Evaluating (and Improving) Estimates of the Solar Radial Magnetic Field Component from Line-of-Sight Magnetograms

3.2 MB·

by  K. D. Leka

Evaluating (and Improving) Estimates of the Solar
Radial Magnetic Field Component from Line-of-Sight
Magnetograms
K. D. Leka1 · G. Barnes1 · E. L. Wagner1 ·
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1 (cid:13)c
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2
Abstract

n Althoughformanysolarphysicsproblemsthedesirableormeaningfulbound-
a
ary is the radial component of the magnetic field B , the most readily available
J r
measurementisthecomponentofthemagneticfieldalongtheline-of-sighttothe
7
observer, B . As this component is only equal to the radial component where
1 los
theviewingangleisexactlyzero,someapproximationisrequiredtoestimateBr
] atallotherobservedlocations.Inthisstudy,acommonapproximationknownas
R
the“µ-correction”,whichassumesallphotosphericfieldtoberadial,iscompared
S
toamethodwhichinvokescomputingapotentialfieldthatmatchestheobserved
.
h B , from which the potential field radial component, Bpot is recovered. We
los r
p demonstrate that in regions that are truly dominated by radially-oriented field
-
o at the resolution of the data employed, the µ-correction performs acceptably
r if not better than the potential-field approach. However, it is also shown that
t
s for any solar structure which includes horizontal fields, i.e. active regions, the
a
potential-field method better recovers both the strength of the radial field and
[
the location of magnetic neutral line.

1
v Keywords: Magnetic fields, Models; Magnetic fields, Photosphere; Active Re-
6 gions, Magnetic Fields
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8
4
1. Introduction
0
.
1
Studies of the solar photospheric magnetic field are ideally performed using
0
the full magnetic field vector; for many scientific investigations which may not
7
1 require the full vector, it is the radial component Br which is often desired,
: as is derivable from vector observations. Yet observations of the full magnetic
v
vector are significantly more difficult to obtain, from both instrumentation and
i
X
data-reduction/analysis points of view, than obtaining maps of solely the line-
r of-sight component of the magnetic field B . Observations consisting of the
a los
line-of-sight component can accurately approximate the radial component only
1 NWRA,3380MitchellLn.,Boulder,CO80301,USA
email:[email protected] email:[email protected]
SOLA: ms.tex; 19 January 2017; 1:16; p. 1
Leka,Barnes,&Wagner
(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
(cid:1) (cid:3)(cid:4)(cid:1) (cid:6)(cid:7)(cid:8)(cid:9)(cid:4)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)
(cid:2) (cid:5)(cid:4)
(cid:1) (cid:3)(cid:4)(cid:1) (cid:6)(cid:7)(cid:8)(cid:9)(cid:4)(cid:10)(cid:15)(cid:14)(cid:13)(cid:16)
(cid:2) (cid:5)(cid:4)
Figure1. Geometryofprojectioneffectswhenobservingaphotosphericmagneticfieldvector
(black) in the image plane, from two different perspectives. Of note regarding the inferred
polarity is that the Blos component (pointed toward/away from the observer) changes sign
betweentheµ=1.0andµ=0.7viewingangles.
along the Sun-Earth line, that is where the observing angle θ = 0, or µ =
cos(θ)=1.0.Awayfromthatline, i.e. atanynon-zeroobservingangle,the line-
of-sight component deviates from the radial component. Observing solely the
line-of-sightcomponentofthe solarphotosphericmagneticfieldimpliesthatthe
observingangleθimposesanadditionaldifficultyininterpretingtheobservations
– it is not simply that the full strength and direction of the magnetic vector is
unknown, but the contribution of these unknown quantities to the B signal
los
changes with viewing angle. In other words, when µ=cos(θ)=1.0, the line-of-
sight component B is equal to the radial component B . Nowhere else is this
los r
true.
