# Global Coronal Magnetic Field Estimation Using Bayesian Inference

Upasna Baweja Aryabhatta Research Institute of Observational Sciences, 263001,
Nainital, India  Mahatma Jyotiba Phule Rohilkhand University, Bareilly-
243006, Uttar Pradesh, India  Vaibhav Pant Aryabhatta Research Institute of
Observational Sciences, 263001, Nainital, India  Iñigo Arregui Instituto de
Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain  Departamento de
Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain

###### Abstract

Estimating the magnetic field strength in the solar corona is crucial for
understanding different physical processes happening over diverse spatio-
temporal scales. However, the high temperatures and low density of the solar
corona make this task challenging. The coronal magnetic field is too weak to
produce a measurable splitting of the spectral lines using the Zeeman effect,
and high temperature causes spectral lines to become weak and broad, making it
difficult to detect the small Zeeman splitting. Coronal magneto-seismology,
which combines the theoretical and observed properties of magnetohydrodynamic
(MHD) waves, can be used to infer the magnetic field strength of oscillating
structures in the solar corona, which are otherwise difficult to estimate. In
this work, we use the Doppler velocity and density data obtained from the
Coronal Multichannel Polarimeter (CoMP) on 2016 October 14 to obtain the
global map of the coronal magnetic field using Bayesian inference. Two priors
are used for plasma density, viz Gaussian and uniform distributions. Bayesian
inference provides us with the probability distribution for the magnetic field
strength at each location from 1.05 to 1.35 $R_{\odot}$. A comparison between
the magnetic field obtained using simple inversion and Bayesian inference is
also drawn. We find that the values obtained using simple inversion do not
always match the maximum posterior estimates obtained using Bayesian
inference. We find that the inferred values follow a power-law function for
the radial variation of the coronal magnetic field, with the power-law indices
for simple and Bayesian inversion being similar.

The Sun (1693) — Magnetohydrodynamics (1964) — Solar coronal waves (1995) —
Bayesian Statistics (1900)

##  1 Introduction

Different layers of the solar atmosphere are coupled by the magnetic field
(Jess et al., 2016). Thus, the information on the magnetic field is essential
to understanding many physical processes of the solar corona, such as coronal
heating. The magnetic field estimation of the solar corona is challenging
primarily for two reasons. First, the magnetic field in the solar corona is
three orders of magnitude lower than in the solar photosphere. This small
magnetic field strength produces a minimal Zeeman’s splitting of the spectral
lines. Second, the million-degree temperature of the solar corona broadens the
spectral lines. Altschuler & Newkirk (1969), and Schatten et al. (1969)
proposed the potential free source surface (PFSS) model, an extrapolation
method to derive the coronal magnetic field strength. This model was later
refined by Wang & Sheeley (1992). Different extrapolation methods: potential
free (Lee et al., 1999), linear force-free field and non-linear force-free
field (Yan & Sakurai, 2000; Régnier, 2007; Su et al., 2009; He et al., 2011;
Xue et al., 2016; Huang et al., 2018) are widely used to reconstruct the
coronal magnetic field. These models utilize the force-free condition by
assuming that the solar corona is current-free or that the currents are not
influencing the global magnetic field structure. Such assumptions are not
always valid. For instance, the solar magnetic field is expected to be
potential, where the dominant force is the magnetic force with no magnetic
helicity, and the field must be in the minimum energy configuration. But,
these conditions are violated in the active regions, where the magnetic fields
are highly sheared or twisted. Also, different extrapolation methods could
lead to different results (Lee et al., 1999; Régnier & Priest, 2007).

Coronal seismology is another method to estimate the magnetic field in the
solar corona (Roberts et al., 1983; Nakariakov & Verwichte, 2005; Aschwanden,
2006; Banerjee et al., 2007; De Moortel & Nakariakov, 2012). Determination of
physical parameters of solar corona such as magnetic field was first suggested
by Uchida (1970); Rosenberg (1970); Roberts et al. (1984). However, this idea
could not be put into practice until the launch of the Solar and Heliospheric
Observatory (SoHO; Domingo et al., 1995) and the Transition Region and Coronal
Explorer (TRACE; Handy et al., 1999). These spacecraft localized and
identified coronal loops, prominences, standing and propagating
magnetohydrodynamic (MHD) waves in the solar corona and the transition region
(Nakariakov et al., 1999; Aschwanden et al., 1999). Soon after, the magnetic
field strength in the solar corona was estimated, using different ground- and
space-based instrumentation by e.g, Aschwanden et al. (1999, 2002); Nakariakov
(2000); Nakariakov & Ofman (2001) using TRACE, Van Doorsselaere et al. (2008),
using Hinode (Kosugi et al., 2007), Aschwanden & Schrijver (2011); Sarkar et
al. (2016); Jess et al. (2016), using the Solar Dynamic Observatory (SDO;
Pesnell et al., 2012) and Tomczyk et al. (2007); Long et al. (2017); Yang et
al. (2020a), using the Coronal Multichannel Polarimeter (CoMP; Tomczyk et al.,
2008).

These observations help us infer the average magnitude of the magnetic field
in these structures. Extracting the information of the physical parameters,
such as magnetic field, from the observations is an inversion problem. To
infer parameters from such inversion problems, one has to deal with ill-posed
mathematical problems where the number of unknowns may exceed the number of
observables. Moreover, these observables cannot be measured with high
accuracy. For inversion problems, the use of Bayesian analysis is increasing
rapidly in solar physics (Arregui & Ramos, 2011; Arregui, 2012; Arregui et
al., 2012, 2014; Arregui, 2015; Scherrer & McKenzie, 2017; Pascoe et al.,
2017; Arregui, 2018; Arregui et al., 2018, 2019; Arregui & Goossens, 2019;
Pascoe et al., 2020; Arregui, 2021, 2022; Pascoe et al., 2022). The
utilization of Bayesian inference in coronal seismology is driven by the need
for probabilistic conclusions due to the presence of uncertainties and limited
information available. This makes the Bayesian analysis method a more
appropriate approach compared to other methods that provide a single numerical
estimate.

