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Multi-Scale Probabilistic M o deling in Geospace Science

Multi-Scale Probabilistic M o deling in Geospace Science. Zach Thomas The Ohio State Universi t y Ment o rs: T omo k o Matsuo, Doug Nych k a Ellen Cousins , Mi k e Wilt b erger August 1, 2014. Outline of Summer W o rk.

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Multi-Scale Probabilistic M o deling in Geospace Science

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  1. Multi-ScaleProbabilisticModelinginGeospaceScience ZachThomas TheOhioStateUniversity Mentors:TomokoMatsuo,DougNychkaEllenCousins,MikeWiltberger August1,2014

  2. OutlineofSummerWork .,. Application:Modelinghigh-latitudeionosphericconvection fromsparseradarobservations

  3. OutlineofSummerWork .,. Application:Modelinghigh-latitudeionosphericconvection fromsparseradarobservations .,. StatisticalMethodology:Techniqueforspatialmodelingonthesphere

  4. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses

  5. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses .., Numericalmodeling

  6. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses .., Numericalmodeling .., Dataanalysis

  7. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses .., Numericalmodeling .., Dataanalysis .,. Practical/Societal:Understandchangesinelectromagneticenergyassociatedwithauroras

  8. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses .., Numericalmodeling .., Dataanalysis .,. Practical/Societal:Understandchangesinelectromagneticenergyassociatedwithauroras .., Disturbancesintelecommunication

  9. ScientificBackground Motivation .,. Scientific:ObtainbetterunderstandingofcomplexinteractionbetweensolarwindandEarth’smagneticfieldbystudyingvariousionosphericprocesses .., Numericalmodeling .., Dataanalysis .,. Practical/Societal:Understandchangesinelectromagneticenergyassociatedwithauroras .., Disturbancesintelecommunication .., Disturbancesinpowergrids

  10. ScientificBackground InteractionbetweenSolarWindandEarth’sMagneticField Figure:OutputfromtheLyon-Fedder-Mobarrymodelcapturingcomplexinteraction betweensolarwindand Earth’smagneticfield.Imagecourtesyofhttps://www.dartmouth.edu/physics/cism/science/lfmmodel.html

  11. ScientificBackground IonosphericElectricPotentialandPlasmaConvectionPatterns Figure:IMFeffectonionosphericconvection.ImagecourtesyofCousins etal.(2010)

  12. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. LatticeKrigisanewR packageforfast/flexiblespatialmodelingbasedonastatisticalmethodologyinNychkaetal. (2014).

  13. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. LatticeKrigisanewR packageforfast/flexiblespatialmodelingbasedonastatisticalmethodologyinNychkaetal. (2014). .,. Keyideaistoexpressthespatialprocessasamulitresolution basisfunctionexpansionwithrandomcoefficients

  14. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. LatticeKrigisanewR packageforfast/flexiblespatialmodelingbasedonastatisticalmethodologyinNychkaetal. (2014). .,. Keyideaistoexpressthespatialprocessasamulitresolution basisfunctionexpansionwithrandomcoefficients .,. RandomcoefficientsmodeledviaacertainMarkovrandom fieldmodelcalledasimultaneousautoregression(SAR)model

  15. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. Observeaprocessatnlocationswithinaspatialdomain D...callthemy(s1),...,y(sn).

  16. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. Observeaprocessatnlocationswithinaspatialdomain D...callthemy(s1),...,y(sn). .,. Foranarbitrarys∈D,wewouldliketomakeinferenceabouttheprocessy(s)fromtheobservations.

  17. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. Observeaprocessatnlocationswithinaspatialdomain D...callthemy(s1),...,y(sn). .,. Foranarbitrarys∈D,wewouldliketomakeinferenceabouttheprocessy(s)fromtheobservations. .,. Commonspatialmodel:Foranys∈D(observedornot): y(s) =m(s)+g(s)+e(s) Process=Mean+SpatialProcess+Ind.Error

  18. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. Observeaprocessatnlocationswithinaspatialdomain D...callthemy(s1),...,y(sn). .,. Foranarbitrarys∈D,wewouldliketomakeinferenceabouttheprocessy(s)fromtheobservations. .,. Commonspatialmodel:Foranys∈D(observedornot): y(s) =m(s)+g(s)+e(s) Process=Mean+SpatialProcess+Ind.Error .,. LatticeKrigdiffersfromothermethodsinitsconstructionof thespatiallydependentprocessg(s).

  19. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. CenterbasisfunctionsatgridlocationsonL gridsovertheobservationregion.

  20. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. CenterbasisfunctionsatgridlocationsonL gridsovertheobservationregion. .,. TheL gridsareobtainedbysequentiallydoublingthe resolutionofthepreviousgrid.

