Algorithm Theoretical Basis Document (ATBD) for Ocean ... · ICESat-2 Algorithm Theoretical Basis...

ICESat-2 Algorithm Theoretical Basis Document for Ocean Surface Height (ATL12)

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ICE, CLOUD, and Land Height Satellite (ICESat-2) Project

Algorithm Theoretical Basis Document

(ATBD) for

Ocean Surface Height

Prepared By: ICESat2 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

GoddardSpaceFlightCenterGreenbelt,Maryland


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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.


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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:TomNeumann,ICESat-2ProjectScientistMailStop615GoddardSpaceFlightCenterGreenbelt,Maryland20771


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Review/Approval Page

Prepared by: Jamie Morison Applied Physics Laboratory University of Washington

Reviewed by:

Bea Csatho University of Buffalo

Tom Neumann NASA GSFC, Code 615


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Change History Log

Revision Level Description of Change

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]


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7/30/2019InTable6changedfrom70010-mbinsto71010-mbinstoaccountforoceansegmentslongerthan7,000m7/30/2019Change5.3.4.1(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.Also


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establishthevector,Cntrpt,ofbincenterpointswithlengthequaltoN.InpracticeBfequals1cm,Cntrptvaluesarewholecentimeters,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_effecttoTable6.03/24/2020Parametersregardingharmonicsandeffectivedegreesoffreedom,whilementionedinthisdocument,willnotbepresentonATL12granulesuntilrelease004.


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List of TBDs/TBRs

Item No.

Location Summary Ind./Org. Due Date


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Table of Contents

Abstract .......................................................................................................................... 2-iCM Foreword .................................................................................................................... iiPreface ............................................................................................................................ ivReview/Approval Page .................................................................................................... viChange History Log ........................................................................................................ viiList of TBDs/TBRs .......................................................................................................... xiiiTable of Contents ........................................................................................................... 14List of Figures ................................................................................................................. 18List of Tables .................................................................................................................. 211.0 INTRODUCTION ................................................................................................... 232.0 OVERVIEW AND BACKGROUND INFORMATION .............................................. 25

2.1 Open Ocean Background ................................................................................... 262.2 The Importance of Waves .................................................................................. 28

2.2.1 Waves and Reflectance ............................................................................... 282.2.2 Waves and Sea State Bias ......................................................................... 302.2.3 ICESat2 Height Statistics and Sea State Bias ............................................. 36

3.0 Open Ocean Products ........................................................................................... 443.1 Open Ocean Surface Height (ATL12/L3A) ......................................................... 44

3.1.1 Height Segment Parameters ....................................................................... 453.1.2 Input from IS-2 Products .............................................................................. 473.1.3 Corrections to height (based on external inputs) ......................................... 47

3.2 Gridded Sea Surface Height - Open Ocean (ATL19/ L3B) ............................... 493.2.1 Grid Parameters .......................................................................................... 49

4.0 ALGORITHM THEORY ......................................................................................... 534.1 Introduction ........................................................................................................ 534.2 ATL12: Finding the surface-reflected photon heights in the photon cloud ......... 53

4.2.1 Selection of Signal Photon Heights ............................................................. 554.2.2 A Priori Sea State Bias Estimation and Significant Wave Height ................ 59


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4.3 ATL12: Correction and Interpretation of the Surface Height Distribution ........... 604.3.1 Separating the Uncertainty due to Instrument impulse response ................ 604.3.2 Characterizing the Random Sea Surface .................................................... 634.3.3 Maximum Likelihood Versus Expectation Maximization .............................. 64

4.4 Calculation of Uncertainty .................................................................................. 644.5 ATL19: Gridding the DOT and SSH ................................................................... 65

5.0 Algorithm Implementation ...................................................................................... 665.1 Outline of Procedure .......................................................................................... 665.2 Input Parameters ................................................................................................ 66

5.2.1 Parameters Required from ICESat-2 Data .................................................. 665.2.2 Parameters Required from Other Sources .................................................. 695.2.3 Control Parameters for ATL12 Processing .................................................. 69

5.3 Processing Procedure ........................................................................................ 69 Coarse Surface Finding and Setting of Segment Length .................................... 695.3.1 ........................................................................................................................ 695.3.2 Processing Procedure for Classifying Ocean Surface Photons, Detrending, and Generating a Refined Histogram of Sea Surface Heights ................................ 725.3.3 Processing to Characterize Long Wavelength Waves, Dependence of Sample Rate on Long Wave Displacement, and A Priori Sea State Bias Estimate 785.3.4 Processing Procedure for Correction and Interpretation of the Surface Height Distribution ................................................................................................... 835.3.5 Applying a priori SSB Estimate .................................................................... 99 Expected Uncertainties in Means of Sea Surface Height .................................... 995.3.6 ........................................................................................................................ 99

5.4 Ancillary Information ......................................................................................... 1035.4.1 Solar Background Photon Rate and Apparent Surface Reflectance (ASR) 1035.4.2 Additional Ancillary Data ............................................................................ 103

5.5 Output Parameters ........................................................................................... 1055.6 Synthetic Test Data .......................................................................................... 1085.7 Numerical Computation Considerations ........................................................... 115


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5.8 Programmer/Procedural Considerations .......................................................... 1155.9 Calibration and Validation ................................................................................ 115

6.0 Browse Products .................................................................................................. 1166.1 Data Quality Monitoring .................................................................................... 116

6.1.1 Line plots (each strong beam) ................................................................... 1166.1.2 Histograms (each strong beam) ................................................................ 116

7.0 Data Quality ......................................................................................................... 1177.1 Statistics ........................................................................................................... 117

7.1.1 Per orbit statistics ...................................................................................... 1178.0 TEST DATA ......................................................................................................... 120

8.1 In Situ Data Sets .............................................................................................. 1208.2 Simulated Test Data ......................................................................................... 120

