# Effects of spectral resolution on simple magnetic field diagnostics of the
Mg II h & k lines

Rebecca Centeno High Altitude Observatory (NCAR), 3080 Center Green Dr.,
Boulder, CO, USA  Matthias Rempel High Altitude Observatory (NCAR), 3080
Center Green Dr., Boulder, CO, USA  Roberto Casini  High Altitude Observatory
(NCAR), 3080 Center Green Dr., Boulder, CO, USA  Tanausú del Pino Alemán
Instituto de Astrofísica de Canarias, C/ Vía Láctea S/N, 38204 La Laguna,
Spain

(Received July 1, 2016; Revised September 27, 2016)

###### Abstract

We study the effects of finite spectral resolution on the magnetic field
values retrieved through the weak field approximation (WFA) from the cores of
the Mg ii h&k lines. The retrieval of the line-of-sight (LOS) component of the
magnetic field, $B_{\rm LOS}$, from synthetic spectra generated in a uniformly
magnetized FAL-C atmosphere are accurate when restricted to the inner lobes of
Stokes V. As we degrade the spectral resolution, partial redistribution (PRD)
effects that more prominently affect the outer lobes of Stokes V, are brought
into the line core through spectral smearing, degrading the accuracy of the
WFA and resulting in an inference bias, which is more pronounced the poorer
the resolution. When applied to a diverse set of spectra emerging from a
sunspot simulation, we find a good accuracy in the retrieved $B_{\rm LOS}$
when comparing it to the model value at the height where the optical depth in
the line core is unity. The accuracy is preserved up to field strengths of
B$\sim$1500 G. Limited spectral resolution results in a small bias toward
weaker retrieved fields. The WFA for the transverse component of the magnetic
field is also evaluated. Reduced spectral resolution degrades the accuracy of
the inferences because spectral mixing results in the line effectively probing
deeper layers of the atmosphere.

solar magnetic fields

††journal: ApJ

##  1 Introduction

Inference of the photospheric magnetic field vector has been routinely carried
out from ground-based observatories for decades. With the advent of
spectropolarimeters in space, these observations (and their interpretation in
terms of the vector magnetic field) have reached the consistent (and
continuous) high data quality required for precise analysis of the magnetic
field evolution. While the HMI instrument (Scherrer et al., 2012) on board the
Solar Dynamics Observatory (Pesnell et al., 2012) provides full disk
measurements at good cadence and moderate spatial resolution, the Hinode
spectro-polarimeter (Kosugi et al., 2007; Lites et al., 2013) offers a higher-
resolution counterpart within a much smaller FOV and limited temporal cadence.
The Solar Orbiter PHI instrument (Solanki et al., 2020) will soon bring the
first ever magnetic field measurements at the solar poles with significant
less foreshortening than observations made from the ecliptic plane. While each
one of these observational benchmarks has furthered our knowledge of the Sun’s
magnetism in a step-wise manner, the inferred magnetic field is still confined
to a thin atmospheric layer next to the Sun’s surface.

Between the dynamically driven photosphere and the magnetically dominated
corona, lies the ever-elusive chromosphere, where the transition from the
high- to low-$\beta$ regime takes place. This interface region is also the
connective tissue between the Sun’s surface and its outer envelope. The
coronal heating conundrum manifests indeed in the chromosphere, where the
radiative losses due to its proximity to outer space need to be compensated by
efficient heating mechanisms; and magnetic reconnection is a promising
candidate to explain said heating (see, for instance, the discussion in Lagg
et al., 2017).

Chromospheric magnetometry is challenging at many levels, from the limits
imposed by instrument sensitivity to the complexity of the interpretation of
the observations. Ground-based observations of magnetically-sensitive
chromospheric spectral lines (most notably He i 10830 Å, Ca ii 8542 Å and He i
D3 at 5876 Å) have started to bridge the knowledge gap of what happens in this
interface region (e.g. Morosin et al., 2020; Libbrecht et al., 2019; Kleint,
2017; Martínez González et al., 2015; Centeno et al., 2010; Merenda et al.,
2006, among many others!). Trujillo Bueno et al. (2017) make a case for
spectropolarimetric observations in the ultraviolet part of the spectrum for
probing the magnetic field in the upper chromosphere and the transition
region. Due to their high line core opacity as well as the relative brightness
of the UV spectral range where they sit, the Mg ii resonance lines at 280 nm
are considered to be excellent probes, not only of the thermodynamics
Leenaarts et al. (2013a, b), but also the magnetic conditions in the high
chromosphere del Pino Alemán et al. (2016); Alsina Ballester et al. (2016);
Manso Sainz et al. (2019); Judge et al. (2021).

Leading up to the launch of the Interface Region Imaging Spectrograph (IRIS De
Pontieu et al., 2014), a series of papers analyzed the properties and modeling
requirements of Mg ii h&k. Leenaarts et al. (2013a) determined the minimum
atomic model necessary to reproduce the intensity spectrum of these lines, and
studied the effects of 1D vs 3D radiative transfer modeling, and including vs
ignoring the effects of partial frequency redistribution (PRD). In a followup
paper, Leenaarts et al. (2013b) carried out a detailed study of the
morphological properties of Mg ii h&k intensity profiles and their correlation
with the thermodynamical properties in the numerical simulation used to
generate them. They found that the positions and relative amplitudes of the
spectral features serve as proxies of the velocity and the temperature in the
higher layers of the chromosphere. Even after applying spectral and spatial
smearing to the synthetic data, some of these inferences remain robust Pereira
et al. (2013). de la Cruz Rodríguez et al. (2016) show that non-LTE inversions
of the spectral line intensity allow for the retrieval of the full temperature
stratification from the mid-photosphere all the way up to the transition
region, whilst the velocity gradients can be estimated throughout the
chromosphere. This is a rather computationally demanding approach, however,
and for this reason, Sainz Dalda et al. (2019) developed a machine-learning
approach that speeds up traditional inversion methods by a factor of
$10^{5}-10^{6}$, while still reproducing the thermodynamical state of the
atmosphere with comparable quality to traditional minimization-based
techniques.