  21. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. CenterbasisfunctionsatgridlocationsonL gridsovertheobservationregion. .,. TheL gridsareobtainedbysequentiallydoublingthe resolutionofthepreviousgrid. .,. Theprocessg(s)isthenexpressedasabasisfunctionexpansion: L nl )) l=1j=1 g(s)= c W[d(s,s)] ∗ l,jl,jl,j

  22. ModificationofLatticeKrigfortheSphere WhatisLatticeKrig? .,. CenterbasisfunctionsatgridlocationsonL gridsovertheobservationregion. .,. TheL gridsareobtainedbysequentiallydoublingthe resolutionofthepreviousgrid. .,. Theprocessg(s)isthenexpressedasabasisfunctionexpansion: L nl )) l=1j=1 g(s)= c W[d(s,s)] ∗ l,jl,jl,j .,. Thecl,j’sarerandomcoefficientsfollowingacertainMarkovrandomfieldmodelcalledaSimultaneousAutoregression(SAR)model.

  23. ModificationofLatticeKrigfortheSphere TheGeodesicGrid .,. ModifyingLatticeKrigforuseonthespherewasdoneintwosteps:

  24. ModificationofLatticeKrigfortheSphere TheGeodesicGrid .,. ModifyingLatticeKrigforuseonthespherewasdoneintwosteps: 1. Insteadoflocatingbasisfunctionsonarectangulargridovera plane,wecenterthemonaspecializedgridonthesphere...calledthegeodesicgrid

  25. ModificationofLatticeKrigfortheSphere TheGeodesicGrid .,. ModifyingLatticeKrigforuseonthespherewasdoneintwosteps: Insteadoflocatingbasisfunctionsonarectangulargridovera plane,wecenterthemonaspecializedgridonthesphere...calledthegeodesicgrid Modifytheconstructionofthe SARmodelfortherandomcoefficients(whichnowliveonthegeodesicgrid)

  26. ModificationofLatticeKrigfortheSphere TheGeodesicGrid .,. ModifyingLatticeKrigforuseonthespherewasdoneintwosteps: Insteadoflocatingbasisfunctionsonarectangulargridovera plane,wecenterthemonaspecializedgridonthesphere...calledthegeodesicgrid Modifytheconstructionofthe SARmodelfortherandomcoefficients(whichnowliveonthegeodesicgrid) .,. Resultisatoolforspatialmodelingovergeneralregionsonthesphere...ortheentiresphere...tobeincludedinfutureversionsofLatticeKrig.

  27. ModificationofLatticeKrigfortheSphere TheGeodesicGrid .,. ModifyingLatticeKrigforuseonthespherewasdoneintwosteps: Insteadoflocatingbasisfunctionsonarectangulargridovera plane,wecenterthemonaspecializedgridonthesphere...calledthegeodesicgrid Modifytheconstructionofthe SARmodelfortherandomcoefficients(whichnowliveonthegeodesicgrid) .,. Resultisatoolforspatialmodelingovergeneralregionsonthesphere...ortheentiresphere...tobeincludedinfutureversionsofLatticeKrig. .,. Thissummer,focusonusingtheprocedureformodelingelectromagneticprocessesintheionosphere.

  28. Figure:LowResolutionBasisFunctions:CapturingLarge-ScaleDependenceFigure:LowResolutionBasisFunctions:CapturingLarge-ScaleDependence

  29. Figure:AddMediumResolutionBasisFunctions:Capturing Medium-ScaleDependence

  30. Figure:AddHighResolutionBasisFunctions:CapturingSmall-ScaleDependenceFigure:AddHighResolutionBasisFunctions:CapturingSmall-ScaleDependence

  31. TrySomeSpatialModeling SpatialInterpolationofElectricPotentialfromLFM-MIXModelOuput .,. Keymodificationforionosphereproblem:varianceofthe electricpotentialisclearlynonstationary.Wecanembedinformationfromnumericalmodeloutputintothestatisticalmodel. Figure:(Left)Regionofhighestvariability;courtesyofMinjieFan,UCDavis(Right)Weightsusedtoinducenonstationaryvarianceinspatialmodel.

  32. TrySomeSpatialModeling SpatialInterpolationofElectricPotentialfromLFM-MIXModelOuput .,. Experiment:UseLFM-MIXmodeloutputofelectricpotentialtostudypredictiveskillofourmodel

  33. TrySomeSpatialModeling SpatialInterpolationofElectricPotentialfromLFM-MIXModelOuput .,. Experiment:UseLFM-MIXmodeloutputofelectricpotentialtostudypredictiveskillofourmodel 1. Randomlyselectsampleof1000points(outof16920)uniformlyoverthepolarregion.

