ICESat-2 Algorithm Theoretical Basis Document for Ocean Surface Height
Release004
Release 004 – Winter 2021
January 21, 2021
ICE, CLOUD, and Land Height Satellite (ICESat-2) Project
Algorithm Theoretical Basis Document
(ATBD) for
Ocean Surface Height
Prepared By: ICESat-2 Science Definition Team Ocean Working Group
Contributors /Code: James Morison David Hancock Suzanne Dickinson John Robbins Leeanne Roberts Ron Kwok Steve Palm Ben Smith Mike Jasinski Bill Plant Tim Urban
GoddardSpaceFlightCenter
Greenbelt,Maryland
ICESat-2 Algorithm Theoretical Basis Document for Ocean Surface Height
Release004
Release 004 – Winter 2021
Abstract
This document describes the theoretical basis of the ocean processing algorithms and the products that are produced by the ICESat-2 mission. It includes descriptions of the parameters that are provided in each product as well as ancillary geophysical parameters, which are used in the derivation of these ICESat-2 products.
ICESat-2 Algorithm Theoretical Basis Document for Ocean Surface Height
Release004
Release 004 – Winter 2021
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CM Foreword
ThisdocumentisanIce,Cloud,andLandHeight(ICESat-2)ProjectScienceOfficecontrolleddocument.ChangestothisdocumentrequirepriorapprovaloftheScienceDevelopmentTeamATBDLeadordesignee.ProposedchangesshallbesubmittedintheICESat-IIManagementInformationSystem(MIS)viaaSignatureControlledRequest(SCoRe),alongwithsupportivematerialjustifyingtheproposedchange.Inthisdocument,arequirementisidentifiedby“shall,”agoodpracticeby“should,”permissionby“may”or“can,”expectationby“will,”anddescriptivematerialby“is.”Questionsorcommentsconcerningthisdocumentshouldbeaddressedto:ICESat-2ProjectScienceOfficeMailStop615GoddardSpaceFlightCenterGreenbelt,Maryland20771
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Preface
ThisdocumentistheAlgorithmTheoreticalBasisDocumentfortheprocessingopenoceandatatobeimplementedattheICESat-2ScienceInvestigator-ledProcessingSystem(SIPS).TheSIPSsupportstheATLAS(AdvanceTopographicLaserAltimeterSystem)instrumentontheICESat-2SpacecraftandencompassestheATLASScienceAlgorithmSoftware(ASAS)andtheSchedulingandDataManagementSystem(SDMS).ThesciencealgorithmsoftwarewillproduceLevel0throughLevel4standarddataproductsaswellastheassociatedproductqualityassessmentsandmetadatainformation.TheICESat-2ScienceDevelopmentTeam,insupportoftheICESat-2ProjectScienceOffice(PSO),assumesresponsibilityforthisdocumentandupdatesit,asrequired,asalgorithmsarerefinedortomeettheneedsoftheICESat-2SIPS.Reviewsofthisdocumentareperformedwhenappropriateandasneededupdatestothisdocumentaremade.Changestothisdocumentwillbemadebycompleterevision.ChangestothisdocumentrequirepriorapprovaloftheChangeAuthoritylistedonthesignaturepage.ProposedchangesshallbesubmittedtotheICESat-2PSO,alongwithsupportivematerialjustifyingtheproposedchange.Questionsorcommentsconcerningthisdocumentshouldbeaddressedto:TomNeaumann,ICESat-2ProjectScientistMailStop615GoddardSpaceFlightCenterGreenbelt,Maryland20771
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Review/Approval Page
Prepared by: Jamie <Enter Position Title Here> Org/Addresss
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Bea <Enter Position Title Here> <Enter Org/Code Here>
Tom <Enter Position Title Here> <Enter Org/Code Here>
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***Signaturesareavailableon-lineat:https:///icesatiimis.gsfc.nasa.gov***
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Change History Log
Revision Level Description of Change SCoRe
No. Date
Approved
Initial Release 7/27/2019Section5.3.4.1codestepsA:replacesnoothrechistwithsmoothrechist7/30/2019–globallyreplacetermMaximumLikelihoodwithExpectationMaximizationtoclarifythattheMatlabdevelopmentalcodeandtheASASFORTRANcodebothuseexpectationmaximization.7/30/2019Changethethirdparagraphof5.3.4.2(H)to:-Toapplytheexpectationmaximizationapproach,wefirstderiveaseasurfaceheightspatialseriesfromtheheightdistributionY,whichisdefinedoverarangeofheightsthesameasthereceivedheighthistogram(i.e.,jlowtojhighfromsurfacefinding).ThisisaccomplishedbyfirstmultiplyingYby10,000,roundinganddeletinganyvalueslessthanorequaltozero(afewunrealisticweaklynegativevaluestooccurinYassociatedwiththeWeinerdeconvolutionprocess)toproduceanintegerdistribution,YI.Aseries,XY,isassembledbyconcatenatingforeachvalueofYI(i)forindexicorrespondingtoheightsshx(i),YI(i)valuesequaltosshx(i).Theshifttoheightsincludingmeanoffit2wasaccomplishedindevelopmentalcodebyaddingmeanoffit2toeachvalueinXY.(ASASFORTRANcodedoesnotaddmeanofffit2atthispointbutaftertheGaussianMixturecalculation.)7/30/2019Addxbindasanoutputvariablein5.3.3(14)andattheendof5.3.3ToTable6add:xbind=1x710elementarrayof10-mbinaveragesofalong-trackdistanceAndchangethedescriptionofxbinto:“Centerof1x710elementarrayof10-mbins.Notethismaybeincludedasadatadescriptionorotherstaticarrayequalto[5,15,25,35…..7095m]7/30/2019InTable6changedfrom70010-mbinsto71010-mbinstoaccountforoceansegmentslongerthan7,000m
GGG
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7/30/2019Change5.3.4.1(G)points1-4to:“Thesurfacehistogramshouldbethesamelengthandhavethesamehorizontal(height)axisasthereceivedhistogram,rechist.Inthispart,wecomputethedistributionofsurfaceheightasaprobabilitydensityfunction(PDF),Y,startingwiththeFouriertransformofthatPDFoutputbytheWeinerdeconvolution.Thestepsare:
1. ComputeYfi,astheinverseFourierTransformofYfdividedbybinsize.YfiasitcomesfromtheinverseFouriertransformisNFFTpointslongwiththefirsthalfappearingdisplacedtotheendofthearray.WereorderthisbyfirstputtingthelastNFFT/2pointsaheadofthefirstNFFT/2points.Toestablishthesurfaceheightprobabilitydensityfunction,Y,wethenremovethefirst(rzbin-1)pointswhererzbinistheindexofthecentroid(heightequalszero)indexofthereceivedhistogram.WekeepasYtheremainingpointsequaltothelengthoftherechist.InthecaseoftheASASFORTRANcode,whichpadseachendofrechisttoreachheightsof±15m,rzbinisthecenterbinofrechist.
2. Setthex-axisofY,derivedsshxequaltothex-axis,xrechist,ofthereceivedhistogramrechist.”
7/30/2019Clarifybincenteringin5.3.1(B)bychanging(B)to:“Establishaninitialcoarsehistogramarray,Hc,spanning±15mwithbinsizeB1equalto0.01mwithcenterbincenteredatzeroandbincentersatwholecentimeters.Alsoestablishadataarray,Acoarse,forupto10,000photonheightsandassociatedinformation(index,geolocation,time)plusnoisephotoncounts.Thiswillbepopulatedwithdataaswestepthrough14-geobinsegmentssearchingforanadequatenumberofsurfacephotoncandidates.“andchange5.3.2(A)to:“Setupforthesurfacefindinghistogram,N,byestablishingthevector,Edges,(withalengthonegreaterthanN)ofhistogrambinedgeswithbinsBfwidebetween-15mto+15m.Alsoestablishthevector,Cntrpt,ofbincenterpointswithlengthequaltoN.InpracticeBfequals1cm,Cntrptvaluesarewhole
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centimeters,andEdgesareathalfcentimetervalues.Alsoestablishavectorarray,BinB,aslongasht_initialforbinassignments.”8/6/2019Change5.3.4.1(G)neartheendfrom“Yisdefinedoveravectorofbincenters,sshx,butYwillbeoutputasa1001elementvectorforeverysegmentwiththe501stcenterbinbeingforheightzero,bin501-YlowX/binsizebeingthelowestbinwithnonzeroYandbin501+YlowX/binsizebeingthehighestbinwithnonzeroY.“To“Yisdefinedoveravectorofbincenters,sshx,butforbinsize=1cmYwillbeoutputasa3001-elementvectorfrom-15mto+15mforeverysegmentwiththe1,501stcenterbinbeingforheightzero.Valueswillbezeroforbinsoutsidetherangeofsshx.OwingtonoiseandtheeffectoftheFFT-deconvolution-inverse-FFTprocess,smallnegativevaluesoccurinYwithintherangeofsshx.ThesewillbesettozerointheoutputYvector.”8/28/2019Replacesection5.3.3tousesimplifiedandmorerobustmethodofcomputingphotonreturnrateinSSBcalculation:xrbin=nbind/109/9/2019in5.3.37.-Addclausetoendofsentencetoclarifyusingonlybinswithphotonsforssbcalculation,i.e.,
7. Compute,binAVG_Xr,theaverageofbinneddatarate,xrbin,overthe10-mbinswithphotons.
11/5/2019Modified5.3.2tochangesurfacefindingbasedonthedistributionofphotonheightstosurfacefindingbasedonthephotonheightanomalyrelativetoamovingbinaverageofhighconfidencephotonheights.Thisisdonetoexcludesubsurfacereturnsunderthecrestsofsurfacewavesthatotherwisefallinsidethehistogramoftruesurfaceheights.AddedtoTable5conf_lim,thelimitingconfidenceleveltobeincludedinthemovingbinaverage,nphoton,thenumberofphotonseithersideofacentralphotontobeincludedinthemovingbinaverage,e.g.,fornphoton=10,a21pointaverageisused
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11/18/2019Addedsection5.3.3.2tocomputenharms+1harmoniccoefficients,a,ofsurfaceheightforsurfacewavesandpossiblerelationtodegreesoffreedomanduncertainty.AddedtoTable5nharmsandaddedtoTable6wnanda.wnisavectorofnharmswavenumbers.a(1)isthecoefficientforthezerowavenumbercomponentofthefittotestifthisisamorestableestimateofaverageheights.a(3,5,7,..2*nharms+1)aresinecoefficientsforwn(1,2,3,..nharms).a(2,4,6,..2*nharms+1)arecosinecoefficientsforwn(1,2,3,..nharms).11/27/2019Addedasectiontotheendof5.3.6.1toestimatetheeffectivenumberofdegreesoffreedom,NP_effect,andassociateduncertainty,h_uncrtn,inseasurfaceheightoveranoceansegment.Theestimatesarebasedontheintegralofautocorrelationof10-mbinaveragesurfaceheight.Addedh_uncrtnandNP_effecttoTable6.12/19/2019Removed(orcorrespondinght_initial,really?)from5.3.2(D)12/19/2019Addedthesentence:“Thereceivedhistogram,Nrs,isthenthehistogramofht_initial2_surfonlyincludingfromthelowestheightwithcount>0tothehighestheightwithcount>0.”to5.3.2(P)toclarifythatthoughthehistogramofheightanomalies,N,isusedforsurfacefinding,welooktothecorrespondinghistogramofheights,Nrs,fortherestoftheanalysis.Inalatersectionweformtheprobabilitydensityfunction,rechist,fromNrsandpassitontohavetheinstrumentimpulseresponseremovedandstatisticsofseasurfaceheightcomputed.12/19/2019Pursuanttothechangeimmediatelyabovewechangedthefirstsentenceof5.3.4.1(A)“Startwiththehistogramofsurfacereflectedphotonsheights,N,fromsection(5.3.2(M))versusthe…”to,”Startwiththehistogramofsurfacereflectedphotonsheights,Nrs,fromsection(5.3.2(P))versusthe…”02/06/2020:Corrected5.3.3.2paragraph4)toproperlyincludeht_initial2_surfinthederivationofharmoniccoefficients:
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4)Foryequaltothecolumnvectorofht_initial2_surfwithmeanoffit2addedtoit,computethevectorofharmoniccoefficients,a,bytheVericekleastsquarederrororleftpseudoinversemethod,where
a=F’F\F’y.Themx1vectorofharmoniccoefficients,a,fortheoptimumharmonicfittoht_initial2_surfplusmeanoffit2versustrackdist_intial2_surf:
a(1)+a(2)*sin(wn(1)*2*p*x)+a(3)*cos(wn(1)*2*p*x)+a(4)*sin(wn(2)*2*p*x)+a(5)*cos(wn(2)*2*p*x)+a(6)*sin(wn(3)*2*p*x)+a(7)*cos(wn(3)*2*p*x)+::+::+a(2*nharms)*sin(wn(nharms)*2*p*x)+a(2*nharms+1)*cos(wn(nharms)*2*p*x)BackslashorleftmatrixdivideA\BisthematrixdivisionofAintoB,whichisessentiallythesameasA-1B.02/24/2020RevisedSection5.3.3.2toclearupvariousambiguitiesandincludedfortheharmonicfitonly,thetechniquetofillgapsinthedatagreaterthangaplimit=3.2mwithwhitenoiseaboutmeanoffit202/24/2020AddedgaplimittocontrolparametersTable5andaddedtheharmonicfitsignaltonoiseratioSNR_harmandgapstatistics,DXbar,Xvar,andDXskewtooutputsTable6.02/25/2020Corrected5.3.3.2paragraph7bychanging“trackdist_initial2_surf”to”thegap-filledtrackdist_initial2_surf”intwoplaces3/19/2020Correct5.3.3.2(1)todividethethirdmomentofdataspacingbythe2ndmomenttothe3/2:skewness(DXskew=Sum(DX-DXbar)3/DXvar3/2from1toN-1dividedbyN-2)).3/23/2020Correct5.3.3.2formulaforSNR-harmtoremovemeanfitinthenumeratoroftheformula.Itnowreads:
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SNR_harm=E[(yfit-a(1))*(yfit-a(1))’]/E[(y-yfit)*(y-yfit)’]whereE[]denotesexpectedvalue.4/7/2020Modify5.3.6.1toclarifycalculationofLscaleandNP_effectforuncertaintydetermination.Thisincludesaddingequations(41),(42),(43),and(44),identifyingequation(40)andmodifyingthepseudocodesectionb)toincludecorrectedformulationsofLscale:“StartwithLscale=0andati=1wherethecorrespondinglag,l,iszero,andR_L(i)=1.Incrementiandthecorrespondingiby1andkeepaddingtoLscalewhileR_L(i+1)isgreaterthanzero,i.e., Lscale=Lscale+((1-l/Nbin10)*R_L(i)+(1-(l+1)/Nbin10)*R_L(i+1))/2WhenR_L(i+1)becomeslessthanzero,terminatetheintegrationbyessentiallyassumingR_L(i+1)iszero,i.e.,
Lscale=Lscale+(1-l/Nbin10)*R_L(i))/2“
AlsoaddedLscaleandNbin10toOutputTable6.
4/17/2020–Modified3.2Grif=dedSeaSurfaceHeight(ATL19)toallowformergingATL10withATL12datainthepolaroceans.4/17/2020–Modified4.5andFigure13toaddcomputingaveragesofSSHmomentsweightedbydegreesoffreedom.4/20/2020–InTable6ATL12Outputs,changethenameofphotonnoiserate,photonns_rate,tophoton_noise_ratetoavoidconfusionwithphotonrate,Photon_rate.4/20/2020–Insertedassection5.5,adescriptionthegriddingprocessincluding5.6.1Grids5.6.2InputsbtoGridding,andTable7ofinputsfromATL12.5/2/2020Edited5.3.6.1paragraphsa)andb)tomakeitclearthatweusethelaggedcovarianceCOV_L,andlaggedautocorrelation,R_L,ofbinnedsurfaceheightareusedfromlag,l,equalto0tothelengthoftherecord.Thisincluded
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standardizingthespellingthelag=dependentvariablesasCOV_LandR_L.5/7/2020ChangedSection5.3.2(B)conf_limfrom4to3.Nowitreads“Forexample,intestingweusednphotons=5andconf_lim=3tomakean11-pointrunningaveragecenteredoneachphotonheight,andincludedineachaverage,onlyheightswithconfidencelevel3and4.5/7/2020and5/8/2020Editedsection5.6Griddingtoincludemissingvariables,regularizevariablenamingforaveraged(_avg)anddegree-of-freedomweightedaveraged(_dfw)variables.AddedAppendix,ahierarchyofATL12andATL19variables.5/19/2020IntheATL19sectionchangeddot_sig_albmtodot_sigma_dfwalbmasabetterdescriptor.5/19/2020SubstitutedDickenson’sTable6editedtohavecorrectfolderlabelsandtobeconsistentwithATL12outputnamesandfolders.5/19/2020AddedfolderheadingstoTable7tobeconsistentwitheditedTable6andmadesmalleditstoATL19toemphasizewewouldperformanalysesereachstrongbeamuntiltheall-beamcombination.6/6/2020addedn_ttl_photontoTable7inputstoATL-19gridding.6/6/2020,addedaparagraphto5.19.4.1toexplaincomputationofgridcell-totalsurfacephotons,totalphotons,surfacephotonrateandnoisephotonrate.Alsoaddedaparagraphto5.19.5toexplaincomputationofall-beam-totalsurfacephotons,totalphotons,surfacephotonrateandnoisephotonrate.6/6/2020,addedthegridcelltotalandall-beamgridcelltotalsurfacephotonandallphotoncountsandphotonrateandnoisephotonratetoTable8ATL19Outputs.6/6/2020InresponsetoHancock“Photoselect”emailJune3,2020,edited4.2.1.1onpage57toclearupambiguitiesintheuse
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ofthedownlinkbandandconfidencelevelfromATL03toselectcandidatephotonsfortheATL12surfacefindingroutine.6/29/2020Edit3.1.3.4todistinguishbetweentidecorrectionsalreadyinATL03photonheightsandthosetobemadeinATL12,AlsoemphasizedtheATL03solidearthtidesareinatide-freereferencesystemasopposedtoamean-tidesystem,andforATL12thisisappropriatewhentheoceantidesareadded.
6/29/2020Addedto4.2.1.1and5.2.1sentencestoincludeexclusionofdataforwhichthepointingandorbitdeterminationaresuspect:“AsofRelease3,theATL03includesaflagforeachgroundtractindicatingthequalityoftheprecisionorbitandpointingdetermination:gtx/geolocation/podppd_flag,andwehavefoundunrealisticvaluesforoceansegmentswhenthisflagisnon-zeroindicatingdegradedorbitand/orpointingdata.ConsequentlyATL12processingshouldonlybedonewithdataforwhichgtx/geolocation/podppd_flag=0.”Alsoaddedwhichgpodppd_flagtoTable2.
6/29/2020Addedto5.3.3.1(5)averagedvariables-Setlatbindtothemeanofphotonlatitudeineach10-mbin.-Setlonbindtothemeanofphotonlongitudeineach10-mbinAlsoaddedlatbindandlonbindtoTable6,ATL12Outputs6/29/2020Addedadataeditingparagraphto5.19.2
“Wefindthatforreasonsthatweareinvestigating,ATL12producessomeoceansegmentheightsthatareunrealisticcomparedtothegeoid.Consequently,priortoinclusioninthegriddingprocess,eachATL12oceangranuleistobeeditedofoceansegmentheights,h,thatdonotsurvivea2-pass3-sigmafilterondynamicoceantopographyequaltoh-geoid_seg.Foreachgranulecomputethestandarddeviationandmeanofh-geoid_seg.Onthefirstpassanyoceansegmentforwhichthemagnitudeofh-geoid_seg–mean(h-geoid_seg)>3xstd(h-geoid_seg)shouldbeeliminatedfromtheinputdataandthestandarddeviationoftheremainh-geoid_segre-computedfromthefirst-passediteddata.Theeditingprocessisthenrepeatedwithanyoceansegmentfromthefirstforwhichthemagnitudeofh-geoid_segminusthere-computedmean(h-geoid_seg)>3xre-computedstd(h-
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geoid_seg)shouldbeeliminatedfromthedatatobeinputtothegriddingprocess.“
6/30/2020-Section3.1.3.1AddedexplanationofATL03Release4changingthegeoidprovidedtoatide-freeEGM2008inplaceofthemean-tideEGM2008providedinearlierreleases.AlsonotedATL03allwillnowincludetheconversionfromtide-freetomean-tidegeoid.6/30/2020Table2,InputsfromATL03,Addedgeoidandgeoid_free2meanconversionfactortotable
6/30/2020Table6OutputsandTable9,addedspecificationthatearth_tide_segwouldbeinthetide-freesystem6/30/2020Section4.2.1.2–AddedspecificationthatATL03geoidwouldnowbeinthetide-freesystemandrequireconversiontomean-tidewiththeprovidedconversionfactor.6/30/2020Section5.3.1(A)4)–AddedspecificationthatATL03geoidwouldnowbeinthetide-freesystemandrequireconversiontomean-tidewiththeprovidedconversionfactor.
6/30/2020Section5.4.2–Addedgeoid_free2meanandtide_earth_free2meantolistofancillaryvariablestobeaveragedoverandoceansegment.6/30/2020Table6ofATL12Outputs–Addedgeoid_free2mean_segandtide_earth_free2mean_segassegmentaveragesofgeoid_free2meanandtide_earth_free2meantolistofoutputs.Notedthattide_earth_free2meanwasnotusedinATL12.7/14/2020(withchangeoflayer_switchdefaultvalueto0on7/15/2020)Changed4.2.1.2asrecommendedbyDavidHancock7/8/2020concerningeditingforlayer_flag=1,i.e.,4.2.1.21)isnow
1. Examineoceandepthandcloudparametersagainstcontrolparameterstotestforsuitabilityforoceanprocessing.-Ifcontrolparameterlayer_swtchisequalto1,then remove the ATL03 segment ids from ocean processing that are associated with the ATL09 segment ids where layer_flag is equal to 1.Whenlayer_swtchequals0
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(thedefaultvalue),thesoftwarewillignorethecloudcoverinacceptingdataduringcoarsesurfacefinding.-Ignore data collected over land or too close to land. If ocean depth, depth_ocn, is less than a depth, depth_shore, specifying the effective shoreline of water for which ocean processing is appropriate,thenproceedtonext14geo-bin(400-pulse)segment. The default value for control parameter depth_shore will be 10 m.
7/14/2020ChangedformattingoflinesinTable7,Inputtogriddingto“normal”inorderforautoformattingofheadingsof5.19nottoskip5.19.3andjumpfrom5.1.2to5.19.47/15/2020,Table3inputsfromATL09,addedlayer_flagasRepresentingpresence(1)orabsence(0)ofsignificantcloudlayers.7/15/2020,Section5.4.2Addeddescriptionoflayer_flag_segandcloudcover_percent_seg:Foreachoceansegment,weoutputlayer_flag_segasequalto1foroceansegmentswithmorethan50%ofgeosegshavinglayer_flag=1,indicatingsignificantcloudlayerswerepresent.Inaddition,wecomputecloudcover_percent_segforeachoceansegmentasthepercentageofgeosegsintheoceansegmentwithlayer_flag=1,i.e.,cloudcover_percent_seg=100*(totalnumberofgeosegsusedwithlayer_flag=1)/(totalnumberofgeosegsused).7/24/2020MadenumeroussmalleditsandupdatessuggestedbyPatriciaVornberger,butwithnochangeintheprocessingcode8/4/2020Madenumerousnameupdates,corrections,anddeletionstovariablesinTables2and3inaccordancetoPatricia’scommentsandJohnRobbin’sATBD_Table_2&3_Notes_JMAlsosubstitutedasr_cloud_probabilityforcloud_flag_asrintextindescriptionofATL09processingforthelayer_flag8/11/2020CorrectederrorthatLeeannespottedinequation49.Correcteddot_mtodot_avgi.e.
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8/18/2020Modifiedequationabove5.6.5tocorrectplacementofparenthesestoread:
dot_sigma_dfwalbm=((Sum[dof_grid*(dot_sigma_dfw)2]beams1,3,5)/dof_grid_albm)1/2
8/28/2020Sect4.2.1.2Paragraph4:Addeditingforquality_ph=0i.e.,“Selectonlyphotonheightsforwhichthesignalconfidence(gtxx/heights/signal_conf_ph)isgreaterthanorequalto1andindicatorforsaturationgtxx/heights/quality_phiszero.”
-Similarly,addedquality_phtoinputTable2and-Sec5.3.1A)3.Nowincludes“Selectonlyphotonheightsfor
whichthesignalconfidence(gtx/heights/signal_conf_ph)isgreaterthanorequalto1andphotonquality(gxtx/heights/quality_ph)isequaltozero.”
8/28/2020Sect5.4.2,Addedfull_sat_fractandnear_sat_fracttolistofvariablestobesimplyaveragedoveranoceansegment.Andaddedthecorrespondingoutputsofaveraging,full_sat_fract_prcntandnear_sat_fract_prcnttoTable69/8/2020Insection5.6.2wegavethesecondparagraphonpre-filteringitsownsubsection:5.6.2.1Pre-gridFiltering–Along-Track9/8/2020InTable2InputsfromATL03,eliminatedreferencetosignal_conf_ph=-1,-3,-4andsubstituted,“Withrelease4signal_conf_phequal=-2indicatespossibleTEPphoton”9/10/2020InTable6,ATL12Outputs,changedfull_sat_fract_prcnttpfull_sat_fract_segandnear_sat_fract_prcnttonear_sat_fract_seg.Inthevariabledescriptions“percentage“ischangedto“fraction”.9/15/2020In4.2.1.25)correctedtypoEGM208toEGM2008andHappy2ndBirthdayICESat-2andHappy100thBirthdayW.H.Morison
c = dot _avg−(a∗ grid _ lon+b∗ grid _ lat)
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10/14/2020-InTable9ATL12outputsperorbitchangethenameofphotonnoiserate,photonns_rate,tophoton_noise_ratetoavoidconfusionwithphotonrate,Photon_rate.10/14/2020–InSection5.3.2(Q)changethelastsentenceto:“Thesurfacereflectedphotonratepermeter,photon_rate=rsurf,isequalton_photonsdividedbylength_seg,andthenoisephotonratepermeter,photon_noise_rate=rnoise,isequaltothedifferenceofn_ttl_photonminusn_photonsdividedbylength_seg.”11/9/2020–Section5.6.3.2.1regardingequation48description,changed“crossproduct”to“productsum”
1.1.1.1.1 1/4/2021–InTable6ChangedBinsizetobinsize
1.1.1.1.2 Deletedmentionofslope_seg
1.1.1.1.3 Addednharms(=32)todescriptionofoutputvariableds_wn,thevectorofwavenumbers
Addedlast_geoseg
1.1.1.1.4 Deletedsegment_id1/4/2021–Section5.4.2AncilliaryData–addedlast-geoseg1/4/2021–DeletednearlyallofSection5.6ATL19GriddedProductinfavorofashortparagraphreferringthereadertotheATL19ATBD1/4/2021–InTable6,changedds_wntownandds_atoaandmovedtosegmentsgroup.1/5/2021-InTable6putda_aandds_wnbackinwithcorrecteddescriptions1/5/2021Table6–Putimalphabeticalorderwimgroupsandappliedacomsistentborderscheme.
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1/5/2021–DeletednearlyallofSection3.6aboutplansforATL19GriddedProductinfavorofashortparagraphreferringthereadertotheATL19ATBD.1/15/2021Section2.1,paragraph3–UpdatedandcorrectedthediscussionofTides,Medium-FrequencyChangesinCirculationandAliasingandthefourthparagraphonheGeoid,NarrowingtheWindowonPhotonReturns,andMeasuringMeanDOT.1/15/2021Section2.2.3-UpdatedprevioussentenceandtheparagraphunderAPrioriEstimationofSSBUsingOnlyAltimeterData
1/18/2021Section3.1.1.6–Anotatedwithoutputdirectoriesandvariablese.g.,gtx/ssh_segments/heights/bimd,latbin,lonbin,htybin,xrbin,bin_ssbia,swh.1/18/2021Section3.1.15–Annotaqtedwithoutputdirectoriesandvaraibles,e.g.,(gtx/ssh_segments/stas/n_photons;(gtx/ssh_segments/heights/y,ymean,yvar,yskew,ykurt,meanoffit2.1/18/2021AddedSection3.1.1.9OceanSegmentStatistics1/18/2021Section4.5onATL19griddedproduct:AddedareferencetoATL19ATBDandremovedasmallamountofoutofdatetextleftoverfrompreviousedits.1/18/2021Section5.2.3Controlparametersadded:“Thesecontrolparametersareoutputinancillary_data/oceanforeachgranule/fileATL12”1/18/2021Section3.1neartheendofthelastparagraph,added:“whereageo-binistheATL03geolocationsegmentandamountstoabout20-malong-trackdistance”1/18/20Section5.4.2atendofthefirstparagraph,added:“variablesareloctedin/gtx/statsgroupandinclude“1/18/2021Section5.6attheendcorrectedATL-19toATL19
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1/20/2021Section3.1.1.4–Addedanexplanationofeditingoutphotonsinafterpulsesduetosaturationasindicartebythequality_phflagbeingnonzero.AlsoaddedtoSection5.3.6areferenceto3.1.1.4.AlsoinTable2,notedindescriptionforfpb_param(n)thatitwasnotusedatpresentandrefrredtoSection3.1.1.4.InTable6descriptionsoffpb_corrandfpb_corr_stdevnotedthattheyare”currentlysettoINVALID.See3.1.1.4.”1/21/2021Section4.2.1.32):Addedtextdescribingchangeto5.3.2tochangesurfacefindingbasedonthedistributionofphotonheightstosurfacefindingbasedonthephotonheightanomalyrelativetoamovingbinaverageofhighconfidencephotonheights.1/21/2021Section3.1.1.3:Addedtextdescribingthrelatestsurfacefindinincluding“ starting with Release 4, the surface finding, insteadofbeingbasedonthedistributionofphotonheights,isbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesobviatingtheimmediateneedforasubsurfacereturncorrection.1/21/2021Alsoto5.6.3added:“Also, for Release 4, the surface finding, insteadofbeingbasedonthedistributionofphotonheights,isbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesobviatingtheimmediateneedforasubsurfacereturncorrection.”1/21/2021Table6:Specifiedtheplace-holdervariablesss_corrandss_corr_stdevwouldhavevaluesequaltoINVALIDpendingfurtherdevelopments
1/21/2021UpdatedSection5.2.1summarydiscussionoftheuseofqualityvariablesandflagsfromATL03,quality_ph, gtx/geolocation/podppd_flag, signal_conf_ph,andlayer_flagineditingandsurfacefinding
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1/22/2021ChangedpreviousTable9to7andTable10to8becausepreviousTables7and8wereremovedwithATL19totheirownATBD.AlsoupdastethelistdofFiguresandTables
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List of TBDs/TBRs
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Table of Contents
Abstract .......................................................................................................................... 2-iCM Foreword .................................................................................................................... iiPreface ............................................................................................................................ ivReview/Approval Page .................................................................................................... viChange History Log ........................................................................................................ viiList of TBDs/TBRs ........................................................................................................ xxiiiTable of Contents ........................................................................................................... 24List of Figures ................................................................................................................. 28List of Tables .................................................................................................................. 311.0 INTRODUCTION ................................................................................................... 342.0 OVERVIEW AND BACKGROUND INFORMATION .............................................. 37
2.1 Open Ocean Background .................................................................................. 382.2 The Importance of Waves .................................................................................. 40
2.2.1 Waves and Reflectance .............................................................................. 402.2.2 Waves and Sea State Bias ......................................................................... 422.2.3 ICESat-2 Height Statistics and Sea State Bias ........................................... 47
3.0 Open Ocean Products ........................................................................................... 523.1 Open Ocean Surface Height (ATL12/L3A) ........................................................ 52
3.1.1 Height Segment Parameters (output group gtx/ssh_segments/heights/) .... 533.1.2 Input from IS-2 Products ............................................................................. 553.1.3 Corrections to height (based on external inputs) ......................................... 56
3.2 Gridded Sea Surface Height (ATL19/ L3B) ....................................................... 574.0 ALGORITHM THEORY ......................................................................................... 59
4.1 Introduction ........................................................................................................ 594.2 ATL12: Finding the surface-reflected photon heights in the photon cloud ......... 59
4.2.1 Selection of Signal Photon Heights ............................................................. 614.2.2 A Priori Sea State Bias Estimation and Significant Wave Height ................ 66
4.3 ATL12: Correction and Interpretation of the Surface Height Distribution ........... 674.3.1 Separating the Uncertainty due to Instrument impulse response ................ 67
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4.3.2 Characterizing the Random Sea Surface .................................................... 694.3.3 Expectation Maximization ............................................................................ 70
4.4 Calculation of Uncertainty .................................................................................. 714.5 ATL19: Gridding the DOT and SSH ................................................................... 72
5.0 Algorithm Implementation ...................................................................................... 735.1 Outline of Procedure .......................................................................................... 735.2 Input Parameters ............................................................................................... 73
5.2.1 Parameters Required from ICESat-2 Data .................................................. 735.2.2 Parameters Required from Other Sources .................................................. 765.2.3 Control Parameters for ATL12 Processing .................................................. 76
5.3 Processing Procedure ........................................................................................ 775.3.1 Coarse Surface Finding and Setting of Segment Length ............................ 775.3.2 Processing Procedure for Classifying Ocean Surface Photons, Detrending, and Generating a Refined Histogram of Sea Surface Heights ............................... 805.3.3 Processing to Characterize Long Wavelength Waves, Dependence of Sample Rate on Long Wave Displacement, and A Priori Sea State Bias Estimate 875.3.4 Processing Procedure for Correction and Interpretation of the Surface Height Distribution ................................................................................................... 935.3.5 Applying a priori SSB Estimate ................................................................. 1085.3.6 Expected Uncertainties in Means of Sea Surface Height .......................... 108
5.4 Ancillary Information ........................................................................................ 1135.4.1 Solar Background Photon Rate and Apparent Surface Reflectance (ASR) 1135.4.2 Additional Ancillary Data ........................................................................... 113
5.5 Output Parameters ........................................................................................... 1155.6 Gridding DOT for ATL19. ................................................................................. 1215.7 Synthetic Test Data .......................................................................................... 1215.8 Numerical Computation Considerations .......................................................... 1285.9 Programmer/Procedural Considerations .......................................................... 1285.10 Calibration and Validation .............................................................................. 128
6.0 Browse Products ................................................................................................. 1296.1 Data Quality Monitoring ................................................................................... 129
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6.1.1 Line plots (each strong beam) ................................................................... 1296.1.2 Histograms (each strong beam) ................................................................ 129
7.0 Data Quality ......................................................................................................... 1307.1 Statistics ........................................................................................................... 130
7.1.1 Per orbit statistics ...................................................................................... 1308.0 TEST DATA ......................................................................................................... 133
8.1 In Situ Data Sets .............................................................................................. 1338.2 Simulated Test Data ........................................................................................ 133
9.0 CONSTRAINTS, LIMITATIONS, AND ASSUMPTIONS ...................................... 1349.1 Constraints ....................................................................................................... 1349.2 Limitations ........................................................................................................ 1349.3 Assumptions .................................................................................................... 135
10.0 References ........................................................................................................ 136ACRONYMS ................................................................................................................. 138GLOSSARY.................................................................................................................. 139APPENDIX A: ICESat-2 Data Products ....................................................................... 140 APPENDIX B: Sea State Bias Computations for a Photon Counting Lidar ................. 144APPENDIX C: Expectation-Maximization (EM) Procedure .......................................... 150APPENDIX D: Fitting a Plane to Spatially Distributed Data ......................................... 151APPENDIX E: Hierarchy of ATL12 and ATL10 Variables ............................................ 15311.0 References ........................................................................................................ 154
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List of Figures
Figure Page Figure1-Characteristicsofanidealizeddistribution………………………………………………37Figure2.Geodeticheight…………………………………………………………………………………………39
Figure3.Reflectanceasmodeled……………………………………………..……………………………...41
Figure4.TypicalGram-Charlierdistribution……………….………………………………………….42Figure5.Trochoidalshapeofsurfacewaves…………………………….……………………………..43
Figure6.ShortwaveMSS&standarddeviationMSS……………….…………………………….…44
Figure7.SimulatedsurfacedisplacementandMSS…………………..….…………………………...45Figure8.Heightduetolong-waves&MSS……………………………...…….…………………………..45
Figure9.Heightduetolong-waves&probability……………………..….…………………………..46Figure10.SurfacereturnheightsfromMABEL…………………..……...…………………………….50
Figure11.ATL12processingblockdiagram……………………………......…………………………..60
Figure12Coarse,smoothed,andfinehistograms…………………………………………………….64Figure13.BlockdiagramfortheATL19gridding……………………....…………………………….71
Figure14.Coarse,smoothed,finalMabelhistograms…………………………………………….…83Figure15.Instrumentimpulseresponsehistogram……………………………………………..…..92
Figure16.ProbabilitydensityfunctionsofthesyntheticSSH……………………………….......93
Figure17.Synthesizedreceivedprobabilitydensityfunctions….……………….………….…..94Figure18.Single-sidedspectraofthereceivedPDFs…………….….……………….………….…..97
Figure19.Single-sidedspectrumofimpulseresponse…………………………….…….…….……98
Figure20.Single-sidedamplitudespectraoftheWienerfilters..….……………………………98Figure21.Single-sidedamplitudespectraofthesurfacePDF….……………….………….…….99
Figure22.PDFofSSHandcalculatedPDFfor8000photons..….….…………….….…………...102Figure22.PDFofSSHandcalculatedPDFfor800photons..…...….……………..……………...104
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List of Tables
Table Page Table 1 Reflectance………………………………………………………………………... 41Table 2 Input parameters (Source: ATL03)……………………………………………….. 76Table 3 Input parameters (Source: ATL09) ……………………………………………… 76Table 4 Control Parameters - Surface Finding…………………………………………….. 77Table 5 Control Parameters for Refined Surface Finding and Analysis……………...…….79 Table 6 Output (See Appendix A for full product specifications)…………………….......114 Table 7 ATL12 Output Variables for per Orbit Statistics………………………………….129 Table 8 ATL12 Test Data …..…………………………………………………………...…132
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1.0 INTRODUCTION
ThisATBDwillcovertheretrievalofSeaSurfaceHeight(SSH)fromICESat-2ATLASlaserreturns.Forthepurposeofthisearlydraft,thelevelsofspatialandtemporalaveragingtobeincludedarestilltobedefinitivelydecided,asistheamountofhigh-leveldataprocessingtoproducegriddedfieldsofSSHandsuchthingsasDynamicOceanTopography(equaltoSSHminusthegeoid).