The IRIS observations, however, only measure the intensity of the Mg ii lines.
Measuring the spectral line polarization would open up the possibility of
magnetic field diagnostics in this crucial interface region at the base of the
corona. In a renewed attempt to analyze spectropolarimetric observations of Mg
ii resonance lines from the Ultraviolet Spectrometer and Polarimeter on board
the Solar Maximum Mission (SMM, Bohlin et al., 1980), Manso Sainz et al.
(2019) confirmed that the magnetograph formula can be applied to observations
of the circular polarization in the line core. They obtained line-of-sight
magnetic fields of the order of 500 G from these observations. The authors
also report on observational evidence of reduction of the broad-band linear
polarization with respect to the field-free case due to weak magnetic fields
outside of active regions. This had been theoretically predicted by del Pino
Alemán et al. (2016) and Alsina Ballester et al. (2016) and presents a
promising diagnostic for the weaker chromospheric fields of the quiet Sun.

Decades after the SMM first measured the polarization of the Mg ii UV doublet,
the 2019 launch of the Chromospheric LAyer Spectropolarimeter (CLASP2,
Ishikawa et al., 2021) revisited this part of the spectrum with very promising
findings. The circular polarization profiles of h&k in a plage region away
from disk center were interpreted using the weak field approximation (WFA) and
revealed longitudinal magnetic fields of up to $\sim 300$ G at the top of the
chromosphere, while a nearby Mn I multiplet provided simultaneous information
for lower layers.

Outside of active regions, the linear polarization in the core of Mg ii k (as
well as the surrounding broad-band polarization) is expected to be dominated
by scattering phenomena. Trying to interpret these non-Zeeman signatures via
the WFA relations would lead to erroneous inferences. Furthermore, most
currently available non-LTE spectral line inversion codes do not model the
quantum-mechanical processes responsible for scattering polarization and the
Hanle effect, also rendering them inadequate for the interpretation of these
signals (see Centeno et al., 2021, for an analysis of this effect in Ca ii
8542 Å)). The newly published Tenerife Inversion Code (TIC, Li et al., 2022)
is the first of its kind able to tackle scattering polarization and Hanle-
effect signatures in the Mg ii resonance lines. This is advancement will
enable detailed interpretation of the spectra, but comes at large
computational cost and can only be used sparingly111The computational expense
of inverting a single spectrum is of the order of 1000 core-hours..

While scattering polarization signals dominate the linear polarization in and
around the Mg II doublet (except for the core of the h line, see del Pino
Alemán et al., 2016; Casini et al., 2002), the core of Stokes V remains
agnostic to these effects (del Pino Alemán, private communication), allowing
for a Zeeman interpretation of the LOS component of the magnetic field. In the
presence of stronger fields, even the linear polarization becomes dominated by
the Zeeman effect, and retrieval methods based on it should in principle still
work. In this paper, we study the error incurred by the WFA when applied to
the Mg ii h & k lines to obtain magnetic field inferences from synthetic
spectra emerging from an active region simulation. In particular, we look at
how spectral resolution degrades the accuracy of the results. Section 2
presents a brief overview of the WFA as well as the radiative transfer code
(Hanle-RT) that enables the spectral synthesis. We first explore the effects
of spectral resolution on the WFA applied to spectra emerging from a standard
semi-empirical model atmosphere (Section 3), and then in Section 4 we take the
analysis to a more diverse set of spectra emerging from a radiative magneto-
hydrodynamic (rMHD) simulation of an active region. We discuss the feasibility
of the method and draw some concluding remarks in Section 5.

##  2 Methodology

###  2.1 The Weak Field Approximation

The weak field approximation (WFA; e.g. Landi Degl’Innocenti & Landolfi, 2004)
is a simple, yet relatively robust method, to extract magnetic field
information from the intensity and polarization of spectral lines. As its name
suggests, its validity is limited to the “weak” field regime, or more
precisely, to the range of field strengths ($B$) in which the Zeeman-splitting
($\Delta\lambda_{\rm B}$) of the spectral line is much smaller than its
Doppler width, $\Delta\lambda_{\rm D}$:

$\bar{g}\frac{\Delta\lambda_{\rm B}}{\Delta\lambda_{\rm D}}<<1$ (1)

where $\bar{g}$ is the effective Landé factor of the spectral line and the
Zeeman splitting is proportional to the magnetic field strength ($B$, in
gauss) and the square of the central wavelength of the spectral line under
consideration ($\lambda_{0}$, in Å):

$\bar{g}\Delta\lambda_{\rm B}=4.67\times 10^{-13}\bar{g}\lambda_{0}^{2}B$ (2)

When applying a perturbative scheme to the radiative transfer equation in this
regime, one arrives at a series of relations that connect the circular and the
linear polarization spectra to derivatives of the intensity with respect to
wavelength. Eq. 3 relates Stokes $V$ to the first derivative of Stokes $I$
with respect to wavelength, through the line-of-sight (LOS) component of the
magnetic field vector, $B_{\rm LOS}$, and a constant $C_{\parallel}$ that
depends on atomic properties of the spectral line.

$V(\lambda)=-C_{\parallel}B_{\rm LOS}\frac{\partial I}{\partial\lambda}$ (3)

Eqs. 4 and 5, on the other hand, relate the total linear polarization
($L=\sqrt{Q^{2}+U^{2}}$) to the derivative of Stokes I through the square of
the transverse component of the magnetic field vector, $B_{\rm T}^{2}$, and a
different constant, $C_{T}$, that also depends on properties intrinsic to the
spectral line:

$L(\lambda_{w})=\frac{3}{4}B_{T}^{2}\left|C_{T}\frac{1}{\lambda_{w}-\lambda_{0}}\right|\left|\frac{\partial
I}{\partial\lambda}\right|_{\lambda_{w}}$ (4)

$L(\lambda_{0})=\frac{1}{4}B_{T}^{2}\left|C_{T}\frac{\partial^{2}I}{\partial\lambda^{2}}\right|_{\lambda_{0}}$
(5)

where the first of these two equations is applicable only in the line wing
($\lambda_{w}$) and the latter in the line core (close to the central
wavelength, $\lambda_{0}$). The derivation of these relations as well as the
conditions for their applicability can be found in Landi Degl’Innocenti &
Landolfi (2004).

The WFA has been used in many works to extract quantitative information of the
magnetic field vector from spectropolarimetric observations (see, for
instance, Kriginsky et al., 2021; Vissers et al., 2021; Siu-Tapia et al.,
2020; Shchukina & Trujillo Bueno, 2019, for some recent applications of the
WFA). Chromospheric spectral lines tend to satisfy the WFA requirement of Eq.
1 for larger field strengths than their photospheric counterparts owing to the
fact that they are typically much wider than the latter. Additionally, lines
that sit in the ultraviolet part of the spectrum enjoy smaller Zeeman
splittings due to their shorter wavelengths, and lighter atomic species also
meet the weak field requirement better, since they typically experience larger
Doppler broadenings than their heavier counterparts.

