Issues with time--distance inversions for supergranular flows

0.31 MB·English

by  Michal Švanda

Astronomy&Astrophysicsmanuscriptno.ms_rev3 (cid:13)cESO2015
January28,2015
Issues with time–distance inversions for supergranular flows
(Research Note)
MichalŠvanda1,2
1 AstronomicalInstitute,AcademyofSciencesoftheCzechRepublic(v.v.i.),Fricˇova298,CZ-25165,Ondˇrejov,CzechRepublic
e-mail:[email protected]
2 AstronomicalInstitute,FacultyofMathematicsandPhysics,CharlesUniversityinPrague,VHolešovicˇkách2,CZ-18000,Prague8,
CzechRepublic
Received23October2014;accepted26January2015
5
1
ABSTRACT
0
2 Aims.Recentstudiesshowedthattime–distanceinversionsforflowsstarttobedominatedbyarandomnoiseatadepthofonlyafew
n Mm.Itwasproposedthattheensembleaveragingmightbeasolutiontolearnaboutthestructureoftheconvectiveflows,e.g.,about
a thedepthstructureofsupergranulation.
J Methods. Time–distance inversion is applied to the statistical sample of ∼ 104 supergranules, which allows to regularise weakly
abouttherandom-noisetermoftheinversioncostfunctionandhencetohaveamuchbetterlocalisationinspace.Wecomparethese
6
inversionsatfourdepths(1.9,2.9,4.3,and6.2Mm)whenusingdifferentspatio-temporalfilteringschemesinordertogainconfidence
2
abouttheseinferences.

Results. The flows inferred by using different spatio-temporal filtering schemes are different (even by the sign) even-though the
]
R formalaveragingkernelsandtherandom-noiselevelsareverysimilar.Theinvertedflowsalteratesitssignseveraltimeswithdepth.
Itissuggestedthatthisisduetotheinaccuraciesintheforwardproblemthatarepossiblyamplifiedbytheinversion.Itispossible
S
thatalsoothertime–distanceinversionsareaffectedbythisissue.
.
h
Keywords. Sun:helioseismology–Sun:Convection
p
-
o
r
1. Motivation of solar convection zone from surface measurements of wave
t
s traveltimes.Formallyitmeansthatwereplaceδq in(1)byv ,
aHelioseismology is considered to be a state-of-the-art method whereβ = (x,y,z),andsearchforit.Thiscanonlyβ beachieveβd
[
of solar research to learn about the solar interior (see review
byinversemodelling.
2 by Gizon et al. 2010). Among various helioseismic methods, Tosolvetheinverseproblem,onehastosetupthecostfunc-
vthe local helioseismology plays an important role in studying
tionwithvariousterms,whichcanbecasttoasystemoflinear
0the spatially localised details. The time–distance helioseismol-
equations.Howtoperformsuchtaskwasindetailsdescribedand
6ogy(Duvalletal.1993)isoneoftheapproaches.Itconsistsof
discussedelsewhere.Theresultingflowestimatevinvisgivenby
1tools for measuring and interpretation of travel times of solar
4waves.Sinceitsformulation,itwasusedextensivelytoinferes- (cid:90) (cid:88)
0peciallyinformationaboutflowsandsound-speedanomaliesin viαnv(r0;z0) = Kβα(r−r0,z;z0)vβ(x)d2rdz
1.upperlayersofsolarconvectionzone. (cid:12) β
(cid:88)
0 The method assumes that the travel time τa is bound to the + wα(r −r ;z )na(r), (2)
5plasmaanomaliesδq withrespecttothebackgroundbyalinear a i 0 0 i
1equation α i,a
:
v (cid:90) (cid:88) wherewαa aretheinversionweightstobedeterminedandKβαin-
Xiδτa(r)= d2r(cid:48) Kβa(r(cid:48)−r)δqβ(r(cid:48),z)+na(r), (1) dicatesthecomponentoftheinversionaveragingkernel(alinear
r (cid:12) β combination of the sensitivity kernels with inversion weights).
a The averaging kernel describes the smoothing of the real solar
whereKaisthesensitivitykernelcomingfromtheforwardmod- convectiveflowsandisconstructedtopeakaroundatargetdepth
α
elling, r = (x,y) is a horizontal position vector in a Cartesian z0.Thesecondtermontheright-handsideof(2)indicatesthere-
coordinatesystem(theremainingverticalcomponentwillbede- alisationoftherandomnoise.
noted as z) and na is the travel-time random noise. Index a de- Fortheinversionmethodwhichisthebaseofthisstudy,the
notestheselectionofthewaves(thecombinationofthespatio- reader is referred to Jackiewicz et al. (2012) and Švanda et al.