A common approach to alleviate some of this “projection effect” on the in-
ferredtotalfieldstrengthis to assumethatthe fieldvectoris radialeverywhere;
then by dividing the observed B by µ, an approximation to the radial field
los
maybe retrieved.This is the “µ-correction”.Its earliestuses firstsupportedthe
hypothesis of the overall radial nature of plage and polar fields, and provided a
reasonableestimate of polar fields for heliospheric and coronalmagnetic model-
ing (Svalgaard, Duvall, and Scherrer,1978; Wang and Sheeley, 1992). However,
itwasevidentfromveryearlystudiesusinglongitudinalmagnetographsandsup-
portingchromosphericimagingthatsunspotswerecomposedoffieldswhichwere
significantlynon-radial,i.e.inclinedwithrespecttothelocalnormal.Thisgeom-
etrycanleadtothenotoriousintroductionofapparentfluximbalanceand“false”
magnetic polarity inversion lines (see Figure 1) when the magnetic vector’s in-
◦
clination relative to the line-of-sight surpasses 90 while the inclination to the
local vertical remains less than 90◦ or vice versa (Chapman and Sheeley, 1968;
Pope and Mosher, 1975; Giovanelli, 1980; Jones, 1985). Although this artifact
canbecleverlyusedforsomeinvestigations(Sainz Dalda and Mart´ınez Pillet, 2005)
it generallyposes a hindrance to interpreting the inherent solar magnetic struc-
ture present.
TheinaccuracieswhicharisefromusingB aregenerallyassumedtobenegli-
los
◦
giblewhentheobservingangleθislessthan30 ;ifthefieldisactuallyradial,the
correctionisonlya≈13%error,andintroducedfalseneutrallinesgenerallyap-
pearonlyinthesuper-penumbralareas.Yetthisisastrongassumption,andone
knowntobeinaccurateformanysolarmagneticstructures.Theestimatesofpo-
SOLA: ms.tex; 19 January 2017; 1:16; p. 2
EstimatesofBr fromBlos
larradialfieldstrengthareespeciallycrucialforglobalcoronalfieldmodelingand
solarwindestimations(Riley et al., 2006;Riley et al., 2014),butthesemeasure-
mentsareexceedinglydifficult(Tsuneta et al., 2008;Ito et al., 2010;Petrie, 2015).
The gains afforded by using the full magnetic vector include the ability to
better estimate the B component by way of a coordinate transform of the
r
azimuthally-disambiguated inverted Stokes vectors (Gary and Hagyard,1990).
However, although there are instruments now which routinely provide full-disk
vector magnetic field data (such as SOLIS, Keller and The Solis Team, 2001;
HMI Hoeksema et al., 2014), the line-of-sight component B remains the least
los
noisy, easiest measurement of basic solar magnetic field properties.
Wepresenthereamethodofretrievingadifferent,andinthecaseofsunspots,
demonstrably better, estimate of the radial field boundary, Bpot, the radial
r
component of a potential field which is constructed from B so as to match
los
to the observedline-of-sightcomponent. This approachwasoriginallydescribed
by Sakurai (1982) and Alissandrakis (1981), but is rarely used in the literature.
We demonstrate here its implementation, including in spherical geometry for
full-disk data.Themethodis describedinsection2,the datausedaredescribed
in section 3, and both the planar and spherical results for Bpot are evaluated
r
quantitatively for active-region and solar polar areas in section 4.1 and 4.2. In
section 4.3 we reflect specifically on the different approximations in the context
of “false polarity-inversionlines” artifacts, and in section 4.4 we investigate the
reasons behind both regions of success and areas of failure.
2. Method
Approaches to computing the potential field which matches the observed line-
of-sight component are outlined here for two geometries, with details given in
AppendicesAandB.WhenoneisfocusedonalimitedpartoftheSunsuchthat
curvatureeffectsareminimal,aplanarapproachcanbeusedtoapproximatethe
radialfield(section2.1,appendixA).Theplanarapproachisfast,andreasonably
robustforactive-regionsizedpatches(section4.1).Whenthedesiredradial-field
boundary is the full disk, or covers an extended area of the disk, such as the
polar area, then the spherical extension of the method is the most appropriate
(section 2.2, appendix B); however, depending on the image size of the input,
calculating the radial field in this way can be quite slow.