The circular polarization of the Mg ii resonance lines is known to be amenable
to the WFA (del Pino Alemán et al., 2016), and given strong enough fields,
their linear polarization signals may be interpreted with it as well.

###  2.2 Spectral synthesis

Hanle-RT (see del Pino Alemán et al., 2016, for a more comprehensive
description) is a non-LTE polarized radiative transfer code for 1-D (plane-
parallel) stratified atmospheres. It accepts any multi-level or multi-term
atom (LS-coupling scheme) without hyperfine structure and in the presence of
magnetic fields of arbitrary strength. The code takes into account the effects
of non-isotropic radiative excitation of the atomic system, which are
responsible for the manifestation of scattering polarization. Therefore, the
code can model seamlessly all possible regimes of magnetic-induced effects on
the polarization of spectral lines: the Hanle, Zeeman, and (incomplete)
Paschen-Back effects, as well as the polarization effects associated with
level crossing and anti-crossing interference. The code takes into account the
effects of inelastic and elastic collisions, and also includes a comprehensive
description of partially coherent scattering (partial redistribution, or PRD)
from polarized atoms (limited to 3-term atoms of the $\Lambda$-type; see
Casini & Manso Sainz, 2016), which is important for modeling deep
chromospheric lines.

The atomic model of Mg II used to solve the statistical equilibrium (SE)
includes 3 terms of the singly-ionized species as well as the ground term of
Mg III. This is the smallest atomic model necessary to model the atomic level
polarization of the Mg ii UV doublet around 280 nm as well as its subordinate
triplet overlapping in the same wavelength range. The UV doublet is treated in
PRD whilst the subordinate triplet is computed assuming complete frequency
redistribution.

Figure 1: Mg ii h & k as well as the subordinate UV triplet lines synthesized
in a FAL-C model atmosphere with a magnetic field strength of $B=200$ G
inclined $45^{\circ}$ from the local vertical, as if observed at disk center.
Note the double lobes of the Stokes V profiles for the h & k lines. The x-axis
represents vacuum wavelengths, and Mg ii k sits around 2796.3 Å.

##  3 Modeling in a FAL-C atmosphere

We first synthesize the Mg ii h & k lines in the standard 1-D FAL-C (Fontenla
et al., 1990) semi-empirical model of the Sun’s atmosphere with an ad-hoc
uniform magnetic field of $200$ G inclined $45^{\circ}$ with respect to the
local solar vertical. For a disk center observing geometry, this configuration
results in a LOS component of the magnetic field of $B_{\rm LOS}\approx 141$
G. A uniform magnetic field is chosen in order to isolate the WFA inference
error from other radiative transfer effects resulting from LOS integration.
Fig. 1 shows the synthetic spectrum including the Mg ii resonance lines as
well as the subordinate triplet. Three different syntheses are computed in
this model atmosphere: the first one is a full Hanle-RT calculation; in the
second one we turn off magneto-optical effects; and in the third one we turn
off PRD effects instead.

We then apply Eq. 3 by carrying out a linear regression ($y_{i}=a\cdot
x_{i}+b$) of the data to determine the slope $a$ and the intercept $b$. Here,
$y_{i}$ correspond to the values of $V_{\lambda}$, while $x_{i}$ are taken to
be the values of $-C_{\parallel}dI_{\lambda}/d\lambda$, with the
proportionality between the two given by the fitted slope ($a=B_{\rm{LOS}}$),
and a fitted intercept $b$ that should be compatible with zero. Because the
WFA is just an approximation, the linear relationship between $V_{\lambda}$
and $dI_{\lambda}/d\lambda$ does not hold outside of the Zeeman weak field
assumption. When the data don’t meet the criteria for the WFA (and the
relationship is no longer linear), the linear fit might yield an intercept
that is not compatible with zero, and a slope that is wrongly interpreted as
$B_{\rm LOS}$.

Fig. 2 shows the WFA fit for the LOS component of the magnetic field applied
to the three synthetic profiles of Mg ii k. The blue dots represent the values
of Stokes V (rearranged as a function of increasing wavelength and normalized
by the local continuum intensity), while the orange line shows the derivative
of Stokes I with respect to wavelength, scaled by $C_{\parallel}B_{\rm LOS}$,
where $B_{\rm LOS}$ was obtained from a linear regression applied to Eq. 3.
The relative error incurred by the WFA is reported for each case inside the
corresponding panel. In the case where the WFA is applied to the inner lobes
of Stokes V only (top row) the inference error is $1.4\%$ when the line is
treated in PRD (left and middle panels), compared to 0.1% in the scenario
where PRD is turned off (right panel). However, when the outer lobes are taken
into account in the linear regression (bottom row), the inference errors jump
to $-13\%$ in the two PRD cases (bottom left and middle panels), which is also
evidenced by the poor quality of the fit. When PRD is turned off, the error is
still very small (bottom right) and the retrieved $B_{\rm LOS}$ produces a
good fit. This shows that the inaccuracy incurred by the WFA is mostly due to
PRD effects and can be avoided by excluding the outer Stokes V lobes from the
calculation. On the other hand, magneto-optical effects alone (for this regime
of field strengths) do not have negative impacts on the WFA retrieval of
$B_{\rm LOS}$222according to Landi Degl’Innocenti & Landolfi (2004), M-O
effects yield corrections of the order of
$(\Delta\lambda_{B}/\Delta\lambda_{D})^{4}$ for the derivation of the LOS
magnetic field from the WFA (pp. 488-489)..

In CRD, the WFA corresponds to the lowest-order result of a perturbative
solution of the polarized radiative transfer equation in the presence of a
magnetic field (Landi Degl’Innocenti & Landi Degl’Innocenti, 1973). In PRD,
however, the emissivity vector $\epsilon$ gets modified by a redistribution
correction $\delta\epsilon$ (see, e.g. Casini et al., 2017), which breaks the
symmetry of such a perturbative solution, and is ultimately responsible for
the observed departure of the validity of the WFA in PRD away from the line
core. We believe this is the only plausible physical mechanism that can be
responsible for the WFA breaking down away from the line core.