OtherICESat-2ATBDsdescribetheproductsforicesheets,vegetation,seaiceandinlandwater,thelattertwohavingcloserelationswiththeoceanATBD.TechnicalATBDsincludeorbitandattitudecalculations,correctionsforatmosphericpath-lengthdelays,andcorrectionsforchangesinthesurfaceheightsduetotidaleffects;theseotherdataareneededtoconvertrangesintoabsolutesurfaceheightswithrespecttothegeoid.
Thisdocumentwilladdresssamplingtheice-freeworldocean,butnotice-coveredorinlandwaters,becausetheICESat-2samplingschemeisdifferentfortheopenocean,generallysamplingstrongbeamsonly,andbecausethedefiningcharacteristics,(e.g.,welldevelopedsurfacewaves)areuniquetotheopenocean.Itwillincludecoastaloceanwaters.Becauseoftheincreasingseasonalvariationintheseaicecover,theboundarybetweentheopen-oceanandice-coveredoceandomainswillvaryseasonally.ThisATBDwillsharesomeconsiderationsandfeatureswiththeseaiceATBD(surfacefindingalgorithm,concernfortidesandthegeoid)andthevegetationandinlandwaterATBD(concernwithwavesinlargebodiesofinlandwater).Weconsiderasinputdatalevel2photonheightsforeachofthethreestrongbeamsalongthesatellitetrackalongwiththerequirednavigationinformation.(Incertainoceanregionsnearlandorseaice,theweakbeamsmaybealsobeactive.Inthesecasestheweakbeamdataovertheoceanshouldbeprocessedthesamewayasthestrongbeams.)Asoutput,theprocessingtobedescribedwillproduceATL12/L3A(AppendixA),theheightoftheopenoceansurfaceatalengthscalesbetween70m(100Hz)and7km(1Hz),determinedbyanadaptivesurfacefindingalgorithm.Outputwillincludeestimatesofheightdistributions(decilebins),significantwaveheight,surfaceslope,andapparentreflectance.
Section2providesanoverviewoftheoceanaltimetryissues,Section3discussestheoceanproductstheparametersthatresideineachproductas
wellasancillarygeophysicalparameters,ofinteresttoscienceusers,whichareusedinthederivationoftheseICESat-2products.
Section4providesatheoreticaldescriptionofthealgorithmsusedinthederivationoftheoceanproducts.
Section5describesthespecificimplementationsofthealgorithmsthatarerelevanttothedevelopmentoftheprocessingcode.Includedherearebothalgorithmicdetailsandsomesoftwarearchitecturedetailsonthroughputoptimizationandcomputationalloading.
Section6providestheprocessingrequirementsfordataqualitymonitoringandcontrol.Theseareprovidedtousersasqualityassessmentsoftheindividualparametersoneachfileandtoprovidecriteriaforautomaticqualitycontroltofacilitatetimely
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distributionoftheproducttotheusers.Summarystatisticsorimagesareprovidedthatallowuserstoeasilyevaluate(orBrowse)whetherthedatawouldbeusefulandofadequatequalityfortheirresearch,andasneededtoaidinthequickapprovalordisapprovalofproductspriortopublicdistribution.
Section7describesthetestingandvalidationproceduresthatareplanned.
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2.0 OVERVIEW AND BACKGROUND INFORMATION
TheAdvancedTopographicLaserAltimeterSystem(ATLAS)consistsofbothLIDARandaltimetrysubsystemsthatwillflyonthededicatedplatformcomprisingthemissionreferredtoasICESat-2,theIce,Cloud,andLandHeightSatellite.ThefollowingsubsectionsdiscussthebasicconceptsofhowATLASworksandtheLevel2datathatwewillturnintoseasurfaceheight.ItthendescribesthebasicpropertiesoftheseasurfaceandhowtheserelatetoprocessingtheICESat-2data.
TheprimarypurposeoftheATLASinstrumentontheICESat-2missionistodetect
Figure1-CharacteristicsofanidealizeddistributionofATLASreturnphotonarrivaltimesassumingaGaussianinstrumentimpulseresponseandareflectivesurfacewithGaussianroughness.Ingeneralsurfaceslopeandroughnessbothbroadenthereturndistribution.Fortheopenoceantheslopeeffectislikelynegligiblecomparedtheeffectofroughnessduetosurfacewaves.Animportantcomplicationisthatsurfacewavesproduceanon-Gaussiansurfaceduetotheirbroadtroughsandnarrowpeaks.
1 nsec
~10 m
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surfaceheightchanges.Withrespecttotheocean,theserepresentSeaSurfaceHeight(SSH).Bywayoftechnicalhistory,thepredecessorofICESat-2ATLAS,theICESatGLASinstrument,usedalaseraltimetertomeasuretherangetothesurface.Rangesweredeterminedfromthemeasuredtimebetweentransmissionofthelaserpulseanddetectionofthepulsereflectedfromthesurfaceandreceivedbytheinstrument.TheGLASlaserfootprintdiameteronthesurfaceduetobeamspreadingwasnominally70m,andthedurationofthetransmittedpulsewas4ns.
TheICESat-2ATLASinstrumentwillsampleatahigherrate,10kHzwith1-2nspulsewidth,withasmallerintrinsicfootprint,~10m(Fig.1),andwillrelyondetectingtherangetraveledbyindividualphotons.ExperiencewithMABELsuggeststhattheinstrumentimpulseresponsedistribution,duelargelytothevariationintransmittimesforthephotonsofeachpulse,willbenon-Gaussianwithsignificantskewness.Consequently,fourpointswillbemeasuredontheoutgoingpulsedistributionandthesewillbepartoftheICESat-2rawdatastream.Thedistributionofreceivetimesofthesurface-reflectedphotons,andhencephotonheights,willbebroadenedbythedistributionofsurfaceheightswithinthefootprintasdepictedinFigure1foranidealizedGaussiandistributionofphotonreturntimesorheights.Photonreturn-timeswillbedigitizedin200ps(3cm)rangebins.However,theoriginationtimeofaphotonwithinapulseisunknown,andtheuncertaintyinthetimeofflightofasinglephotonwillbebetween1and1.5nsRMS(30-45cmflighttime)duemostlytothelaserpulsewidth,withsmallercontributionsfromothereffectswithintheinstrument.Thistimeuncertaintycorrespondstobetween15and22cmRMSinrange.
Returnphotonarrivalswillbeaggregatedatascaledependingonthesurfacetypebeingoverflown.Overtheopenocean,shortsegmentaggregateswillberetrievedat100Hz(100pulsesover70moftrack)andlongsegmentaggregatesretrievedupto1Hz(10,000pulsesover7kmcorrespondingto25timestheatmosphericsamplerate.Incontrast,toensureadequatedetectionofleads,overseaicereturnphotonsmaybetakenatthemaximumrateof10kHz(eachpulseevery0.7m).
2.1 Open Ocean Background
Overdistancesofcmtoafewhundredmeters,theseasurfaceisroughenedbywavesandoceanswell,butoverdistancesofmanykm,theseasurfaceisalmostflat.Nevertheless,surfaceslopesandlong-wavelengthundulationsarepresent,causedbyvariationsinEarth'sgravityfieldrepresentedbythegeoid,oceancurrents,andvariationsinatmosphericpressureandseawaterdensity.Satelliteradaraltimetershaveshownremarkablesuccessinmeasuringsea-surfaceheightandtrackingchangesincirculationandmeansealevel.However,themajorsatelliteradaraltimeters,TOPEX/Poseidon/Jasondonotgobeyondabout62°latitude,andthusmissthesub-ArcticseasandsouthernpartsoftheSouthernOceanthatarecriticaltotheglobaloverturningcirculationandthefateofseaiceintheicecoveredArcticOcean.ICESat-2overtheopenoceanwillcoverthisgapbetweenthetemperatelower-latitudeoceanandtheseaicecoveredregionsoftheArctic
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andAntarcticAsdiscussedabove,thedistributionofICESatphotonreturntimes,orapparentheights
fromLevel2processing,willbedeterminedbythemeanSSH,themeansurfaceslopeandasurfaceroughnesswithinthefootprint.Amongthese,theeffectofseasurfaceslopeonthephotonheightdistributionsshouldbesmallcomparedtotheeffectofroughnessduetosurfacewaves.Fortheaggregationscalesabove,theSSHisthesumofthegeoidheight(fixedintimeandontheorderofmetersamplitude),thedynamicoceantopography(DOT,ontheorderofcentimeterstotensofcentimetersamplitude)associatedwithmeansurfacecurrents,tides(mainlydiurnaltosemidiurnalperiodswithamplitudesofuptometers),andseasurfaceatmosphericpressure(SAP,asmuchas10sofcm).Asafirstapproximation,SSHcanbeestimatedasthemeanofthedistributionofphotonheightswithinaverticalwindowabouttheellipsoidoranapproximatedcanonicalseasurfaceheight(e.g.,30mfromtheestimatedgeoid)andoverthealongtrackaggregationscalesto
bedecidedasabove.However,thespecialandrapidlyvaryingnatureoftheseasurfaceposechallengeswiththeinterpretationandaggregationofmeaningfulSSHfromICESat-2photonheights.Itisusefultooutlinetheseissueshere,keepinginmindthatthemagnitudeofthemostsoughtaftercomponentsoftheSSHsignal,changesinDOTandmeanSSH,aresignificantlysmallerthanmostoftheothercomponentsofSSH.Tides,Medium-FrequencyChangesinCirculationandAliasing:ICESat-2willcompleteoneorbitofthe
earthinabout1.5hours.Therepeatperiodis91days,andcommonly,especiallyatlowlatitudes,a25-kmgridcellinagrid-averageofICESat-2mightonlyafewsatellitepassespermonth.Consequently,tidesandmediumtohigh-frequencychangesincirculationwillseriouslyaliasintoslowlysampledregions,makingaggregationintovalidlong-termmeansdifficult.ThisisaddressedbycorrectingthephotonheightstotheSSHrelativetotheWGS84ellipsoidusingmodelsoftidesandthereponseoftheoceantoatmosphericforcingincludingtheinversebarometereffect.,WiththepredictedtidalandcirculationcontributionstoSSH,,theresidualcanbeaveraged,andonlytheerrorinthemodeledresponseisaliased.Aswewillshow,forATL12processingwereducethesizeofthesignal
Figure 2. Geodetic height measured with a geodetic grad dual-frequency GPS with PPP processing at a drifting ice camp near the North Pole plotted with an ad hoc model of ocean tides and longer variations assumed due to the geoid and DOT variations. The geoid variations over the 60 km drift are estimated to 60 cm, tides 10 cm peak to peak, and DOT only a few cm.
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weprocessbysubtractingtheEGM2008meantidegeoidtoprocessfordynamicoceantopography.
TheGeoid,NarrowingtheWindowonPhotonReturns,andMeasuringMeanDOT:ICESat-2photonheightswillbeheavilyinfluencedbythegeoid.AnexampleofthisfortheArcticOceanneartheNorthPole(Fig.2)showschangesinSSHassociatedwithgeoidheightvariationsofabout1cmperkmofhorizontaldistance.Globally,variationsinSSHandthegeoidarehundredsofmeters,whilevariationsinDOT,equaltoSSHminusthegeoid,areonlyacoupleofmeters.GiventhepotentiallysparsecharacterofICESat-2oceanreturns,thevariationsinthegoidfromEGM2008willbeextractedfromthephotonheightstoremovetheinherentvariabilitynotdotooceanprocessesandreducetherangeofvariabilitywehavetodealwithinoursignalprocessing.Asafinalstepinprocessing,theEGM2008geoidwillbeaddedbacktoDOTtoyieldSSHrelativetotheWGS84eillipsoidastheprimaryATL12output.SurfaceAtmosphericPressureandtheInverseBarometerEffect:ChangesinsurfaceatmosphericpressureashorttimescalesdirectlycauseaninversedeflectionofSSH.Aswithtides,oceancirculationchanges,andthegeoid,thesedirectpressuredisturbancesatthetimeandplaceofeachICESat-2photonretrievalshouldbeestimatedandremovedfromestimatedSSHbeforeaggregationtoavoidspatialandtemporalaliasing.ThiswillrequirebringingatmosphericreanalysisproductsordirectSAPobservationsintotheprocessingstream.Knowledgeofthelikelyinvertedbarometereffectwillalsohelpinnarrowingtherangeofreturnphotonheightsandthusidentifyingphotonsreturningfromtheseasurface.
2.2 The Importance of Waves
2.2.1 Waves and Reflectance
SurfaceWavesReflectance,Scattering,andtheSeaStateBias:WeexpectsurfacewavestohavedominanteffectsontheICESat-2returnsfromtheopenocean.Reflectance:Surfacewavesandwindspeedhaveacriticaleffectonreflectance.Menziesetal.[Menziesetal.,1998]indicatereflectanceisgivenby:
R=WRf+(1-W)Rs+(1-WRf)Ru (1)
whereRisthetotalbackscatterorretro-reflectance,Wisthefractionoftheoceancoveredbyfoam(uptoO[1]forwindsaboveabout7m/s)),Rfisthereflectanceoffoam,Rsisreflectanceduetospecularreflectionfromthewaveroughenedsurface,andRuistheapparentreflectanceduetolightthatpenetratesandcomesbackupthroughthesurfaceanywhereitisnotreflecteddownwardbysurfacefoam.
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RfisconsidereddiffuseLambertianscatteringindependentofincidenceangle.Aswewouldexpect,Rsishighlydependentonincidenceangleandisthedominantformofbackscatter.RuisLambertianbutitisusuallysmall,about0.0075maximum.Menziesetal,modelthereflectanceowingtoRsandRf.ThemainpartisaninverserelationbetweenRsandthevarianceinsurfaceslope,Rs~1/<S2>,duetowindgeneratedsurfacewaves.Theyusearelationbetweenwindspeed,U10,and<S2>byWu[Wu,1972]basedonresultsofCoxandMunk[CoxandMunk,1954];<S2>goesupasthelogofwindspeed.Themodel
resultsaregiveninFigure3,whereinthestrongdependenceonnadirangleisapparent.Theexceptionisathighwindspeedswherethebackscatterfromfoamstartstobecomeimportant,flatteningtheRcurvesfornadiranglesgreaterthan30°.Otherresultsin[Menziesetal.,1998]showgoodagreementbetweenthemodelanddatafromtheLidarIn-
spaceTechnologyExperiment(LITE)shuttlelidarmissionofSeptember1994.ThemodeledreflectanceessentiallyagreeswiththevaluesforreflectanceusedinthedesignperformancestudyforATLAS(Table1).However,mostimportantwithrespecttothisATBD,thedependenceofseasurfacedirectionalreflectanceonsurfacewindstresssuggestsamethodforderivingsurfacewindspeedfromICESat-2measurementsofsea
Table 1
Case Description WaterReflectance,Lambertian,Green
WaterReflectance,Lambertian,IR
10a ConicalScan,lowwind 0.28 10b ConicalScan,mediumwind 0.12 10c ConicalScan,highwind 0.07
ModeledReflectanceforZeroNadirAnglefromFigure4,Menziesetal.[1998]Figure1
Case DescriptionWaterReflectance,Green
10a Wind=4m/s 0.25 10b Wind=6m/s 0.20 10c Wind=10m/s 0.1
Figure3.ReflectanceasmodeledbyMenziesetal.[1998]asafunctionofwindspeedandnadirangle.
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surfacebackscatter.WorkingsolelywithICESat-2observations,wecanconceivablyestimateU10fromreflectance.AswediscussbelowthisestimateofU10maybeusedwithSWH,estimatedfromthevarianceofthephotonheightdistributions,tocalculatetheSeaStateBiasinICESat-2SSHmeasurements.
2.2.2 Waves and Sea State Bias
Seastatebiasisacriticalissuethathasbeenfoundtorelatetotheamplitudeofwavesandtowindforcing.TheSSBproblemisfundamentalandhasreceivedconsiderableattentionwithrespecttoradaraltimetry[Elfouhailyetal.,2000;Gasparetal.,1994].ImprovedcorrectionsforSSBintheTOPEXaltimetershavebeendeveloped[Chambersetal.,2003]relatingSSBtowindspeed,U10,andsignificantwaveheight(SWH,thetrough-to-crestheightofthehighestthirdofoceanwaves)measuredwiththeradaraltimeter.Studiesusingbothlaserandradarinstrumentsfindafairdegreeofcommonality[Chapronetal.,2000;Vandemarketal.,2005].UrbanandSchutz[UrbanandSchutz,2005]compareICESat-derivedSSHwithTOPEX/PoseidonSSHandfindanegative10cmbiasinICESatrelativetotheradaraltimeters.Theyindicatethatthebiasisunknown,butthatitmaybe
relatedinparttoseastate,andtheyrecognizethattheradaraltimeteriscorrectedforSSBusingarelationwithSWH.Unfortunately,aSWHparameterwasnotprovidedwiththeICESatGLASdata,ashortcomingwecannotrepeatwithICESat-2.TheICESatATBDdocumentmadetheassumptionthatseasurfaceheightsareGaussiandistributed,sothatwithaGaussiantransmissionpulse,thereflectedpulsereceivedbyICESatcouldbeassumedtobeGaussianalso.Infact,theoceansurfaceisnotGaussianandthisaffectsthereflectionofeitherradarorlightfromthesurface.The
distributionofsurfaceheightinawave-coveredocean(Fig.4)istypicallygivenbytheGram-Charlierdistribution[Kinsman,1965].Forthisthethirdandfourthmomentsofthedistributionexpressedasskewnessandkurtosis(orexcesskurtosis=kurtosis-3)are
Figure4.TypicalGram-Charlierdistributionseasurfaceheightinawaveenvironment.Thepositiveskewnessandexcesskurtosis(Gaussianskewnessandexcesskurtosis=0)isduetotruncationatlowheightsandalongtailathighheights.Fig.7.4-1ofKinsman[1965].
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importantandindicatetruncationofthedistributionatlowheightswithalongtailathighheights.Thisreflectstheshapeofsurfacewaves.
Ingeneral,surfacewaveshavetrochoidalshape,withnarrow,steepsidedpeaksandrelativelybroadflattroughs(Fig.5).Thisappliesparticularlytotheshort-wavelength
wavesproximallyforcedbywind.PhotonsstrikingtheupperportionsofthesewavearelesslikelytobereturnedtoICESat-2thanphotonsstrikingthelowerportionsofwavesurfaces;peaksareundersampledrelativetotroughsresultinginaseastatebias(SSB)inestimatesofSSHbasedonthemeanofanICESat-2photonheightdistributionor,inthecaseofICESat,meanarrivaltimeofareturnpulse.However,commonlythesea
surfaceiscomposedofswellandwindwaves.Swellisthemanifestationoflargelong-wavelengthwavesthatarematureandforcedelsewhere.Thewavelengthsarelongenoughthatinspiteofsignificantamplitude,theslopesaresmallandthewavesarelinearandwellrepresentedbysinewaves.Thewindwaves,forcedbylocalwind,areshorter(downtocapillarywavelengths)andsteeperwithtrochoidalshape.Intheextreme,whentheyreachthelimitofsteepness(Figure5),theybreakandformwhitecaps.Assuch,thesewavescanbytheirshapeaffectseastatebiasinaltimetryreturns,buttheirdirectbiasissmallbecausetheiramplitudesaresmall.However,severalstudiesincludingonedoneforthedevelopmentofthisATBD,suggestthecharacteroftheshortwavesvarieswithpositiononthelargerlinearwavesandthuscancontributetolargeSSB.
WehaveperformedastudyoftheeffectoflongwavesonshortwavesandtheirimplicationsforICESat-2seastatebias.WehavesoughtamodelseasurfacesuitablefordrivingtheNASAGSFCnumericalATLASsimulator.WiththiswewillbeabletotesttheperformanceATLASovertheopenoceanandevaluatesuchthingsastheseastatebiasinmeanseasurfaceheightmeasurements.Therearetwoelementstoourartificialseasurface.Thefirstisarealisticrepresentationoftheheightandsurfaceslopeduetowaveswithwavelengthsgreaterthanthe10-mfootprintofanATLASlaserpulse.Thiscapturesthebulkoftheenergyinthewavefield.Becausethewavesarelong,alinearmodelofthemcanbeused.Thesecondcomponentisameasureofthedistributionofsurfaceslopesduetothewaveswithwavelengthslessthan10-mlaserpulsefootprintofATLAS.Thedistributionofsurfaceslopes,shiftedbytheinstantaneousslopeofthelongwavecomponentdeterminestheprobabilityofATLASfindingaspecularreturnresultingfroma
Figure5.Trochoidalshapeofsurfacewavesatthelimitofsteepness,wavelengthequaltoseventimesheight.Thesharplypeakedtrochoidalshapeiscommontoallsurfacewaveslongerthancapillarywaves,whichhaveaninvertedtrochoidalshape.Atthesteepnesslimitthewavesbegintobreak.(Figurefromhttp://hyperphysics.phy-astr.gsu.edu/hbase/waves/watwav2.html)
Trochoid wave models
wavetanks thatsuggests that a ratio of1:7 for peak height towavelength is themaximum and that anangle of 120° is theminimum angle for apeak. Above this ratiothe peaks becameunstable. The bottomwave sketch is scaledto that 1:7 ratio ofpeak-to-troughdistance compared towavelength.
As waves movetoward a beach, theshallower waterdecreases thewavespeed, so thewavelength becomesshorter and the peakheights increase. Thewavepeaks becomeunstable and, movingfaster than the waterbelow, they breakforward.
Highest ocean waves Tsunami
HyperPhysics***** Mechanics R Nave Go Back
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photonlaserpulse.Knowingthisasafunctionofsurfaceheightandslopeduetothelong-wavewillallowustopredictSSBintheICESat-2measurements.
Wehavetakenadvantageofacodesimulatinglong-wavesurfaceheightbasedonalong-wavespectrumdevelopedbyDonelanetal.(1985)tomodelthelongwavecomponentofourATLASoceansurfacemodel(AOSM).
Ourmainchallengehasbeendeterminingthemeansquareslope(MSS)representingshort-waveroughnessanditsdependenceonlong-waveheightandslope.Wehadthoughtthatwavewireorsomeothertypeoffieldobservationwouldhavebeenanalyzedoratleastreadilyavailabletodeterminethisrelationship,butsurprisinglywefoundnosuchanalysisintheliterature.Insteadwehaveanalyzedoriginaldatafromafour-wirewave-gaugearrayontheUniversityofMiami’sASISbuoy[WillDrennan,personalcommunication].TheworkisdescribedindetailbyW.Plant.[Plant,2015a;b]butweprovideabriefsynopsishere.TheASISBuoydatawerecollectedintheDeepOceanGasExchangeExperimentintheAtlanticOceanonJuly3,2007atawindspeedbetween10and11m/s.Thewave-gaugesmeasuresurfaceheightsatorthogonalpositions20cmapart,soinprincipleanytwogaugesmeasureslopedirectlydownto~2Hzorwavelength~0.8minthiscase.Asinglegaugeyieldsatimeseriesofheight,andwiththedispersionrelation,thespectrumofheightcanbeconvertedtoaspectrumofsurfaceslope.Thesinglegaugeapproachiscomplicatedbytheneedforthedispersionrelationfortheshortwavestobemodifiedtoaccountfortheorbitalvelocitiesofthelongwaves.Wehavecomparedthetwoapproachesandfoundthetwo-gaugeapproachyieldsresultsinconsistentwiththeknowngaugespacing.However,wecanaccountforthecomplicationstothedispersionrelationandproduceconsistentestimatesoftheshort-waveslopespectra.Spectraforhigherfrequencies(wavenumbers)than2Hzareextrapolatedusingtheobservedfrequency-1spectralslopeat2Hz.TheMSSisobtainedbyintegratingthespectrumouttoamaximumfrequency(100Hz),abovewhichincreasesinMSSarenegligible.
Figure6.Left–ShortwaveMSSasafunctionoflong-waveheightandslope.Right–thestandarddeviationofshortwaveMMSasafunctionofthesamevariables.
Long-Wave Height (m)
Long
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e Sl
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Thepreliminaryresultofthisanalysishasbeenthedependenceofobservedshort-waveMSSontheslopeandheightofthelong-waves(Fig.6).MSSisgreatestforhighheightsandnegativeslopes,i.e.,ontheupperpartofthedownwindfaceofthelong-waves.ThecorrelationmapsofFigure6havebeenusedasatabletospecifyMSSina10-mspotcorrespondingtothe2Dmodeloflong-waveslopeanddisplacementderivedfromtheDonelanspectrum.Arealizationoflongwaveheightandslopeandthecorresponding10-mspotMSSisshowninFigure7[Plant,2015a&b].A1-Dsliceinthedownwinddirection(Figure8)throughthecombineddatafromFigure
7illustrateshowMSSslopetendstoincreaseontheupperpartsofthedownwindfaceof
Figure8.Heightduetolong-wavesversusdistancedownstream(fromupperrightcornertolowerleftinFigureA-left)withcolor-codedcorrespondingMSSfromFigureA-right.
Figure7.FromPlant[2015b].Left–Simulated2Dsurfacedisplacement,Right–2Dvariationofshort-wavemeanMSSvaluesthatwerecalculatedupto2Hz.Downwinddirectionisfromupperrighttolowerleft.
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waves.Italsoillustratesinamuch-simplifiedwayhowtheinterplayoftheshort-waveMSSandlong-waveamplitudeandslopecanaffecttheprobabilityofspecularreflectionandconsequentseastatebias.Foreachpointalongtheslice,weassumethesurfaceisnormallydistributedwithameanequaltothelocallong-waveslopeandavarianceequaltotheMSScorresponding(Fig.6)tothelong-waveheightandslope.Theprobabilityofanyonephotonstrikingthespotfindingaspecularpointisestimatedastheintegraloftheidealizedslopedistributionbetweenplusandminus10-5radians(Fig.9).
Theprobabilityoffindingaspecularpointishighestinthetoughsandatthepeaks,likelybecausethemeanslopeiszero.However,theincreasedMSSintheupperpartsofthewavestendstodecreasetheprobabilityofspecularpointsforhigherheights.Weknowthetruemeanheightalongthesliceis1.8cm.Weightingthesamplesofheightbytheprobabilityofaspecularpointyieldsaphotonsampledmeanheightof-2.6cm,a-4.4cmbias.Interestinglythisisabouttwicethetypicalseastatebiasforradaraltimetersforthe10-mwindspeedatthetimeoftheseASISBuoyobservations.Thebiasisnotduetononlinearities
onthelong-waves.Thesimulationusesalinearspectrum-basedrepresentationofthelongwaves;therearenotrochoidallongwaves.WhenwerepeattheprobabilityweightedmeancalculationassumingtheMSSisuniformlyequaltothecut-wisemeanMSS,themeanheightisonlybiased-0.7cm.Thereareimprovementstobemadetothiswaveanalysis,includingdrawingtheMSSvaluesasrandomvaluesfromdistributionsconditionedonlong-waveslopeandheightanddoingtheanalysisfordataatotherwindspeeds.However,thislimitedsamplesuggeststhatthatthesystematicdistributionofsmall-scalesurfaceroughnesstowardstheupperpartsofthewavesproducesanegativeseastatebiasbecausethewavetroughsprovidemorespecularreturnsthanthewavecrests.Thistypeofbiasislikelycompoundedbyshape-inducedbiasesduetoanynonlinearityinthelong-waves.AsaresultisthattheSSHobservationsbyICESat-2willbenon-Gaussianandexhibitseastatebias.
Figure9.Heightduetolong-wavesversusdistancedownstream(fromupperrightcornertolowerleftinFigure6-left)withcolor-codedprobabilityofspecularreflection.
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However,furtheranalysisofthewavegaugedatatakingintoaccountboththeDopplershiftingofthesmallscalewavebylongwaveorbitalvelocityofthelongwavesandtheeffectsofconvergencebylong-waveorbitalvelocityonshortwaveamplituderesultsinalmosttheoppositedependenceofshortwaveslopeonlongwaveheightandslopeandsuggestsapositiveseastatebias.Consequently,lackingfurtherdataaboutthedependenceofshortwavecharacteristicsonlongwaveposition,wehavederivedamethodwherebytheSSBcanbedeterminedfromICESat-2datathemselves.
2.2.3 ICESat-2 Height Statistics and Sea State Bias
SeaStateBias(SSB)istheerrorinaverageseasurfaceheightmeasuredbyanaltimeterduetothedependenceofaltimeterreturnsoverdifferentpartsofwaves.TheSSBforradaraltimetersisnegativebecausethetroughsofwavesreflectmoreradarenergythanthecrestsofwaves.Thiseffectisaveragedovermanysurfacewavesencompassedbythetypicalradarfootprint(e.g.17kmforCryoSat2).ImprovedcorrectionsforSSBintheTOPEXaltimetershavebeendeveloped[Chambersetal.,2003]relatingSSBtowindspeed,U10,andsignificantwaveheight(SWH,thetrough-to-crestheightofthehighestthirdofoceanwaves)measuredwiththeradaraltimeter.AllthesestudieshavemeasuredtheSSBempiricallybycomparingSSHmeasuredbyvariousmeansorcomparingrepeatmeasurementsatthesamelocationbyagivensatellitefordifferseastatesandwindspeeds[HausmanandZlotnicki,2010].Studiesusingbothlaserandradarinstrumentsfindafairdegreeofcommonality[Chapronetal.,2000;Vandemarketal.,2005].UrbanandSchutz[UrbanandSchutz,2005]compareICESat-derivedSSHwithTOPEX/PoseidonSSHandfindanegative10cmbiasinICESatrelativetotheradaraltimeters.Theyindicatethatthebiasisunknown,butthatitmayberelatedinparttoseastate,andtheyrecognizethattheradaraltimeteriscorrectedforSSBusingarelationwithSWH.Unfortunately,aSWHparameterwasnotprovidedwiththeICESatGLASdata,ashortcomingwecannotrepeatwithICESat-2.
APrioriEstimationofSSBUsingOnlyAltimeterData
ThesmallfootprintofICESat-2(17mfor4sdiameterGaussianintensity),shortdistancebetweenpulse/footprintcenters(0.7m)andtheessentialpointsamplingatrandompositionswithinthefootprintofthephotoncountinglidar,presentsauniqueopportunitytocapturetheshapeofthelongwavelength,energycontainingsurfacewaves.ThismakesitpossibletoestimateICESat-2seastatebiassolelyonthebasisofcontemporaneousICESat-2returnswithoutresortingtooutsidedataorevencomparisonwithpastICESat-2returns.Thekeyisevaluatingtherateofsurfacephotonreturnsasafunctionofsurfaceheight.Asimpleexampleofthisconsidersanoceansurfacewithseasurfaceheight,Hss,disturbedbyalongsinusoidalsurfacewavewithamplitudeA,plusrandomnormallydistributedperturbations,N:
2Y = SSH +Asin 2π
Lx+N(0,σ )
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IfICESat-2samplesthissurfaceataconstantrate, surfacereturnspermeter,thesurfacewillbeuniformlysampled,andtherewillbenoseastatebias.Ifthereisavariationinsamplerate,r,thatiscorrelatedwithvariationsinseasurfaceheightsuchthat:
, (3)
whereaisthecovariancebetweenrandynormalizedbythevarianceiny:
. (4)
Thesurfaceheightestimate,ye,overalargedistance,X,istheaverageofthetrueheightweightedbythesamplerate:
(5)
(6)
ForXequaltoanintegralnumberofwavelengthsorverylongcomparedtomanywavelengths:
(7)
Soseastatebias,SSB=ye-HSS,is:
(8)
Similarly,Arnoldetal.[Arnoldetal.,1995]inevaluatingKu-bandSSBmeasuredwithacombinationofradaraltimetersandcapacitivewavewiresmountedonanoilplatformintheGulfofMexico,calculateelectromagneticbias,e,duetothevariationofbackscattercoefficientwithsurfacewavesas
r
r = r +αAsin 2π
Lx
α =
cov(ry)y2
σ
ey =
1Xr
Yrdx =0
X
∫ 1Xr
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∫
=1Xr
( SSH +Asin 2πLx+N)(r +αAsin 2π
Lx)dx
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=1Xr SSH (r +αAsin 2π
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+1Xr
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+1Xr
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0
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∫ →0
ey =
1Xr
HSSXr +
X2α
2A
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+α
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(9)
wherehiisthemeasuredsurfaceddisplacementcorrespondingAsin(2px/L)inouridealizedexampleand isthemeasuredbackscattercoefficientproportionaltotherate
ofphotonreturns,r.OuraexpressedinthetermsofArnoldetal.[1995]is:
, (10)
sotheirexpressionforelectromagneticbiasisthesameastheSSBforthesinewaveexample:
(11)
AppendixBgivesamoredetailedderivationof(9)intermsofrateofphotonreturnforaphotoncountinglidarinplaceofcross-section.ItalsodescribesthecorrespondingEMorseastatebiasinthehighermomentsofthesurfacedistribution(EquationB20inAppendixB).ExampleofSSBDeterminationfortheUnevenDataDistributionofaPhotonDetectingPulsedLidarAltimeterApplicationof8isstraightforward,andisaccurateincasessuchasthatofArnoldetal.[1995],inwhichthespacingofthedataisuniformandtheenergyofthereturnismeasuredandisthemetricweightingthesampledsurfaceheight.Inthatapplication,eachradarpulsewaspowerfulandtherangeandfootprintsizeweresmallsothateverypulsereturnedasurfaceheight.TheenergyreturnedwitheachpulseanditscorrelationwithheightcouldbeexpectedtoaggregateintotheSSBoverthemuchlargerfootprintofsatelliteradaraltimeter.