9.0 CONSTRAINTS, LIMITATIONS, AND ASSUMPTIONS ...................................... 1219.1 Constraints ....................................................................................................... 1219.2 Limitations ........................................................................................................ 1219.3 Assumptions ..................................................................................................... 122

10.0 References ........................................................................................................ 123ACRONYMS ................................................................................................................. 125GLOSSARY .................................................................................................................. 126APPENDIX A: ICESat2 Data Products ......................................................................... 127 APPENDIX B: Sea State Bias Computations for a Photon Counting Lidar ................. 132APPENDIX C: Expectation-Maximization (EM) Procedure .......................................... 13811.0 References ........................................................................................................ 140


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List of Figures

Figure Page Figure1-Characteristicsofanidealizeddistribution……………………………………………….51Figure2.Geodeticheight………………………………………………………………………………………….53

Figure3.Reflectanceasmodeled……………………………………………..……………………………...55

Figure4.TypicalGram-Charlierdistribution……………….………………………………………….56Figure5.Trochoidalshapeofsurfacewaves…………………………….……………………………..57

Figure6.ShortwaveMSS&standarddeviationMSS……………….…………………………….…58Figure7.SimulatedsurfacedisplacementandMSS…………………..….…………………………..59

Figure8.Heightduetolong-waves&MSS……………………………...…….…………………………..59

Figure9.Heightduetolong-waves&probability……………………..….…………………………..60Figure10.SurfacereturnheightsfromMABEL…………………..……...…………………………….64

Figure11.ATL12processingblockdiagram……………………………......…………………………..78Figure12Coarse,smoothed,andfinehistograms…………………………………………………….82

Figure13.BlockdiagramfortheATL19gridding……………………....…………………………….89

Figure14.Coarse,smoothed,finalMabelhistograms…………………………………………….…98Figure15.Instrumentimpulseresponsehistogram……………………………………………..…104

Figure16.ProbabilitydensityfunctionsofthesyntheticSSH………………………………....105

Figure17.Synthesizedreceivedprobabilitydensityfunctions….………………………….…106Figure18.Single-sidedspectraofthereceivedPDFs…………….….………………………….…110

Figure19.Single-sidedspectrumofimpulseresponse…………………………….………….…110Figure20.Single-sidedamplitudespectraoftheWienerfilters..….………………………….111

Figure21.Single-sidedamplitudespectraofthesurfacePDF….………………………….…..112

Figure22.PDFofSSHandcalculatedPDFfor8000photons..….….…………………………...114Figure22.PDFofSSHandcalculatedPDFfor800photons..…...….…………………………...116


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List of Tables

Table Page Table 1 Reflectance…………………………………………………………………. 67Table 2 Input parameters (Source: ATL03)……………………………………………….. 68Table 3 Input parameters (Source: ATL09) ……………………………………………… 92Table 4 Control Parameters - Surface Finding……………… ………………………….. 94Table 5 Control Parameters for Refined Surface Finding and Analysis……………...…….96 Table 6 Output (See Appendix A for full product specifications)……………………......124 Table 7 ATL12 Output Variables for per Orbit Statistics…………………………………135 Table 8 ATL12 Test Data …………………………………………………………………138


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1.0 INTRODUCTION

ThisATBDwillcovertheretrievalofSeaSurfaceHeight(SSH)fromICESat2ATLASlaserreturns.Forthepurposeofthisearlydraft,thelevelsofspatialandtemporalaveragingtobeincludedarestilltobedefinitivelydecided,asistheamountofhigh-leveldataprocessingtoproducegriddedfieldsofSSHandsuchthingsasDynamicOceanTopography(equaltoSSHminusthegeoid).

OtherICESat2ATBDsdescribetheproductsforicesheets,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.Theseareprovidedtousersasqualityassessmentsoftheindividualparameterson


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eachfileandtoprovidecriteriaforautomaticqualitycontroltofacilitatetimelydistributionoftheproducttotheusers.Summarystatisticsorimagesareprovidedthatallowuserstoeasilyevaluate(orBrowse)whetherthedatawouldbeusefulandofadequatequalityfortheirresearch,andasneededtoaidinthequickapprovalordisapprovalofproductspriortopublicdistribution.

Section7describesthetestingandvalidationproceduresthatareplanned.


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2.0 OVERVIEW AND BACKGROUND INFORMATION

TheAdvancedTopographicLaserAltimeterSystem(ATLAS)consistsofbothLIDARandaltimetrysubsystemsthatwillflyonthededicatedplatformcomprisingthemissionreferredtoasICESat2,theIce,Cloud,andLandHeightSatellite.ThefollowingsubsectionsdiscussthebasicconceptsofhowATLASworksandtheLevel2datathatwewillturnintoseasurfaceheight.ItthendescribesthebasicpropertiesoftheseasurfaceandhowtheserelatetoprocessingtheICESat2data.

Figure1-CharacteristicsofanidealizeddistributionofATLASreturnphotonarrivaltimesassumingaGaussianinstrumentimpulseresponseandareflectivesurfacewithGaussianroughness.Ingeneralsurfaceslopeandroughnessbothbroadenthereturndistribution.Fortheopenoceantheslopeeffectislikelynegligiblecomparedtheeffectofroughnessduetosurfacewaves.Animportantcomplicationisthatsurfacewavesproduceanon-Gaussiansurfaceduetotheirbroadtroughsandnarrowpeaks.

1 nsec

~10 m


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TheprimarypurposeoftheATLASinstrumentontheICESat2missionistodetectsurfaceheightchanges.Withrespecttotheocean,theserepresentSeaSurfaceHeight(SSH).Bywayoftechnicalhistory,thepredecessorofICESat2ATLAS,theICESatGLASinstrument,usedalaseraltimetertomeasuretherangetothesurface.Rangesweredeterminedfromthemeasuredtimebetweentransmissionofthelaserpulseanddetectionofthepulsereflectedfromthesurfaceandreceivedbytheinstrument.TheGLASlaserfootprintdiameteronthesurfaceduetobeamspreadingwasnominally70m,andthedurationofthetransmittedpulsewas4ns.