temporal filter, spatial averaging, and the distance between the (2011).Thecostfunctioncontainsfourtermsbalancedbythree
travel-timemeasurementpoints)andgreekindicesαorβselect trade-offparameters:thequalityofthefit(themisfit)oftheav-
theperturbationδq(flowcomponents,soundspeedperturbation, eraging kernel to a user-defined target function localised in the
...). Sun,theleveloftherandomnoise,thelevelofthepollutionof
Oneofthemaingoalsoftime–distancehelioseismologyisto the inverted flow component by other components (the cross-
inferthestructureofconvectiveflowsinthenear-surfacelayers talk), and the term which ensures that the resulting inversion
Articlenumber,page1of8
A&Aproofs:manuscriptno.ms_rev3
weights are spatially localised (which is a requirement in order image capturing mostly the pattern of supergranulation was re-
tofulfilthemathematicalassumptions). moved.Thedatacubesthenunderwentthespatio-temporalfilter-
Oneofthebiggestissuesintime–distancehelioseismologyis ing for a set of filters (f to p ridge filters and eleven standard
4
thepresenceoftherealisationnoise.Pressureandsurfacegrav- phase-speedfilters)toretainonlythewaveswithdesiredproper-
itywavesarerandomlyexcitedbythevigorousconvectionand ties.Thetraveltimeswerethenmeasuredfromthefiltereddat-
hence their statistics inherently contains a large random com- acubesusingthelinearisedGizon&Birch(2004)approachfor
ponent. This acts as a random noise in any of the helioseismic a set of distances for each of the spatio-temporal filters using
observablesandtraveltimesofthewavesarenoexception.Var- thecentre-to-annulusandcentre-to-quadrantaveragingschemes.
ious helioseismic methods deal with the random noise differ- Travel-timemapsofthewavessensitivetowardsthesurface(the
ently.Sometimesthenoiseisignored(e.g.Kosovichev&Duvall f modeandfirstfourphase-speedfilters)inacentre-to-annulus
1997),sometimesthevarianceofthetraveltimesisconsidered geometry show clearly the pattern of divergence centres corre-
(e.g.Zhaoetal.2001)andintheidealcase,thefulltravel-time spondingtosupergranulation.
noisecovariancematrixisconsidered(e.g.Couvidatetal.2005; These travel-time maps were inverted in an inversion
Jackiewiczetal.2008;Švandaetal.2011). pipeline.TheinversioncodeiswritteninMatlablanguageand
Thepreciseknowledgeofthenoisecovariancematrixises- describedindetailsinŠvandaetal.(2011).ItutilisestheBorn-
sential,asthiserrorpropagatesthroughtheinversionprocedure approximation sensitivity kernels (Birch & Gizon 2007)1 and
and translates into the realisation of the random error in the fulltravel-timecovariancematrixmeasureddirectlyfromalarge
inverted maps. The determination of the full noise-covariance set of travel-time maps. The inversion implements the Multi-
matrix is not straightforward. There are two approaches being channelSubtractive-OLAapproach(Jackiewiczetal.2012)with
used. Either the covariance matrix is read out directly from the additional terms of the cross-talk minimisation and inversion
data by measuring a large set of travel-time maps, or it is de- weights localisation. The cost function is regularised strongly
rived from the model using a Monte-Carlo-like approach. Both about these terms, as they both are possible sources of bi-
approaches have their own advantages and disadvantages. The ases.Boththetravel-timeandinversionpipelineswerevalidated
data-driven approach would be probably ideal, however there againstthesurfacemeasurements(Švandaetal.2013).
mightbetime-varyingsystematicerrorsinthetravel-timemea- Three different combinations of sensitivity kernels were
surements,whichwillaffectalsotheestimatederrorlevelofthe used. The first set combined all ridge-filtered kernels (f to p
4
invertedquantity.Themodel-drivenmethodsreliesonhowpre- with a set of annulus radii in the range of 5 to 20 pixels with a
ciselydoestheusedmodelagreewiththeobservedpowerspec- stepof1px,hence240independentkernels),secondallphase-
trum of the waves. So it would be ideal if one could minimise speed-filtered kernels (eleven standard phase-speed filters with
theeffectimpreciseknowledgeoftravel-timenoisetotheknowl- fivedistancesforeachofthemastabulatedinTable1ofCouvi-
edgeofsolarflows. datetal.(2006),together165kernels)andfinallytheinversion
Moreover,severalstudies(Woodard2007;Jackiewiczetal. that combined all these kernels at once (405 kernels together).