Figure 2: WFA fit of Stokes V for Mg ii k for the cases where the spectra were
synthesized with no approximations (left), when magneto-optical effects were
turned off (middle) and when no PRD effects were considered (right). The top
row corresponds to the WFA applied on the inner lobes of Stokes V only, whilst
the bottom row applies the equation to both sets of lobes. The error incurred
by the WFA is reported inside each panel.

###  3.1 Effects of spectral resolution

Most instruments that observe solar spectral lines have limited spectral
resolution. Spectral smearing mixes information across the wavelength domain,
possibly bringing information from the line wings into the line core, and
vice-versa. For the WFA, this means that information captured by the line
wings (which “sense” magnetic fields at lower atmospheric heights) may be
dragged into the core of the line, resulting in line-of-sight smearing of the
retrieved magnetic quantities. But even when the magnetic field is constant
with height, the PRD effects that are prominent in the outer lobes of Stokes V
can be partially transferred to the inner lobes, and “contaminate” the WFA
inference applied to the narrower spectral range.

In order to quantify the latter effect, we apply gaussian point spread
functions (PSF) of different widths to the synthetic spectra described above,
and apply the WFA to the inner lobes of Stokes V in the degraded spectra.
Because the spectral syntheses were carried out in an atmosphere with a
uniform magnetic field, the line-of-sight smearing of the retrieved magnetic
properties is inconsequential.

Figure 3: Application of the WFA fit (orange) to Mg ii k Stokes V spectra
(blue) that have been previously degraded with a gaussian point spread
function corresponding to a spectral resolution of R = 30,000. In the top row,
the WFA has been applied to the inner lobes of Stokes V only, whilst the
bottom panels show the results when applied to a wider wavelength range. PRD
effects in the full calculation (left) result in large inference errors that
are not present when PRD is turned off (right).

The top left panel of Fig. 3 shows the WFA fit (orange lines) for Mg ii k
applied to the inner lobes of Stokes V (blue dots) in the case of the full
calculation with PRD and a spectral resolution of R = 30,000 (which is
approximately the spectral resolution of the CLASP II observations). The
relative error in the retrieved $B_{\rm LOS}$ is $\sim 10.4\%$. Including the
outer Stokes V lobes in the WFA inference (bottom left) drives the relative
inference error to $-18\%$. On the other hand, when PRD is turned off (right
panels), the inference error remains very low ($0.2\%$) in both cases. This
example epitomizes how the PRD effects that are prominent in the outer lobes,
are brought into the inner lobes due to spectral smearing, negatively
impacting the WFA inference.

Fig. 4 shows the relative error of the WFA results for $B_{\rm LOS}$ as a
function of spectral resolution of the observations. As expected, smaller
values of R result in greater retrieval errors. It is important to note that,
without the effects of PRD, spectral convolution alone should (and does) not
impact the performance of the WFA in the case of a uniform magnetic field.

Figure 4: Error incurred by the WFA when inferring for $B_{\rm LOS}$ from the
Mg ii k line as a function of spectral resolution. The WFA was applied to the
inner lobes of Stokes V only.

##  4 The WFA applied to synthetic Mg ii spectra emerging from an active
region simulation

Figure 5: Physical parameters at h=0 km in the subset of the original MURaM
simulation chosen for this study. From left to right and top to bottom:
temperature, $T$, LOS component of the magnetic field, $B_{\rm LOS}$, LOS
velocity, $v_{\rm LOS}$, magnetic field strength, $B$, its inclination with
respect to the local vertical, $\theta$, and its azimuth in the horizontal
plane, $\phi$).

In this section we analyze the diagnostic capabilities of the WFA when applied
to Mg ii lines emerging from an active region simulation, and in particular,
how the accuracy of the WFA is affected by finite spectral resolution.

We start from a radiative magneto-hydrodynamic (rMHD) simulation of an active
region (AR) from the MURaM code with the same setup as the one described in
Rempel (2017). The simulation presents a small active region with a corona
that is self-consistently heated by the Poynting flux resulting from
photospheric magneto-convection. While the treatment of the chromosphere is
simplified (grey RT and LTE ionization), Bjørgen et al. (2019) showed that
synthetic chromospheric emission captures many observationally known features,
at least on a qualitative level. The purpose of this simulation is not to
provide an exact description of an active region chromosphere, but to provide
a test case with known ”ground truth” that samples the typical parameter space
of an active region. The size of the horizontal domain is 98x98 Mm2 with a
spatial sampling of 64 km in each horizontal direction. The vertical domain
spans $\sim 49.1$ Mm (sampled every 32 km), of which approximately $\sim 13.1$
Mm lie below the surface (defined as the height where the average optical
depth at 500 nm, $\tau_{500}$, is unity). We select a small subset of the
simulation box that encompasses a quarter of one of the sunspots and the
strongly magnetized granulation surrounding it to synthesize the Mg ii lines
in the UV. Fig. 5 presents the physical variables of the simulation in the
horizontal slice at h=0 km. The strong and almost vertical magnetic field at
the center of the umbra (top right corner of each panel) decreases rapidly in
strength towards the outer edge of the sunspot, where it also becomes almost
horizontal.

We extract the physical quantities for each column (1D atmosphere) in the
simulation and feed it to Hanle-RT to generate the Stokes spectra. Before
feeding the atmospheres to Hanle-RT some pre-processing is necessary. Each
column of the simulation is truncated 192 km below its $\tau_{500}=1$ level.
The relative shift between this level across individual atmospheres in the
simulation domain can be as large as 800km due to the surface being very
depressed at the center of the umbra compared to the surrounding granulation.
The top of each atmosphere is also truncated at the location where the
temperature reaches $10^{5}$K, since the Mg ii lines should form well below
said temperature (Heinzel et al., 2014). The atmospheres are further
simplified by removing the velocity field. Because there are no velocity
gradients, the emerging spectra will be close to symmetric. This
simplification is necessary for the purpose of isolating the effects of
spectral resolution on the WFA inferences from further vertical smearing that
the velocity gradients may introduce (although Centeno, 2018, showed that
moderate velocity gradients do not negatively impact WFA inferences of $B_{\rm
LOS}$ from Ca ii 8542 Å). Furthermore, an ad-hoc microturbulent velocity
profile is added to each column’s atmosphere. The latter is modeled as a
linear function of height, ranging from 0 kms-1 at h=-192 km, to 7 kms-1 at
the highest possible height (h=1856 km), so that it has qualitatively
comparable values to those in the semi-empirical model atmosphere FALC, as
well as in the inversion results of sunspot/plage IRIS spectra in de la Cruz
Rodríguez et al. (2016). This high microturbulence value still produces
spectra that are much narrower than those in Level 2 IRIS data, as shown in
Fig 6.