Thisisnottotallytrueinthecaseofthepulsedphoton-countingaltimetersuchasMABELorICESat-2;thereturnrateoftheheightsamplesisnotuniformandtheaverageseasurfaceheightisderivedfromthehistogramofphotonheights.Thecorrelationbetweentherateofsurfacereturnsandsurfaceheightisonlyoneofseveralcomponentsinthehistogramskewness.Othercontributorstoskewnessincludethetruedistributiontosurfaceheightandsubsurfacereturns.However,thedominantsurfacewavestypicallyhave
ε =bi0η
ii=1
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∑
bi0
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∑
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ηi2
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∑=
αN
ηi2
i=1`
N
∑
1N
bi0
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2r = SSB
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longwavelengthsandarenearlylinearsotheycanbeapproximatedassinewaves.Withthisinmind,weexperimentedwithfittingmultiplesinewavestotherawhistoryofunevenlyspacedphotonsurfaceheightsandcalculatingthecorrelationofobservedrateofsurfacereturnswiththesinewavefittoyieldanestimateofSSBusingequation7.However,intestingthemethodonGreenlandSeaMABELdataandAirborneTopographicMapper(ATM)overthePacific,wefoundthemethodwassensitivetofindingthecorrectwavenumbersforthespectraldecompositionoftheoceansurface.Amorerobustmethodistoaverageboththerawphotonheightsfromthesurfacefindingstepandtherateofphotonreturnsinevenlyspacebinsofalongtrackdistance.Thecovarianceofthesebeenaveragesisthenusedtoevaluateequation7fortheseastatebias.Thismethodiseasiertoautomateandaccountsformoreoftheenergyinthewavefieldthandoestheharmonicmethod.
Figure10.SurfacereturnheightsingreenfromMABELtransitovertheGreenlandSeaversusdistancealongtrack.Three-sigmaeditedanddetrendedheightsareplottedingreenanda10-mbinaveragesareplottedinblack.ThebinaveragesandaprioriSSBestimateswerecomputedinthree3000-msegmentsandaggregatedtoyieldSSB=-0.00029m.
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MABELdatafromtheGreenlandSeainApril19,2012illustratestheapplicationofequation7.Inthisexample,a9-kmlengthofMABELtrackisbrokeninto3-kmsegments.Each3-kmsegmentisdividedinto30010-malongtrackbins.Oncesurfacefindingselectsthesurfacephotonheights,theseareeditedwithtwopassesofa3-standarddeviationeditor,anddetrendedwithalongtrackdistance.
Therateofphotonreturnsiscomputedfromthealong-trackrecordofsampleintervals.Therecordofsampleintervalsisgivenbythedifferencebetweensuccessivesurfacephotonalong-tracklocations.FortheMABELdatashownhere,thesamplespacingsvarysignificantlybutaveragealittleoveronemeter.Toalignthedata,thesurfaceheightandsamplepositionareinterpolatedtothecenterofthesampleintervalbycomputingtheaverageofpairsofsuccessivesampleheightsandposition.Theresultingrecordsofsampleratearethenaccumulatedin10-malongtrackbinsandaveraged.Thebinaveragedsampleintervalsaretheninvertedtoyieldthebinaveragedsamplerate.Interpolatingheighttothecenterofthesampleintervalandbin-averagingsampleintervalbeforeinvertingavoidsanapparentincreaseinaveragesamplerateinadjacentsampleintervals.
ThebinaveragedsampleratesandheightsarethenusedtocomputetheheightratecovarianceandtheaveragesampleratesusedinX8todetermineSSB.ThemethodisillustratedinFigure10withMABELdatagatheredApril25,2012overtheGreenlandSea.Aggregatedoverthree3000-mintervals,thecorrelationofphotonreturnrateandheightisverylowonaverage,andresultsinaseastatebiasof0.3mm,aresultthatisnotsurprisinggiventhelowenergyofthesurfacewaveswithasignificantwaveheight(4xStdofheight)oflessthanameterfortherawphotonreturnsandthebin-averagedheights.
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3.0 OPEN OCEAN PRODUCTS
3.1 Open Ocean Surface Height (ATL12/L3A)
TheATL12productcontainsseasurfaceheightsovertheice-freeoceansforeachof3ATLASstrongbeams.ItprovidesthemostbasicdatafromICESat-2:theseasurfaceheightatagivenpointontheopenoceansurfaceatagiventimeplusparametersneededtoassessthequalityofthesurfaceheightestimatesandtointerpretandaggregatetheestimatesovergreaterdistances.TheseheightscanbeusedincomparisonsofICESat-2datawithothergeodeticdataandasinputstohigher-levelICESat-2products,particularlyATL19.
Heightsovertheseasurfacearedefinedforoceansegmentswithvariablelengthatvariableintervalsalongthegroundtrack;thisisnecessarytoadapttothereducedphotoncountsfromthelowbutvariablereflectanceoftheopenoceansurfaces.Italsoisconsistentwithsurfacefindingroutineusedforseaicecoveredocean(ATL07/L3A).BasedoninitialdataATLASgetsontheorderofonesurfacereflectedphotonperlaserpulseor0.7moftrack,so100photonretrievalsintheadaptivesurfacedetectionalgorithmwouldresultinsegmentlengthsofabout70m.Giventhe17-mfootprintoftheICESat-2beams,thisisequivalentto4independent(non-overlapping)spatialsamplesoftheseasurface.
Infuturedevelopments,heightsinmarginalicezonesorcoastalzonesmaybedefinedforshortervariablelengthsegmentssampledatvariableintervalsalongthegroundtracktoadapttothepatchesofwaterbetweenseaiceinthefirstinstanceornearlandinthesecondinstance.Theseoverlapregionsarethosethatareclassifiedasbothseaiceandocean(marginalicezone)orlandandocean(coastalzone).ThemethodsofdistinguishingopenoceanfromseaiceandlandandsettingtheoceansegmentlengthsinthesemixedregionsareTBD..
Inpurelyoceanregions,onlystrongbeamswillbeactive,butinthemarginalicezoneandcoastalzoneoverlapregions,thethreeweakbeamswillalsobeactiveandshouldbeprocessedidenticallytothestrongbeamsandoutputinadditiontothestrongbeamresultsaspartoftheATL12oceanproduct.Thisistofacilitatearationalmergingoftheopenoceanproducts(ATL12)andseaiceproducts(ATL07)andcoastalproducts(ATL??)relatedtoseasurfaceheight.
However,thepresenceofoceansurfacewavesintheopenoceanfarfrommarginalicezoneandcoastalregions,presentchallengesandanopportunity.Thenon-GaussiancharacterofoceansurfacewavesandthescatteringandeffectivereflectanceoftheICESat-2photonslikelycauseanegativeseastatebias(SSB).WeanticipatethatthehigherordermomentsofthephotonheightstatisticswillallowustobetterestimateSSB.WewillestimateforeachATL12variablesegmentatminimumthemean,standarddeviation,skewnessandkurtosisoftheheightdistribution.Thesefourmomentscanbedeterminedfromthemeansandstandarddeviationsofa2-componentGaussianmixture,whichwillalsoprovideamixingratioforthetwocomponents.Thesehighermomentsmakeitpossibletocharacterizenotonlythemeanseasurfaceheight,butalsotheseastateandlikelyseastatebiasoftheheightestimate.
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However,itisnecessarytoaccumulatelargenumberofsurfacereflectedphotonstomakemeaningfulestimatesofthehighermomentsoftheheightdistribution.Giventhelownumberofreturnsfromopenwater(e.g.,O[1photon/pulse]),andconsideringthatoceanwavescanhavewavelengthsofhundredsofmeters,togetmeaningfulstatisticsoveropenwaterwithsignificantwaveactivityrequiresaccumulatingdataoverseveralkilometers.Theatmosphericcloudflag(layer_flaginATL03)isusedinpartinsettingtheoceansegmentlengthrequiredtoachieveaminimumnumberofphotons.ItandothergeophysicaldatasuchasthebackgroundphotonratefromATL09andthegeoidarereportedevery400pulses(25Hz)or14geo-binswhereageo-binistheATL03geolocationsegmentandamountstoabout20-malong-trackdistance.Consequently,overtheopenocean,wewillseekaminimumphotoncountof8,000photonsequivalenttoabout5.6kmor28020-mgeo-binsformeaningfulhigherorderstatistics.
3.1.1 Height Segment Parameters (output group gtx/ssh_segments/heights/)
3.1.1.1 Geolocation/Time
Thelocationisthemeanlocationofdesignatedsignalphotonsusedasinputtothesurfacefindingprocedure(gtx/ssh_segments/longitude&latitude,seeTable6).Thetime(gtx/ssh_segments/delta_time)isthemeantimeofdesignatedsignalphotonsusedasinputtothesurfacefindingprocedure.
3.1.1.2 Height
Thisisthemeanheightestimatefromthesurfacedetectionalgorithm(gtx/ssh_segments/heights/h).Theprimaryqualitymetricistheuncertaintyintheaverageseasurfaceheight,(gtx/ssh_segments/heights/h_uncrtn)determinedasafunctionofheightvarianceandlengthoftheoceansegmentdividedbytheheightcorrelationlengthscale.
3.1.1.3 Subsurface-scattering corrections
Subsurface-scattering corrections are yet to be determined. We find subsurface returns in ICESat-2 ocean photon returns. They tend to be minimal for the deep open ocean where the water is relatively clear. Our surface finding is designed to include few subsurface returns in the first place by rejecting photon heights at greater noise levels below the surface than above the surface. More recently, starting with Release 4, the surface finding, insteadofbeingbasedonthedistributionofphotonheights,isbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesobviatingtheimmediateneedforasubsurfacereturncorrection.
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Subsurfacescatteringismoresignificantinnearcoastalregionswheremorescatteringelementsaretobeexpectedinthewater.Theseconditionswillbemostlikelytobeapparentaspositiveskewnessinthereturnheightdistributionsespeciallyascorrelatedwithoutsidemeasuresofsurfacecolororscattering.GuidedbythetreatmentofsubsurfacescatteringfortheInlandWaterATBDandwemaybeabletodeveloprelationsbetweenphotonheightdistributionskewnessandotherevidenceofscattering,andifso,deriveanappropriatecorrectionforsubsurfacescatteringwhereapplicableandprovideitasanoutputofATL12.
3.1.1.4 First-photon-bias corrections
Generally,giventhenormallylowreflectanceoftheoceansurfacefirst-photon-biascorrectionsarenotneededfortheoceanproducts.However,inlookingaton-orbitdata,wefindrareinstancesofquasi-specularreturnsthatwethinkoccurwhentheseasurfaceissmoothbuthasveryshortwavelengthripples.Forthesecasesphotondetectorsaturationcanoccurleadingtofirst-photonbiasandtheappearanceofsubsurfaceafterpulses.InthecurrentATL12processing(Release4),photonsintheseafterpulsesareeditedoutbyconsideringonlyphotonswiththeindexofphotonheightqualityfromATL03(quality_ph)equalto0.Infuturereleases(Release5)wewilldeleteallphotonscomingfromlaserpulsesthatincludeanyphotonswithaqualityindexindicativeofsaturation,orwewillconsiderapplyingaTBDfirst-photon-biascorrectionifthereareoceansegmentsforwhichthefractionofsaturatedphotonsishighasindicatedbyfull_sat_fract_segandnear_sat_fract_seg.
3.1.1.5 Height Statistics
Thephotonstatisticsdescribethedistributionofthepopulationusedinthesurface-trackingalgorithm.Theseparametersincludethe:numberofphotons(gtx/ssh_segments/stats/n_photons),histogramofthepopulation(gtx/ssh_segments/heights/y),and,atminimumthemean,variance,andskewness,andkurtosis(gtx/ssh_segments/heights/ymean,yvar,yskew,ykurt).Notethatymeanshouldbenearlyzeroanddiffersfromtheaverageseasurfaceheightrelativetothegeoidbymeanoffit2(gtx/ssh_segments/heights/meanoffit2)
3.1.1.6 Initial Sea State Bias Correction and Surface Wave Properties
Asdiscussedin2.2.3,ICESat-2givesusauniqueopportunitytomakeanaprioriestimateofseastatebiasusingonlytheheightsfromthesurfacefindingsystem.Theapproachistousethesurfacephotonheightstoestimatethesurfaceheight(gtx/ssh_segments/heights/xrbin)andtherateofphotonreturns(gtx/ssh_segments/heights/htybin)in10-malong-tracksegments(seegtx/ssh_segments/heights/bind,latbin,lonbin).Thecorrelationofheightandreturnratenormalizedbytheaveragephotonreturnrateistheelectromagnetic(EM)seastatebias
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(gtx/ssh_segments/heights/bin_ssbias).Thiswillbeestimatedfromthesurfacephotonheightrecordbeforeseparationoftheimpulseresponseintheheighthistogramandmadeavailableasoutput.ThestandarddeviationofphotonheightsoverthewholesegmentwillbeprovidedasameasureofSWH(gtx/ssh_segments/heights/swh)(=4xStd.)thatcanbecomparedwithreanalysisproducts.
3.1.1.7 Solar Background Thesolarbackgroundphotonrate,backgr_r_200,isestimatedasanaverageover50laserpulsesinATL03.Itisbasedonthephotonsincludedinthealtimetrichistogramlessthephotonsdeemedsignal(surfaceorcloudreflected)photonsbyATL03.Thisisconvertedtoarateinphotonspersecondbydividingbythetotaltimewindowreducedbythedurationofthewindowcontainingsignalphotons.
ForATL12wewillusetheaverageoftheestimatedsolarbackground.Wewillnotusethepredictedsolarbackgroundratebasedonthesolarzenithangle,thesolarfluxinthereceiveretalon’spassband,andtheeffectiveapertureofthedetectors.
3.1.1.8 Apparent Reflectance
ThisisbasedonthecomparisonofexpectedphotoncountsoveraseasurfacewithSWHestimatedfromphotonstatisticsandphotoncountsfromtheactualsurface(see2.2.1.4andFig.2).
3.1.1.9 Ocean Segment Statisitics Importantstatisticsforeachoceansegmentwillalsobeoutput(inthegtx/ssh_segments/starts/group).Theseinclude:thenumberofsurfacephotonsfoundforthesegment,n_photons;thenumberofphotonsinthe±15-moceandownlinkband,n_ttl_photon;solarbackroundratefromATL03averagedoverthesegment,backgr_seg;thethepercentageofgeo-segsintheoceansegmentwithlayer_flag=1,cloudcover_percent_seg;thesegmentaveragedynamicatmosphericcorrectiontoSSH,dac_seg;segementaverageoceandepth,depth_ocn_seg;theoceansegmentaverageoftheconversionfactoraddedtoATL03geoidtoconvertfromthetide-freetothemean-tidesystem;,geoid_free2mean_seandtheoceansegmentaverageofEGM2008geoidinthemean-tidesystemheightabovetheWGS–84referenceellipsoid,geoid_seg.
3.1.2 Input from IS-2 Products
3.1.2.1 Classified photons (Level 2)
Photonsclassifiedastowhethertheheightissignal,noise,orextendedsignal.Thesehaveaconfidenceastotype.
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3.1.2.2 Atmosphere (Level 3)
Relative/calibratedbackscatter,backgroundrates,cloudstatisticsat25Hz.
3.1.3 Corrections to height (based on external inputs)
Inanticipationofhigherlevelprocessing,thestandardheightproductswillincludeanumberofcorrectionsappliedtotherawheightestimates.Forexampletoreducealiasingproblems,correctionsforhighfrequencyandfinespatialscalevariationsinSSH(e.g.,tidesandotherhighfrequencyoceancirculationchanges)thatmaybeinadequatelysampledbyICESat-2shouldbeappliedbeforeaveraging.Estimatesofthesecorrectionswillbegivenhere.Allcorrectionswillbegivenintermsofheightsothattoapplyacorrection,userswilladdittotheheightestimate,andtoremoveittheywillsubtractit.Additionalcorrectionsthatsomeusersmaydecidetoapplywillbeprovidedwiththeproduct.
3.1.3.1 Geoid adjustment (Static)
Heightsareadjustedforlocalgeoidheightusingmean-tideEGM2008.PriortoATL03Release4,theEGM2008wasreportedbyATL03astakenfromthemean-tideEGM2008model.StartingwithRelease4ATL03reportsEGM2008inthetide-freesystemandprovidesaconversion(geoid_free2mean)thatcanbeaddedtothetide-freegeoidtoyieldthemean-tideEGM2008foruseintheATL12oceanproduct.
3.1.3.2 Atmospheric delay corrections
Heightswillbecorrectedbasedonanatmosphericmodel,givingcorrectionsfortotaldelaycorrectionthataccountsforwetanddrywettroposphere.Correctionswillbeavailablefortheforward-scatteringbias,basedonavailableatmospheric-scatteringdataandanestimateoftheattenuationcalculatedfromtheapparentsurfacereflectance.
3.1.3.3 Dynamic Atmospheric Correction and the Inverse Barometer Effect (IBE, time-varying)
Heightsarecorrectedfortheinversebarometereffectduetothedirectapplicationofatmosphericpressuretotheseasurfaceandthedynamicchangesforcedbywind.ICESat-2hasadoptedtheutilizationofglobal,empirical,6-h,AVISOMOG2D,1/4°×1/4°gridstobeusedasanear-realtimeInvertedBarometer(IB)andDynamicAtmosphericCorrection(DAC)[CarrèreandLyard,2003].ThesegridsareforcedbytheEuropeanCenterforMedium-RangeWeatherForecasting(ECMWF)modelforthesurfacepressureand10-mwindfields.Thiscombinedcorrectiontypicallyhasamplitudeontheorderof±50cm[Markusetal.,2016].
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3.1.3.4 Tidal corrections (time-varying)
ThetiderelatedgeophysicalcorrectionsappliedtoATL03ellipsoidalphotonheightsinputtoATL12includesolidearthtides(intidefreesystem),oceanloading,andpoletides(bothoceanandsolidearth):Solidearthtides(tide_earth):ATL03photonheightsinputtoATL12arerelativetosolidearthtidesinthetide-freesystemderivedusingIERS(2010)conventionsare±30cmmax(detailsinATL03ATBDsection6.3.3).TheATL03solidearthtidesareappropriatetoallhigherlevelproducts,buttheyareappropriatetotheoceanandseaiceproductsonlyinsofarastheyarethereferencetowhichoceantidesareaddedforafullaccountingoftidesinoceansurfaceheights.Oceanloadtides(tide_load):Thelocaldisplacementduetooceanloading(-6to0cm)derivedfromoceantidemodelGOT4.8.DetailsinATL03ATBD,section6.3.1
Poletide(tide_pole):Poletidesincludebothsolidearthandoceanpoletides.TheseeacharecomputedviaIERS(2010)conventions.DetailsarefoundinATL03ATBDsections6.3.5and6.3.6,respectively.Oceanpoletide(tide_oc_pole):OceanPoleTide-Loadingduetocentrifugaleffectofpolarmotionontheoceans(2mm,max),subsumedinPoletide(tide_pole)
TidalcorrectionstobeappliedinATL12includeshortperiodoceantidesandlongerperiodequilibriumtides:Oceantides(tide_ocean):OceanTidesincludingdiurnalandsemi-diurnal(harmonicanalysis),andlongerperiodtides(dynamicandself-consistentequilibrium)(±4m)arefromtheGOT4.8.DetailsinATL03ATBD,section6.3.1.Equilibriumtides(tide_equilibrium):Equilibriumlong-periodtidecomputedusingasubroutineattachedtoGOT4.8calledLPEQMT.FbyRichardRay.ItisaFortranroutineinwhichfifteentidalspectrallinesfromtheCartwright-Tayler-Eddentablesaresummed.(Seesection6.3.1oftheATL03ATBD.)
3.1.3.5 Wind and SWH Estimates
Surfacewindsandsignificantwaveheightforforecast/reanalysisproductswillbetakenfromanappropriatesourcesuchastheECMWFandwiththeATL12productforcomparisonwithICESat-2derivationsofSWHaspartoftheseastatebiascalculation.
3.2 Gridded Sea Surface Height (ATL19/ L3B)
TheATL19productincludesgriddedmonthlyestimatesofdynamicoceantopography(DOT)takenfromATL12oceansegmentdata.Oceansegmentsrangeinlengthroughlyfrom3to7km.Theoceansegmentdataareaveragedin¼°or25kmgridcells.Datafromallsixbeams(gtxrandgtxl)areusedfromthebeginningtotheendofeachmonth,althoughcommonlyovermostoftheoceanonlydatafromthestrongbeamswillbeavaialable.The
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IS-2dataareaveragedontothreegrids,calledmid-latitude,north-polarandsouth-polar.TheATL19producthasgroupswiththesamenamescontainingthegriddeddataforeachofthoseregions.Thegrids,griddingschemes,inputvariablesandoutputvariablesaredescribedindetailintheATBDforATL19.
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4.0 ALGORITHM THEORY
Inthissection,wediscussthefollowingtopics:1. Finding the collection of photon heights representing reflection from the sea surface. 2. Separating the surface wave generated variation in photon heights from other sources of
variance. 3. Determining SWH and higher order measures of sea state 4. Formulating the best sea state bias correction
4.1 Introduction
ThemainpurposeofthealgorithmsdevelopedhereisthedeterminationofSeaSurfaceHeight(SSH).Therearetwomainstepstodeterminingseasurfaceheight(SSH)fromICESat-2photonheights(ATL-03)overtheocean.Theseareidentifyingphotonheightsthatlikelyrepresentreflectionfromtheseasurface,looselytermedsurfacefinding(Sec.4.2),anddeterminingthecorrectstatisticsoftheseasurfaceheightdistribution(Sec4.3).MostimportantlyweseektheSSHequaltothemeanoftheheightdistribution.Becauseofthemotionoftheseasurface,surfacefindingoveropenwaterisinherentlyasearchforadistributionofheightsrepresentativeoftheseasurface.Ourmainproductwillbethemeanofthisdistributionovertime,lengthandspacescalestobedeterminedaspartoftestingandanalysis.InadditionotherpropertiesofthedistributionareusefulforassessingthesurfacewaveenvironmentandbiasesintheSSHdetermination.Thoughwefocushereonobtainingmeaningfulstatisticaldistributionsofseasurfaceheights,wehavefounditpossibleintestswithMABELdatatocreatemeaningfulmoving21-photonmeansoftheheightsfromthephotonsdesignatedsignalphotonsbythesurfacefindingroutines.Thisgivesarelativelyhigh-resolutiontime-(orspace-)seriestraceoftheseasurfaceinfairagreementwiththeanalysisusingthehigh-resolutionseaiceanalysis.Thismaybeusedinexperimentalanalysesforsurfacewaves.Figure11showstheblockdiagramfortheATL12processingtogofromATL03photondatatoalong-trackhistogramsandstatisticsofdynamicoceantopography.Figure13illustratesthegriddingproceduretogofromATL12productstoATL19griddedDOTandSSH.
4.2 ATL12: Finding the surface-reflected photon heights in the photon cloud
SurfacefindingovertheopenoceanrestsonourlimitedexperiencewithMABELdataoveropenwater.Withfewsurfacereturnsperpulseandsignificantwaveheightsmuchlessthantherangeofsamples,themajorityofthebinswillcontainonlynoisephotonssothatthemedianwillequalthenumberofnoisephotonsperbin.Thus,thefundamentalideaforsurfacefindingistoidentifyassurfaceheightbins,thosebinswithcountsexceedingthemediannumberofcounts.InextremecaseswhereICESat-2encounterssignificantwaveheightsapproachingthe30-mdownlinkrangeofbins,wemayhavetoadoptaslightlydifferentthresholdsuchasthecountvalueequaltoorexceeding20%ofthebinpopulation.
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Figure11.ATL12processingblockdiagramasdiscussedinSection3and4.µ,s,S,andKdenotemean,standarddeviation,skewness,andkurtosisrespectively.
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Weanticipatealargenumberofsurfacereflectedphotonsmaybeneededtoadequatelyresolvethehighermomentsoftheseasurfaceheightdistribution.Also,finespatialresolutionisprobablynotrequiredinmostopenoceansettings.Therefore,wewilluseasemi-adaptiveschemetoaccumulatesegmentslongenoughtocaptureupto10,000candidatesurfacephotons.Thiswillinvolveacoarsemedian-basedselectionandsegment-lengthsettingprocess(4.2.1.2).Thiswillbefollowedbyafinerscalehistogramconstructionthatincludesmedian-basedselectionofsurfacephotons,adetrendingprocess,andrepeatselection(4.2.1.3)
4.2.1 Selection of Signal Photon Heights
4.2.1.1 Input to Selection of Photons: ATL12willrequireATL03photonheightsforeachpulseintheoceandownlinkbandandassociatedtimegeolocation,geophysicalcorrections(geoid,tides,dynamicatmosphericcorrection),andconfidencelevelinformation,plusthecloudflag,layer_flag,every400pulsesfromATL09.AsofRelease3,theATL03includesaflagforeachgroundtrackindicatingthequalityoftheprecisionorbitandpointingdetermination:gtx/geolocation/podppd_flag, and we have found unrealistic values for ocean segments when this flag is non-zero indicating degraded orbit and/or pointing data. Consequently ATL12 processing should only be done with data for which gtx/geolocation/podppd_flag=0.
Basedonearlydiscussions,thisATBDandthedevelopmentalsoftwaredevelopedfromitwasassumedthatICESat-2overtheoceanwoulddownlinkphotonheightswithin±15moftheEGM2008geoid.Asidefromthe±15-mbandaboutthegeoid,noclassificationastosurface/non-surfacephotonswastobemade.
However,asconfiguredinDecember2018,theFlightScienceReceiverAlgorithms(FSRA)downlinksphotonheightsinbandsaboutheightschosenbysignalfindingproceduredescribedintextin“ATLASFlightScienceReceiverAlgorithmsVersion3.7c”.Thisidentifiescenterheightsover200-shotmajorframesasthosewherethemostphotonheightsrelativetonoisearefoundinadjacentbinsofacoarsealtimetryband.Whenoperatinginlocaldaylightandinthepresenceofeventhinclouds,andperhapsoverpartsoftheoceanwithreducedreflectance,forexamplewaveslopes,themostpopularheightbincanshifttochanceconcentrationsofnoisephotonsupto200to300maboveorbelowthegeoid.Theresulting30-merroneousdownlinkbandsofphotonheightsappearin200-pulsemajorframesandoneortwoheightbandssupposedlybracketedbyheightstoppingat(usingtheexampleofgroundtrack2right)gt2r/bckgrd_atlas/tlm_top_band1andgt2r/bckgrd_atlas/tlm_top_band2,withheightsgivenasgt2r/bckgrd_atlas/tlm_height_band1andgt2r/bckgrd_atlas/tlm_height_band2.
Themostconcerningproblemisthatwhenthedownlinkbandispulledawayfromthetruesurfacewecanarguablylosetruesurfacereflectedphotons,biasingseasurfaceheightsinsomeunknowndirectionandviolatingtheassumptionsoftheATL12seastate
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biasdetermination.InsubsequentrevisionsofATL03,thephotonsfrommajorframesforwhichthecenterofthedownlinkbandisfarremovedfromthegeoidareeliminated.
InpracticewehavefoundthattheerroneousdownlinkbandscanbeeditedoutasthosecontainingnoATL03hjghconfidencephotons.Therefore,weusethesignalconfidenceproductinATL03toeditouttheerroneousdownlinkbands.Thisgradeseachphotonheightreturnastolikelihoodthatitisasurfacereturnbyexaminingheightbinsat50-pulserateforsignaltonoiseratiointhecontextofnoiselevel.EarlyICESat-2oceandatasuggeststhatphotonheightswithhigh,medium,andlow(4,3,2)confidencelevelsveryseldomappearintheerroneousdownlinkbands.Further,theconfidencelevelsoftwareassignsafill-inconfidencelevelof1towhatwouldbezeroconfidenceheightsinordertofilla±15binaboutthehighconfidenceheights.Therefore,inpracticeconfidencelevel1orgreateressentiallycoversthegeoid±15-mbandandcanbeusedtoeditouttheerroneousdownlinkbands.Inprocessing,ATL12considersphotonswithconfidencelevelequalto1orgreaterasthepossiblesurfacephotonssubjecttothesurfacefindingroutine.
Becauseweareusingthisground-basedediting,inthefuturewewillhavetoinvestigatetheeffectontheseasurfaceheightandSSBcalculationofmissingtruesurfacephotonsnotdownlinkedinthepresenceoferroneousdownlinkbands,probablybycomparingdaytimeandnighttimedataoverthesameregion.
4.2.1.2 Coarse Selection and Setting of Segment Length
Theprocessisbasicallytofirstestablishavariablelength(e.g.,upto7km)oceansegmentwithenough(e.g.)photonheightstoyieldahistogramwithreasonablemomentsupto4thorder.
2. Examineoceandepthandcloudparametersagainstcontrolparameterstotestforsuitabilityforoceanprocessing.-Ifcontrolparameterlayer_swtchisequalto1,then remove the ATL03 segment ids from ocean processing that are associated with the ATL09 segment ids where layer_flag is equal to 1.Whenlayer_swtchequals0(thedefaultvalue),thesoftwarewillignorethecloudcoverinacceptingdataduringcoarsesurfacefinding.-Ignore data collected over land or too close to land. If ocean depth, depth_ocn, is less than a depth, depth_shore, specifying the effective shoreline of water for which ocean processing is appropriate,thenproceedtonext14geo-bin(400-pulse)segment. The default value for control parameter depth_shore will be -10 m for depths given by GEBCO as negative elevation values.
3. Correctphotonheightsforshortperiodandlongperiodoceantides(gtxx/geophys_corr/tide_oceanandtide_equilibrium),theinversebarometereffectandthemodeleddynamicresponsetotheatmosphere(dynamicatmosphericcorrection,gtxx/geophys_corr/dac),
4. Selectonlyphotonheightsforwhichthesignalconfidence(gtxx/heights/signal_conf_ph)isgreaterthanorequalto1andindicatorforsaturationgtxx/heights/quality_phiszero.Thiswillselectphotonsinthelikely
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surface±15-mbufferwindow.Alsodeleteanyphotonheightsoutsidethebanddefinedbygeoidplusorminushalftheheightoftheoceandownlinkwindow(presentlygeoid±15-m,tobecomegeoid±20-m).(IstheheightofthedownlinkbandreportedinATL03perhapsatthegeosegrateandifso,whatisthevariablename?Theheightislistedatthemajorframerate.Butitwouldbehandierifitwasatthegeosegrate)
5. SubtracttheEGM2008geoidinthemean-tidesystemfromphotonheightstoreferenceallheightstothegeoid.NotethatasofRelease4,EGM2008providedbyATL03isinthetide-freesystem.However,ATL03alsoprovidestheconversionthatshouldbeaddedtothetide-freegeoidtoyieldEGM2008inthemean-tidesystem.
6. Establishaninitialcoarsehistogramarray,Hc,spanning±15mwithbinsizeB1(TBD,e.g.,0.01m)andadataarray,Acoarse,forupto10,000photonheightsandassociatedinformation(index,geolocation,time)plusnoisephotoncounts.Thiswillbepopulatedwithdataforthecoarseselectionofsignalphotons.
7. Aggregate photon heights over 14 geo-bins (400-pulses) and construct a temporary 14 geo-bin (400-pulse) height histogram spanning ± 15 m with bin size B1 (e.g., 0.01 m). This is used to estimate a running total number of signal and noise photons. We suggest a bin size of 0.01 for consistency with later steps in the processing. The 14 geo-bin (400-pulse) segment should be aligned so that the once per 14 geo-bin (400-pulse, approximately 280 m along track distance) cloudiness flag, layer_flag in ATL09, represents the aggregate effect of cloudiness and background radiation derived from solar zenith, asr_cloud_probability, cloud_flag_atm, and bsnow_con, during the 280-m segment.
8. Photonsinthe14geo-bin(400-pulse)histogrambinswithgreaterthantheTh_Nc_ctimesthemediannumberofphotons,Nmedian,arecandidatesignalphotonsandphotonsinthebinswiththemediannumberofsamplesorlessareconsiderednoisephotons.WehavebeenusingTh_Nc_c=1inourtestswithearlyICESat-2data.Notethatwithfewsurfacereturnsperpulseandsignificantwaveheightsmuchlessthan30m,themajorityoftheofthebinswillcontainonlynoisephotonssothatthemedianwillequalthenumberofnoisephotonsperbin.InextremecaseswhereICESat-2encounterssignificantwaveheightsapproaching30m,wemayhavetoadoptaslightlydifferentthresholdsuchasthecountvalueequaltoorexceeding20%ofthebinpopulation.
9. Addthesignalandnoisephotonsfromthis14geo-bin(400-pulse)segmenttotherunningtotalofcandidatesignalphotonsandnoisephotons,andaddallthephotonstothecoarsehistogram,Hc.
10. Ifthetotalnumberofcandidatesignalphotonsisgreaterthanorequaltoaminimumnumber,Th_Ps(e.g.,8,000),ofphotonsorSegmax(e.g.,25)14geo-bin(400-pulse)segmentshavebeencollectedthisdefinesanoceansegment,andwegoontoFineSelection(Sec4.2.1.3)withthepopulatedcoarsehistogram,Hc,±15mwithbinsizeB1(TBD,e.g.,0.01m),andthedataarray,Acoarse.IfthetotalnumberofsignalphotonsislessthanTh_Ps,repeatsteps4.2.1.2-2to4.2.1.2-5aboveandintakeanother14geo-bin(400-pulse)segmentuntiltheoceansegmentreachesalengthof
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7kmmaximum.Ifthe7-kmlengthisreachedandtherearelessthanathresholdnumberofphotons(e.g.photon_min=4000),ignorethatoceansegmentandrepeat4.2.1.2-2to4.2.1.2-6.Forphotoncountsabovethisinthe7-kmsegment,proceedto4.2.1.3andfinehistogrambasedsignalfinding.
4.2.1.3 Fine Histogram Selection
1. Consideringthecoarsehistogramarray,Hc,performaNbmv=20-cm/B1bin(e.g.,21binsover20-cm)movingbinarithmeticaverageincrementedby1bin.PadtheendsofthesmoothedhistogramwithNbmv/2bins(equalthesmoothedfirstandlastvalues)tomatchthelengthoftheoriginalhistogram.
2. The ASAS 5.1 finds the bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the Jlow bin position to the Jhigh bin, where the smoothed histogram bin photon count is greater than the median of the smoothed histogram photon count. In practice with preliminary ATLAS data the median has usually been zero. Essentially, we have moved out from the middle of the histogram until the histogram levels fall to zero.
As of 6/4/2019, the developmental code finds preliminary bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the preliminary Jlow bin position to the preliminary Jhigh bin, where the coarse bin photon count is greater than the coarse histogram median. Then the average counts in the bins above the preliminary Jhigh are calculated and set equal to tailnoisehigh and the average counts in the bins below the preliminary Jlow are calculated and set equal to tailnoiselow. Then we find the bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the final Jlow bin position to the final Jhigh bin, where the final Jlow bin position is that below which the smoothed histogram bin photon count is less than Th_Nc_f times tailnoiselow, and the final Jhigh bin position is that above which
Figure12.Illustrativecoarse,smoothed,andfinehistogramsfromsteps1and2of4.2.1.3asappliedto10minutesofApril2012MABELdataoverChesapeakeBay.Thecorrespondingfigurefordetrendeddataintheseconditeration(steps4-6)aresimilar.
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the smoothed histogram bin photon count is less than Th_Nc_f times tailnoisehigh. This procedure is better able to differentially remove subsurface noise returns and above surface returns. Th_Nc_f is TBD, but we have used Th_Nc_f = 1.5 in practice with Release 001 ATLAS data and this has resulted in greatly improved rejection of subsurface returns.
Asof11/5/2019forRelease4,wemodifiedthesurfacefindingprocedureasdescribedinsection5.3.2.Insteadofbasingsurfacefindingonthedistributionofphotonheights,itisnowbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesthatotherwisefallinsidethehistogramoftruesurfaceheights.Thisfurtherreducesanyerrorduetosubsurfacereturns,obviatingtheimmediateneedforasubsurfacereturncorrection.
3. ConsiderthetimeseriesofalltheheightsstoredinAcoarsewithheightsanomaliesthatliebetweenanomaliescorrespondingtheJlowandtheJhighbinpositionsinthedistributionofheightanomalies.Thisarrayofheightsandtimesrepresentsthefirstiterationonsurfaceheightstatistics,butthehighermomentsareelevatedbyvariationsofthemeanwithtime(ordistance).Therefore,calculatetheleastsquareslinearfittotheseversustime(ordistance).Storetheconstantandfirstordercoefficients,P0andP1,ofthelinearfitusedfordetrendingasoutputvariablesforthesegmentandsubtractthemfromthedataofAcoarsetoyieldarrayAfineofdetrendedphotonheightsandassociatedgeolocationdata.
4. FromtheheightsofAfine,createthedetrendedcoarsehistogram,Hd,±15mwithbinsizeB1(TBD,e.g.,0.01m).Thenrepeatsteps1and2substitutingHdforHcincludingcreatingthe11-pointmovingaverageofhigh-confidencephotonstheheightanomaliesaboutthismovingaverage,andthehistogramofheightanomalies.
