IX. EMPIRICAL ORTHOGONAL FUNCTIONS

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1 IX. EMPIRICAL ORTHOGONAL FUNCTION A. Eprcl Orthogonl Functons (EOF): T Don Consdr sptl rry of dscrt t srs obsrvtons (for xpl stwrd currnt) t M loctons - u jt, whr j ndcts stton nubr (j =,... M) nd t ndcts th t stp [t =,...N; whr (N-) Δt = T; lngth of srs]. Th zro-lg cross-covrnc for th j th nd k th lnts of th rry s gvn by R = N N (u - u j)(u - u ) jk jt kt k, (IX.) t= whr R s squr, sytrc, rl M x M trx nd ovrbr ndcts dsgntd srs n vlu. Th dgonl lnts of R jk n (IX.) (or whr j = k) r th stton rcord vrncs ccordng to R = N N jj (u jt - u j) (IX.) t= Th totl vrnc of th syst (or rry) t srs (lso clld trc of R or Tr R) s M Tr R = R jj. (IX.) j= A dgonlzton of th M x M trx R (or dtld xplnton blow) ylds st of M prcl orthogonl functons (EOF) or ods; n whch th j th od conssts of Rl EIGENVECTOR j, wth coponnts =,,.M; nd for whch th j th nd k th ods r orthogonl to ch othr n th sns tht M j k = jk, (IX.4) = nd 4 Fbrury 009 Chptr IX EOFs Nots 8Wndll. Brown

2 Postv EIGENVALUE. such tht M R jk j = k. (IX.4b) j= (Not: Th nubr of M-coponnt gnvctors s qul to th nubr of sttons). Undr ths crcustncs, u jt cn b xpndd n trs of ths gnvctors j ccordng to u = M jt t j (IX.5) = whr th pltud t srs of th th gnvctor jcn b xprssd s = M u t j jt. (IX.5b) j= 4 Fbrury 009 Chptr IX EOFs Nots 8Wndll. Brown

3 Th vrnc of t s th vrnc of th th od. Th th od gnvlu s tht od s vrnc (.., nrgy). Th prcntg of th totl "syst" (or rry) vrnc, whch fro (IX.) cn b wrttn M Tr R =, (IX.6) tht s xplnd by tht od s gvn by th rto of /Tr R. j= Th dnsonl frst EOF (or od- EOF), s dfnd by th product of th norlzd ( gnvctor lj nd pltud l) or hs th lrgst vrnc or gnvlu l. lj( l ) (IX.6b) Ths st of gnvctors/gnvlus th soluton - s dtrnd by lst squr ft btwn th soluton nd th cross-covrnc trx R. Th bst ft to th cross-covrnc trx s dfnd by (R jk - l ljlk ) = nu. (IX.7) j k Mthtclly, th fttng-procss prttons th totl syst (or rry) vrblty vrnc nto st of M orthogonl ods; j / ; ch xplnng dffrnt pttrns of corrltd nforton n th rry, undr th rthr rtfcl constrnt tht th gnvctors b orthogonl (.., sttstclly ndpndnt fro ch othr). AN EXAMPLE OF THE APPLICATION OF THE T-EOF TECHNOLOGY Consdr th followng xpl of n rry of nvrtd cho soundr (IE) surnts d cross th contnntl slop nto th dp ocn swrd of th Azon Rvr outflow (Fgur. IX.). Th IE s n coustc projctor whos trvl t π fro th botto to th ocn to th surfc (s Fgur. IX.b) nd bck s rltd to ocn proprts prrly 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

Th vrblty of th IE t-srs of quvlnt dync hght for ch of th oord IE dploynts s prsntd n (Fgur. IX.

4 th dpth of th n throcln - ccordng to = ( /C) C(Z, t) dz, whr C s sound spd nd h s locl wtr dpth. Hydrogrphc surnts t th IE oorng st r usd to coput th dync hghts; whch r thn corrltd wth th corrspondng IE trvl ts. Th vrblty of th IE t-srs of quvlnt dync hght for ch of th oord IE dploynts s prsntd n (Fgur. IX.). h o Fgur. IX.. (-lft) Th locton of four botto-ountd nvrtd cho soundrs (IE) nd thr currnt tr oorngs tht wr dployd n th tropcl Atlntc fro (b-rght) Th dpths of th dffrnt nstrunts r llustrtd n th bthytrc trnsct. Fgur. IX. Dync hght t srs drvd fro IE trvl ts surd by nstrunts loctd n Fgur IX.. Th EOF dcoposton of th cross-covrnc trx of ths four (sclr) IE dync hght t srs (.., qu IX.) ylds four t-don EOFs (TD-EOF). Th norlzd gnvctor structurs of th two ost nrgtc IE TD-EOFs r dpctd n Fgur IX.. Not tht th t vrblty of thr of ths TD-EOFs cn b drvd by coputng th 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 4