TheICESat2ATLASinstrumentwillsampleatahigherrate,10kHzwith1-2nspulsewidth,withasmallerintrinsicfootprint,~10m(Fig.1),andwillrelyondetectingtherangetraveledbyindividualphotons.ExperiencewithMABELsuggeststhattheinstrumentimpulseresponsedistribution,duelargelytothevariationintransmittimesforthephotonsofeachpulse,willbenon-Gaussianwithsignificantskewness.Consequently,fourpointswillbemeasuredontheoutgoingpulsedistributionandthesewillbepartoftheICESat2rawdatastream.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-ArcticseasandsouthernpartsoftheSouthernOceanthatarecriticaltotheglobaloverturningcirculationandthefateof


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seaiceintheicecoveredArcticOcean.ICESat2overtheopenoceanwillcoverthisgapbetweenthetemperatelower-latitudeoceanandtheseaicecoveredregionsoftheArcticandAntarctic

Asdiscussedabove,thedistributionofICESatphotonreturntimes,orapparentheightsfromLevel2processing,willbedeterminedbythemeanSSH,themeansurfaceslopeandasurfaceroughnesswithinthefootprint.Amongthese,theeffectofseasurfaceslopeonthephotonheightdistributionsshouldbesmallcomparedtotheeffectofroughnessduetosurfacewaves.Fortheaggregationscalesabove,theSSHisthesumofthegeoidheight(fixedintimeandontheorderofmetersamplitude),thedynamicoceantopography(DOT,ontheorderofcentimeterstotensofcentimetersamplitude)associatedwithmeansurfacecurrents,tides(mainlydiurnaltosemidiurnalperiodswithamplitudesofuptometers),andseasurfaceatmosphericpressure(SAP,asmuchas10sofcm).Asafirstapproximation,SSHcanbeestimatedasthemeanofthedistributionofphotonheightswithinaverticalwindowabouttheellipsoidoranapproximatedcanonicalseasurfaceheight(e.g.,30mfromtheestimatedgeoid)andoverthealongtrackaggregationscalesto

bedecidedasabove.However,thespecialandrapidlyvaryingnatureoftheseasurfaceposechallengeswiththeinterpretationandaggregationofmeaningfulSSHfromICESat2photonheights.Itisusefultooutlinetheseissueshere,keepinginmindthatthemagnitudeofthemostsoughtaftercomponentsoftheSSHsignal,changesinDOTandmeanSSH,aresignificantlysmallerthanmostoftheothercomponentsofSSH.Tides,Medium-FrequencyChangesinCirculationandAliasing:ICESat2willcompleteoneorbitofthe

earthinabout1.5hours.Therepeatperiodisunderdiscussionbuthasmainlysettledon91days.Moreimportantly,manyofthemostimportantareasandtimestooceanographersarepronetobeingstormyandovercast.Thus,itislikelythatICESatwillonlygetgoodsurfacereturnsinfrequentlyoverlargeandimportantregionsoftheopenocean.Tidesandmediumtohigh-frequencychangesincirculationwillseriouslyaliasintoslowlysampledregions,makingaggregationintovalidlong-termmeansdifficult.AtsomepointinICESat2

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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oceanprocessingthiswillhavetobeaddressed.ThiscanbedonebyreferencingthephotonheightstotheSSHpredictedbyamodelsuchastheOceanModelofCirculationandTides(OMCT)asusedfordealiasinggravitydatafromtheGravityRecoveryandClimateExperiment(GRACE).Essentially,thepredictedtidalandcirculationcontributionstoSSHaresubtractedfromeachICESat2heightobservation,theresidualisaveragedmonthlyandsimilarlyaveragedmodelresultisaddedbackin.Onlytheerrorinthemodeledresponseisaliased.ThisproceduremaybebeyondthescopeofLevel3data,butthesameoceanmodelcouldbeusedtomorenarrowlyandaccuratelydefinethephotonheightbandthatneededtobeconsideredinselectingtrueoceansurfacephotonreturns.TheGeoid,NarrowingtheWindowonPhotonReturns,andMeasuringMeanDOT:ICESat2photonheightswillbeheavilyinfluencedbythegeoid.AnexampleofthisfortheArcticOceanneartheNorthPole(Fig.2)showschangesinSSHassociatedwithgeoidheightvariationsofabout1cmperkmofhorizontaldistance.GiventhepotentiallypatchyandtemporallysparsecharacterofICESat2oceanreturns,thevariationsinthegeoidfromageoidmodel(e.g.,EGM2008oranimprovementthereonusingGOCEandGRACEdata)shouldbeextractedfromthebasicheightdatatoallowmeaningfulspatialmeansofSSHtobemade.ThespatialresolutionofthesegeoidcorrectionsisanimportantissueaffectingthesmallerDOTandSSHsignals.SurfaceAtmosphericPressureandtheInverseBarometerEffect:ChangesinsurfaceatmosphericpressureashorttimescalesdirectlycauseaninversedeflectionofSSH.Aswithtides,oceancirculationchanges,andthegeoid,thesedirectpressuredisturbancesatthetimeandplaceofeachICESat2photonretrievalshouldbeestimatedandremovedfromestimatedSSHbeforeaggregationtoavoidspatialandtemporalaliasing.ThiswillrequirebringingatmosphericreanalysisproductsordirectSAPobservationsintotheprocessingstream.Knowledgeofthelikelyinvertedbarometereffectwillalsohelpinnarrowingtherangeofreturnphotonheightsandthusidentifyingphotonsreturningfromtheseasurface.