2008)showedthattheinversionsforsnapshotofthesolarflows Theinversionusingthecombinedfilteringschemewasusedonly
averagedoverafewhoursstartstobedominatedbytherandom-
recently (Švanda 2013; DeGrave et al. 2014b,a) and the tests
noisecomponentalreadyatveryshallowdepths.
byŠvanda(2013)showedthatforatestingdepththeyprovided
In Švanda et al. (2011) we suggested the use of the ensem- flowestimatesthatwerehighlycomparable.Thesetestshowever
ble averaging approach. The ensemble averaging uses a strong wereperformedinadifferentregime,whenthecostfunctionwas
assumption that the random-noise realisations in the individual regularisedstronglyabouttherandom-noiseterm.
representatives of the ensemble are independen√t, and hence in A different regime was used in this study by running the
theaveragetherandom-noiselevelscalesas1/ N,where N is inversion suitable for the ensemble averaging approach. In this
thenumberofrepresentatives.WhenNislargeenough,itallows case,therequirementontherandom-noisetermoftheinversion
torelaxthenoisetermintheinversioncostfunctionandthusto costfunctionmayberelaxed,whichallowstheinversiontofind
decrease the effect of the possible inaccurate knowledge of the thesolutionwithasmallermisfit.Inanotherwords,theinversion
noisecovariancematrix.Theensembleaveragingapproachcer- resultsintheaveragingkernelthatisbetterlocalisedinspaceand
tainlyisnotusefulforstudyingthesnapshotsoftheflows,how- hasfewersidelobesusuallymakingtheinterpretationdifficult.
everwheninvestigatingthesetoftherepresentativesofthesame
Thetestingsampleconsistedof17474individualsupergran-
phenomenon (supergranules, sunspots,...), it seems extremely
ular cells, identified by a watershed algorithm in the centre-to-
powerful. In recent years it seems to be a standard method of
annulus travel-time maps. The segmentation algorithm and its
helioseismic research (e.g. Duvall et al. 2006; Duvall & Birch
implementationisdescribedelsewhere(Švandaetal.2014).The
2010;Švanda2012;Birchetal.2013;Švandaetal.2014).
resulting flow maps of all three flow components were aligned
about the centres of all detected supergranules and averaged,
however for the following discussion only the v component
2. Inversion x
(thecomponentinthedirectionofsolarrotation)wasused.The
The results presented in this note were obtained using the real
datacomingfromtheHMIarchive.Usingthestandardtracking 1 Allpoint-to-pointsensitivitykernelswerecomputedbyacodekc3
and mapping pipeline (created and maintained within the Ger- written by Aaron Birch, which uses the normal mode summation ap-
proachwithconsideredmodesuptoradialorderof8,quadrupolesource
man Science Center for SDO by H. Schunker and R. Burston),
atthedepthof100km,andtheobservationheightof300km.Thecor-
24-hours long Dopplergram datacubes were tracked on a daily
relationtimewaschosentobe48s.Averyfinegridinboththewave-
basis,from8May2010to12July2010.Thesedatacubescov-
number (6×10−5 Mm−1) and frequency (43 µHz) space was chosen
eredthecentralpartofsolardiscroughlyhaving60degreeson
for a initial computation, however the results do not seem to be ex-
asideinthePostelprojectionwithacadenceof45s.Eachdat- tremelysensitivetothegridselection.Thepoint-to-annulusandpoint-
acube was processed in a standard way in a travel-time mea- to-quadrantkernelswerecomputedsubsequentlyutilisingspatialaver-
surement pipeline. From each frame of the datacube, the mean aging.
Articlenumber,page2of8
MichalŠvanda:Issueswithtime–distanceinversionsforsupergranularflows(RN)
ridge+phase−speed filters ridge filters phase−speed filters
30 30 30
2
20 20 20
m] 10 m] 10 m] 10 1 −1s]
M 0 M 0 M 0 0 m
∆ y [ −10 ∆ y [ −10 ∆ y [ −10 −1v [kx
−20 −20 −20
−2
−30 −30 −30
−20 0 20 −20 0 20 −20 0 20
∆ x [Mm] ∆ x [Mm] ∆ x [Mm]
σ=10 ms−1 σ=8 ms−1 σ=14 ms−1 x 10−3
4
0 0 0 3]
target depth 1.9 Mm −m
3 M
2 2 2
m] m] m] el [
h [M 4 h [M 4 h [M 4 2 kern
ept 6 ept 6 ept 6 1 ng
d d d gi
a
8 8 8 0 er
v
A
10 10 10
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
x [Mm] x [Mm] x [Mm]
ridge+phase−speed filters ridge filters phase−speed filters
30 30 30
20 20 20 0.5
m] 10 m] 10 m] 10 −1s]
M 0 M 0 M 0 0 m
∆ y [ −10 ∆ y [ −10 ∆ y [ −10 v [kx
−20 −20 −20 −0.5
−30 −30 −30
−20 0 20 −20 0 20 −20 0 20
∆ x [Mm] ∆ x [Mm] ∆ x [Mm]
σ=9 ms−1 σ=8 ms−1 σ=14 ms−1 x 10−3
3
0 0 0 3]
target depth 2.9 Mm −m
M
2 2 2
m] m] m] 2 el [
h [M 4 h [M 4 h [M 4 kern
pt 6 pt 6 pt 6 1 g
e e e n
d d d gi
a
8 8 8 er
0 v
A
10 10 10
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
x [Mm] x [Mm] x [Mm]
Fig.1.Theinversionsforthehorizontalflowsinanaveragesupergranuleatdepths1.9and2.9Mm,togetherwiththedisplayofthecorresponding
averagingkernels.Theleftcolumnrepresentstheinversionutilisingcombinedridge+phase-speedfiltering,themiddlecolumnisfortheridge
filtersonlyandtherightcolumnforphase-speedfiltersonly.