Figure 6: Dashed lines: averaged synthetic Mg ii spectra over a 240x240 km2
area in the top right (umbra) and bottom left (“plage”) corners of the
simulation box. For comparison, Level 2 IRIS spectra inside a sunspot and a
plage area are shown with solid lines. Note that the individual synthetic
spectra averaged for this figure were calculated at rest (no Doppler shift),
and were not subjected to spectral degradation. Both of these factors
contribute to them being narrower than the observed spectra. In this figure,
the MURaM and the IRIS specific intensities were scaled arbitrarily.

The spectra are synthesized in the 3-term Mg ii atom described above, for a
disk center observing geometry (i.e. a heliocentric angle of $0^{\circ}$).
Synthesizing the Mg ii h&k lines plus the UV triplet in the 3-term atomic
model with Hanle-RT is a computationally demanding exercise. The calculation
of the atomic level polarization (population imbalances and quantum coherences
amongst magnetic sublevels) as well as the treatment of the Mg ii h&k doublet
in PRD dramatically drive the increase in computing time compared to spectral
line modeling without these effects. The median run time for generating a
single spectrum is 10 core-hours (c-h), and synthesizing the 11,250 spectra in
the FoV in Fig. 5 took a total 145,000 c-h on NCAR’s Cheyenne supercomputer.

In the case of sunspots, magnetic fields are strong enough to significantly
depolarize the atomic levels, leading to polarization signals that exhibit,
almost typical, Zeeman patterns. In this regime, if the spectral lines are
wide enough, the WFA can be a suitable tool for quantifying the magnetic field
vector because the inference error induced by any residual atomic polarization
can typically be ignored for the expected field strengths of active regions
(Kuckein et al., 2009).

###  4.1 Retrieval of BLOS from Mg ii h & k

In this section we apply Eq. 3 to retrieve the LOS component of the magnetic
field from Stokes I and V of the Mg ii h and Mg ii k lines separately, but
first, we implement an automated algorithm that finds the wavelength range
corresponding to the inner lobes of Stokes V based on the position of its
peaks (which appear as zero-crossings in the first derivative of Stokes V with
respect to wavelength). Even in the absence of velocities, the shape of the
spectral line changes significantly across the FoV, and a single pre-chosen
wavelength interval is not adequate to exclude the outer Stokes V lobes in all
the spectra.

After finding the desired wavelength interval, we carry out a linear fit
between the left- and the right-handsides of Eq. 3, in the same way as in
Section 3, for each set of Stokes $I$ and $V$ emerging from the FoV of the
simulation box. Fig. 7 shows density scatter plots of the retrieved $B_{\rm
LOS}$ from Mg ii k (left) and Mg ii h (middle) against the MURaM model values.
The latter are taken at the height where each line center reaches optical
depth unity ($\tau_{\rm h}$ or $\tau_{\rm k}$ = 1). This is taken as a
representative height for comparison with the model values, however, the
emergent spectra are sensitive to a range of heights over which the magnetic
field is not constant. The gradient of the magnetic field around the region of
formation of the lines is responsible for the scatter in the WFA inferences.
The magnetic field bin size is 40 G, and the darkness of the gray-scale is
proportional to the number of samples in the bin. The red line represents the
ideal solution (if the WFA yielded exactly the model value). From these
figures it is clear that the WFA does a good job at retrieving the LOS
component of the magnetic field up to LOS field strengths of $B_{\rm LOS}\sim
1000$ G. Beyond this value, the WFA starts to systematically overestimate
$B_{\rm LOS}$. This is due the fact that we are entering the strong field
regime in which the WFA is no longer valid, according to the requirement of
Eq. 1. Note that the validity of the weak field regime hinges on the value of
$B$, not of $B_{\rm LOS}$. A LOS component of 1000 G necessarily entails a
field strength larger or equal to that. The right panel of Fig. 7 presents the
relative error incurred by the WFA when applied to the Mg ii k line as a
function of the total field strength, B, in the model. This panel clearly
shows that the systematic deviation of the WFA results towards larger values
starts around $B\sim 1500$ G. Fig. 8 shows the values of
$\bar{g}\Delta\lambda_{B}/\Delta\lambda_{D}$ for the Mg ii k line as a
function of height in one of the 1D MURaM atmospheres, for four different
(uniform) magnetic field strengths. Around 1500 G, the Mg ii k line no longer
satisfies the WFA condition. This threshold varies from one atmosphere to
another, since meeting or not the condition depends on the temperature and the
microturbulent velocity as well as the magnetic field strength.

Figure 7: Scatter density plots of the retrieved value of BLOS (y-axis)
against their model counterparts where $\tau_{\rm LINE}=1$ (x-axis) for Mg ii
k (left) and h (middle). The darker the grey-level, the higher the number of
samples in the bin. The bin size is 40 G. The red line represents the ideal
solution, if the WFA retrieved exactly the model value. The panel on the right
shows the relative error incurred by the WFA applied to Mg ii k as a function
of the magnetic field strength in the MURaM model.  Figure 8: Meeting the WFA
condition. The lines show the value of
$\bar{g}\Delta\lambda_{B}/\Delta\lambda_{D}$ for the Mg ii k line in one of
the MURaM atmospheres, where the magnetic field has been replaced by a uniform
one a function of height. Three different magnetic strength values are
considered, showing that, at the typical height of formation of the Mg ii
lines, a field of 1500 G no longer meets the condition of Eq. 1.

###  4.2 Effects of spectral resolution on the retrieval of BLOS

Figure 9: Scatter density plots of the retrieved value of BLOS retrieved from
Mg ii h against their model counterparts for the case of infinite spectral
resolution (left), R = 45,000h (middle) and R = 30,000 (right). The model
values are taken at the height where the core of each line reaches optical
depth unity. The darker the grey-level, the higher the number of samples in
the bin. The bin size is 40 G. The red line represents the ideal solution and
the blue line shows a linear fit through the data. The left panel in this
figure shows the same data than the middle panel of Fig. 7

In order to understand the effects of a finite spectral resolution on the WFA
inferences, we convolve the spectra emerging from the simulation with gaussian
PSFs of different widths.

