A Time-Evolving 3D Method Dedicated to the Reconst

arXiv:0802.0113v1[astro-ph]1Feb2008SolarPhysicsDOI:10.1007/•••••-•••-•••-••••-•ATime-Evolving3DMethodDedicatedtotheReconstructionofSolarPlumesandResultsUsingExtremeUltra-VioletDataNicolasBarbey∗†·FrédéricAuchère∗·ThomasRodet†·Jean-ClaudeVial∗Received:9May2007/Accepted:24January2008cSpringer••••AbstractAnimportantissueinthetomographicreconstructionofthesolarpolesistherelativelyrapidevolutionofthepolarplumes.Wedemonstratethatitispossibletotakeintoaccountthistemporalevolutioninthereconstruction.Thedifficultyofthisproblemcomesfromthefactthatwewanta4Dreconstruction(threespatialdimensionsplustime)whileweonlyhave3Ddata(2Dimagesplustime).Toovercomethisdifficulty,weintroduceamodelthatdescribespolarplumesasstationaryobjectswhoseintensityvarieshomogeneouslywithtime.Thisassumptioncanbephysicallyjustifiedifoneacceptsthestabilityofthemagneticstructure.Thismodelleadstoabilinearinverseproblem.Wedescribehowtoextendlinearinversionmethodstothesekindsofproblems.Studiesofsimulationsshowthereliabilityofourmethod.ResultsforSOHO/EITdatashowthatweareabletoestimatethetemporalevolutionofpolarplumesinordertoimprovethereconstructionofthesolarpolesfromonlyonepointofview.WeexpectfurtherimprovementsfromSTEREO/EUVIdatawhenthetwoprobeswillbeseparatedbyabout60◦.1.IntroductionAmethodknownassolarrotationaltomographyhasbeenusedtoretrievethe3Dgeometryofthesolarcorona(Frazin2000;FrazinandJanzen2002).Thismethodassumesthestabilityofthestructuresduringthetimenecessarytoacquirethedata.Sincewegenerallyhaveonlyonepointofviewatourdisposal,about15daysarerequiredtohavedataforhalfasolarrotationatthepoles.Here,wefocusourstudyonsolarpolarplumes.Theyarebright,radial,coronalraystructureslocatedatthesolarpolesinregionsofopenmagneticfield.Thestudyofplumesisofgreatinterestsinceitmaybethekeytotheunderstandingoftheaccelerationofthefastcomponentofthesolarwind(Teriacaetal.,2003).Howeverthethree-dimensionalshapeofthesestructuresispoorlyknownanddifferentassumptionshave∗Institutd’AstrophysiqueSpatiale,UniversitéParis-Sud,Orsay,Franceemail:[e-mail],[e-mail],[e-mail]†LaboratoiredesSignauxetSyst`emes,Supéléc,Gif-sur-Yvette,Franceemail:[e-mail],[e-mail],e.g.Gabrieletal.,2005;Llebaria,Saez,andLamy,2002.Theplumesareknowntoevolvewithacharacteristictimeofapproximately24hoursonspatialscalestypicalofExtremeultra-violetImagingTelescope(SOHO/EIT)data(2400km)(DeForest,Lamy,andLlebaria,2001).Consequentlythestabilityassumptionmadeinrotationaltomographyfails.Fortunately,theSolarTErestrialRElationsObservatory(STEREO)missionconsistsoftwoidenticalspacecraftSTEREOAandSTEREOBwhichtakepicturesoftheSunfromtwodifferentpointsofview.WiththeSOHOmissionstilloperating,thisresultsinthree,simultaneouspointsofview.Threeviewpointshelptoimprovethereconstructionoftheplumes,buttheyarestillnotenoughtousestandardtomographicalgorithms.Theproblemisunderdeterminedandconsequentlyonehastoaddaprioriinformationinordertoovercomethelackofinformation.Thisleadstochallengingandinnovativesignalanalysisproblems.Therearedifferentwaystodealwithunderdeterminationdependingonthekindofobjecttobereconstructed.Interestinglythefieldofmedicalimagingfacesthesamekindofissues.Incardiacreconstruction,authorsmakeuseofthemotionperiodicityinassociationwithahighredundancyofthedata(Grassetal.,2003;Kachelriess,Ulzheimer,andKalender,2000).Ifonecanmodelthemotionasanaffinetransfor-mation,andifoneassumesthatweknowthistransformation,onecanobtainananalyticsolution(Ritchieetal.,1996;Rouxetal.,2004).Insolartomography,theproposedinnovativeapproachesinvolvetheuseofad-ditionaldatasuchasmagnetic-fieldmeasurementsinthephotosphere(WiegelmannandInhester,2003)ordatafusion(FrazinandKamalabadi,2005).AttemptshavebeenmadebyFrazinetal.(2005)totreattemporalevolutionusingKalmanfiltering.Sincepolarplumeshaveapparentlyalocal,rapid,andaperiodictemporalevo-lution,wedevelopedasinthepreviouslyreferencedwork,amodelbasedonthespecificsoftheobjectweintendtoreconstruct(preliminaryresultscanbefoundinBarbeyetal.,(2007).Plumeshaveanintensitywhichevolvesrapidlywithtime,buttheirpositioncanbeconsideredasconstant.ThishypothesisisconfirmedbypreviousstudiesoftheplumessuchasDeForest,Lamy,andLlebaria(2001).Themodelismadeupofaninvariantmorphologicalpart(x)multipliedbyagainterm(θt)thatvarieswithtime.Onlyonegaintermisassociatedwitheachplumeinordertoconstrainthemodel.Soweassumethatthepositionofeachplumeinthesceneisknown.Thismodelisjustifiedifweconsiderpolarplumestobeslowlyevolvingmagneticstructuresinwhichplasmaflows.Thankstothismodelwecanperformtime-evolvingthree-dimensionaltomogra-phyofthesolarcoronausingonlyextremeultra-violetimages.Furthermore,thereisnocomplex,underlyingphysicalmodel.Theonlyassumptionsarethesmoothnessofthesolution,thearea-dependantevolutionmodel,andtheknowledgeoftheplumeposition.Theseassumptionsallowustoconsideratemporalvariationofafewdays,whileassumingonlytemporalsmoothnesswouldlimitvariationstotheorderofonesolarrotation(about27days).Toourknowledge,theestimationo