5 product of th odl t srs (Fgur IX. 4) nd th norlzd gnvctor stton pltud structur. Th gostrophc trnsport nfrrd fro th gnod IE dffrncs r ndctd to th rght. Fgur. IX. To th lft r th norlzd structurs of th ost nrgtc IE gnods (MODE- bov nd MODE- blow). Also ndctd to th rght r th gostrophc trnsport dstrbutons tht r qulttvly consstnt th IE dync hght vrblty gnod structur. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 5

6 Fgur. IX.4 Th pltud t srs (n dyn-) of th gnods prsntd n Fgur IX.. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 6

7 B. Eprcl Orthogonl Functons: Frquncy Don Consdr -dnsonl vctor t srs V (t), wth coponnts v} {u,, tht hv bn dscrtly spld N, t to for Vn = V(n t) (IX.8) whr n =,,...N. Th coplx Fourr coffcnts r ^ V = t V N t n whr th rfrs to th frquncs whr th vctor dtls of V ^ f = f n - n xp N t = N t (IX.9) t whch ths r coputd. nd cn b xprssd nubr of dffrnt wys ; ^ V vˆ u u (IX.9b) Coplx Fourr coffcnts for th coponnts cn b xprssd n dffrnt wys s dscussd n th followng. Dfn th gnrl Crtsn cross-spctrl nrgy dnsty trx (DM) for n rbtrry nubr of coponnts of = < > j j, (IX.0) whr th ndx, rfrrng to frquncy hroncs, hs bn droppd. Th coplx rprsntton of cross-spctrl nrgy dnsty (DM) s = C - Q j j j (IX.) Th norlzd DM or spctru cohrnc j (IX.) j / (jj) 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 7

8 Th -coponnt j s x Hrtn trx, whr Hrtn pls tht γ uv = γ uv, s uu uv uv (IX.) vv Consdr th followng two spl knds of oton n th Crtsn rprsntton n whch thr hs bn two-fold dcoposton; n () frquncy nd () long utully-orthogonl xs (crtsn.g.).. Osclltory Rctlnr Moton - long n rbtrry ln dscrbd by r = + j, (IX.4) whr α nd β r drcton cosns. Fgur. IX.4 Hodogrph of rctlnr oton Fourr coponnts of th vlocty vctor [t so frquncy f ] r = o( ) vˆ = o( ) (IX.5) whr th phs of both coponnts θ = θ = θ. If θ = 0, thn or = o o (IX.6) ^ v = vˆ = 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 8 o o (IX.6b)

9 nd th DM, whch s rl for ths cs bcus thr s no phs lg btwn coponnts, s = < o > (IX.7) Not: If surd DM s rl, thn r y b dducd. For xpl, f α= nd β=0, thn th oton wll b only long th x-xs nd only uu rns.. Osclltory Ellptcl Moton Th Fourr coponnts = o xp vˆ = o xp ( / ), (IX.8) whr th ndcts th phs for rotton n thr drcton; + = ACW nd - = CW or ^ V = ( o o ) (IX.9) Fgur IX.5 Hodogrph for llptcl oton. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 9

10 Thn th DM s purly gnry bcus of phs lg = < o > (IX.0) ( ) Not tht α = β => crculr oton Iportnt: In gnrl, snc ll lnts of th crtsn DM r non-zro th vlus α nd β dpnd on th crtsn xs orntton; whch s dsdvntg. Thus w sk or gnrl pproch. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 0

11 Gnrl Fourr Coponnt Rprsntton ^ V = +, (IX.) whr ( =,) r coplx pltuds (s bfor), but now r bss vctors rprsntng dffrnt typs of oton (not orthogonl drctons. For xpl, rctlnr oton long utully-orthogonl xs r dscrbd n trs of Crtsn syst s follows: whr () th = ; 0 rprsnt oton n th two dffrnt drctons; 0 = (IX.) () th now tll us th pltud nd phs of th typ of oton dscrbd by unt vctors. ^ Th r found by fndng th projcton of V ^ = V ^ = V on th rspctv ccordng to (IX.) In ths cs t s trvl bcus thr s no phs (.. gnry prt) so tht = (= ) o (IX.4) = (= ) o (IX.4b) In ths cs, th gnrlzd Fourr coffcnt nd th r th s. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

12 Fgur. IX.6 Crtsn coponnts of typcl thr-dnsonl vctor t srs surnt (Cln, 978). 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

13 Fgur. IX.7 Th Crtsn spctru dnsty trx for th dt shown n Fgur. IX.7 (Cln, 978). 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

14 Fgur. IX.8 Rotry rprsntton of th dt shown n Fgur. IX.7 (Cln, 978). 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 4