2.1. Method: Planar Approximation
The line-of-sight component is observed on an image-coordinate planar grid.
We avoidhaving to interpolate to a regularheliographic grid by performing the
analysisusingauniformgridinimagecoordinates,(ξ,η).Restrictingthevolume
ofinterestto0<ξ <L ,0<η <L andz ≥0,andneglectingcurvatureacross
x y
the field ofview,the potential field canbe writtenin terms of a scalarpotential
Bpot =∇Φ, with the scalar potential given by
Φ(ξ,η,z) = A e[2πimξ/Lx+2πinη/Ly−κmnz]+A z, (1)
mn 0
Xm,n
SOLA: ms.tex; 19 January 2017; 1:16; p. 3
Leka,Barnes,&Wagner
where z is the vertical distance above the solar surface. The value of κ is
mn
determined by ∇2Φ=0, namely
m 2 m n
κ2 = (2π)2 (c2 +c2 ) +2(c c +c c )
mn (cid:20) 11 12 (cid:18)L (cid:19) 11 21 12 22 (cid:18)L (cid:19)(cid:18)L (cid:19)
x x y
2
n
+(c2 +c2 ) , (2)
21 22 (cid:18)L (cid:19) (cid:21)
y
wherec arethecoordinatetransformationcoefficientsgiveninGary and Hagyard
ij
(1990), and choose κ >0 so the field stays finite at large heights. The values
mn
of the coefficients A are determined by requiring that the observed line-of-
mn
sightcomponentofthefieldmatchestheline-of-sightcomponentofthepotential
field, which results in
2πij 2πik
FFT(Bl) = L L A (c a +c a )+ (c a +c a )−κ a
x y jk 11 13 12 23 21 13 22 23 jk 33
(cid:20) L L (cid:21)
x y
+L L a A δ δ . (3)
x y 33 0 0j 0k
where a are the elements of the field components transformationmatrix given
ij
inGary and Hagyard(1990),andFFT(Bl)denotestakingtheFourierTransform
of Bl. The value of A is determined by the net line-of-sight flux through the
0
field of view.
Placing the tangent point at the center of the presented field of view (corre-
sponding to evaluating the coordinate transformation coefficients and the field
components transformation matrix at the longitude and latitude of the center
of the field of view) is the default, producing a boundary labeled Bpot,center.
z
The resulting radial component estimation is least accurate near the edges, as
expected, and as such we also present Bpot,all, for which the tangent point is
r
placedateachpixelpresented,thefieldcalculated,andonlythatpixel’sresulting
radial field (for which it acted as the tangent point) is included.
2.2. Method: Spherical Case
The potential field in a semi-infinite volume r ≥R can be written in terms of a
scalar potential Bpot =−∇Ψ, given by
∞ n n+1
R
Ψ = R (gmcosmφ+hmsinmφ)Pm(µ), (4)
(cid:18)r(cid:19) n n n
nX=1mX=0
where µ = cosθ. Defining the coordinate system such that the line of sight
direction corresponds to the polar axis of the expansion results in particularly
simple expressions for the coefficients gm, hm (Rudenko, 2001). However, be-
n n
causeobservationsareonly availablefor the nearside ofthe Sun, itis necessary
to make an assumption about the far side of the Sun. The resulting potential
field at the surface r =R is not sensitive to this assumption except close to the
limb, so for convenience, let B (R,π−θ,φ) = B (R,θ,φ), where the front side
l l
of the Sun is assumed to lie in the range 0<θ <π/2.