claims are exactly the same for the remaining horizontal com- sensitivitykernelswhichcomefromforwardmodelling.Onlya
ponent v and the issues to be revealed get even worse for a slight regularisation is performed by the random-noise term. A
y
weak vertical v component, for which the random-noise con- strong regularisation about the cross-talk and weights localisa-
z
straintwasrelaxedbeyondthereasonablesignal-to-noiseratio. tiontermsshouldguaranteetheminimisationofbiases.
Allthreedifferenttravel-timefilteringschemesintheinver-
sionprovidesimilaraveragingkernelsandsimilarrandom-noise
3. Interpretationissues
levels.Equation(2)impliesthatinthatcasealsotheestimateof
The aim of this study was to test the credibility of the time– the inverted flow should be similar. Such claim was verified by
distanceinversionsforconvectiveflows.Inthemisfit-dominated varying the trade-off parameters for the inversions utilising the
regimethemainburdenoftheinversionqualityiscarriedbythe samesetofmeasurements.
Articlenumber,page3of8
A&Aproofs:manuscriptno.ms_rev3
ridge+phase−speed filters ridge filters phase−speed filters
30 30 30
0.2
20 20 20
m] 10 m] 10 m] 10 0.1 −1s]
M 0 M 0 M 0 0 m
∆ y [ −10 ∆ y [ −10 ∆ y [ −10 −0.1v [kx
−20 −20 −20 −0.2
−30 −30 −30
−20 0 20 −20 0 20 −20 0 20
∆ x [Mm] ∆ x [Mm] ∆ x [Mm]
σ=6 ms−1 σ=7 ms−1 σ=12 ms−1 x 10−3
0 0 0 3]
target depth 4.3 Mm 2 −m
M
2 2 2
m] m] m] 1.5 el [
h [M 4 h [M 4 h [M 4 1 kern
pt 6 pt 6 pt 6 g
e e e n
d d d 0.5 gi
a
8 8 8 er
v
0 A
10 10 10
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
x [Mm] x [Mm] x [Mm]
ridge+phase−speed filters ridge filters phase−speed filters
30 30 30 0.5
20 20 20
m] 10 m] 10 m] 10 −1s]
M 0 M 0 M 0 0 m
∆ y [ −10 ∆ y [ −10 ∆ y [ −10 v [kx
−20 −20 −20
−30 −30 −30 −0.5
−20 0 20 −20 0 20 −20 0 20
∆ x [Mm] ∆ x [Mm] ∆ x [Mm]
σ=6 ms−1 σ=8 ms−1 σ=17 ms−1 x 10−4
0 0 0 153]
target depth 6.2 Mm −m
M
2 2 2
m] m] m] 10 el [
h [M 4 h [M 4 h [M 4 kern
ept 6 ept 6 ept 6 5 ng
d d d gi
a
8 8 8 er
v
0 A
10 10 10
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
x [Mm] x [Mm] x [Mm]
Fig.2.SameasFig.1,butforthedepthsof4.3and6.2Mm.
Theinvertedflowmapsalongsidewiththecorrespondingav- direction of the inversion (equation (2), first term on the right-
eragingkernelsandnoiselevelsarepresentedinFigs.1and2.It handsideforα (cid:44) β)arenegligible(notdisplayed).Thesecond
seemsthattheabovementionedimplicationdoesnotholdwhen apparentissueisthatwithintheinversionsbasedonthesameset
different sets of measurements are used in inversions. For each ofmeasurement,thesignoftheflowreversesrepeatedlyasone
investigated depth (1.9, 2.9, 4.3, and 6.2 Mm) one can obtain goesdeeper.Thisdoesnotseemtobesupportedfromthesideof
different answers (even by a sign) just by selection of the dif- thetheoryofsolarconvection.
ferent set of measurements, when the random-noise levels are Theinversionsforstatisticalensemblesaresortofextreme.