5. Consideringthedetrendedcoarsehistogramarray,Hd,performaNbmv=20-cm/B1bin(e.g.,21bins)movingbinarithmeticaverageincrementedby1bin.PadtheendsofthesmoothedhistogramwithNbmv/2bins(equalthesmoothedfirstandlastvalues)tomatchthelengthoftheoriginalhistogram.
6. TheASAS5.1finds the bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the Jlow bin position to the Jhigh bin, where the smoothed histogram bin photon count is greater than the median of the smoothed histogram photon count. In practice with preliminary ATLAS data the median has usually been zero. Essentially, we have moved out from the middle of the histogram until the histogram levels fall to zero.
As of 6/4/2019, the developmental code finds preliminary the bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the preliminary Jlow bin position to the preliminary Jhigh bin, where the coarse bin photon count is greater than the coarse histogram median. Then the average counts in the bins above the preliminary Jhigh are calculated and set equal to tailnoisehigh and the average counts in the bins below the preliminary Jlow are calculated and set equal to tailnoiselow. Then we find the bin limits of the fine histogram as all the
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coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the final Jlow bin position to the final Jhigh bin, where the final Jlow bin position is that below which the smoothed histogram bin photon count is less than Th_Nc_f times tailnoiselow, and the final Jhigh bin position is that above which the smoothed histogram bin photon count is less than Th_Nc_f times tailnoisehigh. This procedure is better able to differentially remove subsurface noise returns and above surface returns. We have used Th_Nc_f = 1.5 in practice with Release 001 ATLAS data and this has resulted in greatly improved rejection of subsurface returns.
Asof11/5/2019forRelease4,asforthefirstpas,forethesecondpasswemodifiedthesurfacefindingprocedureasdescribedinsection5.3.2.Insteadofbasingsurfacefindingonthedistributionofphotonheights,itisnowbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesthatotherwisefallinsidethehistogramoftruesurfaceheights.Thisfurtherreducesanyerrorduetosubsurfacereturns,obviatingtheimmediateneedforasubsurfacereturncorrection.Notethatalthoughsurfacefindingisbasedontheheightanomalies,thefinalproductofthesestepsisthehistogramoftheactualheightsoftheselectedsurfacephotons,HR.
7. TheresultingHRisthefinehistogramofreceivedsurfaceheightstobeusedin
furtheranalysis.ThetimeseriesofalltheheightsstoredinAfinethatliebetweenheightscorrespondingtheseconditerationJlowandtheJhighbinpositionswillbesavedforwaveandseastatebiasanalysis(Section4.2.2).
4.2.2 A Priori Sea State Bias Estimation and Significant Wave Height
Asasidecalculation,wewillusethespatialrecordofsignalphotonheightsderivedthroughtheprocessdescribedinSections4.2.1.1-4.2.1.3asanapproximationtothetruesurfaceheight.Thisheightrecordhasbeendetrendedduringsurfacefindingandthemeanofthisfit(linearfitcoefficientsP0andP1in4.2.1.3)overthepositionsofthesurfacedetectedpointsmustbeaddedbacktothesurfaceheightstoincludeallSSBintherecord.Withthisheightrecord,weresolvethestructureofsurfacewavesversusalongtrackdistanceandthevariationinphotonreturnratetoestimateseastatebias.TheinherentlyhighspatialresolutionofICESat-2willallowustomeasurethevariationinsurfaceheightoverthelargeenergy-containingsurfacewavesthatleadtoseastatebias.Wewillalsoknowthephotonreturnratealongthewaves.Overtheoceansegments(equivalentto8,000candidatesignalphotonsor7kmwhicheverisless),wewilldeterminesurfaceheightvariationandphotonreturnratein10-malong-trackbins.Tocalculatethephotonreturnratepermeteritwillbenecessarytosortthephotonheightrecordinascendingorderofalong-trackdistance.Fromtheheightandphotonraterecordswewillcomputethecovarianceofsurfaceheightandphotonreturnrate.Combinedwiththeaveragephotonreturnrate,thecovarianceofheightandreturnrateistheseastatebiasaccordingtoequation(11).Wecanapproximatethesignificantwaveheight(SWH)asfourtimesthe
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standarddeviationofthebin-averagedheights.
4.3 ATL12: Correction and Interpretation of the Surface Height Distribution
Therearetwomainissuesinderivingsurfacestatisticsfromtheapparentheightsofsurface-reflectedphotonsidentifiedinthesurfacedetectionphaseandmanifestinthefinehistogram,HR,ofSection4.2.Thesearecorrectingtheheightsfortheinstrumentimpulseresponseandderivinghigherordermomentsofthecorrectedsurfaceheightdistribution.
4.3.1 Separating the Uncertainty due to Instrument impulse response
Thereceivedheightsofsurfacereflectedphotonsarethesumofatrueheightofthesurfaceplusanuncertaintyduetotheinstrumentimpulseresponse.Thelatteristhetotalofallinstrumentuncertainty,butitisdominatedbyuncertaintyinthetimetheindividualphotonsaretransmitted.TheATLASuncertaintyinheightduetoinstrumentimpulseresponsewillhaveastandarddeviationof15-22cm.Becausetheinstrumentimpulseresponseuncertaintyandthetrueheightvariationareadditive,thereceivedsurfacephotondistribution,HR,isequaltotheconvolutionoftheinstrumentimpulseresponseheighterrordistribution,HT,andthetrueheightdistribution,HY: HR=HT*HY (12)
where*denotesconvolution.Consequently,ifwewanttoknowthestatisticsofthesurfacefromICESat-2returns,wemustdeconvolveHTfromHRtogetHY.IfweknowthatHYisaGaussiandistribution,weonlyneedconcernourselvescomputingthemeanandstandarddeviationofthedistribution.However,weexpectreceivedoceansurfaceheightstodepartfromaGaussiandistributionbecauseofacombinationtheshapeofsurfacewavesandthenon-uniformsamplingduewave-inducedvariationsofspecularreflection.Inadditiontothemeanandstandarddeviation,thethirdandsecondmoments(skewness,andkurtosis)arealsoimportant.Ifweknewthetruesampleheights,wecouldcalculatethesemomentsassamplemoments,e.g.,thenthsamplemomentforYifromthetimeseriesofheights:
(13)
orasthediscreteformoftheintegralovertheprobabilitydensityfunction,
,ifweknewH(Y).
However,theinstrumentimpulseresponsedelay,T,andthereceivedheight,R,areindependentandadditive(R=T+Y),soinprincipleitispossibletocalculatesamplemomentsforYfromthesamplemomentsofTandR:
nM (Y ) = 1m Yi −Y( )ni=1
m∑
n
Y−M 1( ) H Y( )−∞
∞
∫ dY
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(14)
(15)
(16)
(17)ThesesamplemomentswillbecalculatedfortheICESat-2heightsasacheckonmoreexactmethods.However,pendingfurtherstudy,itisTBDwhethertheywillbeincludedintheATL12dataoutput.Theuncertaintyinsamplemomentsissignificant[Percival,personalcommunication]andthisuncertaintyiscompoundedwithcombinationssuchas(14)-(16).Theseproblemsareliabletobeparticularlyimportantwhenlookingforsmallvariationinskewness, ,andexcesskurtosis, .Optimizedapproachesfordeterminingthemomentsoftheheightdistributionrequirethatweseparatetheeffectofinstrumentimpulseresponsefromthereceivedheightstogiveusthedistributionofsurfaceheight,HY.TodothiswemustdeconvolveHTfromHR.
Deconvolutioncanbedoneinseveralways.TheInlandWaterATBD(ATL-13)expressestheconvolutionof(12)forthereceivedhistogramasthemultiplicationofamatrixrepresentingtheinstrumentimpulseresponsehistogramtimesavectorrepresentingthesurfaceheighthistogram.Thismatrixisinvertedandmultipliedtimesthereceivedhistogramtoyieldthesurfacehistogram.Thetechniqueisreportedlysensitivetoproperbinningtoproduceamatrixthatcanbeinverted.DeconvolutioncanalsobedonebytakingtheFouriertransformof(12)andnotingthattheFouriertransform,F(),oftheconvolutionoftwovariables,HR(k)=F(HR)= F(HT*HY)isequaltotheproductoftheFouriertransformofthetwovariables,
HR(k)=F(HR)= F(HT*HY)= F(HT) F(HY) (18)
wherekisthewavenumberinunitsofm-1,theFourierspacecounterparttoheightinmeters,thehistogramindependentvariable.From(18),wecancomputeHYasaninverseFouriertransform,F-1()oftheratioofF(HR)andF(HT),
HY= F-1(F(HR)/F(HT)) (19)
1M (Y ) = 1M (R)− 1M (T ) ≈ 1m Rii=1
m∑ −
1m Tii=1
m∑
2M (Y ) = 2M (R)− 2M (T ) ≈ 1m
2Ri−R( )
i=1
m∑ −
1m
2Ti−T( )
i=1
m∑
3M (Y ) = 3M (R)− 3M (T ) ≈ 1m
3Ri−R( )
i=1
m∑ −
1m
3Ti−T( )
i=1
m∑
4M (Y ) = 4M (R)− 4M (T )− 2M (Y ) 2M (T )
≈ 1m
4Ri−R( )
i=1
m∑ −
1m
4Ti−T( )
i=1
m∑ − 2M (Y ) 2M (T )
M 3 M 23/2 M 4 M 2
2 − 3
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Theproblemwiththisapproachisthatthereisinvariablynoiseinthereceivedsignalhistogram,HR,thatproducessignificantvaluesinF(HR)atwavenumberswhereF(HT)hassmallvaluesorzeros.Thismayhavesomethingincommonwithproblemofconditioningofthematrixinthematrixinversiontechnique.Inanyevent,theseunrealisticcombinationsmaketheinverseFouriertransformunstable.ToaccountforthenoiseinHR,weconsiderWienerdeconvolution,inwhichitisassumedthereceivedhistogramiscontaminatedwithnoise,N,and(12)and(18)become
HR=HT*HY+N (20)
andwhereHT(k)=F(HT),HY(k)=F(HY),HR(k)=F(HR),andN(k)= F(N), HR(k)= HT(k)HY(k)+N(k) (21)
AnestimateHY(k)forHYest(k)oftheform
HYest(k)=W(k)HR(k)/HT(k) (22)
issoughtsuchtheexpectedvalueof(HY(k)-HYest(k))2isaminimum.Thisisfoundfor
W(k)= F(HT)2/[F(HT)2+ F(Noise)2/F(HY)2]~F(HT)2/[F(HT)2+ F(Noise)2/F(HR)2](23)and
HYest= F-1(HYest(k))=F-1(W(k)F(HR)/F(HT)) (24)
Forwavenumberswheretheinversesignaltonoiseratio,F(Noise)2/F(HR)2,islowrelativetoHT(k),W(k)goestooneandequation(24)revertsto(19).Wheretheinversesignaltonoiseratioishigh(highnoise),W(k)goestozeroandthenoiseisfilteredoutoftheresultingHYest(k).
4.3.2 Characterizing the Random Sea Surface
Thebestmethodfordeterminingsamplemomentsisthemethodofexpectationmaximization(ML).Foradistributionofknowntype ,theoptimumchoiceofparameters, ,fordatavalues ,arethosewhichmaximizethelikelihood,L,
(25)
ForICESat-2seasurfaceheight,wearenotsureoftheformoftheprobabilitydensityfunctionbutcommonlytheoceansurfacehasbeencharacterizedbytheGram-Charlierdistributionforwhichthefirst4momentsareimportant.SuchadistributioncanberepresenteditbymixtureoftwoGaussiandistributions,NaandNb
f (x \ µ1,2,3,4 )µ1,2,3,4 x1,x2,x3......xm
L µ1,2,3,4 \x1, x2, x3......xm( ) =i=1
m
∏ f (xi \ µ1,2,3,4 )
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(26)
andthetwomomentsofeachdistributionplusthemixingratio,ma/mb.Giventhesurfaceheighthistogram,theparametersofthetwonormaldistributionsandthemixingratiocanbedeterminedbyexpectationmaximization.TheaggregatemomentsoftheGaussianmixtureXwithncomponentGaussiandistributionsarefunctionsofthecomponentmeans,µ�,andvariances,si2,andthemixingratio,mi,
(27)
Forexample,theaggregatemixturemeanis: (28)
andtheaggregatevarianceis: (29)
4.3.3 Expectation Maximization
IntheMatlabcodeusedfordevelopmentoftheATL12software,anexpectationmaximizationcode(gmdistribution.fittermedmaximumlikelihoodbyMatlab))wasusedtoderivea2-componentGaussianMixturefittothederivedsurfacedistribution.ThemeanandvarianceofthetwoGaussiandistributionsalongwiththemixingratioofthetwoGaussiansallowstheaccuratedeterminationofthemostlikelyfirstfourmomentsofthedistributionofsurfaceheight.Thekernelofgmdistribution.fitisessentiallythesameasusedinderivationofATL007(ATL007,AppendixEandAppendixCinthisATBD).TheASASFortrancodedevelopedforATL12usesthesameasthatforATL07AppendixEtodo
f (xi \ µ1,2,3,4 ) = maNa (xi \ µa1,2 )+mbNb (xi \ µb1,2 )ma +mb = 1
�
E X − µmix( ) j[ ] =j!
k! j − k( )!k=0
j
∑i=1
2
∑ µi − µmix( ) j−kmiE Xi − µi( )k[ ]
µmix = m1µ1 +m2µ2
σ mix2 = m1 µ1 − µmix( )2 +σ 1
2( ) +m2 µ2 − µmix( )2 +σ 22( )
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the2-GaussianMixturefit.ForsynthetictestdatatheFortranexpectationmaximizationproducedequivalentresulttotheMatlabgmdistribution.fitroutineapproach.
4.4 Calculation of Uncertainty
Wehavefoundthewaveenvironment,ratherthaninstrumentalfactors,isprobablythebiggestcontributortouncertaintyinestimatesofthemeanSSH.ThisisillustratedbyouranalysisofmultipleoceansegmentsinthesouthernpartofthecentralNorthPacific.Thesesegmentsshowvariabilityintheirmeanseasurfaceheightsofapproximately±0.12,muchgreaterthanwe’dexpectfrominstrumentalfactors.Thestandarderror,sL,intheestimatesofthemeanaregivenby
where sh is the standard deviation of the population and Ndf is the number of degrees of freedom. If every one of the ~ 8,000 photon heights were independent, even with a SWH = 2.5 m (sh = .625 m), the uncertainty in estimates of the mean would be less than 1 cm. The problem is that owing to the presence of long large waves, successive photon heights are far from independent. The fact that the underlying wave signal is periodic makes the estimate of the effective degrees of freedom in terms of correlation length scale more problematic. As a worst case if we set Ndf
σ L =σ h Ndf( )−1/2
Figure13.BlockdiagramfortheATL19griddingproceduretakingATL12oceanproductsasinput.µ,s,S,andKdenotemean,standarddeviation,skewness,andkurtosisrespectively.
DOTTrends
Over L1,3,5Beams
SSB,SWH, SWEBeams
DOTHistograms
Beams
GeoidAvg
Beams
1 3 5
Mixturem1, m2,µ1,µ2
σ1,σ2
Beams µL1,3,5σL1,3,5ΣL1,3,5κL1,3, 5
SSH Moments for Beams 1, 3, 5
1 3 51 3 5 1 3 51 3 5 1 3 51 3 5 1 3 51 3 5
ATL12 Outputs for Input to ATL19
Combinem=m1+m3+m5 DOT histograms& compute combined moments
Choose m1, m3, m5 length segments in a grid cell in one month
Computeaverage of m geoid heights = Gmadd to mean DOT for mean SSHm
µm=DOTm
σm Σm κm SSHmGm
ComputeN-weighted averages anddegree of freedom (NP_effect)- weighted averages,
µa,σa,Σa,κa,ofm=m1+m3+m5 moments:
µL1,3,5, σL1,3,5, ΣL1,3,5, andκ L1,3, 5,
µa=DOTa
σa Σa κa µa=DOTa
σa Σa κa
N-weighted NP_effect-weighted
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equal to the number of wave periods in the ocean segment, 15 in the case of long waves crossed at an acute angle, the uncertainty,sL,intheestimateofthemeanequals0.16m,closetothemeanSSHweseeovermanysimilaroceansegments.Thisproblemisnotinstrumental;itisjustmadeapparentbytheabilityofICESat-2toresolvewaves.Tolowerthisuncertainty,weareexploringusingharmonicanalysisofsurfaceheightovereachoceansegmentanduseofthezerowave-numberamplitudetorepresentSSH(D.Percival,personalcommunication,2019)asawayofremovinglargeperiodicvariationsinheightasacauseofuncertainty.AfurtheradvantageofthisapproachwillbethatitwilladdameasureofwavespectralpropertiestoATL12.Inanyevent,wewillhavetoderivemeasuresoftheeffectivenumberofdegreesoffreedomasanATL12outputforeachoceansegmentinordertoproperlyaccountforuncertaintiesinthegriddedSSHproduct.
4.5 ATL19: Gridding the DOT and SSH
Thisproduct,basedonProductATL12/L3A,containsgriddedmonthlyestimatesofseasurfaceheightfromallIS-2tracksfromthebeginningtotheendofeachmonth.Below60°Nandabove60°S,thedataaremappedonthecurvilineargridoftheTOPEX/Poseidonwithspacingequivalentto0.25°oflongitude.Above60°Nandbelow60°Sthegriddata are mapped onto a planimetric grid using the SSM/I Polar Stereographic Projection equations at a grid spacing of about 24 km.. ATL12 provides the histograms and first four moments of dynamic ocean topography over ocean segments ~ 3-7-km long (means of DOT are converted to mean sea surface heights by adding ocean segment mean geoid height, which is also output by ATL12). It also provides the number of photon heights, n_photons, going into the moments and an effective degrees-of-freedom, NP_effect, based on the correlation length scale of surface heights. Using these, ATL19 will produce monthly aggregate histograms of surface heights and averages of the ocean segment moments weighted by both n_photons and NP_effect (Fig. 13). See the ATL19 ATBD for a complete description.
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5.0 ALGORITHM IMPLEMENTATION
Thissectiondescribesthespecificimplementationofthealgorithmforprogramdevelopment.
5.1 Outline of Procedure
1) Findingthesurface-reflectedphotonheightsinthephotonclouda) CoarseSelectionandSettingofSegmentLength:
Useamedian-basedselectioncriteriontoaccumulatealargenumberofsurface-reflectedphotons.
b) FineHistogramSelection:Useamedian-basedselectioncriteriontoderiveapreliminarysurfaceheighthistogram,computethemeanandtrendinsurfaceheight,removethistrendfromallheights,andrepeatthemedian-basedselectioncriteriontoderiveafinalsurfaceheighthistogram
2) Usingthealongtracksignalphotonheightsfromthefinehistogramselectionstepandsignalphotonreturnratetoestimateseastatebiasandsurfacewaveproperties.
3) CorrectionandInterpretationoftheSurfaceHeightDistributiona) SeparatingtheUncertaintyduetoInstrumentimpulseresponse
UseWienerdeconvolutiontoderivethesurfaceheighthistogramwiththeuncertaintyduetotheinstrumentimpulseresponseremoved.
b) CharacterizingtheRandomSeaSurfaceCharacterizethesurfaceheighthistogramasamixtureoftwonormaldistributionsandcalculateupto4thmomentofthemixture.
c) Computethemean,variance,skewness,andkurtosisfortheaggregateGaussianmixtureusingEquation17;thefinaldeterminationofSSHforATL12isthemeanoftheaggregatemixturedistributioncomputedusing(28)andtheSSHvarianceisgivenby(29).
5.2 Input Parameters
5.2.1 Parameters Required from ICESat-2 Data
SeeTables2and3.ForATL12,theprimaryinputswillbephotonheightsinthedownlinkbandandassociatedtime,geolocation,thegeoid(geoid),oceantides(tide_ocean,tide_equilibrium),anddynamicatmosphericcorrection(dac)informationfromATL03,plusthecloudflag(layer_flaginATL09)every400laserpulsesfromATL09andoceandepth,depth_ocn,frominterpolatedfromGEBCO-2014griddedoceanbathymetry.Normallyovertheopenocean(orseaice)regionsnotoverlappingwithterrestrialoricesheetregions,thedownlinkbandwillbe±15mcenteredaboutthesignalareaidentifiedbytheATLASon-boardflightsoftware.Intheoceanregionsoverlappingwithlandandicesheetregions,thedownlinkbandwillexpandtomatchthebandforthose
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regions.AlsoforATLASspecialoperationssuchasoceanscansandTransmitterEchoPulsedatacollectionthetelemetrybandisexpanded.ForthepurposesofATL12,theEGM2008geoidsuppliedwithATL03(gtx/geophys_corr/geoidpostedatthe20malong-trackrateforeachbeaminATL_03)willbeusedtoestablishthenarrower±15-mbandwithintheexpandeddownlinkbandoftheoverlapandspecialoperationsregions.Inadditiontothe±15-mbandaboutthegeoid,weuseseveralclassificationsandflagsfromATL03toeditphotonheights.
Thephotonqualityflag,quality_ph,detectsvariousconditionssuchassaturatedlaserpulsesforwhichwewanttodiscardphotonheight,soweonlyprocessheightswiththisflagequalto0.
Wehavealsofoundunrealistic values for ocean segments when ATL03 gtx/geolocation/podppd_flag is non-zero indicating degraded orbit and/or pointing data. Consequently ATL12 processing should only be done with data for which gtx/geolocation/podppd_flag=0.
Perhapsmostimportasntly,weusetheATL03confidencethatthephotonissurfacereflecrted,signal_conf_phathighlevels(3or4)todetermineifwehaveaviablesurfaceinadownlinkband,andweuseonlyphotonswithsignal_conf_phgreaterthanorequalto1inoursurfacefindingprocedure
Also,thesurfacefidingnow(asofRelease4)usesthedistributionoftheanomalyofphotonheightsrelativetoa11-pointmovingaverageofphotonswithsignal_conf_phreaterthanorequalto2tosegregatesurfacephotonsfromnoisephotons.Thismethodismoresensitivethaneditingtheheighthistogrsmandshowsbetterrejectionofsubsurfacereturnsunderoceanwaves.
The14geo-bin(400-pulse)oceansegmentarealignedsothattheonceper14geo-bin(400-pulse)cloudflag(layer_flaginATL09)representsthecloudconditionsduringtheoceansegment.Wehavefoundthatwecangetgoodsurfaceheightsevenwhenthelayer_flagindicatesthepresenceofwaves,butwedocomputetheaverageoceansegmentcloudcoverasthefractionoflayer_flag=1overtheoceansegment.
Table 2 Input parameters (Source: ATL03)
Label Description Symboldelta_time Elapsedsecondssincefirstdatapointingranule time_initiallat_ph latitudeofeachreceivedphoton lat_initiallon_ph longitudeofeachreceivedphoton lon_initialh_ph heightofeachreceivedphoton ht_initialdist_ph_along Alongtrackdistance sigma_along Uncertaintyinalong-tackdistance dist_ph_across Across-trackdistance sigma_across Uncertaintyinacross-trackdistance
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segment_id Along-tracksegmentIDnumber segment_id
signal_conf_ph
Confidencelevelassociatedwitheachphotoneventselectedassignal.Zeroequalsnoise.Onemeansaddedtoallowfor±15-mbufferaroundhighconfidence,butalgorithmclassifiesasbackground.Twoequalslowconfidence.Threeequalsmedium.Fourequalshigh.Withrelease4signal_conf_phequal=-2indicatespossibleTEPphoton
signal_conf_ph
quality_ph
Photonqualityastosaturationandafter-pulse• 0=nominal,nosaturation• 1=possibleafter-pulse• 2=possiblelong-tailimpulseresponse• 3=possibleTEP(alsoindicatedbysignal_conf_ph=-2
quality_ph
ref_azimuth
Thedirection,eastwardsfromnorth,ofthelaserbeamvectorasseenbyanobserveratthelasergroundspotviewingtowardthespacecraft(i.e.,thevectorfromthegroundtothespacecraft).Whenthespacecraftispreciselyatthegeodeticzenith,thevaluewillbe99999degrees.40Hz.
ref_elev Co-elevation(CE)isdirectionfromverticalofthelaserbeamasseenbyanobserverlocatedatthelasergroundspot.
solar_azimuth Thedirection,eastwardsfromnorth,ofthesunvectorasseenbyanobserveratthelasergroundspot.
solar_elevation
SolarAngleaboveorbelowtheplanetangenttotheellipsoidsurfaceatthelaserspot.Positivevaluesmeanthesunisabovethehorizon,whilenegativevaluesmeanitisbelowthehorizon.Theeffectofatmosphericrefractionisnotincluded.Thisisalowprecisionvalue.
bckgrd_rate Backgroundcountrate(backgrd_atlas/bckgrd_rate),simplyaveragedoverthesegment
fpb_parm(n) Parameterneededtocomputefirst-photonbiascorrectiontoameanheight.ThisisnotusedasofRelease4.SeeSection3.1.1.4. TBD
dac Dynamicatmosphericcorrection(DAC)includesinvertedbarometer(IB)effect(±20cm)
tide_earth SolidEarthTidesinthetide-freesystem(±30cm,max) tide_equilibrium Equilibriumlong-periodtide
tide_load LoadTide-LocaldisplacementduetoOceanLoading(-6to0cm)
tide_oc_pole OceanPoleTide-Loadingduetocentrifugaleffectofpolarmotionontheoceans(2mm,max)
tide_oceanOceanTidesincludingdiurnalandsemi-diurnal(harmonicanalysis),andlongerperiodtides(dynamicandself-consistentequilibrium)(±4m)
tide_pole PoleTide-Rotationaldeformationduetopolarmotion(-1.5to1.5cm) neutat_delay_total Totalneutralatmosphericdelaycorrection neutat_del
ay_total
surf_type Five-elementarrayinthegeolocationgroupindicatingthesurfacemasksthatapplytothatsegment
geoid Geoidheight,EGM2008inthetide-freesystem(mean-tideinRel.3andearlier)
5.2.1.1.1.1.1.1 geo
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id
geoid_free2mean Conversionfactortobeaddedtogeoidtoconverttoamean-tidesystem
podppd_flag
Indicationofdegradedpointingororbitdetermination.Possiblevaluesare:0=NOMINAL;1=POD_DEGRADE;2=PPD_DEGRADE;3=PODPPD_DEGRADE.
5.2.1.1.1.1.1.2 podppd_flag
Table 3 Input parameters (Source: ATL09)
Label Description Symbol
delta_timeElapsedGPSsecondssincestartofthegranule.Usethemetadataattributegranule_start_secondstocomputefullgpstime.
latitude Latitude,WGS84,North=+,Latofsegmentcenter longitude Longitude,WGS84,East=+,Lonofsegmentcenter asr_cloud_probability CloudprobabilitybasedonApparentSurfaceReflectivityp=(1-asr/t)*100
apparent_surf_reflec ApparentSurfaceReflectivity(25Hz)
Layer_flag Representspresence(1)orabsence(0)ofsignificantcloudlayers.
5.2.2 Parameters Required from Other Sources
Bathymetry–Becausetheoceanmaskextendsintonon-oceanareasweneedawaytodeterminewhichpartsoftheoceandataareoverwaterforwhichouranalysisprocedureisappropriate.Forthiswewilltaketheoceandepth,depth_ocn,fromtheGeneralBathymetricChartoftheOceans(GEBCO)2014griddeddataset(https://www.gebco.net/data_and_products/gridded_bathymetry_data/)evaluatedatthegeo-segmentlatitudeandlongitude.
5.2.3 Control Parameters for ATL12 Processing
Bathymetry–Thecoursesurfacefindingdescribedin5.3.1willignoredatacollectedinoceanwaterswithadepth,depth_ocn,lessthanadepth,depth_shore,definingthe
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effectiveshorelineofwaterforwhichouranalysisprocedureisappropriate.Thedefaultvaluefordepth_shore,willbe10m.
Cloudcover–Inthecoursesurfacefindingdescribedin5.3.1wemayneedtoignoredatacollectedinoceanwaterswhereandwhenthecloudcoverisheavyasindicatedbylayer_flag,equalto1.Thecontrolparameterlayer_swtchwillspecifyifthecloudinesstestistobeapplied.Ifcontrolparameterlayer_swtchisequalto1andlayer_flagisequalto1,photonheightswillnotbeincludedintheATL12processing.Whenlayer_swtchequalsthedefaultvalue,0,thesoftwarewillignorethecloudcoverinacceptingdataduringcoarsesurfacefinding.Thesecontrolparametersareoutputinancillary_data/oceanforeachgranule/fileATL12.
5.3 Processing Procedure
5.3.1 Coarse Surface Finding and Setting of Segment Length
Theprocessisbasicallytofirstestablishavariablelength(e.g.,upto7km)oceansegmentwithenough(e.g.)photonheightsthatwillyieldahistogramwithreasonablemomentsupto4thorder.Westartwith14geo-bin(400-pulse)segmentsofATL03photonheights(see5.2.1)(A) Ocean Data Selection and Basic Geophysical Corrections
---------------------------------1) Steppingthrough14-geobinsegments,examineoceandepthandcloudparameters
againstcontrolparameterstotestforsuitabilityforoceanprocessing. If ocean depth, depth_ocn, is less than a depth, depth_shore, specifying the effective shoreline of water for which ocean processing is appropriate,thenproceedtonext14geo-bin(400-pulse)
Table 4 Control parameters – Coarse surface finding
Parameter Description ValueBc Binsizeofcoarsehistogram 5cm
Th_Nc_c Numberofphotonstimesmedianrequiredtoclassifyassurfacebin 1typical
Th_Ps Minimumnumberofcandidatesurfacephotonspersegment 8,000
SegmaxMaxnumberof400-pulsesegmentsinasurfacefindingsegment
25
layer_swtch
Flagdeterminingiflayer_flagwill(layer_swtch=1)orwon’t(layer_swtch=0)beeditingphotonheights.
Default=0
depth_shorePrescribeddepthdefining(depth_ocn>depth_shore)boundaryofoceanprocessing
Default=10m
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segment. The default value for control parameter depth_shore will be 10 m.Similarly,ifcontrolparameterlayer_swtchisequalto1andlayer_flagis equal to
1,thenproceedtonext14geo-bin(400-pulse)segment. When layer_swtch equals the default value, 0, the software will ignore the cloud cover in accepting data during coarse surface finding.
2) Correct for photon heights for short period and long period ocean tides (gtxx/geophys_corr/tide_ocean and tide_equilibrium), the inverse barometer effect and the modeled dynamic response to the atmosphere (dynamic atmospheric correction, gtxx/geophys_corr/dac),
3) Select only photon heights for which the signal confidence (gtx/heights/signal_conf_ph) is greater than or equal to 1 and photon quality (gxtx/heights/quality_ph) is equal to zero. This will select photons in the likely surface ± 15-m buffer window. Also delete any photon heights outside the band defined by geoid plus or minus half the height of the ocean downlink window (presently geoid ±15-m, to become geoid ± 20-m). (Is the height of the downlink band reported in ATL03 and if so, what is the variable name?)
4) To narrow the range of variability, subtract the EGM2008 geoid height (gtxx/geophys_corr/geoid), varying over meters to tens of meters, from the photon heights so that we are dealing dynamic ocean topography varying over centimeters to 10s of centimeters. from photon heights to reference all heights to the geoid. Note that as of Release 4, EGM2008 provided by ATL03 is in the tide-free system, but ATL03 also includes a correction mean-tide system, which should be added to the tide-free EGM2008 to obtain the mean-tide EGM2008.
(B) Establish a Working Histogram Array ---------------------------------------Establishaninitialcoarsehistogramarray,Hc,spanning±15mwithbinsizeB1equalto0.01mwithcenterbincenteredatzeroandbincentersatwholecentimeters.Alsoestablishadataarray,Acoarse,forupto10,000photonheightsandassociatedinformation(index,geolocation,time)plusnoisephotoncounts.Thiswillbepopulatedwithdataaswestepthrough14-geobinsegmentssearchingforanadequatenumberofsurfacephotoncandidates.
(C) Step Through 14-Geo-bin Segments --------------------------------------------------Aggregatephotonheightsover14geo-bins(400-pulses)andconstructatemporary14geo-bin(400-pulse)heighthistogramspanning±15mwithbinsizeB1(e.g.,0.01m).Thisisusedtoestimatearunningtotalnumberofsignalandnoisephotons.Wesuggestabinsizeof0.01forconsistencywithlaterstepsintheprocessing.The14geo-bin(400-pulse)segmentshouldbealignedsothattheonceper14geo-bin(400-pulse,approximately280malongtrackdistance)cloudinessflag,layer_flaginATL09,representstheaggregate
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effectofcloudinessandbackgroundradiationderivedfromsolarzenith,asr_cloud_probability,cloud_flag_atm,andbsnow_con,duringthe280-msegment.
(D) Test for Surface Photons
---------------------------------------Photonsinthe14geo-bin(400-pulse)histogrambinswithgreaterthantheTh_Nc_ctimesthemediannumberofphotons,Nmedian,arecandidatesignalphotonsandphotonsinthebinswiththemediannumberofsamplesorlessareconsiderednoisephotons.Th_Nc_cisTBD,butwehavebeenusingTh_Nc_c=1inourtestswithearlyICESat-2data.Notethatwithfewsurfacereturnsperpulseandsignificantwaveheightsmuchlessthan30m,themajorityoftheofthebinswillcontainonlynoisephotonssothatthemedianwillequalthenumberofnoisephotonsperbin.InextremecaseswhereICESat-2encounterssignificantwaveheightsapproaching30m,wemayhavetoadoptaslightlydifferentthresholdsuchasthecountvalueequaltoorexceeding20%ofthebinpopulation.
(E) Increase the Count of Surface Photons ---------------------------------------
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AddthenumberofsurfacereflectedphotonstoNgood,therunningtotalofcandidatesurfacephotonsandnoisephotons.
(F) Add All Photons to Data Array ---------------------------------------Addtheallthephotons,signalandnoise,andtheirassociatedgeolocation,time,andconfidencelevelinformationfromthis14geo-bin(400-pulse)segmenttothedataarrayHTin.
(G) Test Counter for Sufficient Surface Photons ---------------------------------------
Ifthetotalnumberofcandidatesignalphotonsisgreaterthanorequaltoaminimumnumber,Th_Ps(8,000),ofphotonsorifthenumber,Nseg400,of14geo-bin(400-pulse)segmentscollectedreachesSegmax(e.g.,25equivalentto~7km)),goonto5.3.2Processing Procedure for Classifying Ocean Surface Photons, Detrending, and Generating a Refined Histogram of Sea Surface Heightswiththedataarray,HTin.Thetotalnumberofpulsesforthesegment,n_pls_seg,equals400xNseg400.IfthetotalnumberofcandidatesurfacereflectedphotonsislessthanTh_Ps,intakeanother14geo-bin(400-pulse)segmentandrepeatsteps5.3.1A-Gabove.
5.3.2 Processing Procedure for Classifying Ocean Surface Photons, Detrending, and Generating a Refined Histogram of Sea Surface Heights
TheprocedurebasedondevelopmentofMatlabProgram:ATBDdraftMABEL_reader_PhotonClassifyX2Channel6B_noprint.m.