SOLA: ms.tex; 19 January 2017; 1:16; p. 4
EstimatesofBr fromBlos
With these conventions, the coefficients are determined from
gnm = (cid:26)0(22nπ+(n3+)(mn−+m1))!!R02πdφ cosmφR01dµPnm+1(µ)Bl(R,µ,φ) nn++mm oevdedn(5)
and
hmn = (cid:26)0(22nπ+(n3+)(mn−+1m)!)!R02πdφ sinmφR01dµPnm+1(µ)Bl(R,µ,φ) nn++mm oevdedn(6)
and the radial component of the field is given by
∞ n n+2
∂Ψ R
B = − = (n+1) (gmcosmφ+hmsinmφ)Pm(µ). (7)
r ∂r (cid:18)r(cid:19) n n n
nX=1mX=0
When evaluated at r =R, this produces a full-disk radial field boundary desig-
natedBpot,sph fromwhichextractedHARPsandpolarsub-regionsareanalyzed
r
below. Our implementation of this approach uses the Fortran 95 SHTOOLS li-
brary(Wieczorek et al., 2016)forcomputingthe associatedLegendrefunctions.
The routines in this library are considered accurate up to degrees of n ≈ 2800.
For the results presented here, a value of n = 2048 was used, corresponding to
a spatial resolution of about 2Mm.
3. Data
For this study we use solely the vector magnetic field observations from the
Solar Dynamics Observatory (Pesnell, 2008) Helioseismic and Magnetic Imager
(Scherrer et al., 2012;Hoeksema et al., 2014).Twosetsofdatawereconstructed:
afull-disktestandasetofHMIActiveRegionPatches(“HARPs”;Hoeksema et al.,
2014,Centeno et al.,2014,Bobra et al.,2014)over5years.Forboth,inorderto
keep comparisons as informative as possible, we construct line-of-sight compo-
nentdatafromthevectordatabytransformingthemagneticvectorcomponents
into a B map: B =Bcos(ξ) where ξ is the inclination of the vector field in
los los
theobservedplaneofthe skycoordinatesystem,asreturnedfromtheinversion.
The full-disk data target is 2011.03.06 15:48:00 TAI; this date was chosen
due to its extreme B0 angle such that the south pole of the Sun is visible, the
varietyofactiveregionsvisibleatlowµ=cos(θ)observingangle(awayfromdisk
center),andthepresenceofanorthernextensionofremnantactive-regionplage.
The hmi.ME 720s fd10 full-disk series was used; data are available through the
JSOC lookdata tool1. Because the weaker fields do not generally have their
◦
inherent 180 ambiguity resolved in that series and we will be evaluating the
Bpot method in poleward areas of weaker field, two customizing steps were
r
taken. First, a custom noise mask was generated (ambthrsh = 0, rather than
the default value of 50). Second a custom disambiguation was performed using
the cooling parameters: ambtfctr = 0.998, ambneq = 200, ambngrow = 2,
1jsoc.stanford.edu/lookdata.html
SOLA: ms.tex; 19 January 2017; 1:16; p. 5
Leka,Barnes,&Wagner
ambntx = ambnty = 48 (Hoeksema et al., 2014; Barnes et al., 2017); com-
pared to the default HMI pipeline implementation, these parameters provide
smaller tiles over which the potential field is computed to estimate dB/dz, a
smaller“buffer”ofnoisypixelsaroundwell-determinedpixels,andslowercooling
forthesimulatedannealingoptimization.Disambiguationresultsweregenerated
for 10 random number seeds. Pixels used for the comparisons shown herein
are only those for which both the resulting equivalent of the conf disambig
segmentis≥60andtheresultsfromall10randomnumberseedsagreed,aswell.
This requirement translates to 75.3% of the pixels with conf disambig ≥ 60
and88.8%ofthepixelswithconf disambig=90beingincluded.Inthecaseof
the present data, there is a 0.2% chance that the disambiguation solution used
results by chance, even in the weak areas (including the poles).