comparable (within the factor of two, given the magnitudes of Inthisextremeregimewehoweverseeclearlythattheyarenot
theinvertedflowswithverylargesignal-to-noiseratios)andthe robust.Thereisnoreasonsnottobelievethatinthecasewhen
averagingkernelsareverysimilar.Ithastobenotedthatdueto the noise-regularisation term is stronger and the resulting flow
thestrongregularisationofthecostfunctionaboutthecross-talk mapsareseeminglyrealistic,theproblemspersisttoanunknown
term,thecomponentsoftheaveragingkernelsthatarenotinthe (possibly smaller) extent. For instance the number of reversal
Articlenumber,page4of8
MichalŠvanda:Issueswithtime–distanceinversionsforsupergranularflows(RN)
Inversion ρ(vakern,vinv) RMS(vakern) expectednoise derivednoise
x x x
[ms−1] [ms−1] [ms−1]
1.9Mm,combined 0.99 70 14 11
2.9Mm,combined 0.99 59 9 8
4.3Mm,combined 0.99 49 7 6
6.2Mm,combined 0.99 45 6 5
2.9Mm,ridge 0.99 71 14 12
2.9Mm,ridge 0.99 59 12 9
4.3Mm,ridge 0.99 49 10 8
6.2Mm,ridge 0.99 45 8 6
1.9Mm,phase-speed 0.98 70 16 14
2.9Mm,phase-speed 0.98 59 15 13
4.3Mm,phase-speed 0.97 49 15 12
6.2Mm,phase-speed 0.97 44 15 11
Table1.Testsoftheinversionusingthesyntheticdata.Foreachinversion,thecorrelationcoeficientbetweentheexpectedandinvertedflowis
given,theRMSoftheexpectedflow,theexpectedRMSoftherandom-noisecomponent,anditsmeasuredvalue.
may be manipulated by the strength of the noise-regularisation 100
term. Generally speaking, the stronger the regularisation, the
fewer reversals. It is interesting to point out that in the past
thedepthofthesupergranulationwasestimatedfromthedepth
where the inverted flow reversed its sign. Duvall et al. (1997); 50
Duvall(1998);Zhao&Kosovichev(2003);Švandaetal.(2009)
reportedthereversalsdepthwithinafewMmbelowthesurface. 1]
−
Recently,DeGraveetal.(2014b)usedsimilartime–distancein- ms 0
version as this study to validate it by using the state-of-the-art [
n
simulation of solar convection and saw the supergranular flow akerx
reversal, even when it was not present in the simulation. The v
authorsclaimedthatitwasduetotheshortcomingsofthehelio- −50
seismicinversionmethods.
3.1. Verificationoftheinversionusingsyntheticdata −100
−100 −50 0 50 100
A natural explanation of the issues described above would be vxinv [ms−1]
anerror(aprogrammer’sbug)intheinversioncode,oramath-
ematical problem in the inversion itself (e.g., a degeneracy of Fig.3.Adirectcomparisonoftheexpectedhorizontalflowvaluestothe
invertedonesonthepixel-to-pixelbasis(onlyeach10thpixelisplotted
the matrix to be inverted). To eliminate such possibility, a test
tomaketheplotsimpler).Thesolidlinerepresentsthelinearfittothe
involving the synthetic data was applied, similarly to the tests
crosses,thethindottedlinestheintervalsofthepredictedrandom-noise
performedbyŠvandaetal.(2011).
(1σinterval),andthethickdashedlineisthelinewiththeunityslope.
A snapshot from the simulation of the convective Sun-like
flows (Ustyugov 2006) was convolved with the appropriate set
of the sensitivity kernels, in accordance with Eq. (1), to create Mr+p Mr Mp
χ2 [s] 0.24 0.24 0.32
asetofsynthetictravel-timemaps.Arandomrealisationofthe r+p
travel-timenoisehavingthecovariancematrixthatoftheinver- χ2r [s] 0.26 0.30 0.33
sion was added to these maps to mimic the random excitation χ2[s] 0.20 0.16 0.32
p
of solar waves. In total, three different sets of travel-time maps Table2.Matchoftheforward-modelledtraveltimesforthreeconsid-
werecomputed,onefortheridgefilters,oneforthephase-speed eredmodelstotheobservedones.
filters and one for all filters combined. These travel-time maps
were inverted by using exactly the same inversion weights as
usedabove.
tentwiththepredictedonefromtheinversion.Theworstscase
Theresultingflowmapswerecomparedonthepixel-to-pixel
(i.e. depth 6.2 Mm for the phase-speed kernels, which has the
basis to the flow maps obtained by the smoothing of the simu-
lowestcorrelationcoefficientandthesignal-to-noiseratio)isdis-
lationwiththeinversionaveragingkernels.Essentially,theleft-
playedexplicitlyinFig.3.Eventhisworstcasescenariodemon-
hand side of Eq. (2) was directly compared to the first term of
theright-handsideofthesameequation,denotedasvakernhence- stratesthattheinversionbehaveswellandnosignofaobvious
α bugisfound.
forth.Theroot-mean-squared(RMS)valueoftherandomnoise
term(thesecondtermontheright-handsideofEq.(2))wasthen
comparedtotheexpectedleveloftherandomnoise,anotherout-
3.2. Doestheinvertedflowfittheobservedtraveltimes?
putoftheinversion.TheresultsaresummarisedinTable.1.