2.2 The Importance of Waves

2.2.1 Waves and Reflectance

SurfaceWavesReflectance,Scattering,andtheSeaStateBias:WeexpectsurfacewavestohavedominanteffectsontheICESat2returnsfromtheopenocean.

Reflectance:Surfacewavesandwindspeedhaveacriticaleffectonreflectance.Menziesetal.[Menziesetal.,1998]indicatereflectanceisgivenby:

R=WRf+(1-W)Rs+(1-WRf)Ru (1)

whereRisthetotalbackscatterorretro-reflectance,Wisthefractionoftheoceancoveredbyfoam(uptoO[1]forwindsaboveabout7m/s)),Rfisthereflectanceoffoam,


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Rsisreflectanceduetospecularreflectionfromthewaveroughenedsurface,andRuistheapparentreflectanceduetolightthatpenetratesandcomesbackupthroughthesurfaceanywhereitisnotreflecteddownwardbysurfacefoam.

RfisconsidereddiffuseLambertianscatteringindependentofincidenceangle.Aswewouldexpect,Rsishighlydependentonincidenceangleandisthedominantformofbackscatter.RuisLambertianbutitisusuallysmall,about0.0075maximum.Menziesetal,modelthereflectanceowingtoRsandRf.ThemainpartisaninverserelationbetweenRsandthevarianceinsurfaceslope,Rs~1/,duetowindgeneratedsurfacewaves.Theyusearelationbetweenwindspeed,U10,andbyWu[Wu,1972]basedonresultsofCoxandMunk[CoxandMunk,1954];goesupasthelogofwindspeed.Themodel

resultsaregiveninFigure3,whereinthestrongdependenceonnadirangleisapparent.Theexceptionisathighwindspeedswherethebackscatterfromfoamstartstobecomeimportant,flatteningtheRcurvesfornadiranglesgreaterthan30°.Otherresultsin[Menziesetal.,1998]showgoodagreementbetweenthemodelanddatafromtheLidarIn-

spaceTechnologyExperiment(LITE)shuttlelidarmissionofSeptember1994.Themodeledreflectanceessentiallyagreeswiththevaluesforreflectanceusedinthedesign

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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performancestudyforATLAS(Table1).However,mostimportantwithrespecttothisATBD,thedependenceofseasurfacedirectionalreflectanceonsurfacewindstresssuggestsamethodforderivingsurfacewindspeedfromICESat2measurementsofseasurfacebackscatter.WorkingsolelywithICESat2observations,wecanconceivablyestimateU10fromreflectance.AswediscussbelowthisestimateofU10maybeusedwithSWH,estimatedfromthevarianceofthephotonheightdistributions,tocalculatetheSeaStateBiasinICESat2SSHmeasurements.

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,ashortcomingwecannotrepeatwithICESat2.TheICESatATBDdocumentmadetheassumptionthatseasurfaceheightsareGaussiandistributed,sothatwithaGaussiantransmissionpulse,thereflectedpulsereceivedbyICESatcouldbeassumedtobeGaussianalso.Infact,theoceansurfaceisnotGaussianandthisaffectsthereflectionofeitherradarorlightfromthesurface.The

Figure4.TypicalGram-Charlierdistributionseasurfaceheightinawaveenvironment.Thepositiveskewnessandexcesskurtosis(Gaussianskewnessandexcesskurtosis=0)isduetotruncationatlowheightsandalongtailathighheights.Fig.7.4-1ofKinsman[1965].


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distributionofsurfaceheightinawave-coveredocean(Fig.4)istypicallygivenbytheGram-Charlierdistribution[Kinsman,1965].Forthisthethirdandfourthmomentsofthedistributionexpressedasskewnessandkurtosis(orexcesskurtosis=kurtosis-3)areimportantandindicatetruncationofthedistributionatlowheightswithalongtailathighheights.Thisreflectstheshapeofsurfacewaves.

Ingeneral,surfacewaveshavetrochoidalshape,withnarrow,steepsidedpeaksandrelativelybroadflattroughs(Fig.5).Thisappliesparticularlytotheshort-wavelength

wavesproximallyforcedbywind.PhotonsstrikingtheupperportionsofthesewavearelesslikelytobereturnedtoICESat2thanphotonsstrikingthelowerportionsofwavesurfaces;peaksareundersampledrelativetotroughsresultinginaseastatebias(SSB)inestimatesofSSHbasedonthemeanofanICESat2photonheightdistributionor,inthecaseofICESat,meanarrivaltimeofareturnpulse.

However,commonlytheseasurfaceiscomposedofswellandwindwaves.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.Thesecondcomponentisameasureofthedistributionofsurfaceslopesdue

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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tothewaveswithwavelengthslessthan10-mlaserpulsefootprintofATLAS.Thedistributionofsurfaceslopes,shiftedbytheinstantaneousslopeofthelongwavecomponentdeterminestheprobabilityofATLASfindingaspecularreturnresultingfromaphotonlaserpulse.Knowingthisasafunctionofsurfaceheightandslopeduetothelong-wavewillallowustopredictSSBintheICESat2measurements.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.Spectraforhigher

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

-2 -1 0 1 2-20

-10

0

10

20

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0.015

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Long

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e Sl

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(deg

)

Std of High Pass MSS1

-2 -1 0 1 2-20

-10

0

10

20

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frequencies(wavenumbers)than2Hzareextrapolatedusingtheobservedfrequency-1spectralslopeat2Hz.TheMSSisobtainedbyintegratingthespectrumouttoamaximumfrequency(100Hz),abovewhichincreasesinMSSarenegligible.


