FOURIER TRANSFORMS AND WAVES: in four long lectures

1 FOURIERTRANSFORMSANDWAVES:infourlonglect uresJonF. Cl rboutCecilandIdaGreenProfessorofGeophysi csStanfordUniversityc March1, .. ANDZ- TRANSFORMS .. RESTORATION.. two generalthemes,First,insteadofdrillingdow nintoanalyticaldetailsofone-dimensionalF ourieranaly-sis,theselecturesscanthebasi cdefinitionsandconceptsfocusingontheconc rete,namely, ontothebasicprinciplesofmultidimensional spectra,a , (FGDP) ,ProcessingversusInversion(PVI) (BEI) (GEE)Thesebooksareallfreelyavailableonth ewebat ofthecartesianvariables(t,x,y,z) thenitis usefultoFouriertransform(FT) , theearthdoesnotchangewithtime(theoceando es!)sofortheearth,wecangenerallygainbyFo uriertransformingthetimeaxistherebyconve rtingtime-dependentdifferentialequations (hard)to algebraicequations(easier)in frequency (temporalfrequency).Inseismology, theearthgenerallychangesratherstronglywi thdepth,sowecannotuse-fullyFouriertransf ormthedepthzaxisandwearestuckwithdiffere ntialequationsinz. OniiiCONTENTS theotherhand,wecanmodela layeredearthwhereeachlayerhasmaterialpro pertiesthatareconstantinz.

2 , ,x,ybutnotz, sowereducethepartialdifferentialequation sofphysicstoordinarydifferentialequation s(ODEs).AbigadvantageofknowingFTtheoryis thatit enablesustovisualizephysicalbehaviorwith outusneedingtousea (t,x,y,z) wehavea correspondingfrequency ( ,kx,ky,kz).Thek s arespatialfrequencies, is :A seismicwave fromthefastearthgoesintotheslow (inversespatialwavelength)?Ina layeredearth,thehorizonalspatialfrequenc y is a willfindthistobeSnell s sphericalcoordinatesystemora cylindricalcoordinatesystem,Fouriertrans formsareuselessbutthey arecloselyrelatedto sphericalharmonicfunctions andBesseltransforma-tionswhichplaya ,we llseehow tousewave theorytotake theseobservationsmadeontheearth s surfaceand down-wardcontinue them,to a centraltoolin theoryfoundinphysicsbooksandgeophysicsbo oksis succeedingchapterconsiderstwo-dimensiona lspectraofany function,how suchfunc-tionscanbemodeled,whatit meanstodeconvolve 2-Dfunctions,andanall-purposemethodoffil linginmissingdataina uncertaintyprinciple, ,it meansthe earthresponse nowisthesameastheearthresponselater.

3 Switchingourpointofviewfromtimetospace,t heapplicabilityofFouriertransformationme ansthatthe impulseresponse a columnvectorfullofzeroswithsomewherea one,say(0,0, 1, 0, 0, ) (wheretheprime() meanstransposetherowintoa column.)Animpulseresponseis a columnfromthematrixq= q0q1q2q3q4q5q6q7 = b000000b1b00000b2b1b00000b2b1b00000b2b1b 00000b2b1b00000b2b100000b2 p0p1p2p3p4p5 =Bp( )Theimpulseresponseis theqthatcomesoutwhentheinputpis typicalap-plication,thematrixwouldbeabou t1000 1000andnotthesimple8 6 examplethatI showyouabove. Noticethateachcolumnin thematrixcontainsthesamewaveform(b0,b1,b 2). Thiswaveformis calledthe impulseresponse .Thecollectionofimpulseresponsesinequati on( ) ,buteachrowlikewisecontainsthesamething, andthatthingis thebackwardsimpulseresponse(b2,b1,b0).Su ppose(b2,b1,b0) werenumericallyequalto(1, 2, 1)/1t2. Thenequation( )wouldbelike thedifferentialequationd2dt2p=q. Equation( )wouldbea finite-differencerepresenta-tionofa importantideasareequivalent;eitherthey arebothtrueorthey a ( )withFouriertransformsisthatthek-throwin ( )isthek-thpowerofZina polynomialmultiplicationQ(Z)=B(Z)P(Z).

4 Therelationshipofany polynomialsuchasQ(Z) toFourierTransformsresultsfromtherelatio nZ=ei 1t, ANDZ-TRANSFORMST imeandspaceareordinarilythoughtofasconti nuous,but alsocalled sampling or digitizing. Youmightworrythatdiscretizationis a ,physicalconceptshave [ER]signalina computer, it is necessarytoapproximateit insomewaybya toevaluateorobserveb(t) at a uniformspacingofpointsintime,callthisdis cretizedsignalbt. ,sucha discreteapproximationtothecontinuousfunc tioncouldbedenotedbythevectorbt=(..0, 0, 1, 2, 0, 1, 1, 0, 0,..)( )Naturally, if timepointswereclosertogether, done,then,is representa signalis asa polynomial,wherethecoefficientsofthepoly no-mialrepresentthevalueofbtat successive ,B(Z)=1+2Z+0Z2 Z3 Z4( )Thispolynomialis calleda Z-transform. Whatis themeaningofZhere?Zshouldnottakeonsomenu mericalvalue;it is insteadtheunit-delayoperator. Forexample,thecoefficientsofZ B(Z)=Z+2Z2 Z4 Z5areplottedin :ThecoefficientsofZ B(Z) aretheshiftedversionofthecoefficientsofB (Z).

