It uses an L 2 hybrid norm characterized by a residual of transition between 1 and 2 for data 64257tting and another for model styling Both the steepest descent and conjugate direction methods are included The 1D blind decon volution problem is form ID: 62210
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Claerbout2Blockymodels:L1/L2hybridnorm IwasrstattractedtostrictL1byitspotentialforblockymodels.ButthenIrealizedforeachnonspike(zero)onthetimeaxis,theorysaysIwouldneeda\basisequation".Thatimpliesanimmensenumberofiterations,soitisunacceptableinimagingapplications.Withthehybridsolvers,insteadofexactzeroswehavealargeregiondrivendownbytheL2normandasmallL1regionwherelargespikesarewelcomed.MODELDERIVATIVESHereistheusualdenitionofresidualrioftheoreticaldataPjFi;jmjfromobserveddatadiri=(XjFi;jmj)�diorr=Fm�d:(1)LetC()beaconvexfunction(C000)ofascalar.Thepenaltyfunction(ornormofresidualsisexpressedbyN(m)=XiC(ri)(2)Wedenoteacolumnvectorgwithcomponentsgibyg=vec(gi).WesoonrequirethederivativeofC(r)ateachresidualri:g=vec@C(ri) @ri(3)Weoftenupdatemodelsinthedirectionofthegradientofthenormoftheresidual.m=@N @mk=Xi@C(ri) @ri@ri @mk=Xig(ri)Fi;k=F0g(4)Deneamodelupdatedirectionbym=F0g.Sincer=Fm�d,weseetheresidualupdatedirectionwillber=Fm.Tondthedistance tomoveinthosedirectionsm m+ m (5) r r+ r (6) wechoosethescalar tominimizeN( )=XiC(ri+ ri)(7)Thesuminequation(7)isasumof\dishes",shapesbetweenL2parabolasandL1V's.Thei-thdishiscenteredon =�ri=ri.Itissteepandnarrowifriislarge, SEP{139 Claerbout4Blockymodels:L1/L2hybridnorm PLANESEARCHThemostuniversallyusedmethodofsolvingimmenselinearregressionssuchasimag-ingproblemsistheConjugateGradient(CG)method.Ithastheremarkablepropertythatinthepresenceofexactarithmetic,theexactsolutionisfoundinanitenumberofiterations.AsimplermethodwiththesamepropertyistheConjugateDirectionmethod.Itisdebatablewhichhasthebetternumericalroundo properties,sowegenerallyusetheConjugateDirectionmethodasitissimplertocomprehend.Itsaysnottomovealongthegradientdirectionline,butsomewhereintheplaneofthegradientandthepreviousstep.Thebestmoveinthatplanerequiresustondtwoscalars,one toscalethegradient,theothertoscalethepreviousstep.ThatisallforL2optimization.Weproceedhereinthesamewaywithothernormsandhopeforthebest.Sohereweare,embeddedinagiantmultivariateregressionwherewehaveabivariateregression(twounknowns).Fromthemultivarateregressionwearegiventhreevectorsindataspace.ri,giandsi.Youwillrecognizetheseasthecurrentresidual,thegradient(ri),andthepreviousstep.(Thegradientandpreviousstepappearingherehavepreviouslybeentransformedtodataspace(theconjugatespace)bytheoperatorF.)Ournextresidualwillbeaperturbationoftheoldone.ri=ri+ gi+si(20)Weseektominimizebyvariationof( ;)N( ;)=XiC(ri+ gi+si)(21)Letthecoecients(Ci;C0i;C00i)refertoaTaylorexpansionofC(r)aboutri.N( ;)=XiCi+( gi+si)C0i+( gi+si)2C00i=2(22)Wehavetwounknowns,( ;)inaquadraticform.Wesettozerothe derivativeofthequadraticform,likewisethederivativegetting00=XiC0igisi+C00i@ @ @ @( gi+si)( gi+si)(23)resultingina22setofequationstosolvefor and.(XiC00igisi(gisi)) =�XiC0igisi(24)Thesolutionofany22setofsimultaneousequationsisgenerallytrivial.Theonlydicultiesarisewhenthedeterminantvanisheswhichhereiseasy(luckily)to SEP{139 Claerbout6Blockymodels:L1/L2hybridnorm scalars(independentlyforeachz).loopoveralltimepointsfm=d=(1+2)#Thesearescalars!loopovernon-lineariterationsfrd=m�drm=mGetderivativesofhybridnormB0(rd)andB00(rd)fordatagoal.GetderivativesofhybridnormC0(rm)andC00(rm)formodelgoal.