Fig. 9 shows scatter plots of the WFA inference against the corresponding
model value for the cases of “infinite” spectral resolution (left), R =
$\Delta\lambda/\lambda=$45,000 (middle) and R = 30,000 (right). The red line
represents the one-to-one solution while the blue line shows a linear
regression of the data points in each scatter plot. As discussed earlier, the
WFA tends to over-estimate the field above $B_{\rm LOS}=1000$ G in the
spectrally resolved case (left panel). As the value of $R$ decreases, the
method starts to underestimate $B_{\rm LOS}$, counteracting the effect of the
WFA inaccuracy in the strong field regime. This systematic bias towards lower
fields (seen in the middle and right panels) cannot be attributed to the core
of the line sensing lower atmospheric layers due to spectral smearing (i.e.
along the LOS) mixing. Spectral information probing lower atmospheric layers
would typically lead to a bias towards larger inferred field strengths, not
weaker, because there is an almost monotonic decrease of the magnetic field
strength as a function of height across the entire FoV of Fig. 5. This points
to PRD effects being brought into the core of the line due to spectral
smearing and thus impacting (decreasing) the WFA inferences, which should
otherwise increase with spectral mixing (i.e. mixing along the line-of-sight).
The larger the value of R, the greater the impact on the inferred $B_{\rm
LOS}$.

###  4.3 Retrieval of BT through the WFA

The retrieval of the transverse magnetic field through the WFA relies on Eqs.
4 and 5. While Eq. 4 is applicable far away from the central wavelength of the
spectral line, $\lambda_{0}$, Eq. 5 is technically valid only in the core. For
this reason alone, the former can only be used to sample lower atmospheric
layers from the wings of the Mg ii lines, whilst the latter should be able to
probe the transverse component of the magnetic field in the higher layers of
the atmosphere. A practical caveat to both relations is that, when the
magnetic field is very weak, the linear polarization signals tend to be
dominated by scattering polarization, the Hanle effect, and magneto-optical
effects, rendering the WFA a useless tool in these cases. The magnetic field
needs to reach a few hundred gauss in strength in order to significantly
depolarize the atomic energy levels of the upper level of the Mg ii UV
doublet, and allow the linear polarization signals to enter the Zeeman-
dominated regime, where the WFA is valid. In the case at hand, the weakest
magnetic field in the entire FoV is still $\approx 500$ G at the top of the
atmosphere333The top of the atmosphere in this context is defined by the
height at which the temperature reaches $10^{5}$ K.

Since the retrieval of the magnetic field at the top of the chromosphere is of
most interest to this exercise, we will focus on testing the accuracy of the
WFA through Eq. 5. Three different applications of this equation are tested
within the core of the line, here defined as the wavelength range occupied by
the inner lobes of Stokes V. Method #1 performs a standard linear regression
of the left and right handsides of Eq. 5. In the linear regression analogy
described in Section 3, $y_{i}$ now corresponds to $L_{\lambda}$, while
$x_{i}$ takes the values of $1/4C_{T}|d^{2}I_{\lambda}/d\lambda^{2}|$. The
slope of the fit corresponds to $B_{\rm T}^{2}$, while the intercept should be
close to zero if the model represents the data adequately. In method #2, the
intercept $b$ is forced to be zero and only the slope is fitted, which results
in the following expression for the linear regression:

$B_{\rm T}^{2}=\frac{4}{C_{\rm
T}}\frac{\sum_{\lambda}(\big{|}\frac{\partial^{2}I_{\lambda}}{\partial\lambda^{2}}\big{|}L_{\lambda})}{\sum_{\lambda}\big{(}\frac{\partial^{2}I_{\lambda}}{\partial\lambda^{2}}\big{)}^{2}}$
(6)

In method #3, we calculate the wavelength-average of the left- and the right-
hand sides of Eq. 5 and obtain $B_{\rm T}^{2}$ as the ratio between the two:

$B_{\rm T}^{2}=\frac{4}{C_{\rm
T}}\frac{\sum_{\lambda}L_{\lambda}}{\sum_{\lambda}\big{|}\frac{d^{2}I_{\lambda}}{d\lambda^{2}}\big{|}}$
(7)

Additionally, we test the application of Eq. 13 in Martínez González et al.
(2012) (method #4). This approach, estimates the magnetic field vector by
minimizing a merit function that explicitly uses Stokes $Q$ and $U$, instead
of $L$. This results in the following equation to obtain $B_{T}$:

$B_{T}^{2}=\frac{4}{C_{\rm
T}}\frac{\sqrt{\sum_{\lambda}(Q_{\lambda}\frac{\partial^{2}I_{\lambda}}{\partial\lambda^{2}})^{2}+\sum_{\lambda}(U_{\lambda}\frac{\partial^{2}I_{\lambda}}{\partial\lambda^{2}})^{2}}}{\sum_{\lambda}(\frac{\partial^{2}I_{\lambda}}{\partial\lambda^{2}})^{2}}$
(8)

Fig. 10 shows the results of applying the WFA to Mg ii h for inferring the
transverse component of the field from the synthetic spectra using these four
different methods (from left to right, the columns correspond to methods #1,
#2, #3 and #4, respectively). Each panel presents a density scatter plot of
the retrieved transverse magnetic field against its model counterpart at the
height where $\tau_{\rm MgIIh}=1$. The three rows show the results for
different spectral resolutions, R.

Let’s take a look at the first row of results, for the spectrally resolved
case. The first and second panels show the WFA results from the linear
regressions of Eq. 5 with and without intercept, respectively (methods #1 and
#2). We find that, fixing the intercept to zero yields slightly more accurate
results than allowing it to be a free parameter in the regression. This is a
symptom of the WFA model not representing the data accurately (i.e. the best
linear fit does not naturally cross the origin). The third panel shows the
case where the linear fit is forgone and only a ratio of the wavelength-
averaged quantities is used. The accuracy is comparable to that of the
previous methods. The last panel shows the results from the application of Eq.
8, which systematically underestimates the values of $B_{T}$.