Table 5 Control parameters – Fine surface finding and analysis
Bf Binsizeoffinehistogram
5cm,MABEL1cm,ICESat-2
Th_Nc_f Surfacequalified=smoothhist>Th_Nc_f*tailnoise
1.5,ICESat-2
pts2bin Binsinboxcarsmoother
21,ICESat-2
conf_lim Minimumconfidenceleveltobeincludedinmovingaverage
Typically,3or4forICESat-2
nphoton Numberofphotonseithersideofcentralphotontoconsideraveraging
5foran11pointaverage
nharms Numberofharmonicstofittoselectedsurfacephotons
32default
gaplimit Gapbetweenphotonsabovewhichgapsarefilledwithwhitenoiseforharmonicfit
3.2mdefault
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WestartwiththearrayofphotonheightdatainHTinfrom5.2.4.Thesearecandidateheightsaccumulatedoverasegmentofsufficientlengthtoprovideanadequatenumberofsignalphotons.Inviewofthelowreflectanceoftheoceansurfacethismayrequireasmanyas10,000laserpulses.BreakHTinintoavectorofinitialphotonheights,ht_initial,avectorofinitialtimes,time_initial,andvectorsofinitiallocation,lat_initial,lon_initial,avectorofalongtrackdistance,trackdist_initial.,andthevectorofconfidencelevel,conf_initial,foreachphotonheight.(A)EstablishorSummontheBinEdgeVector---------------------------------------Setupforthesurfacefindinghistogram,N,byestablishingthevector,Edges,(withalengthonegreaterthanN)ofhistogrambinedgeswithbinsBfwidebetween-15mto+15minstepsof1cm.Alsoestablishthevector,Cntrpt,ofbincenterpointswithlengthequaltoN.InpracticeBfequals1cm,Cntrptvaluesarewholecentimeters,andEdgesareathalfcentimetervalues.Alsoestablishavectorarray,BinB,aslongasht_initialforbinassignments.(B)ComputeMovingAverageofHighConfidenceSurfacePhotonHeights---------------------------------------Herewewantamovingaverageofhighconfidencesurfaceheights,mvng_avg,withavaluecorrespondingtoeachphotonrawphotonheightinht_initial.Forthevalueofmvng_avg,computetheaverageofthephotonheightsinht_initialwithinnphotoneithersideofthecentralphoton,includingintheaverageonlyheightswithconf_initialgreaterthanorequaltoconf_lim.Forthefirstnphotonphotons,padmvng_avgwiththeaverageforthenphoton+1photoninthesequence.Forthelastnphotonphotons,padtherunningaveragewiththeaverageforthelastphotonminusnphoton.Forexample,intestingweusednphotons=5andconf_lim=3tomakean11-pointrunningaveragecenteredoneachphotonheight,andincludedineachaverage,onlyheightswithconfidencelevel3and4.C)ComputetheAnomalyinHeightAbouttheMovingAverage---------------------------------------Foreachheightinht_initial,computeanom_initialequaltoht_initialminusmvng_avg.(D)ComputeInitialHistogramandKeepTrackofBinAssignmentforEachPhotonHeight---------------------------------------PopulatethehistogramarrayNsuchthateachelementequalsthenumberofvaluesfromanom_initialthatareinthecorrespondingbinboundariesofEdges.Also,foreachvalueinanom_initial,populatethevectorBinBwiththebinnumbertowhichanom_initialisassignedfromonetothelength,LN,ofN(inMatlab,[N,BinB]=histcounts(anom_initial,Edges).
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(E)FindMedianofInitialHistogram---------------------------------------Findthemedian,medianN,ofN.(F)SmoothInitialHistogramwithBoxcarSmoother---------------------------------------- Createasmoothedversion,smoothN,ofthehistogram,N,withaboxcarsmootherincrementedinone-binstepsoverpts2binwherepts2binisacontrolparameterinTable5.Itshouldequalnbin+1wherenbinisanevennumber.Wehavechosennbins=20andpts2binsetat21basedonsuccesswiththedevelopmentalcode.InthisATBDwith1-cmbinsize,werefertothisnominallyasthe“20-cmsmoother”althoughitactuallysmoothsover21bins.(IntheASAS5.1codeusedforICESat-2Rel.001,nbinequals100correspondingtopts2bin=101anda100-cmsmoother).- Ifneeded,padtheendsofthesmoothedhistogramtomatchlengthofN.Forexample,asmootherthatstartsnbin/2+1pointsinfromthebeginningoftheoriginalarrayandstopsatanequalnumberofpointsbeforetheend,willbenbinpointsshorterthantheoriginalarray.Inthiscaseaddnbin/2pointsequaltothefirstsmoothedvaluetothebeginningofthearrayandaddnbin/2pointsequaltothelastsmoothedvaluetotheendofthearray.(G)FindLimitsofValidHistogram---------------------------------------TheASAS5.1findsthelimitsofthehistogramabove(jlow)andbelow(jhigh)forwhich,searchingfromtheindexofthemaximuminthe20-cmsmoothedhistogramsmoothN,thevalueinsmoothNisgreaterthanmedianofsmoothN.
- First find the index, imax, where smoothN(imax) equals the maximum in smoothN. - Increment the index, i, downward from imax until smoothN(i) is less than the
median of smoothN.. At this point jlow = i+1. - Increment the index, i, upnward from imax until smoothN(i) is greater than the
median of smoothN. At this point jhigh = i-1.
Asof6/4/2019,thedevelopmentalcodefindsthelimitsofthehistogramaboveapreliminaryjlowandbelowapreliminaryjhighforwhich,searchingfromtheindexofthemaximuminthe20-cmsmoothedhistogramsmoothN,thevalueinNisgreaterthanTh_Nc_f*medianofN.
- First find the index, imax, where smoothN(imax) equals the maximum in smoothN. - Increment the index, i, downward from imax until N(i) is less than the median of N..
At this point preliminary jlow = i+1. - Increment the index, i, upnward from imax until N(i) is greater than the median of N.
At this point preliminary jhigh = i-1.
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- Then the average counts in all the bins above the preliminary Jhigh are calculated and set equal to tailnoisehigh and
- The average counts in all the bins below the preliminary Jlow are calculated and set equal to tailnoiselow.
- Increment the index, i, downward from imax until smoothN(i) is less than the Th_Nc * tailnoiselow. At this point jlow = i+1.
- Increment the index, i, upnward from imax until smoothN(i) is greater than Th_Nc_f * tailnoisehigh. At this point jhigh = i-1.
(H)ChooseFirstCutSignalPhotons---------------------------------------Thefirst-cutsignalorsurfacephotonsarethoseinthebinsofhistogramNbetweenjlowandjhigh.Thenoisephotonsarethosefrombinsbelowthejlowbinandabovethejhighbin.ThevectorBinBliststhebinassignmentforeachphotonelevation.Therefore,theindicesiiforwhichBinB(ii)islessthanorequaltojhighandgreaterthanorequaltojlow,aretheindicesofthesurfacereflectedphotonheightsinht_initial.- Populatethevectorofsurfacephotonheights,ht_initial_surf,withtheheightsatindicesiiinht_initial(inMatlabthesesurfaceheightswouldbeht_initial_surf=ht_initial(ii);whereiiisthevectorofsurfacephotonindices).- Identifythecorrespondingtrackdistances,trackdist_initial_surf,intrackdist_initialattheindicesinii.
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(I)DoaLinearFitofHeightVersusTrackDistance---------------------------------------Weneedtodoalinearfittosurfaceheightsversustrackdistancebecausethetrendinsurfaceheightcontributessubstantialvariancetothehistogram,whichcancloudthesurfacefindingalgorithm.Therefore,performaleastsquareslinearfitoftheformP1xtrackdist_initial_surf+P0toht_initial_surf.(J)SubtractSurfaceTrend---------------------------------------Wemakeanewvectorarrayofdetrendedheights,ht_initial2,bysubtractingthelinearsurfacetrendderivedaboveatstep5.2.4(G)fromht_initial,i.e.,ht_initial2=ht_initial-P1xtrackdist_initial+P0.----------------------------------------------------------------------------------------REPEATSURFACEFINDINGAFTERREMOVINGTREND----------------------------------------------------------------------------------------Insteps(K)through(P)belowwerepeatSteps(B)Through(H)ActingontheDetrendedElevationsStartingWith:ComputeMovingAverageofHighConfidenceSurfacePhotonHeights(K)ComputeMovingAverageofHighConfidenceDetrendedSurfacePhotonHeights---------------------------------------Herewewantamovingaverageofhighconfidencesurfaceheights,mvng_avg2,withavaluecorrespondingtoeachphotonrawphotonheightinht_initial2.Forthevalueofmvng_avg2,computetheaverageofthephotonheightsinht_initial2withinnphotoneithersideofthecentralphoton,includingintheaverageonlyheightswithconf_initialgreaterthanorequaltoconf_lim.Forthefirstnphotonphotons,padmvng_avg2withtheaverageforthenphoton+1photoninthesequence.Forthelastnphotonphotons,padtherunningaveragewiththeaverageforthelastphotonminusnphoton.Forexample,intestingweusednphotons=5andconf_lim=4,tomakean11-pointrunningaverage
Figure14.Coarse(blue),smoothed(cyan),andfinalproduct(magenta)histogramsofdetrended,10minutesofApril2012MABELdataoverChesapeakeBay.Themagentahistogramisultimateoutputofthesurfacefindingphaseoftheoceananalysissteps3-6of4.2.1.3.ThecorrespondingfigurewithouttrendremovalisshowninFigure12.
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centeredoneachphotonheight,butincludedineachaverage,onlyheightswithconfidencelevel4.L)ComputetheAnomalyinDetrendedHeightAbouttheMovingAverageofDetrendedHeight---------------------------------------Foreachheightinht_initial2,computeanom_initial2equaltoht_initial2minusmvng_avg2.PopulatethehistogramarrayNsuchthateachelementequalsthenumberofvaluesfromanom_initial2thatareinthecorrespondingbinboundariesofEdges.Also,foreachvalueinanom_initial2,populatethevectorBinBwiththebinnumbertowhichanom_initial2isassignedfromonetothelength,LN,ofN[inMatlab,[N,BinB]=histcounts(anom_initial2,Edges)].(M)FindMedianofInitialHistogram---------------------------------------Findthemedian,medianN,ofN.(N)SmooththeInitialHistogramwithBoxcarSmoother---------------------------------------- Createasmoothedversion,smoothN,ofthehistogram,N,withaboxcarsmootherincrementedinone-binstepsoverpts2binwherepts2binisacontrolparameterinTable5.Itshouldequalnbin+1wherenbinisanevennumber.Wehavechosennbins=20andpts2binsetat21basedonsuccesswiththedevelopmentalcode.InthisATBDwith1-cmbinsize,werefertothisnominallyasthe“20-cmsmoother”althoughitactuallysmoothsover21bins.(IntheASAS5.1codeusedforICESat-2Rel.001,nbinequals100correspondingtopts2bin=101anda100-cmsmoother).- Ifneeded,padtheendsofthesmoothedhistogramtomatchlengthofN.Forexample,asmootherthatstartsnbin/2+1pointsinfromthebeginningoftheoriginalarrayandstopsatanequalnumberofpointsbeforetheend,willbenbinpointsshorterthantheoriginalarray.Inthiscaseaddnbin/2pointsequaltothefirstsmoothedvaluetothebeginningofthearrayandaddnbin/2pointsequaltothelastsmoothedvaluetotheendofthearray.(O)FindLimitsofValidHistogram---------------------------------------TheASAS5.1findsthelimitsofthehistogramabove(jlow)andbelow(jhigh)forwhich,searchingfromtheindexofthemaximuminthe20-cmsmoothedhistogramsmoothN,thevalueinsmoothNisgreaterthanmedianofsmoothN.
- First find the index, imax, where smoothN(imax) equals the maximum in smoothN.
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- Increment the index, i, downward from imax until smoothN(i) is less than the median of smoothN.. At this point jlow = i+1.
- Increment the index, i, upnward from imax until smoothN(i) is greater than the median of smoothN. At this point jhigh = i-1.
Asof6/4/2019,thedevelopmentalcodefindsthelimitsofthehistogramaboveapreliminaryjlowandbelowapreliminaryjhighforwhich,searchingfromtheindexofthemaximuminthe20-cmsmoothedhistogramsmoothN,thevalueinNisgreaterthanmedianofN.
- First find the index, imax, where smoothN(imax) equals the maximum in smoothN. - Increment the index, i, downward from imax until N(i) is less than the median of N..
At this point preliminary jlow = i+1. - Increment the index, i, upnward from imax until N(i) is greater than the median of
N. At this point preliminary jhigh = i-1. - Then the average counts in all the bins above the preliminary Jhigh are calculated and
set equal to tailnoisehigh and - The average counts in all the bins below the preliminary Jlow are calculated and set
equal to tailnoiselow. - Increment the index, i, downward from imax until smoothN(i) is less than the
Th_Nc_f * tailnoiselow. At this point jlow = i+1. - Increment the index, i, upward from imax until smoothN(i) is less than Th_Nc_f *
tailnoisehigh. At this point jhigh = i-1.
(P)ChooseSecond-CutSignalPhotons---------------------------------------Thesecond-cutsignalorsurfacephotonsarethoseinthebinsofhistogramNbetweenjlowandjhigh.Thenoisephotonsarethosefrombinsbelowthejlowbinandabovethejhighbin.ThevectorBinBliststhebinassignmentforeachphotonelevation.Therefore,theindicesiiforwhichBinB(ii)islessthanorequaltojhighandgreaterthanorequaltojlow,aretheindicesofthesurfacereflectedphotonheightsinht_initial2.- Populatethevectorofsurfacephotonheights,ht_initial2_surf,withtheheightsatindicesiiinht_initial2(inMatlabthesesurfaceheightswouldbeht_initial2_surf=ht_initial2(ii);whereiiisthevectorofsurfacephotonindices).- Identifythecorrespondingtrackdistances,trackdist_initial2_surf,intrackdist2_initialattheindicesinii,[i.e.,inMatlabcodetrackdist_initial2_surf,=trackdist2_initial(ii)].ht_initial2_surfisthemainproductofthesurfacefindingroutine.Inaddition,thesecond-passsurfaceindicesiniiaretheindicesforthetime,geolocation,trackdistanceandallotherdatacorrespondingtoht_initial2_surf.Thereceivedhistogram,Nrs,isthenthe
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histogramofht_initial2_surfonlyincludingfromthelowestheightwithcount>0tothehighestheightwithcount>0.AnexampleofthesecondpassandthehistogramofsurfaceheightsforMABELdataareshowninFigure14.(Q) CalculateNumberofPhotons,MeanTime,andGeolocationforSegment---------------------------------------Thenumberofsurfacereflectedphotonsintheoceansegment,n_photons,isthelengthoftheindexvectorii.Thetime,geolocation,andtrackdistanceofthesurfacereflectedphotonsareequaltothevaluesinthevectorstime_initial,lat_initial,lon_initial,andtrackdist_initialattheindexpositionsgivenbyii(e.g.,time_initial(ii)).Computethesegmenttime,t_seg,asthemeanoftime_initial(ii),andthedurationofthesegment,delt_seg,asthelastvalueoftime_initial-thefirstvalueoftime_initial.Computethesegmentlatitude,lat_seg,andsegmentlongitude,lon_seg,asthemeansoflat_initial(ii)andlon_initial(ii),andthesegmentlength,length_seg,asthelastvalueoftrackdist_initial2(ii)-thefirstvalueoftrackdist_initial2(ii).Thetotalnumberofphotons,n_ttl_photon,equalsthelengthoftime_initial,andasabove,thenumberofphotonsreflectedfromthesurfaceequalsn_photonsequaltothelengthofii.Thesurfacereflectedphotonratepermeter,photon_rate=rsurf,isequalton_photonsdividedbylength_seg,andthenoisephotonratepermeter,photon_noise_rate=rnoise,isequaltothedifferenceofn_ttl_photonminusn_photonsdividedbylength_seg.
5.3.3 Processing to Characterize Long Wavelength Waves, Dependence of Sample Rate on Long Wave Displacement, and A Priori Sea State Bias Estimate
5.3.3.1ProcessingStepstoEstimateSeaStateBiasInthispartwecomputetheSeaStateBiasproportionaltothecovarianceofphotoreturnrateandsurfaceheightin10-malong-trackbins.ThisconstituteswhatiscommonlyreferredtoastheElectro-MagneticorEMbiasthatisexpectedtodominateICESat-2SSB.Thestepsare:
1. Addmeanoffit2toht_initial2_surf.2. Converttrackdist_initial_surftometersifnecessaryandsubtracttheminimumof
trackdist_initial_surftoobtaindistancefromstartoftheoceansegment.3. Sortheightdatainascendingorderofalong-trackdistance,trackdist_initial_surf,
toallowcalculationofphotonreturnratepermeter4. Definexasthesortedtrackdist_initial_surfanddefinehtytheassociatedheights
fromht_initial2_surf.5. Dothefollowinganalysisforeach10-mbinintheoceansegment.Inpractice710
binstocoverthemax7-kmoceansegmentlength.Forbinswithnophotonreturns,thebin-averagedvariables(xbind,htybin,xrbin,slopebinfit,latbind,lonbind)willbesettoNaNs(notanumber)
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a) Setxbintomiddleof10-mbin.b) Setxbindtothemeanofxineach10-mbin.c) Setlatbindtothemeanofphotonlatitudeineach10-mbin.d) Setlonbindtothemeanofphotonlongitudeineach10-mbine) Sethtybintomeanofhtyineach10-mbin.f) Computethephotondataratepermeter,xrbin,bydividingthenumberof
photonsineach10-m,nbind,binby10.g) Establishslopebinfit:
I. =NaNifonlyonephotonina10-mbinII. =changeinhtyoverchangeinxifonlytwophotonsina10-mbinIII. =linearfitifthreeormorephotonsina10-mbin
6. Computecovarianceofbinaveragedheightanddatarate,binCOVHtXr,astheaverageoverall10-mbinsoftheproductofthedetrendedbinnedheights,htybin,andthedetrendedbinneddatarate,xrbin.
7. Compute,binAVG_Xr,theaverageofbinneddatarate,xrbin,overthe10-mbinswithphotons.
8. Computetheseastatebias,binSSBias,asbinCOVHtXrdividedbybinAVG_Xr.9. Iftherearemorethanone10-mbininanoceansegmentwithavalidslopebinfit:
a) ComputeslopeXreturnastheproductofthedetrendedslope,slopebinfit,andthedetrendedbinneddatarate,xrbin.
b) Computethecovarianceofthephotonrateandtheslope,binCOVslope_Xr,asthesumoftheslopeXreturndividedbythenumberof10-mbinswithavalidslopebinfit.
c) Computetheseastateslopebias,binSlopeBias,asthecovarianceofthedatarateandtheslope,binCOVslope_Xr,dividedbythemeanofthedatarate.
10. Savevectorarraysxbin,xbind,htybin,andxrbinforsegmentoutput.Sincetherationaleforsomeofthesestepsmaynotbeobvious,weprovidethefollowingexplanationsforthesteps:
TodeterminetheSSBusingEq.11requiresthatweknowthecovarianceofthesurfacereturnrateandtheheightanomalyassociatedwiththedominantsurfacewaves.Todothiswemustestablishtheheightvariationsandphotonreturnrateinevenlyspaced10-mbinsalongeachoceansegmentfoundduringthecoarsesignalfindingstep.Section5.3.1.
EstimatingseastatebiasoveroceansegmentsAtthispoint,ifnotearlier,becausewewanttoestimatethepropertiesofwavestypicallyhundredsofmeterslong,itwillbenecessarytoaggregateht_initial2_surfandtrackdist_initial_surfalongeachoceansegmentfoundduringthecoarsesignalfindingstep.Section5.3.1
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AddmeanfromdetrendingusedafterfirstcutsurfacefindingThemeansfromthefirstcutofsurfacefindingarecomputedfromphotonheightsoversurfacewavesandarethereforecontaminatedbythecorrelationofphotonreturnratewithheightofthesurfaceoversurfacewaves.Ifthesearenotaddedbackintotheheightrecord,thatcontaminationbySSBwillbelost.Therefore,weaddmeanoffit2,themeanofP0+P1xtrackdist_initial_surf(thealong-trackpositionof2nditerationsurfacephotons),backintothephotonheightrecord,hty=ht_initial2_surf+meanoffit2CalculatethePhotonReturnRateAfterexperimentingwithmorecomplicatedschemesusingphotonspacingtodeterminephotonreturnrate,wehaveconcludedthereturnratethatmostdirectlyrepresentstheEMseastatebiasistosimplydividethenumberofphotonreturnsineach10-mbinby10toyieldthephotonreturnratepermeterineachbin.Quantifysurfaceheightandslopeandreturnratein10-malong-trackbins.Establishavectorxbin,themiddleof10-mbinsalongalengthof7,100m,i.e.,5:10:7095,where7,100mistheunderstoodmaximumofouroceansegments.Assigneachsampleinterval(ii)inthe,xi,andhtyi,tooneofthesebinswithabinindexibinwhereibin(ii)=round-up(x(ii)-min(x))/10)andcumulativelyaddthehty(ii)tohtybin(ibin(ii))andincrementabincounter,nbinc(ibin(ii)),byoneforeachaddedsample. Findaverages,photonreturnrateandstraight-linefitsineach10-mbinForibinequalto1throughNbin10,thenumberofphotonsineachbin,nbind(ibin)=nbinc(ibin).Computebinaveragesofsurfaceheight,htybin,anddatarate,xrbin,as:Computethebinaveragesofheightanddataspacing:htybin(ibin)=htybin(ibin)/nbinc(ibin)=htybin(ibin)/nbind(ibin)andxrbin(ibin)=nbind(ibin)/10.Setxbindtothemeanofthealong-trackpositionsineach10-mbin=x(ibin)/nbind(ibin).Alsocomputefitsofsurfaceheightslope,slopebinfit,ineachbin-Inthecaseswithonly1pointinabin,slopebinfitwillbesettoNaN(notanumber).-Inthecaseoftwopointsinabin,\slopebinfitwillbeequaltothechangeinhtyoverchangeinx-Ifthenumberofpointsinabinisgreaterthanorequalto3,slopebinfitwillequaltheslopeofalinearfitwithdistanceoverthepoints.ComputeSeaStateBiasusingallviablebins.Computecovarianceofbinaveragedheightanddatarate,binCOVHtXr,astheaverageoverall10-mbinsoftheproductofthedetrendedbinnedheights,htybin,andthedetrendedbinneddatarate,xrbin,i.e.,thesumoverviablebins(binswithnonzeronbind),iii,from1toNbin10of(detrended(htybin(iii))*detrended(xrbin(iii)))/Nbin10.
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Similarly,computebinAVG_Xr,theaverageofbinneddatarate,xrbin,i.e,theaverageofxrbin(iii)fromiii=1toNbin10.
ComputethebinaverageseastatebiasestimateaccordingtoEquationX7:binSSBiasequalsbinCOVHtXrdividedbybinAVG_Xr.
ComputeSignificantWaveHeight,SWHComputesignificantwaveheight(SWH)equaltofourtimesthestandarddeviationofthebin-averagedsurfaceheightsinallthe10-malong-trackbinsinthesegment.Computephotonrateslopecorrelation.Thebinaverageslopebiasestimateisanexperimentalparameterthatindicatesifphotonsarebeingpreferentiallyreturnedfromthebacksidesorfrontfacesofwaves.Iftherearemorethanone10-mbininanoceansegmentwithavalidslopebinfit,computeslopeXreturnastheproductofthedetrendedslope,slopebinfit,andthedetrendedbinneddatarate,xrbin.
Thencomputethecovarianceofthephotonrateandtheslope,binCOVslope_Xr,asthesumoftheslopeXreturndividedbythenumberof10-mbinswithavalidslopebinfit.Savevectorarraysxbin,xbind,htybin,andxrbinforsegmentoutput.
5.3.3.2HarmonicAnalysisofAlong-trackHeightsWithRequisiteGapDeterminationandFillingStartingwiththen_photonspointsinthealongtracksortedtrackdist_initial2_surf,andcorrespondingoceanheightsinht_initial2_surfwithmeanoffit2addedbackin,ouraimistocomputetheharmoniccoefficientsfornharmharmonicsplusthezero-frequencycoefficientbestrepresentingtheseasurface.TheVaníčekmethodtobeusedisaleastsquarederrortechniquethatisnotdependentonevenlyspacedata.However,wehavefoundthatsignificantgapsinthedatacanresultinwideandunrealisticvariationsintheresultingharmoniccoefficients.Thisisbecauseunderthetechnique,thecoefficientscanvaryunrestrainedtofittheexistingdata,whileproducingwildlyunrealisticFourierseriesinthegapswhereunconstrainedbydata.ThesolutiontothisproblemistofindandfillthedatagapswithwhitenoisefromaGaussiandistributionwithmeanequaltomeanoffit2andstandarddeviationequaltothatoftheexistingdata.Thefirststepisidentifyingsignificantdatagapsinanoceansegment.
1) LettingX(j)beeachvalueoftrackdist_initial2_surf(j),ifwetakethedifference,DX(1toN-1)betweenX(2toN)andX(1toN-1),itwillbethevectorofspacesbetweenpoints,andwewillknowthatthereisnodataforDX(j)metersafterX(j).WecanthinkoftheDXasarandomvariablesocomputethesamplemedian,mode,mean(DXbar=SumDX
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from1toN-1dividedbyN-1),samplevariance(DXvar=Sum(DX-DXbar)2from1toN-1dividedbyN-2),andsampleskewness(DXskew=Sum(DX-DXbar)3/DXvar3/2from1toN-1dividedbyN-2)).Iftherearenomajorgaps,themeanwillbe0.4to0.7m.Iftherearemanysmallgaps,thevarianceandskewnesswillincreasebutthemean,mode,andmedianwillprobablystaycloseto1m.Ifthereisonebiggap,theskewnessshouldbecomelarge.Itwillbehelpfulonmanyfrontstocomputethesestatisticsforoceansegmentsasindicationsofgapsintheoceansegmentdata.Forexample,inourtestswithoneoceangranule,wefoundunreasonableharmoniccoefficientscamefromtheVaníčekmethodonlywhenDXskew>10.Theusercouldignoreoceansegmentharmoniccoefficientsand/or ocean heights altogether if these indicators suggestedthereweremanygapsinthedata.
2) Alternatively,forthepurposesofcomputingharmoniccoefficientswewillfillanygapsgreaterthangaplimit(e.g.,gaplimit=3.2m)withrandomnumbersevery0.7mdrawnfromaGaussiandistributionwithmeanequaltomeanoffit2andstandarddeviationequaltothestandarddeviationofht_initial2_surf.Todothis:
3) IdentifytheDX(l)>gaplimitandforeachcreategap-fillingvectorsofcorrespondingdistancesvectors,xgap,every0.7mfromX(l)toX(l)+DX(l)/0.7andofheights,hgap,chosenasrandomnumbersfromaGaussiandistributionwithmeanequaltomeanoffit2andstandarddeviationequaltothestandarddeviationofht_initial2_surf.ForeachtheDX(l)>gaplimitconcatenatethexgapandhgapontothepreviousxgapandhgapvectorsuntilthefinalDX(l)>gaplimitisreached.
4) WhenthefinalDX(l)>gaplimitisreached,concatenatexgapontotheendoftrackdist_initial2_surftomakextempandhgapontotheendofht_initial2_surftomakehtemp.Sortbothxtempandhtemponascendingorderinxtemptoinsertthefilledgapsintheirproperorderversusalong-trackdistance.Setx=xtempandy=htemp,andtheresultinggap-filledrecordswillbethexandyintheharmoniccoefficientdeterminationbelow.Thelengthofthegap-filledxiscalledn_photontemp.Referencextothestartbysubtractingx(1)fromeachxvalue.Notethatthegap-filledxandywillonlybeusedforcomputingharmonicsandwillnotbeusedfortheothercalculationsof10-mbinaveragesandseasurfaceheightstatistics.
5) Fornharmharmonics,computethewavelengthsstartingwiththefundamentalwavelengthequaltothemaximumofxminustheminimumofx.Thevectorofharmonicwavelengths,lwave,isgivenbylwave(i)=length_seg/i,i=1tonharms,andthevectorofharmonicwavenumbersisgivenbyinverseofthewavelengthswn(i)=1/lwave(i),i=1tonharms.
6) CreatematrixFwithn_photontemprowsand1+2*nharmscolumns.PopulateFasfollows:
F(j=1ton_photontemp,i=1)=1F(j,2*i)=sin(2*p*wn(i)*x(j))forj=1ton_photontempandi=1tonharmsF(j,2*i+1)=cos(2*p*wn(i)*x(j))forj=1ton_photontempandi=1tonharms,Andwherex(j)iseachvalueofgap-filledtrackdist_initial2_surf(j).
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7) Formleftpseudoinverseandcomputecoefficients.Forx=thegap-filled
trackdist_initial2_surf,an_photontempx1columnvector,Fisan_photontemprowsbym=1+2*nharmscolumnsmatrix.ThetransposeofF,herecalledF’,isambyn_photontempmatrix.Thematrixproduct,F’Fismbym.TheproductofF’andx=thegap-filledtrackdist_intial2_surf,isamby1vector.
8) Foryequaltothecolumnvectorofgap-filledht_initial2_surfwithmeanoffit2addedto
it,versusx,thegap-filledalongtrackdistance,computethevectorofharmoniccoefficients,a,bytheVaníčekleastsquarederrororleftpseudoinversemethod,where
a=F’F\F’y.Themx1vectorofharmoniccoefficients,a,fortheoptimumharmonicfitofytox:
yfit=a(1)+a(2)*sin(wn(1)*2*p*x)+a(3)*cos(wn(1)*2*p*x)+a(4)*sin(wn(2)*2*p*x)+a(5)*cos(wn(2)*2*p*x)+a(6)*sin(wn(3)*2*p*x)+a(7)*cos(wn(3)*2*p*x)+::+::+a(2*nharms)*sin(wn(nharms)*2*p*x)+a(2*nharms+1)*cos(wn(nharms)*2*p*x)
BackslashorleftmatrixdivideA\BisthematrixdivisionofAintoB,whichisessentiallythesameasA-1B.
9) Comparisontestsbetweengap-freecompleterecordsandidenticalrecordstowhichgapshavebeenaddedandthenfilledwithwhitenoiseaboutmeanoffit2asin(1)-(4)haveshownthatthevectorharmoniccoefficients,a,arealsotheoptimumharmonicfitofht_initial2_surfplusmeanoffit2versustrackdist_intial2_surfalongtrackdistance,0tosegmentlength(length_seg).Inthegapregions,theharmonicfitmakessmalloscillationsconstrainedtobenearmeanoffit2.Asatest,withyandyfitasrowvectors,computethesignaltonoiseratioSNR_harm=E[(yfit-a(1))*(yfit-a(1))’]/E[(y-yfit)*(y-yfit)’]whereE[]denotesexpectedvalue.OvermanyoceansegmentswewillcompareSNR_harmwiththemeasuresthenumberandsizeofrecordgaps,DXbar,DXvar,andDXskew,todeterminethepointatwhichrecordgapsaresodominantthattheharmonicfitisnotuseable.
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5.3.4 Processing Procedure for Correction and Interpretation of the Surface Height Distribution
Thissection,5.3.4,describesprocessingstepsthatanalyzesthehistogramofreceivedsurfacephotonheights,NfromSection5.3.2.TheprocedurehasbeentestedinadevelopmentalMatlabcode(WienerTest_GaussMixandNoise2.m).Matlabexecutablelineshavebeenreplacedbydetailedtextdescriptions.
AsdescribedinSection5.6,toillustrateandvalidatetheprogramstepsinthedevelopmentalcodeweconvolvedarealimpulseresponsedistributiontakenfromMABELdata(Fig.15)withasyntheticseasurfaceheight,2-Gaussianmixturedistributionwithaspecifiedstatisticalproperties(Fig.16).Arepresentativeamountofnoisewasthenaddedtothistoyield
receivedsignalphotonheightdistribution(Fig.17)withaknowntrueparentsurfacedistribution.Hereparts(A)through(G)gothroughtheanalysisprocessdescribedinpart4.3.1todeterminethetruesurfacedistribution,namelyWienerdeconvolutiontoremovetheinstrumentimpulseresponsedistribution.ThisiswheretheprocessingofATL03histogramsbegins.Part(H)in5.2.5(CharacterizingtheRandomSeaSurface)breakstheprocessedhistogramintoaGaussianMixtureandcomputestheaggregatedistributionstatisticsforATL12usingequation17.
5.3.4.1 Separating the Uncertainty due to Instrument impulse response
Parts(A)Startingwithhistogramofreceivedphotonheightsfromtheprevioussection,computeaconstantsignaltonoiseratio(SNRc)
(B)ImpulseResponseDistribution
(C)Fouriertransformofthereceivedhistogram(D)Fouriertransformoftheinstrumentimpulseresponsehistogram
(E)ComputetheWienerfilter(F)ApplyWienerDeconvolutiontoYieldtheFourierTransformoftheSurfaceDistribution
(G)ComputetheSurfaceHeightDistribution
Figure15.Instrumentimpulseresponsehistogram(XmitHist00)
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CodeSteps(A) Noise Ratio Determination
----------------------------------------------------------------
- Start with the histogram of surface reflected photons heights, Nrs, from section (5.3.2 (P)) versus the sea surface height vector, sshx, with height bin size equal to Bf (e.g., Bf= 0.01 m). Convert the histogram to a vector array of probability density function (pdf) values, rechist. The dimensions of rechist and sshx are the same.
- Smooth the received pdf with a 12th order, low-pass Butterworth filter with cutoff wavenumber corresponding to 0.1/binsize. Run the filter forward and backward over rechist to yield the smoothed histogram, smoothrechist. Figure 17 shows rechist in blue and smoothrechist in red for an 8000 photon sample (left) and 800 photon sample (right)
- Compute a signal to noise ratio (SNRc) equal to the standard deviation (std) of smoothrechist divided by the standard deviation of the difference between the received histogram and the smoothed histogram, SNRc= std(smoothrechist)/std((rechist-smoothrechist).
----------------------------------------------------------------(B)ImpulseResponseDistribution----------------------------------------------------------------Obtainthecurrentbestestimatefortheinstrumentresponsedistribution,XmitHist000,avectorarrayofprobabilitydensityfunctionvaluesfortheimpulseresponse.
Figure16.Left-ProbabilitydensityfunctionoftheSyntheticSSHasaGaussianmixture,4000samplesfromN(0,,1)and4000fromN(1,2).Right–Samebutwith800photonsforwhichtherawsyntheticPDFisfairlynoisy,buttwopeaksandskewnessarefaintlydiscernable.
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Theinstrumentimpulseresponsedistributioniscalculatedfromthemostrecenttransmitechopulse(TEP)passedfromATL03in/atlas_impulse_response/pce1_spot1or/pce2_spot3.
TheTransmitEchoPulse
TheTEParederivedfromphotonsdetectedviathetransmitterechopath(TEP,seeSection7.2.2oftheATL03ATBD).Thesephotonsarepickedofffromthetransmittedlaserpulse,androutedintotheATLASreceiver,forATLASstrongbeams1and3.ThesedatamonitortheinternaltimingbiasofATLAS,aswellastheinstrumentimpulseresponseforthebeam.WewillassignTEP1orTEP3toeachbeamweprocess,mainly1,3,and5,accordingtotherecommendationsintep_valid_spot,a6x1arrayindicatingwhichTEPtouseforeachspot.Orderofthearrayisbygroundtrack(gt1l;gt1r;gt2l;gt2r;gt3l;gt3r),fromATL03.(Notealso,thatwhenthespacecraftisflyingforwardbeam1withaTEPiscoveringspotgt1randbeam3withaTEPiscoveringspotgt2r.Whenthespacecraftisflyingbackwardbeam1withaTEPiscoveringspotgt3landbeam3withaTEPiscoveringspotgt2l.)AccordingtoATL03,theparametersaregeneratedasoftenassufficientTEPdataexistsinthegranule,typicallyabouteveryfivegranules(granuleseachspanabout1/14thofanorbit.).Otherwisedatafromthepriorgranuleisretainedinthecurrentgranule.TheTEPphotoneventarrivaltimesareprovidedinahistogram,tep_hist_times.
TheTEPphotoneventsaredownlinkedwhentheTEPphotoneventshappentooccurwithinthetelemetrybandapproximatelytwiceperorbit,foraboutthreehundredseconds.ThenumberofTEPphotonspershotforaparticularbeamvaries.However,theratioofthenumberofTEPphotonsbetweenbeams1and3appearsstablewithavalueof0.7,with
Figure17.Synthesizedreceivedprobabilitydensityfunctions(blue)asaconvolutionoftheinstrumentimpulseresponsePDF(Fig.15)andthesynthesizedtrueSSHPDF(Fig.16)pluspulsenoise.ThePDFfor8000photonsisatleftandfor800photonsisontheright.ThesePDFsrepresentthedataastheywouldcomefromthesurfacefindingpartoftheprocessing.TheredlineisthesmoothedPDFusedtoestimatethesignalenergyforthesignaltonoiseratioofthereceiveddata.