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Thepreliminaryresultofthisanalysishasbeenthedependenceofobservedshort-waveMSSontheslopeandheightofthelong-waves(Fig.6).MSSisgreatestforhighheightsand

negativeslopes,i.e.,ontheupperpartofthedownwindfaceofthelong-waves.ThecorrelationmapsofFigure6havebeenusedasatabletospecifyMSSina10-mspotcorrespondingtothe2Dmodeloflong-waveslopeanddisplacementderivedfromtheDonelanspectrum.Arealizationoflongwaveheightandslopeandthecorresponding10-mspotMSSisshowninFigure7[Plant,2015a&b].A1-Dsliceinthedownwinddirection(Figure8)throughthecombineddatafromFigure7illustrateshowMSSslopetendstoincreaseontheupperpartsofthedownwindfaceof

Figure8.Heightduetolong-wavesversusdistancedownstream(fromupperrightcornertolowerleftinFigureA-left)withcolor-codedcorrespondingMSSfromFigureA-right.

Figure7.FromPlant[2015b].Left–Simulated2Dsurfacedisplacement,Right–2Dvariationofshort-wavemeanMSSvaluesthatwerecalculatedupto2Hz.Downwinddirectionisfromupperrighttolowerleft.

Distance East, m

Dis

tanc

e N

orth

, m

Surface Displacement in m (White shows breaking regions)

200 400 600 800 1000 1200200

300

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1000

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1200

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Dis

tanc

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orth

, m

Mean-Square-Slope (White shows breaking regions)

200 400 600 800 1000 1200200

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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.{SEEBILL’SLATESTWRITEUP],Consequently,lackingfurtherdataaboutthedependenceofshortwavecharacteristicsonlongwaveposition,weseekamethodwherebytheSSBcanbedeterminedtomICESat-2datathemselves.

2.2.3 ICESat2 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,ashortcomingwecannotrepeatwithICESat2.

APrioriEstimationofSSBUsingOnlyAltimeterData

ThesmallfootprintofICESat2(17mfor4sdiameterGaussianintensity),shortdistancebetweenpulse/footprintcenters(0.7m)andtheessentialpointsamplingatrandompositionswithinthefootprintofthephotoncountinglidar,maypresentanopportunitytocapturetheshapeofthelongwavelength,energycontainingsurfacewaves.IfthisistrueitwillbepossibletoestimateICESat2seastatebiassolelyonthebasisofcontemporaneousICESat2returnswithoutresortingtooutsidedataorevencomparisonwithpastICESat2returns.Thekeyisevaluatingtherateofsurfacephotonreturnsasafunctionofsurfaceheight.Asimpleexampleofthisconsidersanoceansurfacewithseasurfaceheight,Hss,disturbedbyalongsinusoidalsurfacewavewithamplitudeA,plusrandomnormallydistributedperturbations,N:


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2

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-bandSSBmeasuredwithacombinationofradaraltimetersandcapacitivewavewiresmountedonanoilplatformin

Y = SSH +Asin

2πLx+N(0,σ )

r

r = r +αAsin 2π

Lx

α =

cov(ry)y2

σ

ey =

1Xr

Yrdx =0

X

∫ 1Xr

( SSH +Asin2πLx+N)rdx

0

X

∫

=1Xr

( SSH +Asin2πLx+N)(r +αAsin 2π

Lx)dx

0

X

∫

=1Xr SSH

(r +αAsin 2πLx)dx→

0

X

∫ 1Xr SSH

r dx0

X

∫

+1Xr

(Asin 2πLx)(r +αAsin 2π

Lx)dx

0

X

∫ → 1Xr

(Asin 2πLx)(αAsin 2π

Lx)dx

0

X

∫

+1Xr

N(r +αAsin 2πLx)dx

0

X

∫ →0

ey =

1Xr

HSSXr +

X2α

2A

⎡

⎣⎢

⎤

⎦⎥=HSS +

α2A2r

eSSB = y −HSS =

α2A2r


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theGulfofMexico,calculateelectromagneticbias,e,duetothevariationofbackscattercoefficientwithsurfacewavesas

(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-countingaltimetersuchasMABELorICESat2;thereturnrateoftheheightsamplesisnotuniformandtheaverageseasurfaceheightisderivedfromthehistogramofphotonheights.Thecorrelationbetween

ε =bi0η

ii=1

N

∑

bi0

i=1

N

∑

σ i0

α =

1N

bi0η

ii=1

N

∑

1N

ηi2

i=1`

N

∑

ε =bi0η

ii=1

N

∑

bi0

i=1

N

∑=

αN

ηi2

i=1`

N

∑

1N

bi0

i=1

N

∑=αA2

2r = SSB


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therateofsurfacereturnsandsurfaceheightisonlyoneofseveralcomponentsinthehistogramskewness.Othercontributorstoskewnessincludethetruedistributiontosurfaceheightandsubsurfacereturns.However,thedominantsurfacewavestypicallyhavelongwavelengthsandarenearlylinearsotheycanbeapproximatedassinewaves.Withthisinmind,weexperimentedwithfittingmultiplesinewavestotherawhistoryofunevenlyspacedphotonsurfaceheightsandcalculatingthecorrelationofobservedrateofsurfacereturnswiththesinewavefittoyieldanestimateofSSBusingequation7.However,intestingthemethodonGreenlandSeaMABELdataandAirborneTopographicMapper(ATM)overthePacific,wefoundthemethodwassensitivetofindingthecorrectwavenumbersforthespectraldecompositionoftheoceansurface.Amorerobustmethodistoaverageboththerawphotonheightsfromthesurfacefindingstepandtherateofphotonreturnsinevenlyspacebinsofalongtrackdistance.Thecovarianceofthesebeenaveragesisthenusedtoevaluateequation7fortheseastatebias.Thismethodiseasierto


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automateandaccountsformoreoftheenergyinthewavefieldthandoestheharmonicmethod.

MABELdatafromtheGreenlandSeainApril19,2012illustratestheapplicationofequation7.Inthisexample,a9-kmlengthofMABELtrackisbrokeninto3-kmsegments.Each3-kmsegmentisdividedinto30010-malongtrackbins.Oncesurfacefindingselectsthesurfacephotonheights,theseareeditedwithtwopassesofa3-standarddeviationeditor,anddetrendedwithalongtrackdistance.

