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

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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

Reviewed by: ?????

Bea <Enter Position Title Here> <Enter Org/Code Here>

Tom <Enter Position Title Here> <Enter Org/Code Here>

Approved by: ????

Thorsten <Enter Position Title Here> <Enter Org/Code Here>

??????

***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

-Wav

e Sl

ope

(deg

)

High Pass MSS1

MSS1: Max = 0.046, Mean = 0.021, Tot = 0.029

03-Jul-2007 07:57:00, U = 10.52 m/s, Small Array, fmax = 2.5 Hz, d = 0.2 m, k is mean wavenumber, fmax = 2.5 Hz

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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.

Distance East, m

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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

( SSH +Asin 2πLx+N)rdx

0

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∫

=1Xr

( SSH +Asin 2πLx+N)(r +αAsin 2π

Lx)dx

0

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=1Xr SSH (r +αAsin 2π

Lx)dx→

0

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∫ 1Xr SSH r dx

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+1Xr

(Asin 2πLx)(r +αAsin 2π

Lx)dx

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Lx)dx

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∫

+1Xr

N(r +αAsin 2πLx)dx

0

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∫ →0

ey =

1Xr

HSSXr +

X2α

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+α

2A2r

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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

N

∑

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1N

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αN

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i=1`

N

∑

1N

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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.

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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

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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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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.

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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