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beam1beingthestrongerofthetwo.Averagingacrossarangeofinstrumentconfigurations,beam1generatesapproximately0.09TEPphotonsperlaserpulse,andbeam3generatesapproximately0.07TEPphotonsperlaserpulse.AssumingoneTEPphotoneventpertwentypulsesasthelowerlimitfortheTEPstrength,downlinkingTEPphotonsforapproximatelythreehundredsecondsshouldyieldabout150,000TEPphotons.TEPphotoneventsarecapturedanytimetherearemorethanthirtysecondsofcontiguouspossibleTEPphotoneventsidentifiedinATL02(e.g.,15,000TEPphotons),andthereareatleast2000possibleTEPphotonsintheset.AllavailableTEPphotonswithinagivensetareused;thereisnoupperlimitonthedurationornumberofTEPphotons.ThesetisendedwhentheTEPphotonsarenolongerpresentinthetelemetryband,orfivesecondshavepassedwithnoadditionalTEPphotons.WithasufficientnumberofpossibleTEPphotoneventsidentified,thesewilleitherbebackgroundphotonevents,misclassifiedsignalphotonevents,orTEPphotonevents.ThetimeofdayofthestartoftheTEPdatausedinsubsequentanalysesisreportedonATL03astep_tod,whilethedurationofTEPdataisreportedastep_duration.
ThehistogramofthesepossibleTEPphotoneventsisformedusing50picosecondsbinwidths.ThereismorethanasinglemeanarrivaltimeofTEPphotons;withtwobandswiththeprimarybandbeingat~20nanosecondsandthesecondarybandbeingat46nanoseconds.TheactualtimebandlimitsforeachTEPwillbeindicatedbytheATL03variabletep_range_prim.Additionalnoisephotonsarealsopresent.Themeanbackgroundphotonrateisdeterminedbyexcludingthebinsbetween17and24ns(theregionoftheprimaryreturn)andtheregionbetween43and50ns(theregionofthesecondaryreturn)andcalculatingthemeanoftheremainingbins.ThisvalueisreportedontheATL03outputproductastep_bckgrd,andissubtractedfromallbincounts.Thenumberofcountsintheremaininghistogramisreportedastep_hist_sum.Thenumberofcountsineachbinisnormalizedbybytep_hist_sumandreportedforeachbinastep_hist.Timeismeasuredrelativetothecentroidtransmitpulse,andonaveragefortheTEPrepresentsthetraveltimebackandforththroughtheATLASopticalpath.Thetimeofthehistogrambincenters(themeanofthetimesoftheleadingandtrailingedgesofthebins)isrecordedastep_hist_time.Theparametertep_todindicateswhentheTEPdataonagivenATL03granulewasgenerated.TheparametersderivedfromTEPdataarepartoftheATL03dataproductinthegroups/atlas_impulse_response/pce1_spot1/and/atlas_impulse_response/pce2_spot3/.TurningtheTEPhistogramintotheInstrumentImpulseResponse
Weconvertfromtep_histexpressedasnormalizedcountsintepbinsize=50picosecondwidebinstoanimpulseresponseexpressedastheprobabilitydensityfunctionwithbinsizewidebinsofsurfaceheight.ForthepurposeofcalculatingthelengthofthederivedsurfacedistributionitisconceptuallyusefultointerpolatetovaluesinXmitHist000sothatthereareanoddnumberofbinswiththeoriginbinatzerodelay(orelevation)andthesamebinsize(binsize)asisusedforthereceivedelevationhistogram.TheprocessofderivingthisimpulseresponsefromaTEPfirstrequiresidentifyingtheappropriateTEPtouseforaparticularbeamspecifiedbytep_valid_spot,
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isolatingtheprimarybandoftimesspecifiedbytep_range_prim,trimmingtheTEPtoprimaryhistogramlevelsabovethebackgroundnoiserate,convertingfromtraveltimetosurfacephotonheightoffset,convertingtobinsizewidebinsstartingfromnegativeoffsets,andnormalizingtoaprobabilitydensityfunctionbydividingbythesumofthehistogramlevelsmultipliedbybinsize.ForthepurposeofcalculatingthelengthofthederivedsurfacedistributionitisconceptuallyusefultointerpolatevaluesinXmitHist000sothatthereareanoddnumberofbinswiththeoriginbinatzerodelay(orelevation)andthesamebinsize(binsize)asisusedforthereceivedelevationhistogram.InpartthispseudocodecomesfromtheMatlabprogramBinconvertWF_JM3.m.- TheappropritateTEPhistogram,tep_hist,asspecifiedbytep_valid_spotforaparticular
beamincludesboththeprimaryandsecondaryarrivals.ChoosetheprimaryTEParrivalsinbinsbetweentep_range_prim(1)andtep_range_prim(2)(typicallyaround17and24ns)foruseinATL12processing.e.g.,(iprimary=find(tepT>= tep_range_prim(1)&tepT<= tep_range_prim(2);tepT=tepT(jprimary);tepP=tepP(jprimary);
- Becausetep_bckgrdissubtractedfromthefromtheTEPhistogram,thetailsoftheTEPwillincludebinswithnegativevalues.WetrimoffthesetailswherenegativevaluesappearindicatingtheTEPisbelowthenoisefloor.WorkingfromtimeofTEPmaximumout,settheTEPvaluetozeroatthefirstbinswhereTEPislessthanzero,anddeleteallbinsbeyondthosepoints.
- Convertfromtraveltimetosurfaceheightcoordinateforthehistogrambymultiplyingittimesminushalfthespeedoflight.
- Reversetheorderofthehistogramanditsdistanceaxissothatthearraystartsatnegativeheightoffsets.
- CentertheTEPhistogramsothatithasheightoffsetequalszeroatthecentroidofthedistribution.
- ConfirmtheTEPbinsize,bin_TEP,asthedistancebetweenbincenters50picoseconds/(2xspeedoflight).
- ConvertTEPbincenterstobinboundariesinmeters- Establishanarrayofimpulseresponsehistogrambinboundarieswithanoddnumber
ofbins,abinsizeequaltobinsize,interiorbincenteredonzero,andallwithintherangeoftheTEPbinboundaries.
- ComputethecumulativehistogramoftheTEPversustheTEPbinboundariesandinterpolatetotheimpulseresponsehistogrambinboundaries.
- Differentiatetheresultingcumulativeimpulseresponsehistogramandformacorrespondingarrayofbincenterstoproducetheinstrumentimpulseresponsehistogram.
- Dividetheimpulseresponsehistogrambytheproductofbinsizeandthesumofvaluesinthehistogramtoyieldtheimpulseresponseprobabilitydensityfunction.
Atthispoint,wehavedescribedtheaveragedpulseshapecharacteristicsfromthethresholdcrossingtimespostedatthenominal~20metersalong-tracksegmentrate.We
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alsohavedescribedtheTEPphotoneventsandhaveformedahistogramandrelatedparametersbasedonTEPevents.Sincethetimesoftheserespectivegroupsareknown,itispossibletoexaminehowtheSPD-basedstatisticsvaryduringtheperiodspannedbytheTEPdata.However,combiningthesetwogroupsquantitativelyiscomplicatedbytwoaspects.First,theSPD-baseddataandtheTEP-baseddatahaveuniquepathsthroughtheinstrumentandsothereportedtimesofthesedatawillbesubstantiallydifferent.Whilethesedifferencesarecharacterizedwithpre-launchdata,wehavelittleinsightintohowthetemporalbiasbetweenthesedatawillevolvethroughtime.Secondly,theSPD-baseddataisderivedfromthereportedtimeswhenafilteredversionofthetransmittedlaserpulsecrossesaparticularamplitudethreshold(measuredinvolts).TheTEP-baseddataontheotherhandisahistogramofcountsofamassivelyattenuatedversionofthetransmittedlaserpulse.Assuchaquantitativecorrespondencebetweenvoltagesandcountsisproblematic.ArelationshipcouldpossiblybemadebetweenthestandarddeviationofthethresholdcrossingtimesandtheapparentwidthoftheTEP-basedhistogramatseveralpointsalongtheprofile.ThismayallowausertobetterpredicthowthetransmittedpulseischangingbetweenrealizationsoftheTEP.
(C)FourierTransformofReceivedHeightDistribution---------------------------------------------------------------------------
- Findlengthofrecord(NFFT)equaltothenextpoweroftwogreaterthanthelengthofrcvhist.Thisshouldcoverthelongestoftheimpulseresponsedistribution,XmitHist000,andrcvhist.
Figure18.Single-sidedamplitudespectraofthereceivedPDFs(Fig.17).Thetransformfor8000photonsisontheleftand800photonsisontheright.Notethehigherwavenumbervariabilityfor800photons.
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- PadtheendofrcvhistwithzerostoextendittolengthNFFT- ComputethefastFouriertransform,Rf,ofthereceivedheightdistribution,rcvhist
- Ensureconsistentscalingsothatenergyispreserved(e.g.,withMatlabFFT,wemultiplytheresultbythebinsizetoreflectintegrationinthespacedomain- Thehighestwavenumber(cyclespermeter)istheNyquistwavenumberequaltohalfthesamplingwavenumber,wvns,whichequalstheinverseofbinsize(wvns=1/binsize).Usingthis,wecomputethewavenumbervector(wvn)correspondingtotheFouriertransformofthereceivedhistogram.ExamplesofRfplottedversuswvnareshowninFigure18.(D)FourierTransformofInstrumentImpulseResponse----------------------------------------------------------------
- PadtheendofXmitHist000withzerostoextendittothelengthNFFT- ComputethefastFouriertransform,Tf,oftheimpulseresponsedistribution,XmitHist000
- Ensureconsistentscalingsothatenergyispreserved(e.g.,withMatlabFFT,wemultiplytheresultbythebinsizetoreflectintegrationinthespacedomain
- ExamplesofTfplottedversuswvnareshowninFigure19.
(E)ComputetheWienerFilter----------------------------------------------------------------
Figure19.Single-sidedamplitudespectrumoftheinstrumentimpulseresponsePDF(Fig.15).
Figure20.Single-sidedamplitudespectraoftheWienerfilterscomputedfromtheFouriertransformoftheinstrumentimpulseresponsePDFs(Fig.19)andthesignaltonoiseratioestimatedfromthereceivedheightPDFsandsmoothedversionsofthem(Fig.17).Owingtoalowersignaltonoiseratio,theWienerfilterfor800photons(right)hasanarrowerlowfrequencypassbandthanthe8000photonfilter(left).
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- ComputetheWienerfilter,WienerF,equaltoTfxTfcc/(TfxTfcc+SNRc-2),wheresubscriptccdenotescomplexconjugate.
- Notethatultimately,wemaycomputetheWienerfilterwithasignaltonoisevariation,SNRf,thatvarieswithwavenumberwavenumber,e.g.,SNRfequaltheabsolutevalueoftheFouriertransformofthereceiveddistribution(Rf)dividedbytheperceivednoise(NoiseRec),buttestssofarusingtheconstantSNRchaveworkedwell.
- ExamplesofWienerFplottedversuswvnareshowninFigure20.(F)ApplyWienerDeconvolutiontoYieldtheFourierTransformoftheSurfaceDistribution----------------------------------------------------------------
- ComputetheFouriertransformofthesurfacedistribution,Yf,astheWienerdeconvolutionoftheinstrumentimpulseresponsedistributionandthereceivedheightdistribution.InFourier(wavenumber)spacethisistheWeinerfilter,WeinerF,multipliedtimestheratioofthetransformofthereceiveddistribution,Rf,andthetransformoftheimpulseresponse,Tf,i.e.,Yf=WeinerFxRf/Tf.
- ExamplesofYfplottedversuswvnareshowninFigure21.
G)ComputetheSurfaceHeightDistributionThesurfacehistogramshouldbethesamelengthandhavethesamehorizontal(height)axisasthereceivedhistogram,rechist.Inthispart,wecomputethedistributionofsurfaceheightasaprobabilitydensityfunction(PDF),Y,startingwiththeFouriertransformofthatPDFoutputbytheWeinerdeconvolution.Thestepsare:
Figure21.Single-sidedamplitudespectraofthetruesurfacePDFestimatedastheWienerfilter(Fig.20)timestheFouriertransformsofthereceivedPDFs(Fig.18)dividedbytheFouriertransformoftheinstrumentimpulseresponsePDFs(Fig.19).Theresultfor8000photonsisontheleftand800photonsisontheright.
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8. ComputeYfi,astheinverseFourierTransformofYfdividedbybinsize.YfiasitcomesfromtheinverseFouriertransformisNFFTpointslongwiththefirsthalfappearingdisplacedtotheendofthearray.WereorderthisbyfirstputtingthelastNFFT/2pointsaheadofthefirstNFFT/2points.Toestablishthesurfaceheightprobabilitydensityfunction,Y,wethenremovethefirst(rzbin-1)pointswhererzbinistheindexofthecentroid(heightequalszero)indexofthereceivedhistogram.WekeepasYtheremainingpointsequaltothelengthoftherechist.InthecaseoftheASASFORTRANcode,whichpadseachendofrechisttoreachheightsof±15m,rzbinisthecenterbinofrechist.
9. Setthex-axisofY,derivedsshxequaltothex-axis,xrechist,ofthereceivedhistogramrechist.
10. NormalizeY,bydividingYbythesumoftheproductofYfiandbinsize.11. ComputethemeanorcentroidofY,meanY,bysummingtheproductofYand
derivedsshxanddividingthatsumbythesumofY.12. ComputethestandarddeviationofY,stdY:
a) squarederivedsshxminusmeanYb) Multiplya)byYc) Takesumofb)d) Dividec)bysumofYe) Takesquarerootofd)
13. ComputetheskewnessofY,skewnessY:a) CubederivedsshxminusmeanYb) Multiplya)byYc) Takesumofb)d) Dividec)bysumofYe) Divided)bythecubestdY.
14. ComputethekurtosisofY:a) ComputederivedsshxminusmeanYtothefourthpowerb) Multiplya)byYc) Takesumofb)d) Dividec)bysumofYe) Divided)bythestdYtothefourthpowerf) Subtractthreefrome)
Sincetherationaleforsomeofthesestepsmaynotbeobvious,andthestepsmayhavetobemodifieddependingonthedifferencesbetweentheMatlab(usedinthedevelopmentalcode)andFortran(usedintheASAScode),weprovidethefollowingexplanationsforthesteps:
- ComputeYfi,equaltotheinverseFouriertransformoftheoutputoftheWienerfilteringprocess,Yf.Ensurethattheresultisnormalizedtoconserveenergyanddimensionalconsistency.Forexample,theoutputoftheMatlabinverseFouriertransform(ifft.m)mustbedividedbybinsizefordimensionalconsistency.
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Computex-axisderivedfromxrechistandcharacteristicsofinstrumentimpulseresponse.ConvolvingonefiniteseriesoflengthNwithanotheroflengthMresultsinaseriesoflengthN+M-1.Weknowfurtherthatiftheinstrumentimpulseresponsehistogramhaszeromeanwiththezerobinatindexpositionzbinthiswilldictatewherethepointsareaddedtothetrueheighthistogramtolengthenthereceivedheighthistogram.Iftheimpulseresponsedistributionweresymmetric,equalpointswouldbeaddedtoeachend,butthisisnotgenerallytrue.Infact,assuminglengththelengthoftheimpulseresponsehistogramislxmithist,andthezeropointisatindex,zbin,zbin-1pointswillbeaddedtothebeginningofthetruesurfaceheighthistogramandlxmithist-zbinpointswillbeaddedtotheendofthesurfaceheighthistogramtoproducetheconvolvedreceivedhistogram.
- Inprocessing,wereversethistotrimthelengthofthevectorx-axisofthereceivedheighthistogram,xrechist,totheappropriatex-axisfortheunderlyingsurfaceheighthistogram,bydeletingthefirstzbin-1pointsandthelastlxmithist-zbinpointsfromxrechisttoyieldthevectorofx-axisofthesurfacehistogram,derivedsshx,withlength,derivedlsshhistequaltolength(xrechist)-(lxmithist-1).
TrimmingthevectorofsurfacehistogramvaluesiscomplicatedbythenecessityofpaddingreceivedhistogramwithzerostoalengthequaltoapoweroftwoforcomputationofitsfastFouriertransform.Asaresult,atleastintheprototypeMatlabcode,theinverseFouriertransformoftheoutputoftheWeinerdeconvolutionprocess,Yfi,alsohasalengthequaltothissamepoweroftwoandshouldbetruncatedtotheappropriatelength.
- Consequently,truncateYfikeepingonlythefirstnumberofpointsintheoriginalreceivedhistogramless(lxmithist-1),i.e.,setderivedlsshhistequaltolength(xrechist)-(lxmithist-1).Thisyieldsthesurfacehistogram,Y,definedoverderivedsshx,thatwillhavethenumberofpointsconsistentwithdeconvolvingtheimpulseresponsefromthereceivedhistogram.
- Togivethesurfaceheightprobabilitydensityfunction,Y,normalizetheYfitomakeupforenergylostintheWienerfilterapplication.TheintegralofYshouldequalunity,sotakeYequaltoYfidividedbytheintegralofYfi,i.e.,Y=Yfi/(sumofYfixbinsize).
- Asacheckonthedeconvolutioncomputation,thecentroidofrechistandYshouldbothbenearzeroonaccountofthedetrendingofthesurfaceheightsandtheassumedzeromeanoftheinstrumentimpulseresponsecomputation
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- ExamplesofYandthedifferencebetweenYandthesyntheticsurfaceexamplesthatwestarted
witharegiveninFigure22for8000photonsandFigure23for800photons.- ForsubsequentuseinthederivationoftheATL19griddedproduct,thesurfacepdf,Y,willalsobeoutputforeachsegmentalongwiththetotalnumberofsurfacephotons,N,andbinsize,binsize.
- Yisdefinedoveravectorofbincenters,sshx,butforbinsize=1cmYwillbeoutputasa3001-elementvectorfrom-15mto+15mforeverysegmentwiththe1,501stcenterbinbeingforheightzero.Valueswillbezeroforbinsoutsidetherangeofsshx.OwingtonoiseandtheeffectoftheFFT-deconvolution-inverse-FFTprocess,smallnegativevaluesoccurinYwithintherangeofsshx.ThesewillbesettozerointheoutputYvector.
- AsasimplecheckontheoptimumGaussianMixturedeterminationofthehighermomentoftheseasurfacedistributionwewillcomputeandoutputthemean,variance,skewness,andexcesskurtosis(Ymean,Yvar,Yskew,Ykurt)relatedtothehighermomentsofYdirectly.
-
(30)
(31)
Ymean=µ
Y= X
iYi
Yi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟ Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟
Yvar = (X
i−µ
Y)2Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟ Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟
Figure22.Forthe8000photonrealization:Onleftinred,thePDFoftrueSSH(Fig.16)withaGaussiancurvefit(dashed-red).AlsoinbluethePDFofSSHcalculatedbytheWienerdeconvolutionprocess(inversetransformofFig.21)withaGaussiancurvefit(dashed-blue).OntherightisthedifferencebetweenthetrueSSHandthecalculatedSSH.
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(32)
(33)
5.3.4.2 Characterizing the Random Sea Surface
Thispartgoesthroughtheanalysisprocessdescribedinpart4.3.2todeterminethetruesurfacedistribution,namely:ExpectationMaximizationanalysistodeterminethemeans,variance,andmixingratioofthetwoparentnormaldistributions.(H)Computethe2-componentGaussianmixtureEquivalentoftheSurfaceDistribution------------------------------------------------------------------------
- At this point we have the probability density function, Y, of detrended DOT over height bins with centers at sshx. The detrending process used in surface finding removed an average, meanoffit2, from the original heights. Once we have determined the mean and higher moments of Y, we could add meanoffit2 to the mean of Y to learn the mean DOT over the ocean segment. However, preliminary analysis suggests if we add meanoffit2 back into the distribution before the optimal Gaussian Mixture determination, the resulting higher moments are less noisy over many ocean segments. This can be done by shifting sshx by adding meanoffit2 to each value in sshx. Then the true average DOT will be a derived as the aggregate mean of the optimally derived Gaussian mixture. For the developmental Matlab code a slightly different but equivalent approach was used.
- The Matlab development code used gmdistribution.fit, an expectation maximization approach, to compute the 2-component Gaussian mixture corresponding to the observed surface height distribution, Y. The ASAS Fortran code uses the expectation maximization routine from ATBD ATL07 Appendix E.
- To apply the expectation maximization approach, we first derive a sea surface height spatial series from the height distribution Y, which is defined over a range of heights the same as the received height histogram (i.e., jlow to jhigh from surface finding). This is accomplished by first multiplying Y by 10,000, rounding and deleting any values less than or equal to zero (a few unrealistic weakly negative values to occur in Y associated with the Weiner deconvolution process) to produce an integer distribution, YI. A series, XY, is assembled by concatenating for each value of YI(i) for index i corresponding to height sshx(i), YI(i) values equal to sshx(i). The shift to heights including meanoffit2 was accomplished in developmental code by adding meanoffit2 to each value in XY. (ASAS FORTRAN code does not add meanofffit2 at this point but after the Gaussian Mixture calculation.) Although it is not strictly necessary, the values of XY are randomly permutated to produce a realistic distribution of heights with the probability density function Y on heights given in sshx shifted by adding meanoffit2.
Yskew = (X
i−µ
Y)3Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟ Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
3/2Yvar
Ykurt = (X
i−µ
Y)4Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟ Y
iYi>0∑⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
2Yvar −3
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- We then compute the 2-compnent Gaussian mixture variables, of the two-Gaussian mixture using expectation maximization, Matlab GMfit=gmfitdistribution.fit (XY’,2).
- The result is Gaussian mixture distribution with 2 components:
-Withexampleoutputfor8000photonsGaussianmixturedistributionwith2componentsin1dimensionsComponent1:Mixingproportion,m1=0.457026Mean,mu1=1.0488StandardDeviation,Sig1=2.0062Component2:Mixingproportion,m2=0.542974Mean,mu2=0.0081StandardDeviation,Sig2=1.0533-Forthe800photoncasetheresultsare:Gaussianmixturedistributionwith2componentsin1dimensionsComponent1:Mixingproportion,m1=0.533033Mean,mu1=0.8430StandardDeviation,Sig1=2.0559Component2:Mixingproportion,m2=0.466967Mean,mu2=-0.0553StandardDeviation,Sig2 =1.0533
Figure23.Forthe800photonrealization:Onleftinred,thePDFoftrueSSH(Fig.13)withaGaussiancurvefit(dashed-red).AlsoinbluethePDFofSSHcalculatedbytheWienerdeconvolutionprocess(inversetransformofFig.21)withaGaussiancurvefit(dashed-blue).OntherightisthedifferencebetweenthetrueSSHandthecalculatedSSH.
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(I)ComputetheAggregateMean,Variance,SkewnessandKurtosisofSeaSurfaceHeight------------------------------------------------------------------------
- Usethefiveparametersofthe2-Gaussianmixturetocomputetheaggregatemoments,firstthroughfourthoftheaggregatemixture.From (27), for a mixture of two Gaussians with a fraction m1 (m1) from the first Gaussian with mean mu1 (µ1) and standard deviation sig1 (s1) and a fraction m2 (m2) (note: m1+m2=1) from the second Gaussian with mean mu2 (µ2) and standard deviation sig2 (s2) the jth moment of the mixture is calculated according to:
(34)
Notethatincomputingthemoments(34),thehistogramdatatheexpectationoperation(E[])isnotactualexecuted,becauseobviouslywedonothavethesamplepopulationsofthemixturecomponents.Theexpectationoperatorappearsin(34)toindicatethemixturemomentsandthemixturecomponentmomentscomingfromSection5.2.5.2(H)above.Theseareappliedsequentiallyusingonlyfirstandsecondmomentsofthemixturecomponentsandthemixingproportionstogetuptothefourthmomentofthemixture.
- Thefirstmomentormeanofthemixture, (28)isthemeanofthedetrendedsurfacephotonheightsrelativetothegeoidwiththemeanofthedetrendingprocess,meanoffit2(equaltothemeanofP0+P1*trackdist_initial_surf),addedbackinpriortotheGaussianmixturedetermination.ThusitrepresentsthemeanDOTuncorrectedforseastatebias,alongtheoceansegment.Therefore,themeanseasurfaceheight,SSH,forthesegmenttobeoutputinATL12isequaltothemeanofthemixture,µmix , plus meangdht equal to the mean of geoid height, geoid, over the segment: (35)
- Thesecondmomentofthemixtureistheseasurfaceheightvariance,SSHvar,forthesegmenttobeoutputinATL12andusedtoestimatesignificantwaveheight.Herewedon’tincludethevarianceduetothesegmenttrendinseasurfaceheightduetounderlyingtrendsintheDOTorgeoidheight.From(29),thesecondmoment,orsquaredstandarddeviation,s2mix,ofthemixtureis:
�
E X − µmix( ) j[ ] =j!
k! j − k( )!k=0
j
∑i=1
2
∑ µi − µmix( ) j−kmiE Xi − µi( )k[ ]
!!E X − µmix( ) j⎡⎣⎢
⎤⎦⎥= j!
k!( j−k)!k=0
j
∑i=1
L
∑ µi − µmix( ) j−k miE Xi − µi( )k⎡⎣⎢
⎤⎦⎥
µmix =m1µ1+m2µ2
SSH =µmix +meangdht =m1µ1+m2µ2+meangdht
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(36)
- Using (34), compute third and fourth moments, we obtain the sea surface height skewness, SSHskew, and excess kurtosis SSHkurt,
(37)
(38)
andoutputtheseaspartofATL12tobettercharacterizetheseasurface.Herewedon’tincludetheskewnessandkurtosisduetothesegmenttrendinsurfaceheight.
5.3.4.3 Conclusions for Section 5.3.4:
OursynthetictrueSSHdistributionwasassumedtobeanevenmixofheightsdrawnfromtwoGaussiandistributions,onewithameanofzeroandastandarddeviationof1m,andtheotherwithameanof1mandastandarddeviationof2m.TheresultoftheWienerdeconvolutionofthesynthesizedreceiveddistribution,madebyconvolvingthetrueSSHdistributionwiththeinstrumentimpulseresponsedistributionandaddingnoise,producedwhatisessentiallyasmoothedversionofthetrueSSHdistribution(Fig.22leftfor8000photonsandFig.23leftfor800photons).TheGaussianfitstothetrueandcalculateddistributionsmeansandstandarddeviationsarevirtuallyidentical,butofcourse,bothmisstheslightbi-modalityandskewnessofthetrueandcalculatedSSHdistributions.UsingtheMatlabgmdistribution.fitfunctiononthecalculatedSSHdistributionandassumingmixturedistributionwith2Gaussiancomponentsin1dimensionyieldsfor8000photonsmeansof0.0081mand1.0488mversusthetruevaluesof0mand1m.TheExpectationMaximization(EM)ofgmdistributin.fityieldsstandarddeviationsforthetwoGaussiancomponentsof1.1481mand2.0062mversusthetruevaluesof1mand2m.Thegmdistribution.fitvaluemixtureratiosare0.5430and0.4570versusthetruevaluesof0.50and0.50.For800photonstheEMprocessyieldsmeansof-0.0553mand0.8430mversusthetruevaluesof0mand1m.TheExpectationMaximizationofgmdistribution.fityieldsstandarddeviationsforthetwoGaussiancomponentsof1.0533mand2.0559mversusthetruevaluesof1mand2m.Thegmdistribution,fitvaluemixtureratiosare0.4670and0.5330versusthetruevaluesof0.50and0.50.
TheseresultswithWienerconvolutionandanalysisbyfittinga2-componentGaussianMixturesarepromising.Clearly8000photonsproducesmuchbetterfidelitythan800photons.Thismodelingapproachgivesusconfidencethattheanalysisprocedureiscapableofanaccurateestimateofsurfacestatistics.Furthertestingwithsyntheticdata,MABELdataandATLASsimulatordatawillbeneededfullyunderstandthecapabilitiesof
SSHvar =σ mix2 = m1 µ1 − µmix( )2 +σ 1
2( ) +m2 µ2 − µmix( )2 +σ 22( )
SSHskew = E Y − µmix( )3⎡⎣
⎤⎦ σ mix
3
SSHkurt = E Y − µmix( )4⎡⎣
⎤⎦ σ mix
4 − 3
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thesetools.Forexample,amoreprecisetestofcapabilitymightbetouseanidealizedsyntheticsurfacehistogramthatwasmorenearlyperfect,havingbeenderivedanalyticallyordrawnaswehavehere,butfromaverylargenumberofsamples(e.g.,800,000).Pulsenoisewouldthenbeaddedcorrespondingtovarioussamplesizes.Wienerdeconvolutionresultscouldthenbecomparedtotheperfectlyidealizedsurfacehistogramanderrorsattributedsolelytotheprocessingprocedure,withnoerrorassociatedwiththeinitialsynthesizedsurface.
InrunningtheMatlabWienerdeconvolutionprototypescheme,wefoundittobestableoverawiderangeofimposednoisevaluesandseveralGaussianmixtures.Forveryhighassumedsignaltonoiseratios,itmorecloselyreproducesthetrueSSHPDF(e.g.,Figs.22and23)seeminglyacceptingassignalwhatlookstouslikenoiseduetolowsamplecounts.Forlowsignaltonoiseratios,theWienerdeconvolutionschemeproducedasmoothedversionofthesyntheticSSHdistribution,throwingoutthenoiseduetolowsamplecounts.Whilewethinkthemethodofsignaltonoiseestimationusedhereisreasonable,itmustbetestedwithrealdata.Inwhateverwayweestimatethesignaltonoiseratio,theWienerdeconvolutionschemeappearstobearobustwayofhandlingit.ItdoesanexcellentjobofreproducingtheoverallmeanandvarianceofthetrueSSHPDF(Figs.22and23).FittingaGaussianmixturemodeltothedataalsoappearstoworkwellbutweshouldexperimentwithlargernumbersofsyntheticphotoncountstoseewhatimprovementscanbemadetotheaccuracyofthecomponentmeansandvariances.Wealsoneedtoseehowthemixturemeansandhighermoments(Eqn.17)improvewithhigherphotoncounts.BecausewehavetreatedtheMatlabroutinegmdistribution.fitasablackbox,wealsoneedtolearnmoreabouthowtheEMmethodworksandwouldbeappliedinastandalonecode.TheschemecanbecomparedtotheminimumsquarederrorapproachusedforGLASandwecanseeifthatexistingroutinecanbeusedforATLAStoderiveGaussianmixturesrepresentingtheoceansurfaceheightdistribution.
5.3.5 Applying a priori SSB Estimate
TheaprioriSSBestimateofSection5.3.3canbeappliedtothemeanSSHandDOT(=SSH-EGM2008Geoid)forcomparisontoinsitucal/valorotheraltimeters.TheimpactofthisSSBontheothermomentsisTBD.
5.3.6 Expected Uncertainties in Means of Sea Surface Height
TheoceanproductsofATL12arethedistributionsofseasurfaceheightoveroceansegmentstypically5to7kmlongandincludingabout8,000candidatesurfacereflectedphotons.Theheightdistributionsarecharacterizedupthroughtheir4thmoments(mean,standarddeviation,skewness,andkurtosis)usingthe2-Gaussianmixtureapproach.Inthissubsection,weexaminethemagnitudeoftheexpecteduncertaintiesinthemeanofthesurfaceheightdistribution.Fortheopenocean,weconsideruncertaintyinthemeanheightdueto:
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1.Signal-to-noiseconsiderationsinadistributionofphotonheights.2.Uncertaintiesinthemeanimpulseresponse.3.Subsurfacescattering.TheseareborrowedfromandconsistentwiththeATL07SeaIceATBDexceptduethelowrateofphotonreturns(about1photonperpulse)overtheoceansurface,forthemostpartweexcludefirst-photonbias(SeeSection3.1.1.4).WealsodonotconsiderseastatebiasdiscussedinpreviouschaptersbecausewearecalculatingSSBdirectlyandtheerrorsinthiscalculationwillhavetobedeterminedduringtheICESat-2cal/valprocess.
5.3.6.1 Signal-to-noise considerations and the dominant effect of surface wave roughness
Weexpectthelargestcontributionstouncertaintyinthemeanseasurfaceheighttobeduetosurfaceroughness,andatleastovershortaveraginglengthsindaytime,backgroundnoise.Thetotalnumberofphotons(NPtot)inaheightwindow,W,isthesumofthenumberofsignalphotons,thebackgroundrate(Bs)andthenumberofpulses(Npulses)withinthatwindow:
NPtot=NPsignal+NPbkg
Backgroundphotonsareuniformlydistributedwithintheheightwindowandthusthestandarddeviation,σbkg,canbewrittenas,
σbkg=W(12)-1/2=0.2887W,wherethefactor.2887isthestandarddeviationofauniformrandomvariableonaunitinterval.ThenumberofbackgroundphotonsNPbkg)inthewindow(W)isaPoissonvariablewithameanvaluegivenby:
NPbkg=2NpulsesBs(W/c)IfthedistributionofthereceivedsignalphotonsisapproximatelyGaussian,thevariancedependsonthetransmit-pulsewidth(σpulse)andthesurfaceroughness(σrough):
σsignal2=σrough2+σpulse2Theexpectedcompositevarianceofsurfaceheightcanthenbewrittenas:
Exceptfortheroughness,allofthequantitiesinthisequationcanbeestimatedfromthedata:thebackgroundandsignalphotoncountscanbeestimatedfromthetotalnumberofphotonsandthebackgroundrate.Forarelativelysmoothflatsurface,theexpectedvarianceintheestimationofsurfaceheightinawindowofNpulsescanbecalculatedusing:
σ
surf2 =
NPsignal
σsignal2 +NP
bkgσbkg2
NPsignal
+NPbkg
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UltimatelyICESat-2oceandatawilllikelyextendintothemarginalicezone(MIZ)forwherethedatawillbeanalyzedforshortalong-trackdistancesandsmoothwater.FromtheATBDforSeaIce(ATL07),overaflatleadwithmeanreturnsof~1photonperpulsefor100pulses(70-mofsmoothopenwater)withanominalreflectanceof0.2andnobackgroundnoise,theexpecteduncertaintyis1.0cmdueentirelytouncertaintyduetopulsewidth.Witha3MHzbackgroundratetheuncertaintyincreasesto1.1cm.Evenatshortdistances,thecontributionofbackgroundnoiseissmall.
Intrueopenoceanconditionsthedominantsourceofuncertaintyinthemeanheightoftheseasurfaceovereachoceansegmentwillbetheroughnessofthesurfaceduetowaves.
(39)
Foratypicalsituationwithasignificantwaveheight(SWH=4σwaves)equalto2m,σwavesequals0.5m,andoveratypicaloceansegmentwith8,000photons,theuncertaintyintheoceansegmentmeanseasurfaceheight, ,wouldbe0.56cmiftheheightswere
uncorrelated,andover8,000pulses, associatedwithsurfaceroughnesswouldremainlessthan1cmforSWHlessthan3.6m.Unfortunately,analysisofinitialICESat-2datasuggeststhatforlargewaves,theoceansegmenttosegmentaverageheightsvarymuchmorethanthis.Thisisbecausesuccessiveheightmeasurementsarecorrelatedduetotheharmoniccharacterofthesurface.Infact,theuncertainty, ,of(39)appearstomoreapplicabletotheICESat-2segment-to-segmentheightvariabilityifwetakethedegreesoffreedom,NPtotal,equaltothenumberofwavelengthsofthedominantwavesintheoceansegment.Forlongwaves,especiallyatanangletothegroundtrack,thisnumbercaneasilybe15sothattheuncertaintycanbeinthe20to30cmrange.ThisisnotacharacteristicofICESat-2,butratheracharacteristicoftheever-changingoceansurface.Toaddressthisnaturaluncertainty,wewillcalculatetheapproximatedegreesoffreedomfromtheautocorrelationof10-mbinaveragedheights(5.3.3)
FromBrockwellandDavis[BrockwellandDavis,2016],consideringndiscretesamplesofh,themeansquarederrorofthesamplemean, ,fromthetruemean, ,is
where istheautocovarianceofhatlag,i.Intermsofdistance,foradistanceincrement,
,thelengthofrecord,L,equals andtaking
σ̂
surf2 =
σsurf2
NPtotal
σ̂
surf=
σsurf
NPtotal
≈σwaves
NPtotal
σ̂ surf
σ̂ surf
σ̂f
µh H
E µh −H( )2 = n−1 1−
in
⎛
⎝⎜⎜
⎞
⎠⎟⎟i=−n
n
∑ γ i( )
γ i( )
δ x nδ x λ = iδ x
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Theautocovarianceissymmetricwithlag,l,andisequaltotheautocorrelationofhtimesthevarianceinh.Therefore
Thiscorrespondsto(39)for:
(40)
Sothecorrelationlengthscale,Lscale,andeffectivedegreesoffreedomNPeffectbecome:
& . (41)&(42)
Equations(41)and(42)forLscaleandNP_effectarewrittenintermsofarecordlength,L.WecanthinkofthisalengthLinmeters.Itisequivalentandmoreconvenienttothinkoftherecordlengthintermsofthenumberof10-mbinsintheoceansegment,N=Nbin10,forwhich(41)and(42)become:
& . (43)&(44)
PseudoCodeforDeterminingNP_effectandh_uncrtnFirst,estimatethedegreesoffreedomusinganapproachinspiredbytheMatlabfunctiondegrees_freedom.mbyKathyKelly(personalcommunication,2020).