Therateofphotonreturnsiscomputedfromthealong-trackrecordofsampleintervals.Therecordofsampleintervalsisgivenbythedifferencebetweensuccessivesurfacephotonalong-tracklocations.FortheMABELdatashownhere,thesamplespacingsvarysignificantlybutaveragealittleoveronemeter.Toalignthedata,thesurfaceheightandsamplepositionareinterpolatedtothecenterofthesampleintervalbycomputingthe

Figure10.SurfacereturnheightsingreenfromMABELtransitovertheGreenlandSeaversusdistancealongtrack.Three-sigmaeditedanddetrendedheightsareplottedingreenanda10-mbinaveragesareplottedinblack.ThebinaveragesandaprioriSSBestimateswerecomputedinthree3000-msegmentsandaggregatedtoyieldSSB=-0.00029m.


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averageofpairsofsuccessivesampleheightsandposition.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-mfootprintoftheICESat2beams,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-GaussiancharacterofoceansurfacewavesandthescatteringandeffectivereflectanceoftheICESat2photonslikelycauseanegativeseastatebias(SSB).WeanticipatethatthehigherordermomentsofthephotonheightstatisticswillallowustobetterestimateSSB.WewillestimateforeachATL12variablesegmentatminimumthemean,standarddeviation,skewnessandkurtosisoftheheightdistribution.Thesefourmomentscanbedeterminedfromthemeansandstandarddeviationsofa2-componentGaussianmixture,whichwillalsoprovideamixingratioforthetwocomponents.Thesehighermomentsmakeitpossibletocharacterizenotonlythemeanseasurfaceheight,butalsotheseastateand


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likelyseastatebiasoftheheightestimate.However,itisnecessarytoaccumulatelargenumberofsurfacereflectedphotonsto

makemeaningfulestimatesofthehighermomentsoftheheightdistribution.Giventhelownumberofreturnsfromopenwater(e.g.,O[1photon/pulse]),andconsideringthatoceanwavescanhavewavelengthsofhundredsofmeters,togetmeaningfulstatisticsoveropenwaterwithsignificantwaveactivityrequiresaccumulatingdataoverseveralkilometers.Theatmosphericcloudflag(layer_flaginATL03)isusedinpartinsettingtheoceansegmentlengthrequiredtoachieveaminimumnumberofphotons.ItandothergeophysicaldatasuchasthebackgroundphotonratefromATL09andthegeoidarereportedevery400pulses(25Hz)or14geo-bins.Consequently,overtheopenocean,wewillseekaminimumphotoncountof8,000photonsequivalenttoabout5.6kmor28020-mgeo-binsformeaningfulhigherorderstatistics.

3.1.1 Height Segment Parameters

3.1.1.1 Geolocation/Time

Thelocationisthemeanlocationofdesignatedsignalphotonsusedasinputtothesurfacefindingprocedure.Thetimeisthemeantimeofdesignatedsignalphotonsusedasinputtothesurfacefindingprocedure.

3.1.1.2 Height

Thisisthemeanheightestimatefromthesurfacedetectionalgorithm.Qualitymetricsinclude: Confidencelevelinthesurfaceheightestimatebasedonthenumberofphotons,thebackgroundnoiserate,andtheanalysisfromthedetectionalgorithm.

3.1.1.3 Subsurface-scattering corrections

Subsurface-scatteringcorrectionsareyettobedetermined.Theyareexpectedtobeminimalforthedeepopenoceanwherethewaterisclear,butmaybesignificantinnearcoastalregionswheremorescatteringelementsaretobeexpectedinthewater.Theseconditionswillbemostlikelytobeapparentaspositiveskewnessinthereturnheightdistributionsespeciallyascorrelatedwithoutsidemeasuresofsurfacecolororscattering.GuidedbythetreatmentofsubsurfacescatteringfortheInlandWaterATBDandasweareabletodeveloprelationsbetweenphotonheightdistributionskewnessandotherevidenceofscattering,wewillderiveanappropriatecorrectionforsubsurfacescatteringwhereapplicableandprovideitasanoutputofATL12.


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3.1.1.4 First-photon-bias corrections

Pendingfurtheranalysis,giventhelowapparentreflectanceoftheoceansurfacefirst-photon-biascorrectionswilllikelynotbeneededfortheoceanproducts.

3.1.1.5 Height Statistics

Thephotonstatisticsdescribethedistributionofthepopulationusedinthesurface-trackingalgorithm.Theseparametersincludethe:numberofphotons,histogramofthepopulation,and,atminimumthemean,variance,andskewness,andkurtosis.

3.1.1.6 Initial Sea State Bias Correction and Surface Wave Properties

Asdiscussedin2.2.3,ICESat-2givesusauniqueopportunitytomakeanaprioriestimateofseastatebiasusingonlytheheightsfromthesurfacefindingsystem.Theapproachistousethesurfacephotonheightstoestimatethesurfaceheightandtherateofphotonreturnsin10-malong-tracksegments.ThecorrelationofheightandreturnratenormalizedbytheaveragephotonreturnrateistheEMorseastatebias.Thiswillbeestimatedfromthesurfacephotonheightrecordbeforeseparationoftheimpulseresponseintheheighthistogramandmadeavailableasoutput.ThestandarddeviationofphotonheightsoverthewholesegmentwillbeprovidedasameasureofSWH(=4xStd.)thatcanbecomparedwithreanalysisproducts.Thestandarddeviationaboutthe10-mbinaverageswillalsobeprovidedasanestimateoftheenergyofshortwavesthatarguablyhasanimportanteffectonreflectanceandseastatebias.

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).