The global map of magnetic field strength was recently inferred using simple
inversion by Yang et al. (2020a). In this study, we inferred the global map of
the coronal magnetic field using Bayesian inference on the same data set used
by Yang et al. (2020a). A comparison is also made between the magnetic field
obtained using Bayesian inference and the magnetic field obtained using simple
inversion. We also obtained the radial variation of the coronal magnetic field
using simple inversion and Bayesian inference. Magnetic field variation along
and across the coronal loops have also been obtained and it is found that the
coronal magnetic field strength is more inside the coronal loops as compared
to the ambient medium. Additionally, the magnetic field strength is found to
be more near the loop footpoints than at the loop apex.

This paper is structured as follows. Section 2 describes the data used. The
global map of the coronal magnetic field using simple inversion is obtained in
Section 3. The magnetic field strength in the solar corona is estimated using
Bayesian inference, and the magnetic field’s radial distribution is described
in Section 4. Section 5 outlines the summary.

Figure 1: Images of the solar corona: (A) Map of Doppler velocity of the Fe
xiii@ 1074.7 nm line at 20:39:09 UT on 2016 October 14. A 3.5 mHz Gaussian
filter has been applied to the Doppler shift image sequence. An animated
version of this panel is available, featuring the 3.5 mHz Gaussian-filtered
Doppler velocity from 20:39:08 UT to 21:26:38 UT on the same day. The
animation has a real-time duration of 10 seconds. (B) Map of the derived wave
propagation direction. In all panels, the brown dashed circle marks the edge
of the solar disc (solar limb), and the black dashed circle indicates the
inner boundary of the CoMP FOV. The X and Y coordinates represent spatial
positions in the east-west and south-north directions, respectively. Figure 2:
Density diagnostics (A and B): Maps of the electron density and the associated
uncertainty. The circles are as in Figure 1.  Figure 3: Phase speed
determination: (A and B) Maps of the derived phase speed and the associated
uncertainty. The circles are as in Figure 1. Figure 4: Coronal magnetic field:
(A and B) Maps of plane-of-sky component of the coronal magnetic field and the
associated uncertainty. The circles are as in Figure 1. This coronal magnetic
field map corresponds to the magnetic field obtained using simple inversion.
The number of pixels in this map, after applying all the filters used in
evaluating the phase speed, is 40,306.

##  2 Instrument and Data

The Coronal Multichannel Polarimeter (CoMP: Tomczyk et al., 2008) is a 20 cm
aperture coronagraph mounted at Mauna Loa Solar Observatory in Hawaii and can
perform spectroscopic observations at infrared wavelengths. Its spatial
sampling is $\sim$$4.35\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, and its
field-of-view (FOV) is about 1.05-1.35 $R_{\odot}$ from the solar center. This
instrument measures the complete polarization state of the Fe xiii@ coronal
emission lines at 1074.7 and 1079.8 nm and He i@ chromospheric line at 1083.0
nm. CoMP observations have measured periodic Doppler velocity disturbances
that have been interpreted as the presence of transverse propagating MHD waves
in the corona (Tomczyk et al., 2007; Tomczyk & McIntosh, 2009; Morton et al.,
2015, 2019).

De Pontieu et al. (2007); Goossens et al. (2009); Tian et al. (2012); Long et
al. (2017) identified these waves as kink waves, which have Alfvénic nature.
Doppler velocity dataset is used to determine the phase speed of kink waves
ubiquitously present in the solar corona (Tomczyk et al., 2007; Tomczyk &
McIntosh, 2009; Morton et al., 2015, 2016). The Doppler velocity dataset is
taken from 20:39 UT to 21:26 UT on 2016 October 14. (Animation of Doppler
velocity data is also available.) It should be noted that no flare or Coronal
Mass Ejection (CME) happened during this time. Different structures present
during the observations are shown in Appendix C.

##  3 Magnetic field estimation using Simple Inversion

To determine the phase speed, we consider 96 frames corresponding to Fe xiii@
1074.7 nm from 20:39 UT to 21:26 UT on 2016 October 14. A constant time
cadence of $\sim$30 s is ensured between these frames. The dataset obtained is
usually associated with a certain degree of uncertainty, i.e. noise. Dataset
is first aligned to remove the noise. Then a Gaussian filter corresponding to
3.5 mHz (corresponding to 5-minute oscillations) (Tomczyk et al., 2007;
Tomczyk & McIntosh, 2009; Morton et al., 2015, 2019) is applied to this
Doppler velocity data. A filtered Doppler velocity dataset is obtained as
shown in Figure 1(A). Furthermore, to reduce the effect of low signal-to-noise
ratio in the data, only those pixels are used which have a peak intensity of
Fe xiii@ 1074.7 nm higher than 1ppm. The magnetic field in the solar corona
acts as the waveguide as these transverse waves propagate along the magnetic
field. Hence, the wave propagation direction is also required. Wave
propagation direction is obtained using a cross-correlation technique as
explained in Yang et al. (2020b). Only those pixels which have a cross-
correlation value higher than 0.5 are considered. The wave propagation
direction map is shown in Figure 1(B). The wave propagation direction is
determined with respect to east-west orientation. However, an inherent 180∘
ambiguity limits the wave propagation direction map to a range between -90∘
and +90∘. The abrupt shifts in wave propagation angles at specific pixels are
a consequence of this 180∘ ambiguity, where wave angles near -90∘ and +90∘
essentially indicate the same wave propagation directions.