Onecanseethattheinversionbehavesasexpected.Thein- In SOLA inversions it is not guaranteed that the inverted flow
vertedflowmapsvinv arehighlycorrelatedtotheexpectedflow model provides a reasonable fit to the observed travel times.
mapsvakern.Thedeterminedleveloftherandomnoiseisconsis- It is obvious already from Eq. (1): in order to obtain forward-
Articlenumber,page5of8
A&Aproofs:manuscriptno.ms_rev3
M M M
P+R R P
0 0 0
0.4
2 2 2
m] m] m] 0.21]
M 4 M 4 M 4 −s
h [ h [ h [ 0 m
dept 6 dept 6 dept 6 −0.2v [kx
8 8 8
−0.4
10 10 10
−20 0 20 −20 0 20 −20 0 20
x [Mm] x [Mm] x [Mm]
Fig.4.Estimatedcontinuoushorizontalvelocitiesobtainedfromfourtomographicmapsatdepthsof1.9,2.9,4.3,and6.4Mm.Themagnitudes
aresaturatedatlevels±0.5kms−1.
modelled travel times, the knowledge of the continuous flow 4. Possiblecauses
model is needed, which is not returned by SOLA. A proper
There are several suspicious contributors possibly responsible
reconstruction is probably possible (a work in progress), how-
everforasimpleillustrationamuchrougherestimatevestcanbe fortheissues.
α Incompatibility of the travel time maps with the inver-
made:
(cid:88) sion weights. As it was pointed out many times in the past by
vest(r,z)= vinv(r;z )K1D(z;z ), (3)
α α 0 α 0 various authors, it is crucial that any step in the data process-
z0 ing (mapping, filtering, the way the travel times are measured)
(cid:82)
where K1D = d2rKα(r,z;z ) is a horizontally averaged av- must be taken into account when computing the corresponding
α α 0
eraging kernel. Such an estimate approximately fulfills Eq. (2) sensitivitykernels.Inourcase,thisneedwasenforcedasmuch
with the random-noise term excluded. The vest components of aspossible.Thepixelsizewasexactlythesameincaseofboth
x
three different models from inversions based on different fil- thedataprocessingandthecomputationsensitivitykernels,the
tering schemes are shown in Fig. 4 – MP+R for the combined spatio-temporalfilteringwasdonenotonlyusingthesamecode,
ridge+phase-speed filtering scheme, M for ridge and M for but moreover using the exactly same files, in which the filters
R P
phase-speedfiltersbasedinversionsrespectively.Thesuspicious werestored.TheBornsensitivitykernels(Birch&Gizon2007)
alterations of the sign of the horizontal flow with depth are are consistent with the Gizon & Birch (2004) definition of the
prominentlyvisible.Models MP+R and MP aresimilaratdepths travel times. The issue with unknown impact are the possible
less than 5 Mm and differ by sign deeper down. Model M is non-linearitiesinboththeforwardandinverseproblems.
R
significantlydifferentfromtheothertwo. Difference of the power spectrum of the real data and
From these models, forward-modelled travel-time maps the model. The sensitivity kernels are computed using the
werecomputedfollowingEq.(1)withthenoisetermneglected. power spectrum, which comes from the model, which is con-
Such neglection was fully justified by the ensemble averaging sidered Sun-like (Christensen-Dalsgaard et al. 1996). However,
technique, which increased the signal-to-noise of the measured theeigenfrequenciesdeviatefromthedataeigenfrequenciesfor
travel-time maps by more than two orders. A match to the ob- higherordermodes(alreadya p6 modefrequenciesareconsid-
servedtraveltimesτa totheforward-modelledonesτa wases- erablyoff),theeigenfrequenciesarenotavailableforverysmall
timatedfrom fm wave numbers (k < 0.1 Mm−1). To investigate the influence of
1 (cid:88) (cid:16) (cid:17) thisissue,Irecomputedsensitivitykernels,traveltimes,andin-
χ2 = RMS τa (r)−τa(r) , (4) versionswithanadditionalspatio-temporalfilter,whichlimited
fm M fm
a thepowerspectrumtotheregionwherethemodelpowerspec-
where M wasthetotalnumberoftravel-timemeasurementsin- trummatchedwelltothepowerspectrumoftheuseddatacubes.
dexed by a. The larger χ2 , the worse the forward-modelled Hence low wave numbers were filtered out, as was all the sig-
fm
traveltimesfittheobservedones. nalofoscillationsbeyondthe p ridge.Theintroductionofthis
6
Thefitwasevaluatedforeachconsideredvector-flowmodel additionalfilterhadonlyminorimpactontheresults.