E µh −H( )2 = nδ x( )−1 1−iδ x
nδ x
⎛
⎝⎜⎜
⎞
⎠⎟⎟iδ x=−nδ x
nδ x
∑ γ iδ x( )
= L−1 1−λL
⎛
⎝⎜⎜
⎞
⎠⎟⎟λ=−L
L
∑ γ λ( )
E µh −H( )2 =2L−1 1−
λL
⎛
⎝⎜⎜
⎞
⎠⎟⎟λ=0
L
∑ γ λ( ) =σ h22L−1 1−
λL
⎛
⎝⎜⎜
⎞
⎠⎟⎟λ=0
L
∑ R λ( )
σ̂ surf =σ waves
NPeffect=
σ waves
L
2 1−λ
L
⎛⎝⎜
⎞⎠⎟λ=0
L
∑ R λ( )
Lscale = 1−
λ
L
⎛
⎝⎜⎞
⎠⎟λ=0
L
∑ R λ( )
NPeffect =L
2Lscale= L
2 1−λ
L
⎛⎝⎜
⎞⎠⎟λ=0
L
∑ R λ( )
Lscale = 1−
λ
N
⎛
⎝⎜⎞
⎠⎟λ=0
N
∑ R λ( )
NPeffect =L
2Lscale= L
2 1−λ
N
⎛⎝⎜
⎞⎠⎟λ=0
N
∑ R λ( )
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a)Computeasfunctionoflag,l,thelaggedautocorrelationvectorR_L(l)ofhtybin,whichisdefinedfor10-mbinswithcentersxbin.Lagswillbeincrementsof10-mbinindicesandextendfroml=0tolequalthenumberofbinsintheoceansegment,Nbin10=L/10m.(Note:Toconsiderlagsuptothelengthoftherecord,itmaybeadvisabletodoublethelengthofhtybinbyzeropaddingeachendNbin10/2.Thiswouldnotusuallybenecessarybecauseweshouldonlyneedtousethecovarianceoverthefirst10to20lags,butincaseswheretheoceansurfaceisnotdominatedbylongwaves,theautocorrelationmayneverbelessthanzeroanditmaybenecessarytoperformtheintegrationof(41)outtoalagequaltotherecordlength).LaggedcovarianceCOV_Latlaglwillbethesumoveralli=1tolength,Nbin10,oftheproducthtybin(i)*htybin(i+l).Whereeitherhtybin(i)orhtybin(i+l)doesnotexist,noadditionismadetothesum.Theautocorrelation,R_Lasafunctionoflag,l,equalsCOV_L(l)dividedbythecovariance,COV_L,atzerolagsotheR_Latzerolag,l=0,isequalto1,R_L(0)=1.b)Integrate(1-l/L)*R_Lfromzerolagtothefirstzerocrossingtogetdecorrelationscale,Lscale.StartwithLscale=0andati=1wherethecorrespondinglag,l,iszero,andR_L(i)=1.Incrementiandthecorrespondinglby1andkeepaddingtoLscalewhileR_L(i+1)isgreaterthanzero,i.e., Lscale=Lscale+((1-l/Nbin10)*R_L(i)+(1-(l+1)/Nbin10)*R_L(i+1))/2WhenR_L(i+1)becomeslessthanzero,terminatetheintegrationbyessentiallyassumingR_L(i+1)iszero,i.e.,
Lscale=Lscale+(1-l/Nbin10)*R_L(i))/2c)Degreesoffreedom,NP_effect,equalslengthoftheoceansegement,Nbin10,dividedbytwotimestheLscale.NotethatMatlabcodefromKathyKelly(personalcommunication,2020)thathasinformedthiscodedevelopmentdeletesthedivisionby2andthe(1-l/Nbin10)termintheintegrationandwillgivelargerdegreesoffreedomthanthisroutine.Thiscodeismoreconservative.d)h_uncrtnequalsthestandarddeviationofseasurfaceheightdividedbythesquarerootofNP_effect.
5.3.6.2 Uncertainties in the mean impulse response
Inourprocessingwe,accountfortheimpactoftheinstrumentimpulseresponse(IIP)onthedistributionsofsurfaceheightsbydeconvolvingfromthereceivedheightdistributionstheimpulseresponseestimatedfromthetransmitechopulse(TEP).TotheextentthattheTEPrepresentstheimpulseresponsethisshouldeliminateuncertaintydue
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totheIIP.UncertaintiesduetoamismatchbetweentheTEPandtrueIIPwillhavetobeinferredbyexaminationoftheTEPintheon-orbitdata,whichremainstobedone.
5.3.6.3 Uncertainties due to subsurface scattering.
Becausetheblue-greenlaserofATLASpenetrateswater,truesubsurfacereturnshavealwaysbeenaconcern,andthehighersubsurfacedensityofphotonsmaybedueinparttotruesubsurfacescatteringintheocean.However,weseesimilarlyenhancedsubsurfacedensitiesover,cleardeepoceanwatersandevenoverlandwherepenetrationandbackscattershouldn’toccur.ConsequentlypresentthinkingexpressedintheATL03KnownIssuesisthatthesubsurfacenoiselevelisduetoforwardscatteringdelaysintheatmosphereofsurfacereflectedphotons.
Whatevertheircause,somesubsurfacephotonheightsarebeingincludedinthesurfaceheighthistogram,creatingwhatwethinkmaybeanorder1-3cmbiasinaverageSSH.Toreducethesensitivitytosubsurfacereturns,theprocessingcodefirstmakesasimpleestimateofthehighandlowhistogramlimitsandthenusesthesetodetermineseparateabovesurfaceandsubsurfacenoiselevels,tailnoisehighandtailnoiselow.TheultimatehighlimitisthenchosenwherethesmoothedhistogramfallsbelowafactorTh_Nc_f(e.g.,Th_Nc_f=1.5)timestailnoisehighandlowlimitischosenwherethesmoothedhistogramfallsbelowthesamefactortimestailnoiselow.Also,forRelease4,thesurfacefinding,insteadofbeingbasedonthedistributionofphotonheights,isbasedonthedistributionofthephotonheightanomaliesrelativetoamoving11-pointbinaverageofhigh-confidencephotonheights.Thisexcludessubsurfacereturnsunderthecrestsofsurfacewavesobviatingtheimmediateneedforasubsurfaceretur.
5.4 Ancillary Information
5.4.1 Solar Background Photon Rate and Apparent Surface Reflectance (ASR) Seesection3.1.1.5.ThesolarbackgroundrateistakenfromATL03(backgrd_atlas/bckgrd_rate)andsimplyaveragedoverthelengthoftheheightsegmenttoproducebackgr_seg.TheapparentsurfacereflectanceshouldbetakenfromATL09(/profile_x/high_rate/apparent_surf_reflec)atthe25Hzrateandsimplyaveragedoverthelengthoftheheightsegmenttoproduceasr_seg.
5.4.2 Additional Ancillary Data VariablesfromtheATL03geolocation/andgeophys_corr/groupswillalsosimplybeaveragedovereachoceansegmentandsaved.Segmentaveragesofthefollowingvariables,correspondingtotheheightsstoredinAfinewillbereturnedinATL12.Thosevariablesareloctedin/gtx/statsgroupandinclude
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neutat_delay_totalref_azimuthref_elevseg_dist_xsolar_azimuthsolar_elevationgeoidgeoid_free2meandactide_earthtide_earth_free2meantide_loadtide_oc_poletide_oceantide_equilibrium
tide_polefull_sat_fractnear_sat_fract
Also,thefirstandlastofthegeosegmentids(segment_id)foreachoceansegmentwillbereturnedinATL12asfirst_geosegandlast_geoseg.Thepercentagesofeachsurf_typeofthephotonsfromFineSelectintheoceansegmentwillalsobereturnedassurf_type_prcnt.Thiswillbea5-elementvariableforeachoceansegmentwitheachelementcorrespondingtothepercentageofphotonsfromeachofthe5surfacetypes.Foreachoceansegment,weoutputlayer_flag_segasequalto1foroceansegmentswithmorethan50%ofgeosegshavinglayer_flag=1,indicatingsignificantcloudlayerswerepresent.Inaddition,wecomputecloudcover_percent_segforeachoceansegmentasthepercentageofgeosegsintheoceansegmentwithlayer_flag=1,i.e.,cloudcover_percent_seg=100*(totalnumberofgeosegsusedwithlayer_flag=1)/(totalnumberofgeosegsused).
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5.5 Output Parameters
Table 6 ATL12 Output (See Appendix A for full product specifications)
ProductLabel Units Description Symbolancillary_data/
ds_a
Dimensionscale(1:65)fortheharmoniccoefficients(a)inthegtx/ssh_segments/heightsgroup.a(1)isthecoefficientforwavenumberequalzero(constantpart),Abovea(1),a(evenindexi)isthesinecoefficientforwn(i/2),anda(oddindexi)isthecosinecoefficientforwn((i-1)/2)
a
ds_surf_type land(1),ocean(2),seaice(3),landice(4),inlandwater(5)
ds_wn
Dimensionscale(1:32)forthewavenumbers(wn)inthegtx/ssh_segments/heightsgroup.
wn
ds_xbin meters
Centerof1x710elementarrayof10-mbins.Notethismaybeincludedasadatadescriptionorotherstaticarrayequalto[5,15,25,35…..7095m]
xbin
ds_y_bincenters Meters -15to+15,by1cmbins Hist_bin_size meters BinsizeforYandsshx binsize
gtx/
gtx/ssh_segments/
delt_seg seconds Timedurationsegment
delta_time seconds Meantimeofsurfacephotonsinsegment t_seg
latitude degrees Meanlatitudeofsurfacephotonsinsegment lat_seg
longitude degrees Meanlongitudeofsurfacephotonsinsegment lon_seg
gtx/ssh_segments/heights
a meters Vectorof2xnharms+1coefficientsforeachharmonic a
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componentinharmonicanalysisofheights(5.3.3.2).a(1)isthecoefficientforwavenumberequalzero(constantpart),Abovea(1),a(evenindexi)isthesinecoefficientforwn(i/2),anda(oddindexi)isthecosinecoefficientforwn((i-1)/2)
bin_ssbias meters
Seastatebiasestimatedfromthecorrelationofphotonreturnratewithalongtrack10-mbinaveragedsurfaceheight(4.3.1)
binSSBias
dxbar m Meandistancebetweensurfacereflectedphotons DXbar
dxskew Skewnessofthedistributionofdistancebetweensurfacereflectedphotons
DXskew
dxvar m2Varianceofthedistributionofdistancebetweensurfacereflectedphotons
DXvar
h meters MeanseasurfaceheightrelativetotheWGS84ellipsoid SSH
h_kurtosis Excesskurtosisofseasurfaceheighthistogram SSHkurt
h_skewness Skewnessofphotonseasurfaceheighthistogram SSHskew
h_uncrtn mUncertaintyinthemeanseasurfaceheightoveranoceansegment
h_uncrtn
h_var meters2 Varianceofbestfitprobabilitydensityfunction(meters2) SSHvar
htybin meters1x710elementarrayof10-mbinaverageheightsfromSSBcalculation
htybin
l_scale
Correlationlengthscaleexpressedinunitsofnumberof10-mbins(maybefractional,i.e.,decimeters)
Lscale
latbind °N 1x710elementarrayof10-mbinaveragesoflatitude
latbind
length_seg meters Lengthofsegment(m) length_seg
lonbind °E 1x710elementarrayof10-mbinaveragesoflongitude
lonbind
meanoffit2 meters Meanoflinearfitremovedfromsurfacephotonheight meanoffit2
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mix_m1 Fractionofcomponent1in2-componentGaussianmixture m1
mix_m2 Fractionofcomponent1in2-componentGaussianmixture m2
mix_mu1 meters Meanofcomponent1in2-componentGaussianmixture mu1
mix_mu2 meters Meanofcomponent1in2-componentGaussianmixture mu2
mix_sig1meters Standarddeviationof
component1in2-componentGaussianmixture
Sig1
mix_sig2meters Standarddeviationof
component1in2-componentGaussianmixture
Sig2
n_pulse_seg Numberoflaserpulsesinsegment n_pls_seg
nbin10 Numberof10-mbinsinanoceansegment Nbin10
np_effect Effectivedegreesoffreedomoftheaverageseasurfaceheightfortheoceansegment
NP_effect
p0 meters
Constantoflinearfitversusalong-trackdistancetosurfacephotonheightovertheoceansegment
PO
p1
Slopeoflinearfitversusalong-trackdistancetosurfacephotonheightovertheoceansegment
P1
snr_harm
Signaltonoiseratioofharmonicfitwithcoefficientsinatothesurfacereflectedphotonsincludingmeanoffit2andwithdatagapsgreaterthangaplimitfilledwithGaussianwhitenoiseaboutmeanoffit2
SNR_harm
swh meters
Significantwaveheightestimatedas4timesthestandarddeviationofalongtrack10-mbinaveragedsurfaceheight
SWH
wn
meter-
1
nharms(=32)wavenumbersequaltotheinverseofwavelengthsforeachharmoniccomponentinharmonicanalysisofheights(5.3.3.2)
wn
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bind meters1x710elementarrayof10-mbinaveragesofalong-trackdistance
xbind
xrbin Photonm-1
1x710elementarrayof10-mbinaveragephotonratefromSSBcalculation
xrbin
y meters-1
Probabilitydensityfunctionofphotonsurfaceheight Y
ykurt ExcessKurtosis=(fourthmomentofY)/Yvar2-3 Ykurt
ymean meters Mean=firstmomentofY,shouldbe~0=h-meanoffit2 Ymean
yskew Skewness=(thirdmomentofY)/Yvar3/2 Yskew
yvar meters2 Variance=secondmoment(m2)ofY Yvar
gtx/ssh_segments/stats
backgr_seg m-1backgrd_atlas/bckgrd_ratefromATL03averagedoverthesegment
backgr_seg
cloudcover_percent_seg %Thepercentageofgeosegsintheoceansegmentwithlayer_flag=1.
cloudcover_percent_seg
dac_seg meters
Oceansegmentaverageofdynamicatmosphericcorrection(DAC)includesinvertedbarometer(IB)affect
dac_seg
depth_ocn_seg metersTheaverageofdepthoceanofgeo-segmentsusedintheoceansegment.
depth_ocn_seg
first_geoseg Thefirstofthegeosegmentids(segment_id)foreachoceansegment
first_geoseg
first_pce_mframe_cnt FirstMajorFrameIDintheSSHsegment first_pce_mframe_cnt
first_tx_pulse FirstTransmitpulseinalong-tracksegment first_tx_pulse
fpb_corr meters
Estimatedfirst-photonbiascorrectiontomeansegmentheightcurrentlysettoINVALID.See3.1.1.4.
fpb_corr
fpb_corr_stdev metersEstimatederrorinfpb_corrcurrentlysettoINVALID.See3.1.1.4.
fpb_corr_stdev
full_sat_fract_seg fractionofallpulsesinallgeosegsusedthatwerefullysaturated
full_sat_fract_seg
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geoid_free2mean_seg meters
OceansegmentaverageoftheconversionfactoraddedtoATL03geoidtoconvertfromthetide-freetothemean-tidesystem
geoid_free2mean_seg
geoid_seg meters
Oceansegmentaverageofgeoidinthemean-tidesystemheightabovetheWGS–84referenceellipsoid(range-107to86m)
geoid_seg
last_geoseg Thelastofthegeosegmentids(segment_id)foreachoceansegment
last_geoseg
last_pce_mframe_cnt LastMajorFrameIDintheSSHsegment Last_pce_mframe_cnt
last_tx_pulse LastTransmitpulseinalong-tracksegment last_tx_pulse
layer_flag_seg
ThelayerflagfromATL09thatisineffectover50%oftheoceansegment,0indicatingabsenceofcloudsandforwardscattering,and1indicatingpossibilityofforwardscatteringasinATL09
layer_flag_seg
n_photons Numberofsurfacephotonsfoundforthesegment n_photon
n_ttl_photon Numberofphotonsinthe±15-moceandownlinkband n_ttl_photon
near_sat_fract_seg fractionofallpulsesinallgeosegsusedthatwerenearlysaturated
near_sat_fract_seg
neutat_delay_total_seg metersOceansegmentaverageoftotalneutralatmospheredelaycorrection(wet+dry)
neutat_delay_total_seg
orbit_number UniqueidentifyingnumberforeachplannedICESat-2orbit orbit_number
photon_noise_rate m-1 Noisephotoncountrate,averagedoverthesegment rnoise
photon_rate m-1 Photoncountrate,averagedoverthesegment rsurf
ref_azimuth_seg deg
OceansegmentaverageofazimuthoftheunitpointingvectorforthereferencephotoninthelocalENUframeinradians.TheangleismeasuredfromNorthandpositivetowardsEast
ref_azimuth_seg
ref_elev_seg degOceansegmentaverageofelevationoftheunitpointingvectorforthereference
ref_elev_seg
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photoninthelocalENUframeinradians.TheangleismeasuredfromtheEast-NorthplaneandpositivetowardsUp
seg_dist_x_seg meters
Oceansegmentaverageofthealong-trackdistancefromtheequatorcrossingtothestartofthe20-mgeolocationsegmentsincludedintheoceansegment
seg_dist_x_seg
solar_azimuth_seg deg
OceansegmentaverageoftheazimuthofthesunpositionvectorfromthereferencephotonbouncepointpositioninthelocalENUframe.TheangleismeasuredfromNorthandispositivetowardsEast.Theaverageisprovidedindegrees.
solar_azimuth_seg
solar_elevation_seg deg
OceansegmentaverageoftheelevationofthesunpositionvectorfromthereferencephotonbouncepointpositioninthelocalENUframe.TheangleismeasuredfromtheEast-NorthplaneandispositivetowardsUp.Theaverageisprovidedindegrees.
solar_elevation_seg
ss_corr meters
Subsurfacescatteringcorrection,placeholder=INVALIDpendingfurtherfindingstothecontrary
ss_corr
ss_corr_stdev meters2
Estimatederrorofsubsurfacescatteringcorrection,placeholder=INVALIDpendingfurtherfindingstothecontrary
ss_corr_stdev
surf_type_prcnt
Thepercentagesofeachsurf_typeofthephotonsintheoceansegmentasa5-elementvariablewitheachelementcorrespondingtothepercentageofphotonsfromeachofthe5surfacetypes
surf_type_prcnt
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tide_earth_free2mean_seg meters
Oceansegmentaverageofthefactorthatneedstobeaddedtotide_earth_segtoconvertfromthetide-freetothemean-tidesystem.ThisisnotusedinATL12.
tide_earth_free2mean_seg
tide_earth_seg metersOceansegmentaverageofsolidearthtidesinthetide-freesystems
tide_earth_seg
tide_equilibrium_seg meters Longperiodequilibriumtides tide_equilibrium_seg
tide_load_seg metersOceansegmentaverageoflocaldisplacementduetooceanloading(-6to0cm)
tide_load_seg
tide_oc_pole_seg meters
Oceansegmentaverageofoceanicsurfacerotationaldeformationduetopolarmotion(-2to+2mm)
tide_oc_pole_seg
tide_ocean_seg meters
Oceansegmentaverageofoceantidesincludingdiurnalandsemi-diurnal(harmonicanalysis)
tide_ocean_seg
tide_pole_seg meters
Oceansegmentaverageofsolidpoletide-Rotationaldeformationduetopolarmotion(-1.5to1.5cm)
tide_pole_seg
5.6 Gridding DOT for ATL19.
TheATL19productincludesgriddedmonthlyestimatesofdynamicoceantopography(DOT)takenfromATL12oceansegmentdata.Oceansegmentsrangeinlengthroughlyfrom3to7km.Theoceansegmentdataareaveragedin¼°or25kmgridcells.Datafromallsixbeams(gtxrandgtxl)areusedfromthebeginningtotheendofeachmonth,althoughcommonlyovermostoftheoceanonlydatafromthestrongbeamswillbeavaialable.TheIS-2dataareaveragedontothreegrids,calledmid-latitude,north-polarandsouth-polar.TheATL19producthasgroupswiththesamenamescontainingthegriddeddataforeachofthoseregions.Thegrids,griddingschemes,inputvariablesandoutputvariablesaredescribedindetailintheATBDforATL19.
5.7 Synthetic Test Data
Thefirsttaskinthecreationofthedevelopmentalsoftwarehasbeenthecreationofasyntheticdataset.ThisisnotpartofATLASprocessingperse,butisagoodtoolforalgorithmtestinganddevelopment.TheprocedureisimbeddedinadevelopmentalMatlabcode(WienerTest_GaussMixandNoise2.m).Here,thepercentsignindicatesaplanetextdescriptionoftheproceduralstepandisanon-
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executablecommentintheMatlabcode.Matlabexecutablelinesarehighlightedinyellow.Theprogrambeginswithparts(A)through(E)whichestablishanartificialsurfaceheighthistogramconsistingofamixtureof2normaldistributionsconvolvedwitharealimpulseresponsedistributiontakenfromMABELdatabyRonKwok.Arepresentativeamountofnoiseisthenaddedtothistosynthesizeareceivedsurfaceheightwithaknownunderlyingtruesurfacedistribution.
(A)EstablishtheinstrumentimpulseresponsedistributionfromMABELdata(XmitHist000)(B)MakeaGaussianmixturerepresentingthesshdistribution(sshhist)
(C)Computenoise(Noise)astherRMSdifferencefromtheanalyticGaussianmixtureandsshistinthetails(+-2sigto+-3sig)(D)ConvolvetheSSHdistributionwiththeinstrumentimpulseresponsedistributiontoproducethereceiveddistribution(rechist)(E)AddrandomPoisson(orpulse)noiseduetodiscretenatureofphotoncountsrepresentativeofwhatwemightexpectforagivenbinsizeandtotalnumberofphotoncounts.%------------------------------------------------------------------------------------
%BeginMatlabcode:%Inthistestprogramwewill:%(A)EstablishtheinstrumentimpulseresponsedistributionfromMabeldata(XmitHist000)%(B)MakeaGaussianmixturerepresentingthesshdistribution(sshhist)%(C)Computenoise(Noise)asthermsdifferencefromtheanalyticgaussianmixture%andsshistinthetails(+-2sigto+-3sig)%(D)Convolvethesshditributionwiththeinstrumentimpulseresponsedistributiontoproduce%thereceiveddistribution(rechist)%(E)AddrandomPoisson(orpulse)noiseduetodiscretenatureofphoton%countsrepresentativeofwhatwemightexpectforagivenbinsizeand%totalnumberofphotoncounts(NptsMix)%photons.%%---------------------------------------------------------%(A)loadandplotInstrumentimpulseresponseHistogram%---------------------------------------------------------%---------------------------------------------------------cd/Users/jamie/Documents/ICESat-2_SDT/ICESat-2_OceanATBD_Development/PhotonSourceDistributionloadXmitHist000meanXmit00=sum(XmitHist000(:,1).*XmitHist000(:,2))/sum(XmitHist000(:,2))
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stdXmit00=sqrt(sum(XmitHist000(:,1).*XmitHist000(:,1).*XmitHist000(:,2))/sum(XmitHist000(:,2)))%XmitHist000isarawhistogramandXmitHist000(:,2)arethetotalnumberofphotonsineach%0.01-mbinconvert.Weconvertittoaprobabilitydensityfunction,theintegral%ofwhichis1.0.Binsizeinallhistograms/pdfswillbe0.01m.%XmitHist000binsarerelativetothemeandelayiszero.binsize=0.01;totalxmit=sum(XmitHist000(:,2));XmitHist_fraction=(XmitHist000(:,2)/totalxmit);XmitHist000(:,2)=XmitHist_fraction/binsize;checkintegralXMIT=sum(XmitHist000(:,2)*binsize);meanXmit00=sum(XmitHist000(:,1).*XmitHist000(:,2))/sum(XmitHist000(:,2))stdXmit00=sqrt(sum(XmitHist000(:,1).*XmitHist000(:,1).*XmitHist000(:,2))/sum(XmitHist000(:,2)))%plotfigureplot(XmitHist000(:,1),XmitHist000(:,2))title('XmitPulseDelayReltoMeaninEquivSurfaceHeight')ylabel('OccurencesasFractionofTotalpermeter')xlabel('HeightDelay(m)')print-dpngXmitHistogram_GM_8k.png%------------------------------%(B)Makeasequenceseasurfaceheightconistingofamixoftwonormally%distributedrandomnumberswithMeansMuMix,standarddeviationsSigmaMix%andpercentmixingratiosRatioMix,andnumberofpointstotalNptsMix%---------------------------------------------------------%---------------------------------------------------------MuMix=[1,0]SigmaMix=[2,1]RatioMix=[50,50]NptsMix=8000ssh=makeGaussianMix(MuMix,SigmaMix,RatioMix,NptsMix);meanssh=mean(ssh)MeanfromMixValues=sum(RatioMix.*MuMix/sum(RatioMix))stdssh=std(ssh)StdfromMixValues=sqrt(sum(RatioMix.*(SigmaMix.^2)/sum(RatioMix)))lssh=length(ssh)%Computehistogramofsshusingbinsize=0.01mrunningfrom-3Stdto+3
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%Stdofthesshmix.Theoutputofhistisarawhistogramandweconvertitto%aprobabilitydensityfunction,theintegralofwhichis1.0bydividingbythe%totalnumberofpoints(=lengthofssh)andthebinsize%Nsigs=3;sshx=round(meanssh)-Nsigs*round(stdssh):binsize:round(meanssh)+Nsigs*round(stdssh);sshhist=hist(ssh,sshx);sshhist_fraction=(sshhist/lssh);sshhist=sshhist_fraction/binsize;checkintegralSSH=sum(sshhist*binsize)%meansshhist=sum(sshx.*sshhist)/sum(sshhist)stdsshhist=sqrt(sum((sshx-meansshhist).*(sshx-meansshhist).*sshhist)/sum(sshhist))totalssh=sum(sshhist)Nbins=length(sshhist);%---------------------------------------------------------%(C)Computenoise(Noise)asthermsdifferencefromtheanalyticGaussianmixture%(gaussfit)andsshistinthetails(+-2sigto+-3sig)%---------------------------------------------------------%---------------------------------------------------------gaussfit=(1/stdssh)*(1/sqrt(2*pi))*exp(-0.5*((sshx-meanssh).^2)/stdssh^2);NoiseAll=std(sshhist-gaussfit)%becausegaussfitisapdfwedividehistogramvaluesbybinsizefor%comparison.Noisevaluesshouldprobablybemutltipliedbybinsizefor%computatiuionasnoiseinhistograms?Noise=binsize*NoiseInTails?DiffFromNorm=(sshhist)-gaussfit;DiffFromNormInTails=[DiffFromNorm(1:round(0.5*Nbins/Nsigs)),DiffFromNorm(end-round(0.5*Nbins/Nsigs))];NoiseInTails=std(DiffFromNormInTails')figureplot(sshx,sshhist,sshx,gaussfit,'--g')title(['SimulatedTrueSeaSurfaceHeightDistribution,2-GaussMix',num2str(NptsMix),'Photons'])ylabel('OccurencesasFractionofTotalpermeter')xlabel('Height(m)')print-dpngSim_SSH_Distriburtion_GM_8k.png%---------------------------------------------------------
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%(D)Convolvetheinstrumentimpulseresponseandsimulatedtrueheighthistograms%togetasimulatedreceivedheightsshdistributionrcvhist%---------------------------------------------------------%---------------------------------------------------------%--------------MatlabRoutineconv.m------------------%convConvolutionandpolynomialmultiplication.%C=conv(A,B)convolvesvectorsAandB.Theresultingvectoris%lengthMAX([LENGTH(A)+LENGTH(B)-1,LENGTH(A),LENGTH(B)]).IfAandBare%vectorsofpolynomialcoefficients,convolvingthemisequivalentto%multiplyingthetwopolynomials.%%C=conv(A,B,SHAPE)returnsasubsectionoftheconvolutionwithsize%specifiedbySHAPE:%'full'-(default)returnsthefullconvolution,%'same'-returnsthecentralpartoftheconvolution%thatisthesamesizeasA.%'valid'-returnsonlythosepartsoftheconvolution%thatarecomputedwithoutthezero-paddededges.%LENGTH(C)isMAX(LENGTH(A)-MAX(0,LENGTH(B)-1),0).%----------------------------------------------------------------------%NOTE:conv.mdoesarawconvolution,i.e.,simplesumsofproducts.To%replicateatrueconvolutionintegralwithproperscaling,theresultsof%conv.mmustbemultipliedbythebinsizesothatthescaleofthe%resultingconvolutionisofthesameorderasthescaleonthetwo%histogramsconvolved.%Alsothelengthoftheconvolutionshouldequallengthofthessh%histogramplusthelengthXmithistogramminus1%-----------------------------------------------------------------------ticrcvhist=conv(sshhist,XmitHist000(:,2)')*binsize;cleanrcvhist=rcvhist;toclrcvhist=length(rcvhist)lxmithist=length(XmitHist000(:,2))lsshhist=length(sshhist)halfwidthrcvhist=(lxmithist+lsshhist-2)/2if2*halfwidthrcvhist+1==lrcvhistmessage=['lengthofrcvhistisconsistent']end
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%Iftheimpulseresponsedistributionweresymmetricconv.mwouldpadthelengthofthesshhistby(lxmithist-1)/2ateachend.TheXmithistogramisnotsymmetric.Theminimuimx-axisindexofthereceivedhistogramshouldbetheminimumindexofthetruesurfaceheighthistogramplus%theminimumnegativebinindexoftheimpulseresponsehistogramandthemaximux-axis%indexofthereceivedhistogramshouldbethemaximumindexofthetruehistogramplus%themaximu(positive)binindexoftheXmithistogram.%Therefore,wefindtheindexofthezerobin,zbin,%ofXmitHist000(whichistheoldXmitHist00histograminterpolatedtoevencmbinindices)and%addthebinswithindex0tozbin-1beforethefirstbinofthesshhistogramand%addthebinswithindexzbin+1toendafterthelastbinofthesshhistogramzbin=find(XmitHist000(:,1)==0);xaddend=XmitHist000(zbin+1:end,1)'+sshx(end)*ones(1,length(XmitHist000(zbin+1:end,1)));xaddbegin=XmitHist000(1:zbin-1,1)'+sshx(1)*ones(1,length(XmitHist000(1:zbin-1,1)));xrechist=[xaddbegin,sshx,xaddend];figureplot(xrechist,rcvhist)title('SimulatedConvolvedSeaSurfaceHeightDistribution')ylabel('OccurencesasFractionofTotalpermeter')xlabel('Height(m)')%----------------------------------------------------------------%----------------------------------------------------------------%----------------------------------------------------------------%(E)AddrandomPoisson(orpulse)noiseduetodiscretenatureofphoton%countsrepresentativeofwhatwemightexpectforagivenbinsizeand%totalnumberofphotoncounts(NptsMix)%----------------------------------------------------------------sumrchhist=sum(rcvhist*binsize)noisey=rcvhist;%makePoissondistributedphotoncountsinhistogramtorepesentreceived+pulsenoiseforinoise=1:length(rcvhist)noisey(inoise)=randpoisson_JM(NptsMix*binsize*cleanrcvhist(inoise));
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end
%ConvertbacktoPDFNoise=std(noisey/(NptsMix*binsize)-rcvhist);rcvhist=noisey/(NptsMix*binsize);sumnoiseyrchhist=sum(rcvhist*binsize)figureplot(xrechist,rcvhist)title(['SimulatedReceivedSeaSurfaceHeightDistribution+PulseNoise=',num2str(Noise)])ylabel('OccurencesasFractionofTotalpermeter')xlabel('Height(m)')%----------------------------------------------------------------%rcvhististhesimulatedreceivedhistogramatthehistogrambinscenteredatxrechist.ItisshowninblueinFigure17for8000photons(left)and800photons(right).
%NOTE%%randpoisson_JM.misaMatlabsubroutinetogeneraterandomnumbersdrawn%fromaPoissondistribution:functionX=randpoisson_JM(np)x=1:round(np+1)*10;p=poisspdf(x,np);m=1;X=zeros(m,1);%Preallocatememoryfori=1:mu=rand;I=find(u<cumsum(p));ifisempty(I)X(i)=0;elseX(i)=min(I);endend%poisspdfPoissonprobabilitydensityfunction.%Y=poisspdf(X,LAMBDA)returnsthePoissonprobabilitydensity%functionwithparameterLAMBDAatthevaluesinX.%ThesizeofYisthecommonsizeofXandLAMBDA.Ascalarinput%functionsasaconstantmatrixofthesamesizeastheotherinput.%NotethatthedensityfunctioniszerounlessXisaninteger.
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5.8 Numerical Computation Considerations
TBD–asneededspecificconsiderationsonmethodofcodecomputation
5.9 Programmer/Procedural Considerations
TBD-provideinformationrelatedtooutputparametersthatwerenotinthealgorithmdescription
5.10 Calibration and Validation
Therearethreetypesofopenoceancalibrationandvalidation:1) Direct comparison of sea surface height or dynamic ocean topography with satellite radar
altimetry from TOPEX/Poseidon and CryoSat-2. This may be possible to automate so that the project personnel can do it, but funding is being sought outside the ICESat-2 and NASA Cryosphere program for Cal/Val.
2) Comparison of changes in DOT with in situ measurements of dynamic heights from hydrography plus ocean bottom pressure from in situ gauges or GRACE-FO. This will have to be done for the open ocean perhaps funded outside the Cryosphere program.
3) Direct comparison to in situ precision GPS measurements. Buoys are being built with NOPP funding to do this as an add-on to similar measurements for NASA Oceanography’s SWOT cal/val effort.
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6.0 BROWSE PRODUCTS Thesebrowsefiguresshowimagesthatallowusertoeasilyevaluateifthedatawouldbeusefulandofqualityfortheirresearchandwillaidinthequickapprovalordisapprovalofproductspriortopublicdistribution.