After obtaining the wave propagation direction, the phase speed of the
prevailing transverse waves in the FOV of CoMP is determined using the
following procedure. First, we define the track of 31 pixels along the wave
propagation direction. Following that, a space-time diagram is constructed for
each pixel on the track, and then a 2D Fast-Fourier transform (FFT) is applied
to obtain the $k$-$\omega$ diagram. The $k$-$\omega$ diagram separates the
outward and inward propagating waves. The inverse FFT corresponding to the
positive and negative frequency of $k$-$\omega$ diagram is computed to obtain
the space-time diagrams corresponding to outward and inward propagating waves.
These space-time diagrams are used to cross-correlate the time series at each
pixel with the rest of the pixels of the same track. The slope of these
distance-time plots gives the phase speed of the waves at each pixel. In
Figure 3, only those pixels with a phase speed less than 700 km s-1 are shown.
A few pixels also have a phase speed of more than 700 km s-1. These pixels are
near the occulting disc or at the outer edge of the FOV and are thus generally
associated with large uncertainty.

To perform the coronal seismology, density estimates are required. In this
study, we use the Fe xiii@ line pairs and CHIANTI (Dere et al., 1997)
database. The intensity ratio of Fe xiii@ lines at 1079.8 nm and 1074.7 nm is
obtained using CoMP data from 19:24 UT to 20:17 UT corresponding to different
heights in the CoMP FOV. This intensity ratio obtained from CoMP is then
compared with the ratio of nearly the same wavelength range of Fe xiii@ ion
from the CHIANTI database. This period is chosen because it is similar in time
to the wave tracking period. CHIANTI Version 9.0 (Dere et al., 2019) is used.
Then, from the calibration curves obtained using the CHIANTI database,
electron density in the solar corona can be obtained as in Figure 2.

The phase speed of the transverse propagating kink waves and density data are
used to estimate the magnetic field strength in the solar corona, first by
simple inversion and then by Bayesian inference. The phase speed of the kink
waves, $c_{k}$, can be expressed as,

$c_{k}^{2}=\frac{{B_{i}}^{2}+{B_{o}}^{2}}{\mu_{o}(\rho_{i}+\rho_{o})},$ (1)

where B is the magnetic field strength, $\rho$ is the mass density, $\mu_{o}$
is the magnetic permeability of the free space, and the subscripts $i$ and $o$
represent the corresponding physical parameters inside and outside the coronal
structures, respectively. In coronal plasma, magnetic pressure will dominate
the thermal pressure because of the low plasma $\beta$ value, so $B_{i}\sim
B_{o}$. Moreover, individual flux tubes are likely to be unresolved at the
spatial resolution of CoMP, allowing us to take the density averaged inside
and outside these flux tubes $(\langle\rho\rangle)$ within each spatial pixel
and estimate the magnetic field strength via,

$c_{k}=\frac{B}{\sqrt{\mu_{o}\langle\rho\rangle}}.$ (2)

Substituting density data $(\langle\rho\rangle)$ and phase speed data $c_{k}$
at each pixel, the plane of sky component of the magnetic field is estimated
using Equation 2. The magnetic field strength, obtained through a simple
inversion process, is presented in Figure 4. The estimated magnetic field
strength primarily ranges between 1 and 4 G, with a corresponding uncertainty
ranging from 0.1 to 0.5 G.

Also, it should be noted that for wave tracking, observations are taken from
20:39 UT to 21:26 UT, and for density estimations, observations are taken from
19:24 UT to 20:17 UT. Due to the dynamic nature of the solar corona, the
measurement of physical parameters, such as coronal density, is subject to
both statistical and systematic uncertainties. To improve the magnetic field
strength estimation, it is recommended to use an inversion process that
considers the uncertainties in the measurements of physical parameters, such
as coronal density. Bayesian inference provides a suitable approach for this,
as it incorporates the uncertainty in density when determining the range of
plausible magnetic field values.

##  4 Magnetic field estimation using Bayesian Inference

Bayesian analysis enables to perform parameter inference, model comparison,
and model averaging applications to gain information about the magnetic field
in the solar corona. This approach to parameter inference is based on the use
of Bayes Theorem (Bayes & Price, 1763),

$p(\theta|d,M)=\frac{p(d|\theta,M)p(\theta|M)}{\int
p(d|\theta,M)p(\theta|M)d\theta}.$ (3)

In this expression, $p(\theta|M)$ is the prior probability density for the
parameter vector $\theta$, and $p(d|\theta,M)$ is the likelihood function.
Both quantities are conditional on the assumed model M. Their combination
leads to the posterior probability distribution $p(\theta|d,M)$, which
contains all the information that can be inferred from the observed data and
the assumed model.

It should be noted that Bayesian inference does not provide point estimates
for parameters but rather the full posterior probability distribution function
(PDF), $p(\theta|d,M)$, which captures the entire range of plausible values
for the parameters. This is a significant advantage over simple inversion.

In parameter inference, a parameterized model may have multiple parameters,
$\theta={\theta_{1},...,\theta_{i},...,\theta_{N}}$. The marginal posterior
distribution for a parameter of interest, $\theta_{i}$, can be obtained by
integrating the full posterior over the remaining nuisance parameters.

$p(\theta_{i}|d)=\int
p(\theta|d)d\theta_{1}...d\theta_{i-1},d\theta_{i+1},...,d\theta_{N}.$ (4)

This results in the marginal posterior distribution for model parameter
$\theta_{i}$, which includes all the information available in the prior
knowledge and observed data. The uncertainty from the nuisance parameters will
also propagate correctly to the one of interest.