5 Cs-triv2[ER] ,butnow ANDZ-TRANSFORMS3bymultiplyingB(Z) byZn. ThedelayoperatorZisimportantinanalyzingw avessimplybecausewavestake a certainamountoftimetomove thatit impulseresponse. Ifanotherexplosionoccurredatt=10timeunit safterthefirst,wewouldexpectthepressuref unctiony(t) ,thispressurefunctionwouldbeexpressedasY (Z)=B(Z)+Z10B(Z). :Responsetotwo [ER] thefirstexplosionwerefollowedbyanimplosi onofhalf-strength,wewouldhaveB(Z) 12Z10B(Z).If pulsesoverlappedoneanotherintime(aswould bethecaseifB(Z) hadde-greegreaterthan10), wouldjustaddtogetherwithoutany interactionis calledthe linearity property. Inseismologywefindthat althoughtheearthis a heterogeneousconglomerationofrocksofdiff erentshapesandtypes whenseismicwavestravel throughtheearth,they donotinterferewithoneanother. They satisfylinearsuperposition. dominatingfeatureinhydrody-namics,wheref lowvelocitiesarea noticeablefractionofthewave velocity. Nonlinearityisabsentfromreflectionseismo logyexceptwithina superposingis :Crossingplanewavessu-perposingviewedont heleftas wig-gletraces andontherightas raster.

6 Cs-super[ER] supposetherewasanexplosionatt=0, a half-strengthimplosionatt=1,andanother,q uarter-strengthexplosionatt= source timeseries,xt=(1, 12, 0,14).TheZ-transformofthesourceisX(Z)=1 12Z+14Z3. Theobservedytforthissequenceofexplosions andimplosionsthroughtheseismometerhasaZ- transformY(Z), givenbyY(Z)=B(Z) Z2B(Z)+Z34B(Z)=(1 Z2+Z34)B(Z)=X(Z)B(Z)( )Thelastequationshowspolynomialmultiplic ationastheunderlyingbasisoftime-invarian tlinear-systemtheory, namelythattheoutputY(Z) canbeexpressedastheinputX(Z) timestheimpulse-responsefilterB(Z).Whens ignalvaluesareinsignificantexceptina small regiononthetimeaxis,thesignalsarecalled wavelets. a computerwhenwemultiplytwoZ-transformstog ether?Thefilter2+Zwouldberepresentedina computerbythestorageinmemoryofthecoeffic ients(2,1).Likewise,for1 Z, thenumbers(1, 1) theseinputsandproducethesequence(2, 1, 1).Letusseehowthecomputationproceedsina generalcase,sayX(Z)B(Z)=Y(Z)( )(x0+x1Z+x2Z2+ ) (b0+b1Z+b2Z2)=y0+y1Z+y2Z2+ ( )Identifyingcoefficientsofsuccessive powersofZ, wegety0=x0b0y1=x1b0+x0b1y2=x2b0+x1b1+x0b 2( )y3=x3b0+x2b1+x1b2y4=x4b0+x3b1+x2b2= ANDZ-TRANSFORMS5 Inmatrixformthislookslike y0y1y2y3y4y5y6 = x000x1x00x2x1x0x3x2x1x4x3x20x4x300x4 b0b1b2 ( )Thefollowingequation,calledthe convolutionequation, carriesthespiritofthegroupshownin( ):yk=Nb i=0xk ibi( )To becorrectindetailwhenweassociateequation ( )withthegroup( ),weshouldalsoassertthateithertheinputxk vanishesbeforek=0 orNbmustbeadjustedsothatthesumdoesnotext endbeforex0.

7 Theseendconditionsareexpressedmoreconven ientlybydefiningj=k iinequation( )andeliminatingkgettingyj+i=Nb i=0xjbi( )Aconvolutionprogrambasedonequation( )includingendeffectsonbothends,iscon-vol ve().#convolution:Y(Z)= X(Z)* B(Z)#subroutineconvolve(nb, bb,nx, xx, yy )integernb# numberof coefficientsin filterintegernx# numberof coefficientsin input# numberof coefficientsin outputwillbe nx+nb-1realbb(nb)# filtercoefficientsrealxx(nx)# inputtracerealyy(1)# outputtraceintegerib, ix,iy, nyny = nx + nb -1callnull(yy,ny)do ib= 1, nbdo ix=1, nxyy( ix+ib-1)= yy(ix+ib-1)+ xx(ix)* bb(ib)return;endThisprogramis writtenina languagecalledRatfor, a rational is ,but,if youareinterested,moredetailsinthelastcha pterofPVI1, thebookthatI timeNoticethatX(Z) andY(Z) neednotstrictlybepolynomials;they maycontainbothpositiveandnegative powersofZ, suchasX(Z)= +x 2Z2+x 1Z+x0+x1Z+ ( )Y(Z)= +y 2Z2+y 1Z+y0+y1Z+ ( )Thenegative powersofZinX(Z) andY(Z) showthatthedatais definedbeforet= powersofZinthefilteris ( )showsthattheoutputykthatoccursat timekis a linearcombinationofcurrentandpreviousinp uts;thatis,(xi,i k).