for all three sets of travel-time measurements separately. The Inaccuracy of sensitivity kernels. Should the sensitivity
results are summarised in Table 2. One would naively expect kernels forward-modelled from a reference solar model be dif-
the main-diagonal terms be much smaller than the off-diagonal ferent from “those of the real Sun”, we must expect an impact
terms.Fromavisualinspectionboththeobservedandforward- of this deviation to the results. The difference is expected to
modelled travel-time maps it becomes clear that all the fits are besmallifthereferencemodelisSun-like.Unfortunately,there
far from being satisfactory. Dispite being very different (even doesnotseemtobeadirectwaytomeasurethesensitivityker-
byasign),allthreemodelsfittheobservedtraveltimesequally nels from solar data, perhaps except for the iterative inversions
(badly). (Hanasoge&Tromp2014;Hanasoge2014),introducedtohelio-
Thesimplereconstructionofthecontinuousflowfieldprob- seismologyonlyrecently.Theremightbethewaytoatleastver-
ablyalsoaffectstheresults,butgiventhefactthatthealteration ifythetotalintegralofthekernel,whichistheworkinprogress
ofthesignisvisiblealreadyinthetomographicmaps,theaver- and will be reported on in a later paper. This point embodies
agingkernelsareveryconfinedaroundthetargetdepthandhave both the effects of the differences between the reference model
negligiblesidelobes,itcanbeexpectedthatamorerealistices- andtherealSunandthepossibleissueswiththekernelcomputa-
timateofthecontinuousflowfieldwillnotdramaticallychange tion,boththeoreticalandnumerical.Thispointisalsosupported
theconclusions. by DeGrave et al. (2014b), where the authors using a realistic
Articlenumber,page6of8
MichalŠvanda:Issueswithtime–distanceinversionsforsupergranularflows(RN)
10 3
ridge component ridge component
phase−speed component phase−speed component
2
−1m] 5 sum −1m] sum
M M 1
el [ el [
n n
er 0 er 0
k k
g g
n n
gi gi−1
a a
er −5 er
v v
A A−2
2.9 Mm
1.9 Mm
−10 −3
−25 −20 −15 −10 −5 0 −25 −20 −15 −10 −5 0
z [Mm] z [Mm]
0.8 1
ridge component ridge component
0.6 phase−speed component phase−speed component
1] sum 1] sum
−m −m 0.5
0.4
M M
el [ el [
n 0.2 n
r r
e e 0
k k
g 0 g
n n
gi gi
a−0.2 a
er er−0.5
v v
A A
−0.4
4.3 Mm 6.2 Mm
−0.6 −1
−25 −20 −15 −10 −5 0 −25 −20 −15 −10 −5 0
z [Mm] z [Mm]
Fig.5.Thecontributionsofvariousfilteringschemestotheinversionaveragingkernelincaseofthecombinedfilteringscheme.
convection simulation noticed a significantly decreasing corre- depthsof1.9Mmand2.9Mm,whichisnotrealistic.Onehasto
lationbetweenthemeasuredandforward-modelledtraveltimes bearinmindthatthesubtractionsoccurineveryinversion,e.g.
forridgefilterswithhigherorders(p andbeyond)andalsofor thecontributionofpoint-to-annulusmeasurementsissubtracted
3
largerphasespeeds. fromthecontributionofthepoint-to-quadrantmeasurementsetc.
Anotherindicationforthisexplanationisthatthemagnitudeof
Mathematicallyposedproblem.Thispossibilityisclosely theinvertedflowwhichusesthephase-speedfiltersseemstobe
relatedtothepreviouspoint.Let’ssaythatthesensitivitykernels abitunrealistic(around2kms−1forthedepthof1.9Mm).
are accurate (again, by “accurate” it is ment “those of the real Bothlastpointsimplicatethatthe“real”inversionaveraging
Sun”)towithin1%.Thecombinedinversioncombines405such kernels may be very different from those predicted by the in-
kernels.These1%errorstranslatethroughtheinversionandmay version,perhapsevenhavinganegativetotalintegral.Thenalso
beamplifiedtoanunknownextent.Therearesomehintsthatthis thecross-talkcontributionisnotconstrained,whichisestimated
islikelythemostprobablycause.Thecontributionsoftheaver- fromthecross-talkcomponentsoftheaveragingkernel.