Our model M is given by the theoretical prediction for the kink speed in the
extended wave-length approximation, given by Equation (2). This is a two-
parameter function, hence $\theta=\\{\rho,B\\}$ in our Bayesian inference
model. To estimate the probability of the magnetic field strength using
Bayesian inference, we first adopt prior distributions for the two model
parameters, $\rho$, $B$. For the plasma density, we use measured density
values and associated uncertainties from Yang et al. (2020b) instead of
synthetic density values as used by Arregui & Asensio Ramos (2011). Two
different prior distributions are used: a Gaussian prior and a uniform prior.
The uniform prior assigns an equal probability for the density in the range
$[\mu_{\rho_{i}}-3\sigma_{\rho_{i}},\mu_{\rho_{i}}+3\sigma_{\rho_{i}}]$, where
${\mu_{\rho_{i}}}$ represents the measured density value and
${\sigma_{\rho_{i}}}$ represents the uncertainty for the $i^{th}$ pixel. The
uniform prior probability for density is described by the following equation:

$p(\rho)=\begin{cases}\frac{1}{(\mu_{\rho_{i}}+3\sigma_{\rho_{i}})-(\mu_{\rho_{i}}-3\sigma_{\rho_{i}})}&\text{$\mu_{\rho_{i}}-3\sigma_{\rho_{i}}\leq\rho\leq\mu_{\rho_{i}}+3\sigma_{\rho_{i}}$}\\\
0&\text{otherwise}.\end{cases}$ (5)

The Gaussian prior for the density at each pixel in the FOV is given by:

$p(\rho)=\frac{1}{\sqrt{2\pi}\sigma_{\rho_{i}}}\exp{\frac{-(\rho-\mu_{\rho_{i}})^{2}}{2\sigma_{\rho_{i}}^{2}}}.$
(6)

For the magnetic field strength, we restrict the possible range of variation
to the one obtained using simple inversion, i.e., [$B_{\textit{min}}$,
$B_{\textit{max}}$] = [1,8] G, with $B_{\textit{max}}$ only present at a few
pixels. For the magnetic field, we restrict the possible range of variation to
the one obtained using simple inversion. Within this range, we assign a
uniform prior probability for the magnetic field strength described by the
following equation:

$p(B)=\begin{cases}\frac{1}{B_{\textit{max}}-B_{\textit{min}}}&\text{$B_{\textit{min}}\leq
B\leq B_{\textit{max}}$}\\\ 0&\text{otherwise}.\end{cases}$ (7)

For our computations, we constructed a two-dimensional grid over the parameter
space with $N_{\rho}=10,000$ and $N_{B}=71$ points, which covers the ranges of
considered values for plasma density and magnetic field strength.

An additional magnetic field prior, known as the gamma prior, is also
considered. The probability distribution function of the gamma prior is given
by Equation A1. Further details of how the parameters of this distribution are
evaluated can be found in Appendix A.

In the Bayesian framework, the data are fixed, and the model parameters are
unknown to which a probability value can be assigned. The general expression
of the likelihood function is the numerator of Equation (3), $p(d|\theta,M)$,
explicitly indicates that this function is conditional on the model M, which
is considered to be true. It does not represent the likelihood of different
data realisations. Its purpose is to measure the discrepancy between model
predictions and observed data as a function of the model parameters, taking as
reference the uncertainty of the data, thus assigning different levels of
likelihood to alternative parameter combinations. In our case, the phase
velocity is forward modelled using model $M$ in Equation (3) for different
combinations of the parameter values defined by those priors. These
theoretical predictions $v_{p}$ are compared to the observed phase speed
$c_{k}$ at each pixel, and the relative merit (likelihood) of each combination
is assessed by the adoption of a Gaussian likelihood function of the form

$p(c_{k}|B,\rho,M)=\frac{1}{\sqrt{2\pi}\sigma_{c_{k}}}\exp{\frac{-(c_{k}-v_{p}(B,\rho))^{2}}{2\sigma_{c_{k}}^{2}}}.$
(8)

Here, $\sigma_{c_{k}}$ corresponds to the uncertainty in the phase speed
obtained earlier. This uncertainty is different at each pixel and corresponds
to the uncertainty obtained while evaluating the phase speed $c_{k}$. The
joint posterior probability of the magnetic field and density in the FOV,
given our model is true, is obtained as

$p(B,\rho|c_{k})=\frac{p(c_{k}|B,\rho,M)p(B)p(\rho)}{\int\int
p(c_{k}|B,\rho,M)p(B)p(\rho)dBd\rho},$ (9)

where $p(c_{k}|B,\rho,M)$ is the likelihood function and $p(B)$, $p(\rho)$ are
the prior probability distribution functions of the magnetic field and
density, respectively. We have done forward modelling of phase speed with the
measured phase speed and the corresponding error for the likelihood function.
All the posteriors in this analysis are normalized.

Figure 5: Left and middle panels represent the joint probability distributions
of magnetic field and density at the specified location. For Gaussian prior on
density [Left panel], it is maximum for density value for which the prior
probability is also maximum, but for a uniform prior on density. The right
panel represents the marginal magnetic field obtained by integrating the joint
probability distributions with density.

Figure 6: Marginal magnetic field: (A) Marginal magnetic field at two
locations using Gaussian prior on density. (B) Marginal magnetic field at the
same locations using a uniform prior on density. The vertical dot-dash violet
line corresponds to the magnetic field at the same location using simple
inversion, and the vertical solid violet line corresponds to the lower and
upper limits of the magnetic field. Phase speed and associated uncertainty at
location (804.8,-693.8) ((-696,800.4)) is $475\pm
49.\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$($275\pm
4.\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$), and density and associated
uncertainty values are
$1.98(0.44)e-13\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$($4.20(1.16)e-13\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$).
These results in the magnetic field and associated uncertainty obtained using
simple inversion as $2.4(0.4)\text{\,}\mathrm{G}$
($2.0(0.3)\text{\,}\mathrm{G}$).

The relative simplicity of the forward and inverse problems (Equations 2 and
9), which involve a two-dimensional parameter space and a one-dimensional
observable space, makes feasible the computation of our Bayesian posteriors
using direct numerical evaluation and integration over a grid of points in
parameter space. These are performed using routine IDL and Python scripts.
Arregui et al. (2019) have shown that this grid-approximation method and a
Markov chain Monte Carlo (MCMC) sampling of the posterior give essentially the
same results to the solution of kink wave phase speed inversion problem (see
their Appendix A). In our study, we have verified the good correspondence
between the direct integration and the MCMC solutions, described in Appendix
B.