8 If thefilterB(Z) hadincludeda termlikeb 1/Z, thentheoutputykat timekwouldbea linearcombinationofcurrentandpreviousinp utsandxk+1, aninputthatreallyhasnotarrivedat timek. Sucha filteris calleda nonrealizable filter, becauseit , nonrealizablefiltersareoccasionallyusefu lincomputersimulationswhereallthedatais pointona spinningwheelare(x,y)=(cos( t+ ), sin( t+ )),where is theangularfrequency ofrevolutionand is pendulumis nearlysinusoidal, signalis its spectrum. Smallamplitudesignalsarewidespreadinnatu re,fromthevibrationsofatomstothesoundvib rationswecreateandobserve 3to10 6. Inwaterorsolid,thecompressionistypically 10 6to10 9. Amathematicalreasonwhysinusoidsaresocomm oninnatureis (whicharefunctionsofmaterialproperties)a reconstantintimeandspace,theequationshav e builtfromthecomplex exponentiale i t=cos t isin t( )A Fouriercomponentofa timesignalis a complex number, a sumofrealandimaginaryparts,sayB=ReB+iImB ( ) attachedtosomefrequency. Letjbeanintegerand jbea (t) canbemanufacturedbyaddinga collectionofcomplex exponentialsignals,eachcomplex exponentialbeingscaledbya complex coefficientBj, namely,b(t)= jBje i jt( )Thismanufacturesacomplex-valuedsignal.

9 How dowearrangeforb(t) tobereal?We canthrowawaytheimaginarypart,whichis like addingb(t) toitscomplex conjugateb(t), andthendividingbytwo:Reb(t)=12 j(Bje i jt+ Bjei jt)( )Inotherwords,foreachpositive jwithamplitudeBj, weadda negative jwithamplitude Bj(likewise,foreverynegative ).TheBjarecalledthe frequency function, orthe Fouriertransform. Loosely, theBjarecalledthe spectrum, thoughin formalmathematics,theword spectrum isreservedfortheproduct BjBj. Thewords amplitudespectrum universallymean , =j1 sothatb(t)= jBje i(j1 )t( )Representinga signalbya sumofsinusoidsis technicallyknownas inverseFouriertransfor-mation. Anexampleofthisis ,timeis generallymappedintointegerstoo,sayt=n1t. Thisiscalled discretizing or sampling. Thehighestpossiblefrequency expressibleonameshis( , 1, 1,+1, 1,+1, 1, ), whichis thesameasei n. Settingei maxt=ei n, weseethatthemaximumfrequency is max= 1t( )Timeis commonlygivenineithersecondsorsampleunit s,whicharethesamewhen1t= ,frequency isusuallyexpressedincyclespersecond,whic histhesameasHertz, abbreviatedHz.

10 Incomputerwork,frequency is usuallyspecifiedin ,frequency isusuallyexpressedinradians wheretherelationbetweenradiansandcyclesi s =2 f. We useradiansbecause,otherwise,equationsare filledwith2 ,themaximumfrequency hasa name:it isthe Nyquistfrequency, whichis radiansor1/2 :Superpositionoftwo [NR] functionofa delayedimpulseat timedelayt0isei Fouriersum :B( )= nbnei tn= nbnei n1t( )TheFouriersumtransformsthesignalbttothe frequency functionB( ). Timewilloftenbedenotedbyt, ( )insteadofbn, resultinginanimplied1t= functionofa pulseattimetn=n1tisei n1t=(ei 1t)n. Thefactorei 1toccurssoofteninappliedworkthatit hasa name:Z=ei 1t( )WiththisZ, thepulseattimetniscompactlyrepresentedas Zn. ThevariableZmakesFouriertransforms looklike polynomials,thesubjectofa literaturecalled Z- TRANSFORMS . TheZ-transformis a variantformoftheFouriertransformthatis particularlyusefulfortime-discretized(sa mpled) ( ),wehaveZ2=ei 21t,Z3=ei 31t, ,equation( )becomesB( )=B( (Z))= nbnZn( ) , is a realvariable,soZ=ei 1t=cos 1t+isin 1tis a complex hasunitmagnitudebecausesin2+cos2= rangesontherealaxis,Zrangesontheunitcirc le|Z|= timefunctionlike thisddtf(t)=lim1t 0f(t) f(t 1t)1tComputationally, wethinkofa differentialasa finitedifference,namely, a functionis delayeda ,thefinitedifferenceoperatoris (1 Z) withanimplicit1t= ,thetimederivative isthefilter(+1, 1).