FromthisfulldiskmagnetogramthenineidentifiedHMIActiveRegionPatch
(“HARP”) areas are extracted using the keywords harpnum, crpix1, crpix2,
crsize1, crsize2 fromthe hmi.Mharp 720sseries.Two polar regions were also
extracted: a “Northern plage area”, which is an extended remnant field and a
“South Pole Region” that encompasses the entire visible polar area. A context
image is shown in Figure 2, and summary information about each sub-area is
givenin Table 1,including the WCScoordinatesfor the twonon-HARP regions
for reproducibility. Also in Table 1 are summary data for an additional 22 sub-
areas,each2562 pixelsinsizecenteredalongthe midpointsinx, y onthe image
at a variety of µ=cosθ positions.
Throughoutthis study,wedifferentiate betweenwhenplanarapproximations
are invoked and when curvature is accounted for by referring to “B ” and
z
“B ”, respectively. The potential-field approximation is performed three ways,
r
as described in section 2, above: using a planar approximation from the center-
point coordinates Bpot,center of each HARP or extracted sub-region, using a
z
planarapproximationwitheachpoint ofthe extractedregionusedasthe center
point Bpot,all, and using the spherical full-disk approach Bpot,sph. In addition,
r r
we calculate two common µ-correction approximations for each sub-region: the
center point value of µ=cos(θ) used as a tangent point to obtainBµ =B /µ,
z los
and secondly each pixel’s µ value is calculated and applied independently, for
Bµ(s) =B /µ(s) where (s) is the spatial location of the pixel.
r los
The second set of data consists of a subset of all HARPs selected over 5.5
years, selected so as to generally not repeat sampling any particular HARP:
on days ending with ’5’ (5th, 15th, and 25th) of all months 2010.05 – 2015.06,
the first ‘good’ (quality flag is 0) HARP set at :48 past each hour on/after
15:48 was used. HARPs which were defined but for which there were no active
pixels are skipped. The result is 1,819 extracted HARPs without regard for
size, complexity, or location on the disk. Effectively the hmi.Bharp 720s series
was used, including the standard pipeline disambiguation. NWRA’s database
initially began construction prior to the pipeline disambiguation being per-
formed for earlier parts of the mission, thus for some of the data base, the
hmi.ME 720s fd10 data were used, the HARP regions extracted and the dis-
ambiguation performed in-house, matching the implementation performed in
◦
theHMI pipeline.Allanalysisisperformedupto80 fromdisk center,andonly
pointswithsignificantsignal/noiseintherelevantcomponents(S/N >3relative
SOLA: ms.tex; 19 January 2017; 1:16; p. 6
EstimatesofBr fromBlos
Table 1. Extracted AreaDescriptions
I.D. HARP NOAA Locale µ=cos(θ) Description
No. ARNo.
H392 392 11163 N22W54 0.54 smallplage
H393 393 11164 N32W39 0.65 largecomplexactiveregion
H394 394 11165 S17W59 0.49 smallactiveregion
H399 399 N/A N27W13 0.87 smallplage
H401 401 11166 N15E34 0.79 simpleactiveregion
H403 403 11167 N21W03 0.93 bipolarplage
H407 407 11169 N23E62 0.43 smallactiveregion
H409 409 N/A S17E58 0.51 smallplage
H411 411 N/A N22E27 0.92 smallspot
N.Plage N/A N/A N51E01 0.63 northremnantplage
crpix1=1304crpix2=344
crsize1=1516crsize2=428
S.Pole N/A N/A S60E01 0.51 southpolararea
crpix1=1199crpix2=3527
crsize1=1427crsize2=440
QS E350 N/A N/A N00E70 0.35 crpix1=3709crpix2=1921
QS E450 N/A N/A N00E63 0.45 crpix1=3625crpix2=1921
QS E550 N/A N/A N00E57 0.55 crpix1=3514crpix2=1921
QS E650 N/A N/A N00E49 0.65 crpix1=3370crpix2=1921
QS E750 N/A N/A N00E41 0.35 crpix1=3181crpix2=1921
QS E850 N/A N/A N00E32 0.85 crpix1=2923crpix2=1921
QS E950 N/A N/A N00E18 0.95 crpix1=2511crpix2=1921