Using a uniform prior on magnetic field (Equation 7) and two different priors
on density (Equation 5 and 6) and the likelihood function (Equation 8),
Equation 9 helps evaluate the joint posterior probability distribution of the
magnetic field and density. The left and middle panels of Figure 5 represent
the joint probability distribution of the magnetic field and density obtained
at one location using two different priors on density. The shape of the
inferred joint probability distribution differs for different priors on
density. The use of a Gaussian prior on density leads to a more constrained
inference because it provides more information to the inference process. From
the joint posteriors, our interest is in estimating the marginal probability
distributions of the magnetic field strength; this is done by integrating the
joint posterior probability distribution over the density as follows. This
marginalized distribution of the magnetic field is determined using an
equation similar to the Equation 4 as,

$p(B|c_{k})=\int p(B,\rho|c_{k})d\rho.$ (10)

The evaluation of the marginal probability distributions of the magnetic field
also incorporates the uncertainty in density as it involves integrating the
whole joint posterior probability with respect to density. The rightmost panel
of Figure 5 represents the marginal probability distribution of the magnetic
field. For Gaussian prior on density, the marginal probability distribution of
the magnetic field peaks where the joint probability distribution of the
magnetic field and density is maximum. The magnetic field value for which the
marginal probability distribution is maximum is the most probable value of the
magnetic field for that location.

Figure 6(A) represents the marginal posterior for the magnetic field inferred
at two other locations in the FOV, using a uniform prior on the magnetic field
and two different priors on density. The vertical dash-dotted and solid violet
lines indicate the magnetic field estimate and the associated uncertainty
obtained using simple inversion. Our results show that the simple inversion
point estimates and the Bayesian maximum a posteriori estimates do not always
coincide. The magnitude of the discrepancy depends on the particular location
and the type of prior on density employed in the inference. The difference is
larger on location [-696.0,800.4] (bottom panels) than on location [804.8,
-643.8] (top panels). The reason for this is that the relative uncertainty on
density at [-696.0,800.4] is 27% while at location [804.8, -643.8] and
[-1131.0, -208.8], the relative uncertainty in density is 22% and 15%. The
mismatch between the simple and Bayesian estimates is larger when using a
uniform prior on density (right panels) than when a Gaussian prior is employed
(left panels). Here, again, the use of a Gaussian prior is helping in
obtaining a more constrained inference result.

We have explored another option for the prior on magnetic field strength, with
the use of a Gamma distribution. This has the advantage of being strictly
positive and with no upper boundary. A summary of inference results obtained
using this prior for the magnetic field strength is presented in Appendix A.
One comparison between the marginal magnetic field distribution evaluated
using a uniform prior on the magnetic field and gamma prior on the magnetic
field is also made at location [-1131, -208.8] in Figure 14. The violet lines
are as mentioned above. The royal blue curve corresponds to the marginal
magnetic field distribution obtained using a uniform prior on the magnetic
field, whereas the black curve corresponds to the gamma prior on the magnetic
field for both density distributions. The marginal magnetic field distribution
at the other two locations is shown in Figure 13. The area under the plots of
marginal magnetic field versus magnetic field will be unity, for the pixels
having higher values of the marginal distribution of the magnetic field will
result in giving the more constrained plausible range of magnetic field and
vice versa. This can be seen from Figure 14 for the uniform prior on the
magnetic field, which has a more constrained curve as compared to the other
locations as in Figure 14. Furthermore, one can easily check that the magnetic
field distribution obtained using gamma prior on the magnetic field gives more
constrained results than that of a uniform prior on the magnetic field.

Further, we computed the marginal probability distribution of the magnetic
field in the whole FOV of CoMP. For this, we normalise the marginal
probability of the magnetic field at each location with respect to the maximum
value of the marginal probability obtained in the whole FOV. The results are
shown in Figure 7. Here, the height of each bar represents the range of
plausible values of the magnetic field for that location, and the colour bar
represents the colour corresponding to the marginal probability of the most
probable magnetic field at each location. The larger the height of the bar,
the more the values of the magnetic field are plausible. Similar maps using
gamma prior on the magnetic field are shown in 15 in Appendix (A). Further,
the convergence of the magnetic field is also shown by using Markov chain
Monte Carlo (refer Appendix B)

Figure 7: [A] Marginal distribution of magnetic field obtained using Gaussian
prior density distribution and a uniform prior magnetic field distribution in
the whole FOV of CoMP. [B] Marginal magnetic field obtained using a uniform
prior density distribution and a uniform prior magnetic field distribution in
the whole FOV. Figure 8: Coronal magnetic field using Bayesian Inference: (A)
Map of plane-of-sky component of the coronal magnetic field obtained using
Gaussian prior on density. (B) Map of the plane of sky component of the
magnetic field obtained using a uniform prior on density. The circles are as
in Figure 1.  Figure 9: Magnetic field obtained using Bayesian inference
versus simple inversion: (A) Difference between the plane-of-sky component of
magnetic field obtained using Bayesian inference and simple inversion at each
pixel in CoMP FOV using Gaussian prior on density. (B) Difference between the
plane-of-sky component of magnetic field obtained using Bayesian inference and
simple inversion at the same pixel in CoMP FOV using a uniform prior on
density.

The simple inversion approach yields a single-valued magnitude of the magnetic
field at each pixel, along with its associated uncertainty. On the other hand,
Bayesian inference produces a probability distribution corresponding to each
magnetic field value at each pixel. Figure 8 shows the Bayesian analogue of
the map in Figure 4A, representing now the maximum a posteriori estimates for
the magnetic field strength, for the two considered priors in density. As
explained above, the inference results have a dependence on the prior
assumption on density, because the uniform prior is less informative that the
Gaussian prior. These differences can be appreciated at several pixels in the
inference results displayed in Figure 8.

