Center of the Indian Institute of Astrophysics, Bangalore. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (SUNMAG, grant agreement 759548). This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. This research used version 3.1.6 (The SunPy Community et al., 2020) of the SunPy open source software package (Barnes et al., 2020). We have also used the packages h5py (Collette, 2013), matplotlib (Hunter, 2007) and numpy (Harris et al., 2020) to carry out our data analysis.

Appendix A Data reduction

The data are reduced with standard procedures of bias and flat fielding. Calibration data with the procedures described in Socas-Navarro et al. (2006) are used to correct for instrumental polarization. No absolute wavelength calibration is done because of the absence of suitable telluric lines. Instead, we average a few spatial pixels in the quiescent region outside the pore (quiet-Sun profile) and fit the quiet-Sun profile over the full Ca II 8542 Å  spectral range ($I_{\mathrm{obs}}^{\mathrm{mean}}$ ) with the BASS 2000 atlas (Paletou et al., 2009).

The procedures of the spectral veil correction, SI intensity calibration, and estimation of spectral Point Spread Function (PSF) for the Ca II 8542 Å  and H $\alpha$  data are described as follows. We followed the spectral veil correction process described in Borrero et al. (2016) with an additional step correcting the tilt in the spectrum continuum. The continuum of the raw data, because of detector flat-field residuals and pre-filter shape, is tilted. These tilts were corrected in the data reduction pipeline by subtracting a linear fit ($y=a+b\lambda$) with the average spectrum. However, the observed spectral range is not symmetric with respect to the core of the Ca II 8542 Å  and H $\alpha$  lines, and there is an inherent tilt present even in the BASS 2000 atlas. Hence we again corrected this over-correction in the tilts by dividing with a normalized linear fit by matching the continuum intensity levels to that of the reference profile ($I_{\mathrm{ref}}$ ). For the Ca II 8542 Å  data, we used spectra synthesized using the RH code (Uitenbroek, 2001) with FAL-C (Avrett, 1985; Fontenla et al., 1993) model atmosphere at $\mu=0.8$ as $I_{\mathrm{ref}}$ . We ensured that there is a good match of synthesized spectra of the Ca II 8542 Å  line at $\mu=1$ with BASS 2000 atlas Spectrum, which gave us confidence in using $\mu=0.8$ spectra as $I_{\mathrm{ref}}$ . After correcting the continua tilt, we estimated the PSF and straylight fraction. PSF is assumed to be Gaussian ($\sigma$) for the whole Ca II 8542 Å  line spectral range, but straylight fraction ($\nu$) is allowed to vary over the blends of the Si I 8536 Å , the Fe I 8538 Å  and about the Ca II 8542 Å  line core. To estimate $\sigma$ and $\nu$, we minimised the $\chi^{2}$ distance between the $I_{\mathrm{obs}}^{\mathrm{mean}}$  and $I_{ref}^{degraded}$ with $\sigma$ and $\nu$.

[TABLE]

where $I_{\mathrm{ref}}^{\mathrm{c}}$ is the value of intensity at the observed predefined far-wing wavelength point. The absolute SI intensity calibration is done by comparing the intensity of the observed predefined continuum wavelength point with the intensity at degraded $I_{\mathrm{ref}}$ .

We followed a similar process for estimation of the spectral veil and spectral PSF for the H $\alpha$  data, but instead of using synthesized spectrum as $I_{\mathrm{ref}}$ , we inferred $I_{\mathrm{ref}}$  using BASS 2000 atlas H $\alpha$  spectrum (Paletou et al., 2009) and center-to-limb variation calculated from $\mu=1$ to $\mu=0.8$ using the RH code with FAL-C model atmosphere. We used a 6-level Hydrogen atom without a fine structure to synthesize the intensity of H $\alpha$  line with the blend of the Fe I 6569 Å  line. The atomic parameters of the Fe I 6569 Å  line are retrieved from Kurucz’s line lists (Kurucz, 2011) and synthesized in LTE approximation.

Appendix B JK Coupling

In $J_{1}$-$l$ ($JK$) coupling scheme, a ’parent’ level of orbital angular momentum $L_{1}$ and spin $S_{1}$ couples its total angular momentum $J_{1}$ with the orbital angular momentum $l$ of a further electron, to give an angular momentum $K$ which in turn couples with electron’s spin to give total angular momentum $J$ (for more details see page 77 of Landi Degl’Innocenti & Landolfi, 2004).

The Landé factor for level in $J_{1}$-$l$ ($JK$) coupling is

[TABLE]

where

[TABLE]

Appendix C Quality of fits

In Fig. 10 we discuss the match between the observed and synthesized Stokes $I$ and $V$ for the profiles discussed in the paper.

In general, the synthesized Stokes $I$ and $V$ profiles show a good match with the observed profiles. The emission in the blue wing of the Ca II 8542 Å  line and reversal in the Stokes $V$ is very well reproduced in the synthesized profiles (see blue-colored profile). When the Stokes $V$ signal is higher than 1%, a good match is seen in the Si I 8536 Å , Fe I 8538 Å  and Ca II 8542 Å  Stokes $I$ and $V$ profiles, for example, cyan, maroon, pink and khaki colored profiles. The opposite sign is very well reproduced in the Stokes $V$ signal of the Si I 8536 Å  and Fe I 8538 Å  lines and reversal in the Ca II 8542 Å  line far wing Stokes $V$ signal, for example, green and purple colored profile. When the signal in Stokes $V$ is less than 0.5%, the match of the synthesized Stokes $V$ profile is relatively poor compared to the above cases, for example, purple and brown colored profiles.