aging kernel from two different filtering schemes are displayed
inFig.5.Onecanseethatinallfourcasesdiscussedinthisre-
searchnotethetwocomponentsfromdifferentfilteringschemes
5. Lessonslearned
largely subtract from each other. Should one filtering scheme
have an unknown systematical bias, the resulting “real” aver- This work is based on many tens of thousands of CPU-hours
aging kernel (i.e. the averaging kernels obtained by convolving trial-and-errorruns.Incaseoftheinversionssuitablefortheto-
inversionweightswiththesensitivitykernels“oftherealSun”, mographyoftheflowsnapshot(seee.g.Švanda2013)theissues
which are unknown) is different and that would easily explain arenotprominentlyvisible,howeverthequestionis,whetherthe
eventhechangeofsignoftheflow.Acuriousreadermayobject stronger regularisation of the solution (about the random-noise
that the way out is not to combine the two filtering schemes. It term in this case) removes the issues or rather hides them. By
isshowninFig.1thattheinvertedhorizontalflowbasedsolely studying the literature on flow inversions in supergranules one
onthephase-speedfilteringschemereversesitssignbetweenthe has to conclude that all the inversions except for the inversion
Articlenumber,page7of8
A&Aproofs:manuscriptno.ms_rev3
ofWoodard(2007)indicatedthereversalofthehorizontalflow Stein,R.F.,Nordlund,Å.,Georgoviani,D.,Benson,D.,&Schaffenberger,W.
at various depths. It may be seen suspicious, because the state- 2009, in Astronomical Society of the Pacific Conference Series, Vol. 416,
of-the-artsimulations(Steinetal.2009;Rempeletal.2009)do Solar-StellarDynamosasRevealedbyHelio-andAsteroseismology:GONG
2008/SOHO21,ed.M.Dikpati,T.Arentoft,I.GonzálezHernández,C.Lind-
notindicatesuchreversals.Itmayalsoeasilybethatthelarge-
sey,&F.Hill,421
amplitude flows in supergranulation recently reported by time– Švanda,M.2012,ApJ,759,L29
distanceinversions(Švanda2012)areartefactsofsimilarissues, Švanda,M.,Gizon,L.,Hanasoge,S.M.,&Ustyugov,S.D.2011,A&A,530,
asinthatcase f modeand p and p ridgeswereused,necessar- A148
1 2 Švanda,M.,Klvanˇa,M.,Sobotka,M.,Kosovichev,A.G.,&Duvall,T.L.2009,
ilyleadingtosubtractionsintheinversion.
NewAstron.,14,429
Fromcomparisonsoftheinversionresultswiththedirectsur- Švanda,M.,Zhao,J.,&Kosovichev,A.G.2007,Sol.Phys.,241,27
facemeasurements(e.g.Ambrož2005;Georgobianietal.2007; Ustyugov,S.D.2006,inAstronomicalSocietyofthePacificConferenceSeries,
Švanda et al. 2007; Švanda et al. 2013) it seems that the very Vol.354,SolarMHDTheoryandObservations:AHighSpatialResolution
Perspective,ed.J.Leibacher,R.F.Stein,&H.Uitenbroek,115
near-surface inversions (hence involving f modes or acoustic
Švanda,M.2013,ApJ,775,7
waveswithsmallphasespeeds)donotsufferfromthediscussed Švanda,M.,Roudier,T.,Rieutord,M.,Burston,R.,&Gizon,L.2013,ApJ,771,
problems.Asweshowedrecently(Švandaetal.2013),theflow 32
inversion using the f mode ridge is not only highly correlated Švanda,M.,Sobotka,M.,&Bárta,T.2014,ApJ,790,135
Woodard,M.F.2007,ApJ,668,1189
withtheinferencesfromthesurfacegranuletracking,butitalso
Zhao, J. & Kosovichev, A. G. 2003, in ESA Special Publication, Vol. 517,
provides the properly scaled magnitude of the flow. Such vali- GONG+2002.LocalandGlobalHelioseismology:thePresentandFuture,
dations against the independently obtained measurements fully ed.H.Sawaya-Lacoste,417–420
justify the scientific results obtained on surface flow fields. It Zhao,J.,Kosovichev,A.G.,&Duvall,Jr.,T.L.2001,ApJ,557,384
hastobepointedoutthatnolargersubtractionsoccuredinthose
inversions.
Acknowledgements. This work was supported by the Czech Science Founda-
tion (grant P209/12/P568). All computations were performed using the Sun-
quake compute cluster at Astronomical Institute of Academy of Sciences in
Ondˇrejov,thetrackedandmappeddatacubeswereobtainedfromdataprocessing
pipelinesatMax-Planck-InstitutfürSonnensystemforschung(MPS),Göttingen,
Germany,whichisfundedbytheGermanAerospaceCenter(DLR).Thesolar
measurementswerekindlyprovidedbytheHMIconsortium.Thepoint-to-point
travel-timesensitivitykernelswereobtainedusingthecodeofAaronBirchde-
ployed at the MPS. This work was done within the institute research project
RVO:67985815toAstronomicalInstituteofCzechAcademyofSciences.This
workbenefitsfromcouloirdiscussionsfollowingthetalkŠvanda:Currentissues
withtime–distanceinversionsforflowsatAsÚatLWSHelioseismologyWork-
shop#4:SolarSubsurfaceFlowsfromHelioseismology:ProblemsandProspects
whichwasheldatStanfordUniversity,July21–23,2014.
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