  34. TrySomeSpatialModeling SpatialInterpolationofElectricPotentialfromLFM-MIXModelOuput .,. Experiment:UseLFM-MIXmodeloutputofelectricpotentialtostudypredictiveskillofourmodel Randomlyselectsampleof1000points(outof16920)uniformlyoverthepolarregion. Treatthesepointsas’TheData’...trytogetbackthefull 16920pointsusingourspatialmodelonthesphere

  35. Figure:Fullelectricpotentialprocessfrommodeloutput

  36. Figure:1000randomlysampledlocationsusedtoinformtheinterpolationFigure:1000randomlysampledlocationsusedtoinformtheinterpolation

  37. "True"ElectricPotentialProcess(FromLFM-MIX) 12 40 32 24 16 40 30° 20° 8 > e- 18 0 06 -8 -16 -24 -32 -40 min:-25.37 max:41.50 00

  38. SpatialInterpolationofElectricPotential 12 40 32 24 16 40 30° 20° 8 c 0 18 0 06 B -0 -8 0. -16 -24 -32 -40 min:-25.19 max:41.55 00

  39. Error("True"ProcessMinusPredictedProcess) 12 2.0 1.6 1.2 0.8 40 30° 20° 0.4 10· "C 18 0.0-0 0. 06 -0.4 -0.8 -1.2 -1.6 -2.0 min:-2.19 max:0.76 00

  40. Error("True"Process MinusPredictedProcess) 12 .....::: 2.0 ·..·...:.···· : :.·.. ..· .,..· 1.6 ·.· ···. ..· .. ·. .... 1.2 .·--:. ·.... ::.:.... ·... ...:.·...: .:-··....... 0.8 ·:30°:..:. .,...·..·. .·: ,.'·.. '1:::·:....::.·:"·.,;;. ... ....... 0.4 ··: "' 0.0-0 0. ...·.· .·..·· :-. 18 06 o I,-,-• ...·.··....... .......·:·.· ::. :.·.· ··.:.. .: .: .... -0.4 .,.·... ·.·.. ·::.···.· ,.... .. ..... -0.8 :·. .-. .,. . .-.... -1.2 .... .:•'-• .... ...... -1.6 ....· 00 -2.0 min:-2.19 max:0.76

  41. NextStep:ModificationsforRadarObservations GettingfromObservationSpacetoElectricPotentialSpace .,. Inpractice,theelectricpotentialcanonlybeinferredfromsparseobservationsofother(related)processes.

  42. NextStep:ModificationsforRadarObservations GettingfromObservationSpacetoElectricPotentialSpace .,. Inpractice,theelectricpotentialcanonlybeinferredfromsparseobservationsofother(related)processes. .,. WeusethemethodologyinRichmondandKamide(1988)...transformbasisfunctionsintoobservationspace

  43. NextStep:ModificationsforRadarObservations GettingfromObservationSpacetoElectricPotentialSpace .,. Inpractice,theelectricpotentialcanonlybeinferredfromsparseobservationsofother(related)processes. .,. WeusethemethodologyinRichmondandKamide(1988)...transformbasisfunctionsintoobservationspace .,. Theradarsmeasureprojectionsofionosphericplasmadrift velocitiesontotheline-sight-direction:vLOS(s)=✈(s)·❛LOS.

  44. NextStep:ModificationsforRadarObservations GettingfromObservationSpacetoElectricPotentialSpace .,. Inpractice,theelectricpotentialcanonlybeinferredfromsparseobservationsofother(related)processes. .,. WeusethemethodologyinRichmondandKamide(1988)...transformbasisfunctionsintoobservationspace .,. Theradarsmeasureprojectionsofionosphericplasmadrift velocitiesontotheline-sight-direction:vLOS(s)=✈(s)·❛LOS. .,. These LOSvelocitesarerelatedtotheelectricpotentialbythefollowing: 1 ∂Φ(s) ∂Φ(s) vLOS(s)= ,− • ❛LOS · |❇(s)|∂θ ∂φ θ=Latitude,φ=Longitude,❇(s)=MagneticFieldats

  45. NextStep:ModificationsforRadarObservations TransformingBasisFunctions .,. ThisfunctionL:Φ(s)1→vLOS(s)isalinearoperator

  46. NextStep:ModificationsforRadarObservations TransformingBasisFunctions .,. ThisfunctionL:Φ(s)1→vLOS(s)isalinearoperator .,. Before: L nl Φ(s)=))ci,jWi,j[d(s,s∗j)] l=1j=1

  47. NextStep:ModificationsforRadarObservations TransformingBasisFunctions .,. ThisfunctionL:Φ(s)1→vLOS(s)isalinearoperator .,. Before: L nl Φ(s)=))ci,jWi,j[d(s,s∗j)] l=1j=1 .,. ForRadarData: L nl )) l=1j=1 { � L{Φ(s)}=v (s)= c LW[d(s,s)] ∗ i,j i,j LOS j

  48. NextStep:ModificationsforRadarObservations TransformingBasisFunctions .,. ThisfunctionL:Φ(s)1→vLOS(s)isalinearoperator .,. Before: L nl Φ(s)=))ci,jWi,j[d(s,s∗j)] l=1j=1 .,. ForRadarData: L nl )) l=1j=1 { � L{Φ(s)}=v (s)= c LW[d(s,s)] ∗ i,j i,j LOS j .,. Thecoefficientprocessisthesame...usesameestimationprocedurebutwithtransformedbasisfunctions

  49. Figure:FullelectricpotentialfieldfromLFM-MIXmodeloutput.

  50. Figure:RandomlysampledLOS directionsandcorrespondingvelocitiesusedinnumericalstudy.

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