The difference between the magnetic field obtained through simple inversion
and the most probable magnetic field obtained through Bayesian inference is
illustrated in Figure 9. In most regions, the difference is close to zero.
However, the difference is more pronounced near the occulter or at the edge of
the FOV, where the uncertainty in density or phase speed is large. It should
be noted that the difference is larger for the uniform prior on density as
compared to the Gaussian prior on density. The most probable magnetic field
obtained using Gaussian prior on density is not very much different from the
magnetic field estimate obtained using simple inversion.

The Bayesian inference results enable us to compare the probability of a given
value for the magnetic field strength at different locations in the FOV. This
is an advantage with respect to the simple inversion result, with which this
comparison cannot be drawn. Figure 10 represents the marginal probability of
magnetic field = 1.5 G at all the pixels in the field of view of CoMP. For 1.5
G, the magnetic field is more structured and concentrated at the flux tubes.
Lower values of magnetic fields are most probable at the outer edges of FOV,
whereas higher values of the magnetic field are most probable near the inner
edge of the FOV. One can also conclude from the movies that the higher values
of the magnetic field are most probable in the west limb. From the animations,
it is clear that the marginal distribution of the magnetic field, when plotted
in the whole FOV by varying the strength of the magnetic field, gives the
region where those magnetic field values are most probable.

A similar analysis was also performed with a different range of density in
uniform prior where density range is
$[\mu_{\rho_{i}}-\sigma_{\rho_{i}},\mu_{\rho_{i}}+\sigma_{\rho_{i}}]$. With
this constrained prior in density, the marginal distribution of the magnetic
field was also constrained. The most probable magnetic field obtained using
this density range was similar to the one obtained using Gaussian prior on
density. In Section 2, it was noted that phase speeds exceeding 700 km s-1
were observed in a few pixels. To evaluate the impact of these observations,
the analysis was extended to include phase speeds up to 2000 km s-1, resulting
in a revised maximum magnetic field strength of 21 G instead of 8 G as
obtained through simple inversion. This new magnetic field range, B $\epsilon$
[1,21] G, was then considered to estimate the magnetic field strength using
Bayesian inference. The pixels having phase speed less than 700 km s-1 were
found to yield the same results. The same analysis was repeated for a
different range of magnetic field, B $\epsilon$ [1,25] G, to check if it has
any impact; however, again, similar results were obtained.

Figure 10: Variation of marginal probability with magnetic field: Left and
Right panel represent the different marginal probability values of the
magnetic field (= 1.5 G) at different pixels for Gaussian prior on density and
uniform prior on density, respectively. An animation of the variation of
marginal probability values with the variation of the magnetic field is
available. The animation shows the changes from a magnetic field of 1.0 G to
8.0 G in increments of 0.1 G. The animation shows that the lower magnetic
field values are more probable in the outer FOV, while higher magnetic field
values are more probable near the inner edge of the FOV. The animation has a
real-time duration of 5 seconds.

###  4.1 Radial variation of the coronal magnetic field

The radial variation of the coronal magnetic field obtained using simple
inversion and Bayesian inference is shown in Figure 11. The region on the east
limb represents the quiet Sun (QS) region, whereas the region on the west limb
represents the active region (AR). Previous investigations (Kumari et al.,
2019; Yang et al., 2020a) have shown that the radial variation of the coronal
magnetic field follows a power-law function. The power-law function
($Ar^{\alpha}$) is fitted to each curve as shown in Figure 11, and the power-
law index ($\alpha$) is obtained. The power law index value for each prior on
density distribution in Bayesian inference and simple inversion is almost the
same.

Figure 11:  Radial profile of magnetic field: Top panel shows the plane of sky
component of the magnetic field (same as in Figure 8) with four regions marked
in the east and west limb. (A), (B), (C) and (D) show the variation of average
coronal magnetic field strength as a function of radial distance from the
solar centre for the marked regions.

###  4.2 Magnetic field along and across the coronal loops

Figure 12 displays the full FOV of CoMP in terms of intensity. A spline
fitting technique employing 100 spline points is used to obtain smooth arc
coordinates along the loops. Different loops at different locations are
analyzed, and it was found that the magnetic field is maximum at the
footpoints of the coronal loop. This result was also expected from the radial
variation of the magnetic field. A Savitzky-Golay filter of $3^{rd}$
polynomial order and window size of 10 pixels is applied along all the loops
to smoothen out the variation along the distance further. The middle row of
Figure 12 represents the magnetic field variation along the coronal loop. It
should also be noted that the magnetic field variation in going from one
footpoint to the other in the same loop is not smooth but rather, the magnetic
field decreases rapidly and then remains constant for some distance and then
increases rapidly. This trend can be appreciated in almost all the selected
loops. Here, the colour represents the probability of the magnetic field. An
attempt was also made to obtain the magnetic field of coronal loops close to
the occulter. But, as evident from the radial variation of the coronal
magnetic fields, magnetic fields do not vary much as one moves from footpoints
to the loop apex for very small coronal loops near the occulter region.

Along the middle of the fitted arc on the coronal loop, a perpendicular slit
of 80 pixels is defined. The variation of the magnetic field across this slit
is obtained. The distance in the bottom panel of Figure 12 corresponds from
inside to outside of the FOV. The magnetic field is expected to increase as it
encounters the coronal loop and then decrease as one moves out of that coronal
loop and again increases as the slit passes through another coronal loop. A
similar trend can also be seen in the bottom panel of Figure 12.

##  5 Summary and Discussion

A global map of the coronal magnetic field was obtained for the first time by
Yang et al. (2020b), and we have applied Bayesian analysis for the first time
on the whole global map using the density and associated uncertainty values
derived from CoMP observation. We found that the magnetic field obtained using
simple inversion does not always match the most probable magnetic field
obtained using Bayesian inference. We also showed the impact of choosing two
different density priors as well as two different magnetic field priors. The
use of Gaussian density priors leads to a better constrained inference of
magnetic field strength, because it provides more information to the inference
process, in comparison to a uniform density prior over a given range of
values. This underscores the crucial significance of obtaining precise density
estimates. On the other hand, using a gamma function prior for the magnetic
field strength, yields more tightly constrained inferences, although this
requires some prior knowledge to set the parameters of the prior probability
density. This underscores the crucial significance of obtaining precise
density estimates.