#Plantond toupdatem=m+ #TaylorseriesfordatapenaltyN(rd)=B+B0 +B00 2=2#TaylorseriesformodelpenaltyN(rm)=C+C0 +C00 2=2#0=@ @ (N(rd)+N(rm)) =�(B0+C0)=(B00+C00)m=m+ gendofloopovernon-lineariterationsgendofloopoveralltimepointsTohelpusunderstandthechoiceofparametersRd,Rm,and,WeexaminethetheoreticalrelationbetweenmanddimpliedbytheabovecodeasafunctionofandRmatRd!1,inotherwords,whenthedatahasnormalbehaviorandwearemostlyinterestedintheroleoftheregularizationdrawingweaksignalsdowntowardszero.ThedatattingpenaltyisB=(m�d)2=2anditsderivativeB0=m�d.Thederivativeofthemodelpenalty(fromequation(15))isC0=m=p 1+m2=R2m.Settingthesumofthederivativestozerowehave0=B0+C0=m�d+m p 1+m2=R2m(27)Thissaysmismostlyalittlesmallerthand,butitgetsmoreinterestingnear(m;d)0.Theretheslopem=d=1=(1+)whichsaysan=4willdampthesignal(wheresmall)byafactorof5.Movingawayfromm=0weseethedampingpowerofdiminishesuniformlyasmexceedsRm.UNKNOWNSHOTWAVEFORMAone-dimensionalseismogramd(t)isunknownre ectivityc(t)convolvedwithun-knownsourcewaveforms(t).ThenumberofdatapointsNDNCislessthanthenumberofunknownsNC+NS.Clearlyweneeda"smart"regularization.Letusseehowthisproblemcanbesetupsore ectivityc(t)comesoutwithsparsespikessotheintegralofc(t)isblocky.Thisisanonlinearproblembecausetheconvolutionoftheunknownsismadeoftheirproduct.Nonlinearproblemselicitwell-warrantedfearofmultiplesolutionsleadingtousgettingstuckinthewrongone.Thekeytoavoidingthispitfallis SEP{139 Claerbout8Blockymodels:L1/L2hybridnorm Weneedderivativesofeachnormateachresidual.WebasetheseontheconvexfunctionC(r)oftheHybridnorm.LetuscalltheseAiforthedatatting,andBiforthemodelstyling.Ai=C(Rd;ri) (33) Bi=C(Rm;ci) (34) (Actually,wedon'tneedAi(becauseforLeastSquares,A0i=riandA00i=1),butIincludeithereincasewewishtodealwithnoiseburstsinthedata.)Asearlier,expandingthenormsinTaylorseries,equation(32)becomes0=XiA0iri+ XiA00ir2i+ XiB0ici+ XiB00ic2i!(35)whichgivesthe weneedtoupdatethemodelcandtheresidualrd. =�PiA0iri+PiB0ici PiA00ir2i+PiB00ic2i(36)Thisisthesteepestdescentmethod.Fortheconjugatedirectionsmethodthereisa22equationlikeequation(24).Non-linearsolverThenon-linearapproachisalittlemorecomplicatedbutitexplicitlydealswiththeinteractionbetweensandcsoitconvergesfaster.Werepresenteverythingasa\known"partplusaperturbationpartwhichwewillndandaddintotheknownpart.ThisismosteasilyexpressedintheFourierdomain.0(S+S)(C+C)�D(37)LinearizebydroppingSC.0SC+CS+(CS�D)(38)Letuschangetothetimedomainwithamatrixnotation.PuttheunknownsCandSinvectors~cand~s.PuttheknownsCandSinconvolutionmatricesCandS.ExpressCS�Dasacolumnvectord.Itstimedomaincoecientsared0=c0s0�d0andd1=c0s1+c1s0�d1,etc.Thedatattingregressionisnow0S~c+C~s+d(39)Thisregressionisexpressedmoreexplicitlybelow. SEP{139 Claerbout10Blockymodels:L1/L2hybridnorm Thatwassteepestdescent.Theextensiontoconjugatedirectionisstraightforward.Aswithallnonlinearproblemsthereisthedangerofbizarrebehaviorandmultipleminima.Toavoidfrustration,whilelearningyoushouldspendabouthalfofyoure ortdirectedtowardndingagoodstartingsolution.Thisnormallyamountstodeningandsolvingoneortwolinearproblems.Inthisapplicationwemightgetourstartingsolutionfors(t)andc(t)fromconventionaldeconvolutionanalysis,orwemightgetitfromtheblockcyclicsolver. SEP{139