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3.1.2 Input from IS-2 Products

3.1.2.1 Classified photons (Level 2)

Photonsclassifiedastowhethertheheightissignal,noise,orextendedsignal.Thesehaveaconfidenceastotype.

3.1.2.2 Atmosphere (Level 3)

Relative/calibratedbackscatter,backgroundrates,cloudstatisticsat25Hz.

3.1.3 Corrections to height (based on external inputs)

Inanticipationofhigherlevelprocessing,thestandardheightproductswillincludeanumberofcorrectionsappliedtotherawheightestimates.Forexampletoreducealiasingproblems,correctionsforhigh-frequencyandfinespatialscalevariationsinSSH(e.g.,tidesandotherhighfrequencyoceancirculationchanges)thatmaybeinadequatelysampledbyICESat2shouldbeappliedbeforeaveraging.Estimatesofthesecorrectionswillbegivenhere.Allcorrectionswillbegivenintermsofheightsothattoapplyacorrection,userswilladdittotheheightestimate,andtoremoveittheywillsubtractit.Additionalcorrectionsthatsomeusersmaydecidetoapplywillbeprovidedwiththeproduct.

3.1.3.1 Geoid adjustment (Static)

HeightsareadjustedforlocalgeoidheightusingmeantideEGM2008modelbeingreportedbyATL03astakenfromthemean-tideEGM2008model.

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-m


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windfields.Thiscombinedcorrectiontypicallyhasamplitudeontheorderof±50cm[Markusetal.,2016].

3.1.3.4 Tidal corrections (time-varying)

Heightsarecorrectedfor:Solidearthtides(tide_earth):SolidearthtidesarederivedusingIERS(2010)conventionsare±30cmmax(detailsinATL03ATBDsection6.3.3)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)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/reanalysisproductswillbetakenfromanappropriatesourcesuchastheECMWFandwiththeATL12productforcomparisonforcomparisonwithICESat-2derivationsofSWHaspartoftheseastatebiascalculation.


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3.2 Gridded Sea Surface Height - Open Ocean (ATL19/ L3B)

Thisproduct,basedentirelyonProductATL12/L3Awithnoexternaldependence,containsgriddedmonthlyestimatesofseasurfacefromallIS-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. The exact spacing of the Polar Stereographic grid should be adjusted to match the longitudinal spacing of the TOPEX/Poseidon grid at 60°S and 60°N, and the latitudinal spacing adjust to have an integer number of grid cells between 60°N (S) and the North (South) Pole.

3.2.1 Grid Parameters

3.2.1.1 Sea surface height estimate

WithonlyATL12neededasinput,thisderivedproductcontainsthestatisticaldescriptionoftheseasurface(mean;standarddeviation,skewness,kurtosis)withingridcell.Thesewillbecomputedintwoways,usingaggregationofsegmenthistogramsdiscussedinthissection,andaggregationofsegmentmomentsdiscussedinSection3.2.1.2.ThegriddingschemewilltakeasinputfromATL12,thehistogramsofDOTaccumulatedinsegmentslyingwithineachgridcellthesewillbecombinedbyaddingthephotoncountsfromeachsegmentineachheightbinofeachsegmentinthegridcell.

Theaveragegeoidheightfromeachsegmentwillalsobeaveraged,weightedbythenumberofsignalphotonsineachsegment.Similarly,theweightedaverageaprioriseastatebiaswillbecomputed.

Thefourmomentsoftheaggregatehistogramwillbecomputedusingthe2-GaussianExpectationmaximizationapproach.TheseastatebiaswillbesubtractedfromthefirstmomenttoyieldthemeangridcellDOT.ThemeanSSHcanbecalculatedasthemeanDOTplustheweightedaveragegeoidheight.

3.2.1.2 Sea surface statistics histogram within grid

Wewillperformaweightedaverageofthefirstthroughfourthmomentsofheightsinthelengthsegmentsthatgointothegridcellinagivenmonth.Beingmindfulthateachsegmentgoingintoagridaveragemayhaveadifferentunderlyingdistribution,wewillessentiallycomputethemomentsofthesegmentmoments.Thus,themomentsfortheindividualsegmentswillbeweightedbythenumberofsurfacephotonsintheirsegmentsrelativetothetotalnumberofphotonsgoingintothegridcellandthenaddedtoyieldestimatesoftheaveragemomentsinthegridcell.Whereenoughsegmentsareincludedinagridcell(e.g.,morethan20),histogramsofthemomentsgoingintothegridcellcanbeincludedasoutput.


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3.2.1.3 Wave statistics within grid

EstimatesofSWH,shortwaveenergyandSSBfromtheaprioriestimationofseastatebiaswillalsobegrid-averagedwithappropriatenormalizationforthenumberofsurfacephotonsineachsegment.

3.2.1.4 Sea surface segments (Input)

Seasurfaceheightinsurfaceheightsegmentsasdescribedabovewiththetime/location/qualityoftheseasurfaceheightinthesegment.


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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,themajorityoftheofthebinswillcontainonlynoisephotonssothatthemedianwillequalthenumberofnoisephotonsperbin.Thusthefundamentalideaforsurfacefindingistoidentifyassurfaceheightbins,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: ATL12willrequireATL03photonheightsforeachpulseintheocean±15-mdownlinkbandandassociatedtimegeolocation,geophysicalcorrections(geoid,tides,dynamicatmosphericcorrection),andconfidencelevelinformation,plusthecloudflag,layer_flag,every400pulsesfromATL09.Basedonearlydiscussions,thisATBDandthedevelopmentalsoftwaredevelopedfromitassumedthatICESat-2overtheoceanwoulddownlinkphotonheightswithin±15moftheEGM2008geoid.Alsoasidefromthe±15-mbandaboutthegeoid,noclassificationastosurface/non-surfacephotonsismadefortheATL03oceanresults.AsconfiguredinDecember2018,theFlightScienceReceiverAlgorithms(FSRA)downlinksphotonheightsin±15-mbandaboutheightschosenbysignalfindingproceduredescribedintextin“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,biasingseasurfaceheightsinsomeunknowndirectionandviolatingtheassumptionsoftheATL12seastatebiasdetermination.UnlesstheFSRAischangedtosimplydownlinkallphotonheightswithin±15mofthegeoidwhenovertheocean,thephotonsfrommajorframesforwhich


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thecenterofthedownlinkbandismorethan2or3-mfromthetrueseasurfaceshouldbeeliminatedfromATL03.