As we decrease the spectral resolution to 45,000 (middle row) and 30,000
(bottom row) we find that the accuracy of all methods degrades, with method #1
losing all correlation and retaining no diagnostic value at R = 30,000.
Methods #2 and #3 show very similar behaviors as the spectral resolution
worsens. They both result in an increased spread of the WFA solution around
the model value, and they over-estimate the transverse field by a larger
amount the poorer the resolution. The spectral smearing is bringing
information from the line wings into the core and “contaminating” the WFA
inferences. This over-estimate is partly due to the fact that the wings of the
line sense lower layers (and therefore stronger fields). The impacts of PRD
effects and scattering polarization in the line wings could also be
contributing to the error. From the setup of this particular experiment it is
impossible to quantify the relative contributions of each one of these
effects. In any case, this over-estimate is relatively well behaved
(systematic as a function of field strength) and could be corrected with a
proper calibration of the WFA inferences 444This may be MHD model-dependent,
in which case may only be done when MHD models are able to reproduce the
observed spectra.. The blue lines in these panels show a linear fit through
the scatter plot data, suggesting that a simple linear correction to the WFA
inference could bring it closer to the real $B_{\rm T}$ value. Method #4 also
results in an increased spread with systematically larger inferred values as
we degrade the resolution.

Figure 10: Scatter density plots of the values of BT retrieved from Mg ii h
against their model counterparts for the case of infinite spectral resolution
(top row), R = 45,000h (middle row) and R = 30,00 (bottom row). The four
columns correspond to retrieval methods #1, #2, #3 and #4 respectively,
described in the manuscript text. The darker the grey-level, the higher the
number of samples in the bin. The bin size is 40 G and the color scale is the
same for all panels. The red line represents the ideal solution if the WFA
inference were perfect, while blue lines show linear fits through the scatter
plot data.

It is worth noting that, while the amplitudes of the linear polarization
signals emerging from the AR simulation reach values of the order of 1% of the
continuum intensity, we might expect them to be much smaller in real
observations, challenging the detection of Stokes Q and U signals above
typical observational noise levels. A brief analysis of the effects of noise
is shown in the Appendix.

##  5 Conclusions

We study the impact of finite spectral resolution on the retrieval of magnetic
field quantities from the Mg ii h&k lines through the weak field
approximation.

First, we look at the case of the WFA applied to synthetic Mg ii spectra
emerging from a semi-empirical FAL-C model atmosphere with an ad hoc uniform
magnetic field with a LOS component of 141 G. Eq 3 is applied to the
spectrally resolved case in two wavelength ranges: one that contains only the
inner lobes of Stokes V and one that encompasses the outer lobes as well. We
find that the error in the retrieval of $B_{\rm LOS}$ jumps from 1.4% in the
case of the narrower wavelength range, to -13% when the outer lobes are
considered (left panels of Fig. 2). The results remain almost identical when
the spectra are synthesized without magneto-optical effects. It is only when
PRD is turned off that the WFA inferences in the two spectral ranges agree and
yield the smallest error (0.2%), confirming the results from del Pino Alemán
et al. (2016). It is well know than PRD effects affect much more prominently
the wings than the line core (Leenaarts et al., 2013a, b), and we speculate
that the correction to the emissivity term due to PRD is purely responsible
for the WFA breaking down.

Broad spectral point spread functions have the effect of spectral mixing,
bringing information from the line wings into the core and vice-versa. We
degrade the data spectrally by convolving them with gaussian PSFs of different
widths that emulate various spectral resolutions. We find that, when we apply
the WFA to the inner lobes of Stokes V in the degraded spectra, a larger error
ensues. By repeating the experiment turning off PRD effects, we show that the
latter are purely responsible for the inference errors, as the spectral
smearing brings information from the line wings into the inner wavelength
range around the core of the line. Fig. 4 shows the inaccuracy of the inferred
of $B_{\rm LOS}$ as a function of spectral resolution. The relative error
decreases with increasing spectral resolution, from $\sim 12$% for R = 30,000
to $\sim 3$% for R = 90,000.

In order to test this effect in a more diverse set of atmospheres, we
synthesize the Mg ii h&k spectra emerging from a rMHD simulation of an active
region from the MURaM code. We select a portion of the simulation box that
contains a quadrant of a sunspot and the strongly magnetized granulation
surrounding it. When Eq. 3 is applied to the inner lobes of Stokes V, both Mg
ii h and Mg ii k deliver remarkably accurate results for most pixels in the
FoV, when comparing the retrieved value of the LOS component of the magnetic
field to the model value at $\tau_{\rm LINE}=1$. The WFA inference starts to
deviate significantly from the model value starting at a magnetic field
strength of $\sim 1500$ G (right panel of Fig. 7), due to a departure from the
weak field regime. Both spectral lines deliver quantitatively comparable
results. The effect of a reduced spectral resolution results in the WFA
underestimating the LOS magnetic field when compared to the spectrally
resolved case. We justify that this bias is due to PRD effects being
transported into the line core due to spectral smearing, rather than a result
of sensing lower atmospheric layers due to line-of-sight mixing.

The extraction of the transverse component of the magnetic field through Eq. 5
was tested subsequently in the line core (defined here as the region spanned
by the inner lobes of Stokes V). We find that performing a linear regression
of the left and right handsides of the equation yields better results when the
intercept is forced to be zero than when it is a free parameter of the fit
(see first and second panels of the top row of Fig. 10). This is because the
WFA model does not reproduce the behavior of the data with fidelity. A simple
ratio of the wavelength averages of the left and right sides of Eq. 5 (third
panel of the top row) delivers equally good results, while Eq. 8 results in
systematically lower inferences of $B_{\rm T}$ (fourth panel of the top row).
Degrading the spectral resolution results in a systematic over-estimate of the
retrieved $B_{\rm T}$, which grows with decreasing value of $R$. The precision
of the estimate also worsens (see the increased spread in the scatter plots of
Fig. 10), yielding method #1 useless in the case of R=30,000. For methods #2
and #3 , the systematic offset of the WFA retrieval could be calibrated with a
simple linear fit.

One big caveat to the WFA is that the method does not shed information on the
atmospheric heights of the retrieved magnetic quantities. The WFA inferences
will correspond to weighted averages around the height of formation of the
spectral window that is used (typically the line core). Across the domain of
the MURaM simulation used in this work, the geometric height of formation of
the Mg ii cores varies by $\approx 1.4$ Mm, being very depressed at the center
of the umbra and a lot higher in the quieter areas. A similar variation is
expected for the heights probed by the WFA. This could pose a problem for
extrapolations of the vector magnetic field inferred from observations,
especially in the absence of further modeling that constrains the height
probed by the WFA.