QS W 1000 N/A N/A N00W01 1.00 crpix1=1874crpix2=1921
QS W900 N/A N/A N00W26 0.90 crpix1=1076crpix2=1921
QS W800 N/A N/A N00W37 0.80 crpix1=761crpix2=1921
QS W700 N/A N/A N00W46 0.70 crpix1=542crpix2=1921
QS W600 N/A N/A N00W53 0.60 crpix1=377crpix2=1921
QS W500 N/A N/A N00W60 0.50 crpix1=251crpix2=1921
QS W400 N/A N/A N00W66 0.40 crpix1=154crpix2=1921
QS W300 N/A N/A N00W72 0.30 crpix1=82crpix2=1921
QS N375 N/A N/A N68E01 0.375 crpix1=1921crpix2=143
QS N775 N/A N/A N39E00 0.775 crpix1=1921crpix2=708
QS N875 N/A N/A N29E00 0.875 crpix1=1921crpix2=992
QS N975 N/A N/A N13E00 0.975 crpix1=1921crpix2=1495
QS S925 N/A N/A S22E00 0.925 crpix1=1921crpix2=2650
QS S825 N/A N/A S34E00 0.825 crpix1=1921crpix2=3005
QS S725 N/A N/A S44E00 0.725 crpix1=1921crpix2=3242
SOLA: ms.tex; 19 January 2017; 1:16; p. 7
Leka,Barnes,&Wagner
Figure 2. Full-diskimage of the line-of-sightcomponent of the solar photospheric magnetic
field on 2011.03.06 at 15:48:00 TAI, scaled to ±200G. The solar limb is indicated as are the
HMIActiveRegionPatcheslabeledwiththeir“HARPnumber”,andtheadditionaltwopolar
areas used for this analysis. Additional small patches of quiet-Sun, distributed in µ = cosθ
are indicated as grey boxes, occasionally overlapping. Solar north is up, west to the right,
positive/negative directedfieldisshownaswhite/blackrespectively.
to the returned uncertainties from the inversion, and propagated accordingly)
are included in the analyses. For this larger dataset, all calculations are done
with a planar approximation. The “answer” is B , and the boundary estimates
z
calculated are: Bµ = B /µ, Bµ(s) = B /µ(s) (which imparts a spherical
z los r los
accounting due to the variation of µ over the field of view), and Bpot,center.
z
Bpot,all and Bpot,sph are computationally possible but extremely slow, and are
r r
not employed for this second dataset.
4. Results
Fortheresultspresentedhere,the“goldenstandard”istakentobetheradialor
normalfieldascomputedfromthevectordata,andtothisquantitywecompare
resultsofdifferentapproximationsofthe boundary.We docautionthatB data
r
do include the observed B component, which is inherently noisier than the
trans
B component. Additionally we stress that the comparisons are performed
los
against a particular instrument’s retrieval of the photospheric magnetic field
vector, which may not reflect the true Sun as per influences in polarimetric
sensitivity, spectral finesse, spatial resolution, etc.
SOLA: ms.tex; 19 January 2017; 1:16; p. 8
EstimatesofBr fromBlos
3 3
(a) (b)
2 2
1 G) 1
Blos (kG) −01 µos / (s) (k −01
Bl
−2 −2
−3 −3
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
Br (kG) Br (kG)
3 3
(c) (d)
2 2
G)
planar,centerBr (kpot −011 Blos (kG)pot −011
−2 −2
−3 −3
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
Br (kG) Blos (kG)
Figure3. (a)–(c)Non-parametricdensityestimatescomparingthedistributionoftheradial
fieldfromthevectorfielddataBr (x-axis)totheinferredradialfieldstrengthestimatesusing
representativesofthedifferentmodelsdiscussedinthetext(y-axis).Shownherearedistribu-
tionsfromNOAAAR11164(H393).Contourlevelsareequalinlog(probability)rangingfrom
[10−3−102],withx=ylineincludedforreference.ThecomparisonsareforBragainst(a)Blos
(b)Blos/µ(s), (c)Bz potential, planar,center-pointpivotPanel (d)shows,insummary,that
the µ-corrected estimates generally show less bias compared to the Blos fields, but still have
alargerandomerror.Thepotentialfieldcorrectedfieldsshowlargerdifferencesforweakfield
strengths, but less random error.In (d) the scatter plot is between the Blos calculated from
theinversion,andtherecoveredBlos calculatedfromthesphericalpotential vector(Sect. B).