6.1 Data Quality Monitoring
6.1.1 Line plots (each strong beam)
a) meanseasurfaceheightforeachsegment b) standarddeviationofheightdistributionandassociatedSWHforeachsegment c) skewnessofheightdistributionforeachsegment d) kurtosisofheightdistributionforeachsegment
6.1.2 Histograms (each strong beam)
a) Lineplotsofparametersof2-Gaussianfitforeachsegment b) TBDqualitymeasureofeach2-Gaussianfit
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7.0 DATA QUALITY
7.1 Statistics
Calculate the following statistics for each open ocean granule and broken down by strong beam ground track – so there is one set for each ground track. • Aggregate histogram and mean/standard deviation of each of the following parameters
7.1.1 Per orbit statistics
Table 7 ATL12 Output Variables for Per Orbit Statistics
ProductLabel Description Symbol
sea_surface_heights h Meanseasurfaceheight SSH
h_var Varianceofbestfitprobabilitydensityfunction(meters2) SSHvar
h_skewness Skewnessofphotonseasurfaceheighthistogram SSHskewh_kurtosis Excesskurtosisofseasurfaceheighthistogram SSHkurt
mix_m1 Fractionofcomponent1in2-componentGaussianmixture m1
mix_mu1 Meanofcomponent1in2-componentGaussianmixture mu1
mix_sig1 Standarddeviationofcomponent1in2-componentGaussianmixture Sig1
mix_m2 Fractionofcomponent1in2-componentGaussianmixture m2
mix_mu2 Meanofcomponent1in2-componentGaussianmixture mu2
mix_sig2 Standarddeviationofcomponent1in2-componentGaussianmixture Sig2
delt_seg Timedurationsegment t_seglat_seg Meanlatitudeofsurfacephotonsinsegment lat_seglon_seg Meanlongitudeofsurfacephotonsinsegment lon_seglength_seg Lengthofsegment(m) length_seg
P0 Constantoflinearfitversusalong-trackdistancetosurfacephotonheightovertheoceansegment P0
slope_seg Slopeoflinearfitversusalong-trackdistancetosurfacephotonheightovertheoceansegment P1
n_pulse_seg Numberoflaserpulsesinsegment n_pls_segmeanoffit2 Meanoflinearfitremovedfromsurfacephotonheight meanoffit2Y Probabilitydensityfunctionofphotonsurfaceheight YYmean Mean=firstmomentofY,shouldbe~0=h-meanoffit2 YmeanYvar Variance=secondmoment(m2)ofY Yvar
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Yskew Skewness=(thirdmomentofY)/Yvar3/2 YskewYkurt ExcessKurtosis=(fourthmomentofY)/Yvar2-3 Ykurt
binSSBiasSeastatebiasestimatedfromthecorrelationofphotonreturnratewithalongtrack10-mbinaveragedsurfaceheight(4.3.1)
binSSBias
SWHSignificantwaveheightestimatedas4timesthestandarddeviationofalongtrack10-mbinaveragedsurfaceheight
SWH
Height_segment_stat
n_ttl_photon Numberofphotonsinthe±15-moceandownlinkband n_ttl_photon
n_photons Numberofsurfacephotonsfoundforthesegment n_photonPhoton_rate Photoncountrate,averagedoverthesegment rsurfPhoton_noise_rate Noisephotoncountrate,averagedoverthesegment rnoise
backgr_seg backgrd_atlas/bckgrd_ratefromATL03averagedoverthesegment backgr_seg
first_pce_mframe_cnt FirstMajorFrameIDintheSSHsegment first_pce_mframe_cntfirst_tx_pulse FirstTransmitpulseinalong-tracksegment first_tx_pulse
layer_flag_seg
ThelayerflagfromATL09thatisineffectover50%oftheoceansegment,0indicatingabsenceofcloudsandforwardscattering,and1indicatingpossibilityofforwardscatteringasinATL09
layer_flag_seg
depth_ocn_seg Theaverageofdepth_ocnofgeo-segmentsusedintheoceansegment. depth_ocn_seg
orbit_number= UniqueidentifyingnumberforeachplannedICESat-2orbit orbit_number
ss_corr Subsurfacescatteringcorrection,placeholder=zeropendingfurtherfindingstothecontrary ss_corr
ss_corr_stdevEstimatederrorofsubsurfacescatteringcorrection,placeholder=zeropendingfurtherfindingstothecontrary
ss_corr_stdev
neutat_delay_total_seg
Oceansegmentaverageoftotalneutralatmospheredelaycorrection(wet+dry)
neutat_delay_total_seg
seg_dist_x_seg Oceansegmentaverageofthealong-trackdistancefromtheequatorcrossingtothestartofthe20-mgeolocationsegmentsincludedintheoceansegment
seg_dist_x_seg
solar_azimuth_seg
OceansegmentaverageoftheazimuthofthesunpositionvectorfromthereferencephotonbouncepointpositioninthelocalENUframe.TheangleismeasuredfromNorthandispositivetowardsEast.Theaverageisprovidedindegrees.
solar_azimuth_seg
solar_elevation_seg
OceansegmentaverageoftheelevationofthesunpositionvectorfromthereferencephotonbouncepointpositioninthelocalENUframe.TheangleismeasuredfromtheEast-NorthplaneandispositivetowardsUp.Theaverageisprovidedindegrees.
solar_elevation_seg
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geoid_seg OceansegmentaverageofgeoidheightabovetheWGS–84referenceellipsoid(range-107to86m)
geoid_seg
dac_seg Oceansegmentaverageofdynamicatmosphericcorrection(DAC)includesinvertedbarometer(IB)affect
dac_seg
tide_earth_seg Oceansegmentaverageofsolidearthtidesinthetide-freesystem
tide_earth_seg
tide_load_seg Oceansegmentaverageoflocaldisplacementduetooceanloading(-6to0cm)
tide_load_seg
tide_oc_pole_seg Oceansegmentaverageofoceanicsurfacerotationaldeformationduetopolarmotion(-2to+2mm)
tide_oc_pole_seg
tide_ocean_seg Oceansegmentaverageofoceantidesincludingdiurnalandsemi-diurnal(harmonicanalysis)andlongerperiodtides(dynamicandself-consistentequilibrium)
tide_ocean_seg
tide_equilibrium_seg Longperiodequilibriumtides
tide_equilibrium_seg
tide_pole_seg Oceansegmentaverageofsolidpoletide-Rotationaldeformationduetopolarmotion(-1.5to1.5cm)
tide_pole_seg
surf_type_prcnt Thepercentagesofeachsurf_typeofthephotonsintheoceansegmentasa5-elementvariablewitheachelementcorrespondingtothepercentageofphotonsfromeachofthe5surfacetypes
surf_type_prcnt
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8.0 TEST DATA
This section describes the test data sets that have been used to derive and verify the performance of the ATL12 code for surface finding, deconvolution of the instrument impulse response, and Gaussian mixture determination.
8.1 In Situ Data Sets
Table 8 ATL12 Test Data
Instrument Date Location MABEL April 2012 Fram Strait and Greenland
Sea MABEL July 2014 North Pacific
8.2 Simulated Test Data
DevelopmentofATL12usedthesimulateddatasetdescribedinSection5.6SyntheticData.Forthesesyntheticdatawestartedwithasyntheticseasurfaceheightrecordcomprisedofaknown2-Gaussianmixturethatweconvolvedwithaknowninstrumentimpulseresponseandaddedrandommeasurementnoise.WeexercisedthedevelopmentalMatlabcodeandtheASASPGEcodeonthisdatatoseethattheyoutputthesame2-Gaussianmixturethatwasthebasisofthesyntheticoceansurface
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9.0 CONSTRAINTS, LIMITATIONS, AND ASSUMPTIONS
In this section, we list notable constraints, limitations, and assumptions important to ATL12 that affect the coverage, quality and interpretation of the sea surface height retrievals and our objectives for the ATL12. These topics have been discussed throughout the ATBD. In order to compare with other open ocean topography measures, the overarching objective for the ATL12 product is that it should be a solid description of sea surface height probability distribution over each strong beam ground track (and weak beam ground track where available). This objective and the constantly changing wave-covered character of the ocean surface impose the primary constraint on the ATL12 product. To achieve standard errors in mean sea surface height better than 1-cm under typical sea states, we accumulate the distributions over distances of 5 to 7 km. This is far longer than the ultimate spatial resolution of the ICESat-2 instrument, which we in fact use in the ATL12 analysis for sea state bias estimation. Users can aggregate the ATL 12 distributions to obtain greater statistical significance, and we do this to produce the ocean gridded product, ATL19. Where applicable we and other users can go back to ATL03 photon heights and derive specialized ocean products using ATL12 statistics as a guide and benchmark.
9.1 Constraints
The following are constraints imposed by the inherent capability of the instrument. • At 532 nm, clouds will affect visibility of the open ocean. First impressions looking at preliminary ICESat-2 data the lidar appears able detect the ocean surface through at least thin clouds better than expected. • The reflectance of the ocean surface is lower than the other surfaces ICESat-2 operates over with the result that only on the order of one photon per pulse is returned from the surface. Thus, longer path lengths are required to accumulate desired statistical significance of ocean height measurements than would be required for other surfaces. On the other hand, we don’t have to worry about detector saturation and first photon bias over the ocean.
9.2 Limitations
These limitations stem from on our current understanding of the altimetric returns from the sea ice cover. • Subsurface Scattering - Quantification of the impact of subsurface scattering on height retrievals due to multiple scattering from bubbles or particles in the water remains to be addressed. For the central parts of the deep ocean, these effects are likely reduced by the scarcity of scatterers in the water. We might also expect that just as surface waves reduce the reflectance of the surface, they may also reduce the transmittance of subsurface returns up through the ocean surface, a benefit offsetting the generation of bubbles by wind waves. Preliminary analysis of ICESat-2 deep ocean data suggests long, but low energy negative tails in the raw height distribution that are mainly removed in the trimming of the histogram in the fine surface finding step. Subsurface scattering will be most important in coastal regions where sediment and biological productivity increase the density of scatterers. Gaining a better understanding the subsurface return problem will be a high priority for future ICESat-2 oceanography.
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• Sea State Bias – ICESat-2 will be unique in estimating what we think will be the main source of sea state bias by calculating the correlation of photon return rate and surface height at scales resolving the energy containing surface waves. This constitutes what is termed EM-bias in other altimeters. Because we don’t use retracking in the traditional sense or assume a Gaussian surface distribution, we anticipate ICESat-2 will not suffer from sea state bias due to these factors, and the ICESat-2 sea state bias is wholly due to EM-bias. We will assess this assumption as part of the validation process.
9.3 Assumptions
These are assumptions made by the retrieval routines that will have to be tested. • Ocean Surface Height retrievals – As discussed above key assumptions are that our fine surface finding clips off the negative histogram tail due to subsurface return and that what is commonly termed EM-bias is the sole source of sea state bias in our analysis of ICESat-2 over the ocean.
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10.0 REFERENCES
Arnold,D.V.,W.K.Melville,R.H.Stewart,J.A.Kong,W.C.Keller,andE.Lamarre(1995),MeasurementsofelectromagneticbiasatKuandCbands,J.Geophys.Res.,100(C1),969-980.
Carrère,L.,andF.Lyard(2003),Modelingthebarotropicresponseoftheglobaloceantoatmosphericwindandpressureforcing-comparisonswithobservations.
Chambers,D.P.,S.A.Hayes,J.C.Ries,andT.J.Urban(2003),NewTOPEXseastatebiasmodelsandtheireffectonglobalmeansealevel,J.Geophys.Res.,108(C10),3305.
Chapron,B.,V.Kerbaol,D.Vandemark,andT.Elfouhaily(2000),Importanceofpeakednessinseasurfaceslopemeasurementsandapplications,J.Geophys.Res.,105(C7),17195-17202.
Cox,C.,andW.Munk(1954),MeasurementoftheRoughnessoftheSeaSurfacefromPhotographsoftheSun?sGlitter,J.Opt.Soc.Am.,44(11),838-850.
Elfouhaily,T.,D.R.Thompson,B.Chapron,andD.Vandemark(2000),Improvedelectromagneticbiastheory,J.Geophys.Res.,105(C1),1299-1310.
Gaspar,P.,F.Ogor,P.-Y.LeTraon,andO.-Z.Zanife(1994),EstimatingtheseastatebiasoftheTOPEXandPOSEIDONaltimetersfromcrossoverdifferences,J.Geophys.Res.,99(C12),24981-24994.
Hausman,J.,andV.Zlotnicki(2010),SeaStateBiasInRadarAltimetryRevisited,,MarineGeodesy,33(S1),336---347.
Markus,T.,etal.(2016),TheIce,Cloud,andlandElevationSatellite-2(ICESat-2):Science1requirementsconcept,andimplementation.,RemoteSensingoftheEnvironment,inpress.
Menzies,R.T.,D.M.Tratt,andW.H.Hunt(1998),LidarIn-SpaceTechnologyExperimentMeasurementsofSeaSurfaceDirectionalReflectanceandtheLinktoSurfaceWindSpeed,Appl.Opt.,37(24),5550-5559.
Plant,W.J.(2015a),Shortwindwavesontheocean:Wavenumber-frequencyspectra,J.Geophys.Res.Oceans,120,2147–2158.
Plant,W.J.(2015b),Shortwindwavesontheocean:Long-waveandwindspeeddependences,J.Geophys.Res.Oceans,120,6436–6444.
Urban,T.J.,andB.E.Schutz(2005),ICESatsealevelcomparisons,Geophys.Res.Lett.,32(23),L23S10.
Vandemark,D.,B.Chapron,T.Elfouhaily,andJ.W.Campbell(2005),Impactofhigh-frequencywavesontheoceanaltimeterrangebias,J.Geophys.Res.,110(C11),C11006.
Wu,J.(1972),Sea-SurfaceSlopeandEquilibriumWind-WaveSpectra,PhysicsofFluids,15(5),741-747.
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ACRONYMS
AcronymListASA ATLASScienceAlgorithmSoftwareATLAS ATLASAdvanceTopographicLaserAltimeterSystemGSFC GoddardSpaceFlightCenterICESat-2MIS ICESat-2ManagementInformationSystemIIP InstrumentImpulseResponseMIZ MarginalIceZonePSO ProjectScienceOfficePSO ICESat-2ProjectSupportOfficeSDMS SchedulingandDataManagementSystemSIPS ScienceInvestigator-ledProcessingSystemTEP TransmitEchoPulse
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GLOSSARY
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APPENDIX A: ICESat-2 Data Products
ICESat-2DataProducts
File ID/Level
Product Name Concept Short Description Frequency
00/0 Telemetry Data Full rate Along-track with channel info
Raw ATLAS telemetry in Packets with any duplicates removed
Files for each APID for some defined time period
01/1A Reformatted Telemetry
Full rate Along-track with channel info
Parsed, partially reformatted, time ordered telemetry. Proposed storage format is NCSA HDF5.
Uniform time TBD minutes (1 minute?)
02/1B Science Unit Converted Telemetry
Full rate Along-track with channel info
Science unit converted time ordered telemetry. Reference Range/Heights determined by ATBD Algorithm using Predict Orbit and s/c pointing. All photon events per channel per pulse. Includes Atmosphere raw profiles.
Uniform time TBD minutes (1 minute?)
03/2A Global Geolocated Photon Data
Full rate Along-track with channel info
Reference Range/Heights determined by ATBD Algorithm using POD and PPD. All photon events per pulse per beam. Includes POD and PPD vectors. Classification of each photon by several ATBD Algorithms.
Uniform time TBD minutes (1 minute?)
04/2A Calibrated Backscatter Profiles
3 profiles at 25 Hz rate (based on 400 pulse mean)
Along-track backscatter data at full instrument resolution. The product will include full 532 nm (14 to -1.0 km) calibrated attenuated backscatter profiles at 25 times per second for vertical bins of approximately 30 meters. Also included will be calibration coefficient values for the polar region.
Per orbit
05/2B Photon Height Histograms
Fixed distances Along-track for each beam
Histograms by prime Classification by several ATBD Algorithms. By beam
Uniform time TBD minutes (30 minutes?)
06/L3 Antarctica Ice Sheet Height / Greenland Ice Sheet Height
Heights calculated with the ice sheet algorithm, as adapted for a dH/dt calculation
Surface heights for each beam, along and across-track slopes calculated for beam pairs. All parameters are calculated for the same along-track increments for each beam and repeat.
There will be TBD files for each ice sheet per orbit
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File ID/Level
Product Name Concept Short Description Frequency
07/ L3 Arctic Sea Ice Height/ Antarctic Sea Ice Height
Along-track heights for each beam ~50-100m (uniform sampling); separate Arctic and Antarctic products
Heights of sea ice and open water samples (at TBD length scale) relative to ellipsoid after adjusted for geoidal and tidal variations, and inverted barometer effects. Includes surface roughness from height statistics and apparent reflectance
There will be files for each pole per orbit
08/ L3 Land Water Vegetation Heights
Uniform sampling along-track for each beam pair and variable footpath
Heights of ground including inland water and canopy surface at TBD length scales. Where data permits, include estimates of canopy height, relative canopy cover, canopy height distributions (decile bins), surface roughness, surface slope and aspect, and apparent reflectance. (Inland water > 50 m length -TBD)
Per half (TBD) orbit
09/ L3 ATLAS Atmosphere Cloud Layer Characteristics
Based on 3 profiles at a 25 hz rate. (400 laser pulses are summed for each of the 3 strong beams.)
Cloud and other significant atmosphere layer heights, blowing snow, integrated backscatter, optical depth
Per day
10/ L3 Arctic Sea Ice Freeboard / Antarctic Sea Ice Freeboard
Along-track all beams. Freeboard estimate along-track (per pass); separate Arctic/ Antarctic products
Estimates of freeboard using sea ice heights and available sea surface heights within a ~TBD km length scale; contains statistics of sea surface samples used in the estimates.
There will be files for each polar region per day
11/ L3 Antarctica Ice Sheet H(t) Series/ Greenland Ice Sheet H(t) Series
Height time series for pre-specified points (every 200m) along-track and Crossovers.
Height time series at points on the ice sheet, calculated based on repeat tracks and/or crossovers
There will be files for each ice sheet for each year
12/ L3 Ocean Height Along-track heights per beam for ocean including coastal areas
Height of the surface10 Hz/700 m (TBD) length scales. Where data permits, include estimates of height distributions (decile bins), surface roughness, surface slope, and apparent reflectance
Per half orbit
13/ L3 Inland Water Height
Along-track height per beam
Along-track inland ground and water height extracted from Land/Water/ Vegetation product. TBD data-derived surface indicator or mask. Includes roughness, slope and aspect.
TBD files Per day
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File
ID/Level Product Name Concept Short Description Frequency
14/L4 Antarctica Ice Sheet Gridded/ Greenland Ice Sheet Gridded
Height time series interpolated onto a regular grid for each ice sheet. Series (5-km posting interval)
Height maps of each ice sheet for each year of the mission, based on all available ICESat-2 data.
Per ice sheet per year
15/L4 Antarctica Ice Sheet dh/dt Gridded/ Greenland Ice Sheet dh/dt Gridded
Images of dH/dt for each ice sheet, gridded at 5 km.
Height-change maps of each ice sheet, with error maps, for each mission year and for the whole mission.
Per ice sheet for each year of mission, and for the mission as a whole
16/ L4 ATLAS Atmosphere Weekly
Computed statistics on weekly occurrences of polar cloud and blowing snow
Polar cloud fraction, blowing snow frequency, ground detection frequency
Per polar region Gridded 2 x 2 deg. weekly
17/ L4 ATLAS Atmosphere Monthly
Computed statistics on monthly occurrences of polar cloud and blowing snow
Global cloud fraction, blowing snow and ground detection frequency
Per polar region Gridded 1 x 1 deg. Monthly
18/L4 Land Height/ Canopy Height Gridded
Height model of the ground surface, estimated canopy heights and canopy cover gridded on an annual basis. Final high resolution DEM generated at end of mission
Gridded ground surface heights, canopy height and canopy cover estimates
Products released annually at a coarse resolution (e.g. 0.5 deg. tiles, TBD). End of mission high resolution (~1-2km)
19/ L4 Ocean MSS Gridded monthly Gridded ocean height product including coastal areas. TBD grid size. TBD merge with Sea Ice SSH
Monthly
20/ L4 Arctic and Antarctic Gridded Sea Ice Freeboard/
Gridded monthly; separate Arctic and Antarctic products
Gridded sea ice freeboard. (TBD length scale)
Aggregate for entire month for each polar region
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File ID/Level
Product Name Concept Short Description Frequency
21/ L4 Arctic Gridded Sea Surface Height within Sea Ice/ Antarctic Gridded Sea Surface Height within Sea Ice
Aggregate for entire month (all sea surface heights within a grid) separate Arctic and Antarctic products
Gridded monthly sea surface height inside the sea ice cover. TBD grid
Aggregate for entire month for each polar region
Expe
rimen
tal
Arctic Sea Ice Thickness / Antarctic Sea Ice Thickness
PerPassThicknesssamples(from10-100mfreeboardmeans)forevery10km(TBD)segment(allbeams)whereleadsareavailable;(perpass)
Sea ice thickness estimates derived from the sea ice freeboard product. External input: snow depth and density for each pass.
There will be files for each polar region per day
Expe
rimen
tal
Arctic Gridded monthly Sea Ice Thickness / Antarctic Gridded monthly Sea Ice Thickness
Aggregateforentiremonth(allthicknessobservationswithinagrid)plusThickness(correctedforgrowth)
Gridded sea ice thickness product; centered at mid-month. Include thickness with or without adjustment for ice growth (based on time differences between freeboard observation).
Griddedmonthly(allthicknessobservationswithinagrid)foreachpolarregion
Expe
rimen
tal Lake Height Along reference
track per beam in Pan-Arctic basin (>50-60 deg N).
Extracted from Product 08 and 13, for lakes >10 km2, with slope and aspect. Ice on/off flag. TBD water mask developed from existing masks.
Monthly along track product, no pointing
Expe
rimen
tal Snow Depth Along reference
track per beam for Pan-Arctic basin (>50-60 deg N).
Extracted from Product 08 and 13 along track repeat heights, with slope and aspect. Snow detection flag.
Monthly along track product, no pointing
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APPENDIX B: Sea State Bias Computations for a Photon Counting Lidar
ForICESat-2,wearelookingoverallthereceivedsurfacephotonheightstogetareceivedheightdistributionandthendeconvolvingtheinstrumentimpulseresponsetogetthesurfaceheightdistribution.Ifwearegettingadisproportionateshareofphotonsbackfromthewavecrestsortroughsbecauseofadifferenceinreflectancebetweencrestsandtroughs,thecorrelationofheightandreflectance(orinthiscaserateofphotonreturns)andtheseastatebiasitcausesareunknowntous.However,ifasaseparatestepwetakeadvantageofthefinespatialresolutionofICESat-2toobtainalong-trackdistancebinaverageestimatesofheightandphotonreturnrateunbiasedbythenumberofphotonsinthebins,andoverbinsthatareshortcomparedtothedominatesurfacewavelength,wecanestimatetheseastatebiasinthesurfaceheightdistributionusingtherelationofArnoldetal.[1995].TodothisweneedtoconsidertheArnoldetal.relationintermsofbinaveragesratherthanaggregatephotonaverages.
FirstweestablishtherelationbetweenaverageheightcomputedoverallsurfacephotonsandtheaveragetakenoverallphotonsgroupedinKalong-trackdistancebins.ThephotonaverageseasurfaceheightisthesumtheheightsofNsurfacereflectedphotonsdividedbyN:
(B1)
However,ifweexpressNasthesumoverallKalong-trackbinsofthenumberofphotonsgroupedintoeachbin,rj,weseethat:
(B2)
Similarly,wecangrouptheheightsintothecontributionsfromeachbin
(14-B3)
wherehlistheheightofthelthphotoninabinwithatotalofrjphotonsinit.Therefore,theaverageheightoverallphotonheightsisequaltothebinaverageheightsweightedbytherelativeshareofphotonsineachbin,combining(12-14)
(15-B4)
Hphotonavg
=1N
ηi
i=1
N
∑
N = r
jj=1
K
∑
ηi
i=1
N
∑ = ηl
l=1
rj
∑j=1
K
∑
1N
ηi
i=1
N
∑ =1N
ηl
l=1
rj
∑j=1
K
∑ =
ηl
l=1
rj
∑j=1
K
∑
rj
j=1
K
∑
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IntherelationofArnoldetal.,[1995](9),wereplacethebackscattercoefficientof ,by
thenumberofphotonsperbinrlasanexpressionoftheenergyreturnedfromthesurfaceineachbin.
SSBisthephotonaverageorphotonweightedbinaverageminusthetrueaverageofsurfaceheightsoverallthedistancebins, :
(B5)
wherehjisthetrueaverageofheightinthejthbinnotbiasedbythenumberphotons,andsothesecondtermontherightisµh,thetruearithmeticmeanofsurfaceheightaveragedoverKbins.
(B6)
Thenumberofsamplesinalong-trackbinj,isrj,andthoughthesampleratewitheachsample,rj,ispotentiallydifferentforeachsample,wetakerjastheaveragesamplerateineachdxwidebinsuchthat (B7)
Theaverageofphotonheightsineachbinis
(B8)
So
(B9)
and(B5)becomes
σ i0
ηj
j=1
K
∑ K
SSB =1N
ηi
i=1
N
∑ −
ηj
j=1
K
∑
K=
ηl
l=1
rj
∑j=1
K
∑
rj
j=1
K
∑−
ηj
j=1
K
∑
K
µη=
ηj
j=1
K
∑
K
δ xρ j = rj
η j =
1rj
ηll=1
rj
∑
rjη j = ηl
l=1
rj
∑
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(B10)
andwith(B6),(B10)becomes
(B11)
Thislookslikethesamerelationas(9)byArnoldetal.[1995]butitisalittleunclearifArnoldetalmeanttousethebackscattercoefficientinhisequationorthevariationinbackscattercoefficientaboutthemean.Tocheckthisforourrelationwebreakheightandreturnratein intomean,(µ)andtimevarying(primed)components.Speaking
ofthealong-trackbinquantities, =µh+ hj’andri=µ�+rj’.From(B9):
(B12)
Ifthealong-trackbinsaresmallenoughsothattheseasurfaceheightvariationwithinabinismuchsmallerthanthevariationofheightbetweenbins(binsaresmallcomparedtothewavelengthofthedominantwaves),wecanassumethattheaverageofphotonheightswithinbin, ,equalsthetrueaverageofheightwithinabin, ,andthetrueaverage
heightisequaltotheaverageof overallbins.
(B13)
Thisassumptionisequivalenttosayingthebiasduetovariationofthenumberofphotonswithinbinsissmallcomparedtothebiasduetothedifferencesinthenumberofphotonsamongbins.Similarly,theaveragephotonrate,µr,is
SSB =rj η j
j=1
K
∑
rjj=1
K
∑−
η jj=1
K
∑K
SSB =
rjηj
j=1
K
∑
rj
j=1
K
∑−µ
η=
rjηj
j=1
K
∑ − rj
j=1
K
∑ µη
rj
j=1
K
∑=
rjηj−µ
η( )j=1
K
∑
rj
j=1
K
∑
ηl
l=1
rj
∑j=1
K
∑ rj
j=1
K
∑
η j
1N
ηi
i=1
N
∑ =1N
ηl
l=1
rj
∑j=1
K
∑ =
ηl
l=1
rj
∑j=1
K
∑
rj
j=1
K
∑=
rjηj
j=1
K
∑
rj
j=1
K
∑
η j
η j
η j
µη=
ηj
j=1
K
∑
K=
ηj
j=1
K
∑
K
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(B14)
Applying(B9)to(B4)andbreakingintomeanandvaryingparts
(B15)
SoinfacttheproductintheSSBexpressionofArnoldetal.(1995)istheproductofthevariationsindistancebinphotonrateandphotonheight:
(B16)
Alternatively,considerbreakingdown(B11)
(B17)
with, =µh+ hj’andri=µ�+rj’.
µρ=
rj
j=1
K
∑
K
1N
ηi
i=1
N
∑ =
ηl
l=1
rj
∑j=1
K
∑
rj
j=1
K
∑=
rjηj
j=1
K
∑
rj
j=1
K
∑=
rjηj
j=1
K
∑
rj
j=1
K
∑=
(rj'+µ
r)(η
j'+µ
η)
j=1
K
∑
rj
K=1
K
∑
=
(rj'η
j'+r
j'µ
η+µ
rηi'+µ
rµη)
j=1
K
∑
rj
j=1
K
∑
=
(rj'η
j'+µ
rµη)
j=1
K
∑
rj
j=1
K
∑=
(rj'η
j')
j=1
K
∑
rj
j=1
K
∑+Kµ
rµη
rj
j=1
K
∑=
(rj'η
j')
j=1
K
∑
rj
j=1
K
∑+µ
η
SSB =(rj 'ηi ')
j=1
K
∑
rjj=1
K
∑
SSB =
rjηj
j=1
K
∑
rj
j=1
K
∑−µ
η=
rjηj
j=1
K
∑ − rj
j=1
K
∑ µη
rj
j=1
K
∑=
rjηj−µ
η( )j=1
K
∑
rj
j=1
K
∑
η j
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(B18)
consistentwith(B16).Weproposethatthereisaseastatebiasinthehighermomentsalso.Ifweconsiderhighersamplenthmomentsofheight
(B19)
Aswith(B15):
Thus,the ,nthmoment,wouldhaveasimilarderivationfortheirseastatebias,SSBM(n):
SSB =
rjηj
j=1
K
∑
rj
j=1
K
∑−µ
η=
(rj'+µ
r)(η
j'+µ
η)
j=1
K
∑
rj
j=1
K
∑−µ
η=
(rj'η
j')
j=1
K
∑
rj
j=1
K
∑+µ
η−µ
η=
(rj'η
j')
j=1
K
∑
rj
j=1
K
∑
M(n)=
(ηj−µ
η)n
j=1
K
∑
K=
gj(n)
j=1
K
∑
K
M(n)= 1N
gi(n)
i=1
N
∑ =
gj(n)
l=1
rj
∑j=1
K
∑
rj
j=1
K
∑=
rjgj(n)
j=1
K
∑
rj
j=1
K
∑=
rjgj(n)
j=1
K
∑
rj
j=1
K
∑=
(rj'+µ
r)(g
j(n)'+µ
g)
j=1
K
∑
rj
K=1
K
∑
=
(rj'g
j(n)'+r
j'µ
g+µ
rgj(n)'+µ
rµg)
j=1
K
∑
rj
j=1
K
∑
=
(rj'g
j(n)'+µ
rµg)
j=1
K
∑
rj
j=1
K
∑=
(rj'g
j(n)')
j=1
K
∑
rj
j=1
K
∑+Kµ
rµg
rj
j=1
K
∑=
(rj'g
j(n)')
j=1
K
∑
rj
j=1
K
∑+µ
g
(η j−µ
η)n
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(B20)
SSBM(n)=rj'((η
j−µ
η)n −avg(η
j−µ
η)n)
j=1
K
∑
rj
j=1
K
∑
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APPENDIX C: Expectation-Maximization (EM) Procedure FromAppendixEofATBDATL07
EMalgorithmforestimatingtheparametersoftwo-componentnormalmixturesTheEMalgorithmestimatestheparametersofatwo-componentnormalmixturethatbestdescribeadistributionofrandomvariables(Dempsteretal.,1977;Bilmes,1998).Thetwo-componentmixturemodel:
(NormalDistribution)
TheEMstepstocomputetheparametersofthemixturemodelwithTrandomvariates
are:1.Initializewith: 2.Expectation(E)step:compute:
fort=1,...,T.
3.Maximization(M)step.compute:
,
, ,and
IIterateEandMstepsuntilparametersconverge.
f (xi )=αφi(xi ;µ1 ,σ 1)+(1−α )φ2(xi ;µ2 ,σ 2)
φi(x;µi ,σ i )= 1
2πσ i
e− x−µi( )22σ i2
xt(α ,µ1 ,σ 1 ,µ2 ,σ 2)α ,µ1 ,σ 1 ,µ2 ,σ 2
γ i =
αφi(xt ;µ1 ,σ 1)αφi(xt ;µ1 ,σ 1)+(1−α )φ2(xt ;µ2 ,σ 2)
µ1 =γ t x t
t=1
T
∑
γ tt=1
T
∑
σ 12 =
γ t(x t−µ1)2t=1
T
∑
γ tt=1
T
∑
µ2 =(1−γ t )x t
t=1
T
∑
(1−γ t )t=1
T
∑
σ 22 =
(1−γ t )(x t−µ2)2t=1
T
∑
(1−γ t )t=1
T
∑
α = γ t
Tt=1
T
∑
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APPENDIX D: Fitting a Plane to Spatially Distributed Data
ToevaluatetheaverageDOTatthecenterofagridcellwefitaplanetoallthesampleswithinthegridcellandevaluatetheheightoftheplaneatthecenterofthegridcell.TodothiswefirsthavetomakealeastsquaresfitofaplanetoDOTatNlocationswithinthegridcell.FollowingEberly(2019),givendata(DOT)asafunctionofxandy,hi=f(xi,,yi)ati=1toNlocations,findaleastsquaresfitofaplane,coefficientsa,b,andc,tohiwith
mean, .(Alsonote and )
(1)
Andwewanttochoosea,b,andcsuchthattheerror,E,
isminimized.AccordingtoEberle(2019)thesolutionismorerobustandtheequationsaresimplerifweinitiallyeliminatetheneedtodeterminecbytakingtheaverageof(1)andsubtractingitfrom(1),toget:
andsoforthedeviations , ,and (2) (3)
andwewillchooseaandbtominimize:
(4)
Thencwillbegivenby .TominimizeEwithrespecttoaandbwefindaandbforwhich:
or
h = hi
i=1
N
∑x = xi
i=1
N
∑y = yi
i=1
N
∑
hi ≈axi +byi + c
E(a,b,c)= ((axi +byi + c)−hi
i=1
N
∑ )2
hi −h ≈a(xi − x )+b( yi − y)
′hi = hi −h ′xi = xi − x ′yi = yi − y
′hi ≈a ′xi +b ′yi
E(a,b)= ((a ′xi +b ′yi )−hi′
i=1
N
∑ )2
c = h −(ax +by)
∂E(a,b)∂a
=2 ′xi((a ′xi +b ′yi )−hi′i=1
N
∑ )=2 a ′xi ′xi +b ′xi ′yi − ′xihi′i=1
N
∑ =0
∂E(a,b)∂b
=2 ′yi((a ′xi +b ′yi )−hi′i=1
N
∑ )=2 a ′yi ′xi +b ′yi ′yi − ′yihi′i=1
N
∑ =0
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Letting
and (5)
a,b,andcaregivenby
(6)
Eberly,D.,2019,LeastSquaresFittingofDatabyLinearorQuadraticStructuresDavidEberly,GeometricTools,RedmondWA98052https://www.geometrictools.com/ThisworkislicensedundertheCreativeCommonsAttribution4.0InternationalLicense.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by/4.0/orsendalettertoCreativeCommons,POBox1866,MountainView,CA94042,USA.Created:July15,1999LastModi_ed:February14,2019https://www.geometrictools.com/Documentation/LeastSquaresFitting.pdf
a ′xi ′xi +b ′xi ′yii=1
N
∑ii=1
N
∑ = ′xihi′ii=1
N
∑
a ′yi ′xi +b ′yi ′yiii=1
n
∑ = ′yiihi′ii=1
N
∑i=1
N
∑
Lxx = ′xi ′xii=1
N
∑
Lyy = ′yi ′yiii=1
n
∑
Lxy = ′yi ′xii=1
N
∑
Rxh = ′xihi′i=1
N
∑
Ryh = ′yiihi′i=1
N
∑
a= RxhLyy −RyhLxyLxxLyy −Lxy2
b= RyhLxx −RxhLxyLxxLyy −Lxy2
c = h −(ax +by)
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APPENDIX E: Hierarchy of ATL12 and ATL10 Variables
ATL12InputstoATL19SegmentAverages:SSH-geoid_seg,lat_seg,lon_seg,bin_ssbias,geoid_seg,depth_ocn_seg,length_seg,andsurf_type_prcnt.SegmentMoments:SSHvar,SSHskew,SSHkurt,SWHSegmentHistogram:Y,n_photonsSegmentDegrees-of_Freedom:NP_effect
5.6.4.1.1OutputATL19AveragingOvern_segsinbinBinsGridCellAverages:dot_avg,grid_lat,grid_lon,ssb_avg,geoid_avg,depth_avg,length_avg,surf_avgGridCellAverageMoments:dot_sigma_avg,dot_skew_avg,dot_kurt_avg,SWH_avgGridCellTotalHistogram:dot_hist_gridGridCellTotals:n_segsinbin,n_photons_gridttl,length_sum
5.6.4.1.2OutputATL19AveragingWeightedbyDegrees-of-FreedomGridCellDegree-of-FreedomWeightedAveragesdot_dfw,grid_lat_dfw,grid_lon_dfw,,ssb_dfw,geoid_dfw,depth_dfw,length_dfw,surf_dfw,GridCellDegree-of-FreedomWeightedMomentsdot_sigma_dfw,dot_skew_dfw,dot_kurt_dfw,SWH_dfwGridCellDegrees-of-FreedomandDOTUncertainty:dof_grid,dot_dfw_uncertain
5.6.4.2.1OutputATL19InterpolatedtoBinCenters(gridcntr_lonandgridcntr_lat)AverageDOTatGridCellCenter:dot_avgcntr5.6.4.2.2OutputATL19InterpolatedtoBinCenters(gridcntrandgridcntr_lat)DOFWeightedAverageDOTatGridCellCenter:dot_dfwcntr
5.6.4.5.1OutputATL19MergingAll3StrongBeamsofDOTAveragedbyn_segsinbin
AverageDOTatGridCellCenter:dot_allbeam
5.6.4.5.2OutputATL19MergingAll3StrongBeamsofDOTAveragedbydof_gridDegree-of_FreedomWeightedAverageDOTatGridCellCenter:dot_dfwallbeamDegree-of_FreedomWeightedAverageDOTatGridCellCenter:dot_sigma_dfwalbmDegree-of_FreedomWeightedAverageDOFandUncertaintyatGridCellCenter:dof_grid_allbeam,,dot_dfwalbm_uncrtn
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11.0 REFERENCES
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Carrère,L.,andF.Lyard(2003),Modelingthebarotropicresponseoftheglobaloceantoatmosphericwindandpressureforcing-comparisonswithobservations.
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Hausman,J.,andV.Zlotnicki(2010),SeaStateBiasInRadarAltimetryRevisited,,MarineGeodesy,33(S1),336---347.
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Wu,J.(1972),Sea-SurfaceSlopeandEquilibriumWind-WaveSpectra,PhysicsofFluids,15(5),741-747.
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