It is interesting to note that synthesizing the Mg ii spectra in a simulation
box featuring 11,250 1D realizations of the atmospheric properties took
145,000 core-hours on NCAR’s Cheyenne supercomputer. This highlights one of
the difficulties of interpreting the Mg ii polarization spectra (and other
chromospheric resonance lines). The forward modeling alone is computationally
expensive, and traditional inversion methods based on merit function
minimization schemes tend to require many calls of the forward model until the
synthetic profiles converge to the observations, limiting the use of this
approach. That is why there are several ongoing explorations of machine-
learning based inversion algorithms, or of model parameterizations that can
significantly reduce the number of forward-model iterations. Physical
approximations such as the WFA remain valuable to render the magnetic field
retrieval tractable.

In conclusion, the WFA can be applied to the Mg ii h&k lines emerging from
strong field regions (sunspots and strongly magnetized surroundings), but one
has to be aware of the systematic impacts that finite spectral resolution has
on the inferences.

The magnetic field inferences through the WFA must be impacted by photon noise
in the observations, resulting not just in uncertainties, but also in
systematic biases (Martínez González et al., 2012; Centeno, 2018). Even though
it is beyond the scope of this work to study the effects of noise in detail,
we briefly evaluate the relative merits of the different methods used in
Section 4.3 for the inference of $B_{\rm T}$ when gaussian noise is applied to
the spectra. To quantify the effects of noise we add wavelength-independent
gaussian noise to Stokes $I$, $Q$, $U$ and $V$ and repeat the WFA inferences
for the spectrally resolved case. Three different signal-to-noise ratios (SNR)
are considered, $SNR=I_{\rm C}/\sigma=[\infty,1000,100]$, where the reference
signal, $I_{\rm C}$, is the pseudo-continuum intensity 3.3 Å blue-wards of Mg
ii k, and $\sigma$ is the standard deviation of the noise.

Figure 11 shows the WFA inferences of $B_{\rm T}$ from Mg ii h for the
noiseless case in the top row (which is the same as the top row of Fig.10),
for $SNR=1000$ in the middle row, and $SNR=100$ in the bottom row. While for a
$SNR=1000$ the results are still robust, a $SNR=100$ not only increases the
spread of the inferences around the ideal solution, but also introduces
systematic biases in almost all of the cases. While method #1 (linear fit of
Eq. 5 with free-fitting intercept) tends to experience a systematic bias
towards lower inferred values of $B_{\rm T}$, methods #2 and #3 are affected
in the opposite direction, with a tendency to overestimate $B_{\rm T}$. Method
#4 also experiences a bias towards larger inferred values, but is less
dramatically affected than the previous two methods.

Figure 11: Scatter density plots of the values of BT retrieved from Mg ii h
against their model counterparts for three different SNR values: SNR
$\rightarrow\infty$ (top row), SNR = 1000 (middle row) and SNR = 100 (bottom
row). The four columns correspond to retrieval methods #1, #2, #3 and #4
respectively, described in the manuscript text. The darker the grey-level, the
higher the number of samples in the bin, which is 40 G in width. The color
scale is kept constant across all panels. The red line represents the one-to-
one correspondence.

The inference of $B_{\rm LOS}$ is not affected by the noise levels tested
above. Because the amplitude of Stokes $V$ across the FoV of the simulation
domain is large, even a SNR=100 renders circular polarization signals
significantly above the noise level, retaining their diagnostic power with
little to no negative impacts.

This material is based upon work supported by the National Center for
Atmospheric Research, which is a major facility sponsored by the National
Science Foundation under Cooperative Agreement No. 1852977. We would like to
acknowledge high-performance computing support from Cheyenne
(doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information
Systems Laboratory, sponsored by the National Science Foundation. T.d.P.A.
acknowledges the funding received from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (ERC
Advanced Grant agreement No 742265).

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semantics, i.e., at least once, at most once, exactly once as well as ordered
or out-of-order delivery, (4) and tasks related to buffer management and
transport protocols.

The quality of decision-making on migration depends on the data available to
calculate its cost and benefit, as well as the freshness of the data. The
continuous collection and dissemination of monitoring data in a DSPS can be
expensive. Efficient monitoring solutions, and leveraging other sources of
monitoring data that are, for example, used for network and system management
have the potential to reduce the overall cost of a DSPS and ensure good
decision-making.

Leveraging historical data to perform predictions with advanced statistics or
modern machine learning solutions, as is done for traffic prediction in
network management (abbasi2021deep, ) and data prediction in wireless sensor
networks (dias2016survey, ), is another subject that deserves attention in
research. Both proactive migration and the use of amortization time in the
cost model require some form of prediction. The oxymoron of operator
migration, i.e., that the need for migration occurs when the cost of migration
is high, can be avoided with proactive migration. Furthermore, proactive
migration can be used to schedule a migration when the state is still small in
size. However, both traffic and data patterns might be changing during the
deployment of DSPSs, and appropriate and efficient online learning solutions
need to be investigated for operator migration.

##  6\. Conclusions

DSP is becoming increasingly important for handling data with high velocity
and a large variety. The variety of data from different sources and over time
as well as other system dynamics, e.g., resource availability, require
adapting distributed stream processing accordingly. Operator migration is the
mechanism for keeping a DSPS in an ”optimal” configuration over its lifetime.
This survey provided an overview of solutions for operator migration from a
historical perspective and that of the goal of migration. Both perspectives
show that the deployment environments and the purpose of the system have a
strong impact on the design of migration mechanisms. Unfortunately, the
terminology in this area is not always consistent. Therefore, we introduced a
conceptual model of operator migration based on the largest common denominator
in the literature to introduce a common/unified terminology. The model
facilitated the classification of existing solutions and structured their
description with respect to two research questions: (1) Which mechanisms are
used to perform migration? (2) How is the migration decision executed?
Emphasis was placed on its costs and benefits. These aspects are important for
operator migration, but are often only implicitly addressed or are neglected
altogether. The description of existing solutions shall provide the reader
with a good understanding of the design alternatives from an algorithmic
viewpoint. We complemented this with an empirical study to give the reader
some quantitative insights into the impact of different design alternatives
for migration mechanisms (i.e., all-at-once and partial state movements), and
the impact of the choice of data stream processing (i.e., Siddhi and Apache
Flink).

##  7\. Acknowledgments

This work was supported by the Parrot Project (Research Council of Norway,
IKTPluss, 311197). The authors thank Fabrice Starks and Stein Kristiansen for
insightful discussions on mechanisms of operator migration, and Boris
Koldehofe for reviewing and commenting on an earlier version of the
manuscript.

## References

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