Whiletheboundary isthus fairlywell recovered, thedisagreement beyond machine-precision
differencesisduetothesphericalcalculationnotbeingperformedwithhighenoughdegreeto
removeallsmall-scale“ringing”.
4.1. Field Strength Comparisons
To demonstrate the general resulting trends for each of the radial field approxi-
mations,wefirstpresentdensity-histogramsoftheinferredradialfieldstrengths
fortworepresentativesub-regions,NOAAAR11164(HARP#393,Figure3)and
the south pole region (Figure 4). Throughout, we do not indicate the errors for
clarity; a 10% uncertainty in field strength is a fair approximation overall, and
a detailed analysis beyond that level is not informative here.
For both HARP H393 (NOAAAR11164)and the south pole area, the initial
comparison of B to B (panel (a) in Figures 3, 4) shows the expected signa-
los r
ture of underestimated field strengths overall. Note that for both regions, but
especially for the the south pole, there is a strong underestimation of the radial
field strength across magnitudes.
TheBµ(s) correction(panel(b)inFigures3,4;theBµ plotisalmostidentical
r z
and not shown here) shows improvementby eye for both regions,with distribu-
SOLA: ms.tex; 19 January 2017; 1:16; p. 9
Leka,Barnes,&Wagner
1 1
(a) (b)
0.5 0.5
G)
Blos (kG) 0 µs / (s) (k 0
o
−0.5 Bl −0.5
−1 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
Br (kG) Br (kG)
1 1
(c) (d)
G) 0.5 0.5
planar,centerBr (kpot −0.05 Blos (kG)pot −0.05
−1 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
Br (kG) Blos (kG)
Figure 4. Same as Figure 3, but for the south pole area shown in Figure 2. These panels
show,insummary,thattheµ-correctedestimatesforthesouthpolarareashowdistinctspurs
withincorrectpolaritywhilethepotential fieldestimates havelargebiases.
tions systematically deviating less from the x = y line. However, in the case of
the H393 corrections, the stronger-field strengths are often over-corrected, and
the opposite-polarity erroneouspixels are exacerbatedin their error.This trend
isalsotrueforthesouthpolearea:bothanimprovement(especiallyforstronger-
field points) and the appearance of a distinct erroneous opposite-polarity spur
in the Bµ(s) results.
r
In the next panel (panel (c) in Figures 3, 4), a representative potential-field
option, Bpot,center, is shown with regards to B (again, Bpot,sph and Bpot,all
z r r r
plotslook essentiallyidentical);the weakerfieldstrengthsappearto be less-well
corrected than the Bµ(s) approximation, although the strong-field approxima-
r
tions for the sunspots in H393are significantlybetter than the Bµ(s) correction.
r
There appears to be a small number of points for which the opposite polarity
is retained but overall the stronger field points (generally ≥ 1000G) lie close
to the x = y line. There are still significant deviations from the B field; this
r
is expected at some level since a potential-field model is being imposed, and it
cannot be expected that the solar magnetic fields are in fact potential. Addi-
tionally, while a non-linear force-free field model may better represent the true
field(Livshits et al., 2015),thereisinsufficientinformationintheB boundary
los
withwhichtoconstructsuchamodel.Forthesouthpoleregion,thestrong-field
areasare less well correctedthan was seen in the Bµ(s) plot (Figure 4, panels c,
r
µ(s)
b respectively), but the distinct incorrect-polarity spur visible for B is less
r
pronounced in the potential-field-based estimate.
SOLA: ms.tex; 19 January 2017; 1:16; p. 10