We have also obtained 3D maps of the global coronal magnetic field, which
gives the probability of occurrence of a magnetic field value at each location
in the plane of the sky. We have shown the variation of the marginal
probability at each location with the varying magnetic field. We also obtained
the radial variation of the magnetic field using Bayesian inference and simple
inversion. The sharp decrease of the magnetic field with height in AR compared
to the QS region is seen in both Bayesian and simple inversion. This result is
consistent with Yang et al. (2020a). We have also shown the magnetic field
variation along and across the coronal loops. We hope this study will be
beneficial to understanding the magnetic structure of multi-thermal loops by
making use of MHD simulations.

Figure 12: Intensity: Map of the edge enhanced intensity of the FOV at
20:39:09 UT on October 14 2016, along with selected coronal loops and slits
across which the variation of the magnetic field is obtained. Variation of
coronal magnetic field: In the bottom panels, the first row represents the
variation of the coronal magnetic field along the 4 different coronal loops.
The bottom panels represent the variation of the coronal magnetic field across
these loops.

Bayesian inference helps us obtain not just a point estimate of the magnetic
field but gives a whole distribution of magnetic field values along with their
associated probabilities. Our study found that higher magnetic field values
are most probable near the base of the coronal loops, and the probability of a
high magnetic field decreases while moving outwards in the FOV. This is also
expected as the strength of the magnetic field decreases at larger solar
radii.

Bayesian inference has already been used for seismology of coronal
oscillations (Arregui & Asensio Ramos, 2011; Arregui et al., 2019), but with
assumptions on the density estimation. In this work, we used densities derived
from the observations of Fe XIII emission line pairs.

The Bayesian seismology techniques have helped infer parameters in the solar
corona and model comparison methods. Bayesian takes care of uncertainty in the
parameters’ values; this information is passed throughout the analysis from
the prior distribution to the posterior and the marginal distributions. One
can easily change the prior distribution of density as well as the magnetic
field to obtain the more realistic values of the most probable magnetic field.
One can also update the value of the magnetic field, density, and phase speed
in the prior distribution and likelihood functions, as the information about
these parameters keeps on updating with the advancement in observations, as
mentioned in Arregui & Asensio Ramos (2011). The reliability of the obtained
magnetic field distribution is based on the reliability of the estimated
density values. Using Bayesian inference for inversion in upcoming solar
facilities such as the Upgraded Coronal Multichannel Polarimeter (UCoMP) and
Daniel K. Inouye Solar Telescope (DKIST), we may obtain the magnetic field
values with more accuracy.

The Mauna Loa Solar Observatory, operated by the High Altitude Observatory as
a part of the National Center for Atmospheric Research (NCAR), is acknowledged
for providing the data. NCAR is supported by the National Science Foundation.
VP is supported by a SERB start-up research grant (File no. SRG/2022/001687).
IA is supported by project PID2021-127487NB-I00 from Ministerio de Ciencia,
Innovación y Universidades, and FEDER funds. The python code employed for the
MCMC sampling of the posteriors in this study makes use of the emcee algorithm
(Foreman-Mackey et al. 2013) and was developed upon an earlier version created
by M. Montes-Solís. The authors would like to thank Hui Tian and Zihao Yang
for sharing the density data. UB and VP express gratitude to ARIES for
providing computational resources. Additionally, the insightful suggestions of
Dipankar Banerjee and Tom Van Doorsselaere to enhance this work are greatly
appreciated. The valuable comments and assistance provided by Ritesh Patel,
Bibhuti Kumar Jha, and Nitin Vashishtha are sincerely appreciated. Finally,
the authors wish to thank the reviewers for their valuable suggestions.

##  Appendix A INFERENCE WITH A GAMMA PRIOR ON THE MAGNETIC FIELD STRENGTH

For completeness, we show here results on global maps of the coronal magnetic
field, obtained with the use of a Gamma distribution for the prior on magnetic
field strength. The distribution is strictly positive and has no upper
boundary. A parametric form of this distribution can be expressed as:

$p(B)=\begin{cases}\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta
x}}{\Gamma(\alpha)}&\text{$B_{\textit{min}}\leq B\leq B_{\textit{max}}$}\\\
0&\text{otherwise}.\end{cases}$ (A1)

Here, $\alpha$ and $\beta$ represent the shape and rate parameters. The mean
and coefficient of variance of the distribution is given by $\alpha/\beta$ and
$\alpha/\beta^{2}$, respectively. To determine the parameters of the
distribution, we used information from the results of the simple inversion. In
particular, we took the magnetic field obtained from simple inversion as the
mean and the magnetic field’s uncertainty as the coefficient of variance in
the distribution at each location.

We obtained the magnetic field with the gamma prior on the magnetic field and
both uniform and Gaussian priors on density. The marginal magnetic field
distribution at each pixel for both density priors now exhibits a Gaussian
distribution, as depicted in Figure 13.

Figure 13: Marginal magnetic field using gamma prior on magnetic field: (A)
Marginal magnetic field at two locations using Gaussian prior on density. (B)
Marginal magnetic field at the same locations using a uniform prior on
density. The vertical dot-dash violet line corresponds to the magnetic field
at the same location using simple inversion, and the vertical solid violet
line corresponds to the lower and upper limits of the magnetic field. Pixel
location and other parameters like phase speed, density, magnetic field and
uncertainty associated with these parameters are the same as in Figure 6.
Figure 14: Comparison of marginal probability distribution: (A) Marginal
probability distribution of magnetic field at location (-1131.0,208.8) using
Gaussian prior on density. (B) Marginal magnetic field at the same location
using a uniform prior on density. The vertical solid and dot-dash lines are
the same as in Figure 6. The black and royal blue curves correspond to the
gamma prior on the magnetic field and uniform prior on the magnetic field,
respectively. Phase speed and associated uncertainty at this location is is
$408\pm 12.\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and density and
associated uncertainty values are
$1.60(0.24)e-13\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$. These results
in the magnetic field and associated uncertainty obtained using simple
inversion as $1.9(0.2)\text{\,}\mathrm{G}$.