15 ^ Rotry Rprsntton of V Th dcoposton n trs of Crtsn coponnts cn b rwrttn s ^ xp xp u u u V = = = C + C ( - / ) (IX.5) ( + / ) u u xp u xp whr th bss (or unt) vctors now r -/ -/ = = ; - (IX.6) + - Hodogrph Fgur IX.9 Hodogrph for crculr oton Countr-clockws nd clockws rottng crculr rottng unt vctor otons nd r orthonorl.. th sclr products r = (IX.7) = 0. W cn fnd th gnrlzd Fourr coffcnts by projctng th unt vctors onto th orgnl Crtsn Fourr coponnt vctor,.. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 5

16 = + = ^ = V _ ^ = V = = -/ -/ ( - îu ) ( + îu ),.. / (, ) - / (, - ){ } (IX.8) whr = u = u. (IX.8b) Coputng th dgonl lnts of th gnrlzd spctrl nrgy dnsty trx = + = = < _ = < > = > = < < > + < > + < > - Q > + Q cohrnc btwn Not th rltonshp btwn th gnrlzd nd crtsn spctrl dnsts. _ (IX.9) A Gnrlzd pctrl Dnsty Mtrx Thn = j = < j > n whch th gnrlzd cohrncs r: < j > j =. / (IX.9) < >< j j > It cn b shown tht gnrlzd DM cn b drvd fro th Crtsn DM = c ccordng to c = j < j > = j (IX.0) 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 6

17 EMPIRICAL EIGENMODE PECTRA Lt us now gnrlz th prvous pproch stll furthr by dtrnng n orthonorl st of gnvctors g of th Crtsn cross-spctrl nrgy dnsty trx (DM) such tht c g = g, (IX.) whr th frquncy-dpndnt g cn b thought of s th kntcl norl ods for th t srs surnts.in th gnrl cs, th gnvctors g b found by solvng th gnvlu probl for ch frquncy. Expndng -D Fourr coponnt vctor ^ V n th gnvctors of th prtculr crossspctrl dnsty trx for spcfc frquncy: ^ V = + g g (IX.)!! whr / λ / λ r rl gnvlus, snc th DM s Hrtn. H r spctru pltuds r th orgnl lnts nd r = λ. W hv lrdy shown tht for th gnrlzd rprsntton of th Fourr coffcnt vctor ^ V = +... =, (IX.) th cohrnc btwn ods s coputd n trs of th gnrlzd cross-spctrl dnsty trx (or Crtsn DM) ccordng to By lttng j c < j > j. (IX.4) = g nd rcllng th orgnl gnvlu probl bov: = g c g Thn fro th orthonorl proprty of g j j = g g. j j (IX.5) 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 7

18 = j j j (IX.6) Bcus for = j j =, (IX.7) 0 for j nd = λ (IX.8) j 0 (IX.9) Th lttr showng tht th dffrnt ods g r ndpndnt (.., ll cohrncs r zro). But ths splcty cos t th xpns of or coplctd hodogrph for ch gnvctor. Eprcl Orthogonl Functons for -D Lnr Moton ^ Crtsn DM V o = ; o c = o olvng for th gnvlus of th bov probl lds to = < > ( + o = 0 whch shows tht only on od of oton prsnt t ths frquncy. ) Th corrspondng unt gntud gnvctor s g = N ; whr N (( + )). Hr g rprsnts lnr oton long r - th only cohrnt oton. Th scond gnvlu s dgnrt so tht g s ndtrnnt; g = 0. -/ 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 8

19 Eprcl Orthogonl Functons for -D Ellptcl Moton ^ V o = ( / o ) Crtsn DM c = < o > ( ) Egnvlus: λ = o (α + β ) ; λ = 0 Agn only on od of oton Unt Mgntud Egnvctor: = N g N = [( + -/ whr ] ) -/ 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 9

20 Fgur. IX.0 Eprcl od spctr for th dt shown n Fgur. IX.7. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 0

21 Fgur. IX. Th thr-dnsonl Fourr vctor hologrph. Th prtrs whch spcfy th shp of th hodogrph r ndctd. THREE DIMENIONAL VECTOR whr V(t) = R u(t) V (t) = v(t) w(t) N/ - = ^ V f t Coplx Fourr Coffcnts ^ V = [ /(N t)] V n n (t) xp(- n/n T) t f = f = N t wth ^ V vˆ ŵ u u u Crtsn spctrl nrgy dnsty trx (DM) (subscrpt hs bn droppd) j whr < > nsbl vrg. < j > 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

22 Off-dgonl lnts of th DM - coplx rprsntton: j = C j - Q j co- qud- spctr Coplx Cohrnc j [ j / jj] j s x Hrtn ( rl nd coplx ndpndnt lnts) trx hr uu uv uu ww uw vv (Hrtn lowr lft off dgonl lnts r coplx conjugts of uppr rght off dgonl lnts.) For Fourr coponnt vctors fro two surnts (k) = (k) (k) (k) wth k =,,,. Th cross-spctrl dnsty trx (CDM) s 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