WewilluseheightsreferencedtotheEGM2008geoidandcorrectedforoceantides,anddynamicatmosphericcorrectionprovidedinATL03.

WewillusethesignalconfidenceproductinATL03toeditouttheerroneousdownlinkbands.Thisgradeseachphotonheightreturnastolikelihoodthatitisasurfacereturnbyexaminingheightbinsat50-pulserateforsignaltonoiseratiointhecontextofnoiselevel.EarlyICESat-2oceandatasuggeststhatphotonheightswithhigh,medium,andlow(4,3,2)confidencelevelsveryseldomappearintheerroneousdownlinkbands.Further,theconfidencelevelsoftwareassignsafill-inconfidencelevelof1towhatwouldbezeroconfidenceheightsinordertofilla±10-m(tobechangedto±15-m)binaboutthehighconfidenceheights.Therefore,inpracticeconfidencelevel1oressentiallycoversthegeoid±15-mbandandcanbeusedtoeditouttheerroneousdownlinkbands.

Becauseweareusingthisground-basedediting,inthefuturewewillhavetoinvestigatetheeffectontheseasurfaceheightandSSBcalculationofmissingtruesurfacephotonsnotdownlinkedinthepresenceoferroneousdownlinkbands,probablybycomparingdaytimeandnighttimedataoverthesameregion.

4.2.1.2 Coarse Selection and Setting of Segment Length

Theprocessisbasicallytofirstestablishavariablelength(e.g.,upto7km)oceansegmentwithenough(e.g.)photonheightstoyieldahistogramwithreasonablemomentsupto4thorder.

1. Examineoceandepthandcloudparametersagainstcontrolparameterstotestforsuitabilityforoceanprocessing.-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.-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. Correctforphotonheightsforshortperiodandlongperiodoceantides(gtxx/geophys_corr/tide_oceanandtide_equilibrium),theinversebarometereffectandthemodeleddynamicresponsetotheatmosphere(dynamicatmosphericcorrection,gtxx/geophys_corr/dac),

3. Selectonlyphotonheightsforwhichthesignalconfidence(gtxx/signal_conf_ph)isgreaterthanorequalto1.Thiswillselectphotonsinthelikelysurface±15-mbufferwindow.Alsodeleteanyphotonheightsoutsidethebanddefinedbygeoidplusorminushalftheheightoftheoceandownlinkwindow(presentlygeoid±15-m,to


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becomegeoid±20-m).(IstheheightofthedownlinkbandreportedinATL03perhapsatthegeosegrateandifso,whatisthevariablename?Theheightislistedatthemajorframerate.Butitwouldbehandierifitwasatthegeosegrate)

4. SubtracttheEGM2008geoidfromphotonheightstoreferenceallheightstothegeoid.NotethatEGM2008includesacorrectionforthepermanentormeantide.

5. Establishaninitialcoarsehistogramarray,Hc,spanning±15mwithbinsizeB1(TBD,e.g.,0.01m)andadataarray,Acoarse,forupto10,000photonheightsandassociatedinformation(index,geolocation,time)plusnoisephotoncounts.Thiswillbepopulatedwithdataforthecoarseselectionofsignalphotons.

6. 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, cloud_flag_ASR ,cloud_flag_atm, and bsnow_con, during the 280-m segment.

7. Photonsinthe14geo-bin(400-pulse)histogrambinswithgreaterthantheTh_Nc_ctimesthemediannumberofphotons,Nmedian,arecandidatesignalphotonsandphotonsinthebinswiththemediannumberofsamplesorlessareconsiderednoisephotons.WehavebeenusingTh_Nc_c=1inourtestswithearlyICESat-2data.Notethatwithfewsurfacereturnsperpulseandsignificantwaveheightsmuchlessthan30m,themajorityoftheofthebinswillcontainonlynoisephotonssothatthemedianwillequalthenumberofnoisephotonsperbin.InextremecaseswhereICESat-2encounterssignificantwaveheightsapproaching30m,wemayhavetoadoptaslightlydifferentthresholdsuchasthecountvalueequaltoorexceeding20%ofthebinpopulation.

8. Addthesignalandnoisephotonsfromthis14geo-bin(400-pulse)segmenttotherunningtotalofcandidatesignalphotonsandnoisephotons,andaddallthephotonstothecoarsehistogram,Hc.

9. 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)segmentuntiltheoceansegmentreachesalengthof7kmmaximum.Ifthe7-kmlengthisreachedandtherearelessthanathresholdnumberofphotons(e.g.photon_min=4000),ignorethatoceansegmentandrepeat4.2.1.2-2to4.2.1.2-6.Forphotoncountsabovethisinthe7-kmsegment,proceedto4.2.1.3andfinehistogrambasedsignalfinding.


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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 the bin limits of the fine histogram as all the coarse histogram bins on either side of the maximum in the 20-cm smoothed histogram, from the preliminary Jlow bin position to the preliminary Jhigh bin, where the coarse bin photon count is greater than the coarse histogram median. Then the average counts in the bins above the preliminary Jhigh are calculated and set equal to tailnoisehigh and the average counts in the bins below the preliminary Jlow are calculated and set equal to tailnoiselow. Then we find the bin limits of the fine histogram as all the 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. 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.

3. ConsiderthetimeseriesofalltheheightsstoredinAcoarsethat

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