One comparison between the marginal magnetic field distribution evaluated
using a uniform prior on the magnetic field and gamma prior on the magnetic
field is also made at location [-1131, -208.8] in Figure 14. The violet lines
are as mentioned above. The royal blue curve corresponds to the marginal
magnetic field distribution obtained using a uniform prior on the magnetic
field, whereas the black curve corresponds to the gamma prior on the magnetic
field for both density distributions. It is clear from Figure 14 that the
marginal magnetic field distribution is more constrained when gamma prior on
the magnetic field is used as compared to when uniform prior on the magnetic
field is used.

The marginal probability distribution of the magnetic field, as obtained
through a Gamma prior on the magnetic field, demonstrates more tightly
constrained values when compared to the results obtained with a uniform prior
on the magnetic field (Figure 6). Just as in the case of a uniform prior on
the magnetic field (Figure 7), we have also plotted the marginal probability
of the magnetic field, using gamma prior on the magnetic field and both the
density distribution on density, in the whole field of view. These plots are
shown in Figure 15. The height and colour of each bar are as explained
previously in Section 4. The joint probability of magnetic field and density
for both the priors on density and gamma prior on magnetic field is shown in
Figure 16. Again, the joint probability is the maximum for which the priors on
density and magnetic field have a higher probability.

Figure 15: [A] Marginal magnetic field distribution obtained using Gaussian
prior density distribution and a gamma prior magnetic field distribution in
the whole FOV of CoMP. [B] Marginal distribution of magnetic field obtained
using a uniform prior density distribution and a gamma prior magnetic field
distribution in the whole FOV of CoMP. Figure 16: Joint probability
distributions of magnetic field and density at the specified location. For
both the priors on density, it is the maximum for which the prior
probabilities are also maximum.

A comparison between the most probable magnetic field obtained using gamma
prior on magnetic field for two different density distributions and the
magnetic field obtained using simple inversion is also made and depicted in
Figure 17. This also represents that the difference is mainly near the
occulter or at the edge of FOV for the abovementioned reasons.

Figure 17: Magnetic field obtained using gamma prior on magnetic field versus
simple inversion: (A) Difference between the most probable value of magnetic
field obtained using Bayesian inference and simple inversion at each pixel in
CoMP FOV using Gaussian prior on density and gamma prior on magnetic field.
(B) Difference between the most probable value of magnetic field obtained
using Bayesian inference and simple inversion at the same pixel in CoMP FOV
using a uniform prior on density and gamma prior on magnetic field.

Figure 18: [Top panels] Marginal magnetic field distribution for gamma prior
on magnetic field and Gaussian prior on density. [Bottom panels] Marginal
magnetic field distribution for gamma prior on magnetic field and uniform
prior on density. Orange-coloured Histograms are the results obtained using
the emcee algorithm of MCMC, whereas the blue solid curve represents the
marginal magnetic field distribution obtained using Bayesian inference. The
pixel locations are as in Figures 5 and 6. The marginal probability
distributions are not normalized for the sake of comparison with the
histograms of samples obtained using MCMC.

Figure 19: [Top panels] Marginal magnetic field distribution for a uniform
prior on magnetic field and Gaussian prior on density. [Bottom panels]
Marginal magnetic field distribution for a uniform prior on magnetic field and
uniform prior on density. Green-coloured Histograms are the results obtained
using the emcee algorithm of MCMC, whereas the blue solid curve represents the
marginal magnetic field distribution obtained using Bayesian inference. The
pixel locations are as in Figures 5 and 6. The marginal probability
distributions are not normalized for the sake of comparison with the
histograms of samples obtained using MCMC. Figure 20: Notable coronal features
in the CoMP FOV during observation.

##  Appendix B Application of MCMC on marginal magnetic field

We also compared the results obtained using direct numerical integration
(Bayesian inference) with Markov chain Monte Carlo (MCMC) sampling for the
posterior, making use of the emcee algorithm (Foreman-Mackey et al., 2013).
The application of the emcee algorithm is explained in Montes-Solís & Arregui
(2017) and Arregui et al. (2019). Since there are only two parameters involved
as $theta=[B,\rho]$, thus, the number of dimensions in emcee is 2. We used
around 15 walkers with 50,000 steps in each walker for emcee algorithm. Out of
which, we have used an initial 20$\%$ of the iterations for the burn-in phase
to avoid any outliers. The marginal posterior distribution of the magnetic
field results in the same distribution for two different density distributions
when the gamma distribution is taken as the prior distribution on the magnetic
field. Further, one can check from Figure 18 that there is no effect of change
in the prior distribution of density on the marginal probability distribution
of the magnetic field as mentioned in Arregui et al. (2019). Also, the
marginal probability distribution of the magnetic field is the same even if we
consider two different priors of the magnetic field (gamma and uniform),
keeping the density distribution the same (Gaussian prior on density). This
can be seen from the top panels of Figures 18 and 19. We have compared the
results with MCMC only for the locations shown in Figures 5 and 6. The results
obtained using MCMC and direct numerical integration also match the case when
uniform prior is considered for density and the magnetic field.

##  Appendix C Features in FOV

Various structures, including coronal holes, active regions, and more, can be
observed in the CoMP intensity images. Coronal holes are notably present in
polar regions and along the west limb. The locations of these various coronal
features are shown in Figure 20.

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