23 (k) j = < (k) () j > Ths s 6x6 Hrtn trx < >< vˆ >< ŵ >< >< vˆ >< ŵ > < vˆ vˆ >< vˆ ŵ > < ŵ ŵ > < u > nd vctor ELLIPTIC MOTION - n Crtsn rprsntton for on -D vctor whr, f α = β, thn th oton s crculr. In gnrl, lt th Fourr coponnt vctor, wth coplx, b rprsntd ^ V = + + whr r unt vctors - ch rprsntng prtculr typ of oton. For xpl, Crtsn In gnrl, = = 0 ^ = V j = j 0 = 0 orthonorl 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons

24 whr n ths cs, = u Gnrlzd pctrl Dnsty Mtrx Gnrlzd Cohrnc ; whch pls rctlnr oton. j j < [< j > < j j >< > j > ] / It cn b shown tht ny gnrlzd DM cn b found fro th Crtsn DM, (c), by usng th pproprt (orthonorl) unt vctors to trnsfor th spctru ccordng to j < j > (c) j. For ultpl-vctor t srs, th cross-spctru dnsty trx gnrlzs n th s wy n ths cs, lthough () nd () ^ V = = y b dffrnt () () nd ^ U = () () = < ˆ ()() () () Gnrlzd CDM j â j > whr usully ê = ê () (). Rottonl Invrnts Trc Tr = x KE Tr M = x KE = < > = nd KE = (u ) Dtrnnt of DM 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 4

25 = (- - = + R ) whr Γ = γ γ γ nd γ j r coplx cohrncs. For -D, th Dtrnnt s H = (- ( ) ) nd sur of th ncohrnt nos Dgr of Polrzton P - (/ Tr M P H ) s th frcton of non-rndo, non-sotropc nrgy P s rl nd vrs btwn 0 nd. 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 5

26 Ansotropy Rto A P A = - P s th rto of non-rndo, non-sotropc nrgy to th rndo, sotropc nrgy (.., sgnl to nos rto). u of Prncpl Mnors Th -D dtrnnt n on of th coordnt plns M M = j Mn dgr of polrzton M =- (Tr) P jj - Totl qurd Qudrtur pctru Q / ( j j - j ) j Q whch s th nt ount of rottng nrgy n pln. Rotry Coffcnt QH CrH, / Tr H whch s th frcton of th nt rottng Mn Rotry Coffcnt Cr Q / Tr = j j, Multpl Cohrnc - M rd prncpl nor ROTARY REPREENTATION Fourr vctor s dcoposd nto two countr-rottng crculr otons n pln (usully th horzontl pln). In -D, lnr oscllton tht s norl to th pln of rotry vctors s ddd 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 6

27 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 7 u C 0 u u + C 0 u u = C u u u = V ) - ( ) / + ( ^ Ths cn b wrttn n trs of th followng unt vctors 0 = -/ Countr-Clockws 0 - = -/ Clockws 0 0 Vrtcl Gnrlzd Fourr Coffcnts countrclockws (+) - ) - ( = V - = -/ ^ + clockws (-) - ) + ( = V = = -/ ^ - vrtcl -, = u V = ^ whr = u tc. ELEMENT OF THE GENERALIZED DM cn b found usng Crtsn pctrl Dnsty Mtrx > < j j or

28 (c) j Crtsn spctrl dnsty trx = + = < > =/ < > + < > - Q / Tr H Q (- Cr) Cr = Tr uv H = - = < > =/ Tr H (+ Cr) coputd fro th Crtsn DM = < > = < > = w + ; - r th rotry spctr "nnr utospctr" of Moors,97 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 8

29 Rotry coffcnt C r = frcton of nt rottng nrgy Norlzd Cross pctru = = / + - ( ) / (uu - vv) - Cuv = / / (Tr H )(- Cr ) + - tblty = (P - Cr ) /(- Cr ) whr +- tn < + - > = C uv /( Rnng Cohrnc Elnt for -D vctors +-w vv - uu ), (outr utospctru (Moors)) < + > (uw vw) = = / / < >< > ( w ) Rotry pctru Rprsntton for -D Vctor w -w w 4 Fbrury 009 Chptr IX. Eprcl Orthogonl Functons 9

A general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex.

A general N-dimensional vector consists of N values. They can be arranged as a column or a row and can be real or complex. Lnr lgr Vctors gnrl -dmnsonl ctor conssts of lus h cn rrngd s column or row nd cn rl or compl Rcll -dmnsonl ctor cn rprsnt poston, loct, or cclrton Lt & k,, unt ctors long,, & rspctl nd lt k h th componnts