Fractal analysis of deep ocean current speed time series

F ac al Analysis o Deep Ocean Cu en Speed Time Se ies LAURA CABRERA-BRITO Depa amen o de Fisica, Uni e sidad de Las Palmas de G an Cana ia, Las Palmas de G an Cana ia, Las Palmas, Spain GERMAN RODRIGUEZ,LUIS GARCÍA-WEIL,MERCEDES PACHECO,AND ESTHER PEREZ Applied Ma ine Physics and Remo e Sensing G oup, Ins i u e o En i onmen al S udies and Na u al Resou ces, and Depa amen o de Fisica, Uni e sidad de Las Palmas de G an Cana ia, Las Palmas de G an Cana ia, Las Palmas, Spain JOANNA J. WANIEK Leibniz Ins i u e o Bal ic Sea Resea ch, Ros ock, Ge many (Manusc ip ecei ed 24 May 2016, in inal o m 26 Janua y 2017) ABSTRACT F ac al p ope ies o deep ocean cu en speed ime se ies, measu ed a a single-poin moo ing on he Madei a Abyssal Plain a 1000- and 3000-m dep h, a e explo ed o e he ange be ween one week and 5 yea s, by using he de ended luc ua ion analysis and mul i ac al de ended luc ua ion analysis me hodologies. The de ended luc ua ion analysis e eals he exis ence o wo sub anges wi h di e en scaling beha io s. Long- ange empo al co ela ions ollowing a powe law a e ound in he ime-scale ange be ween ap- p oxima ely 50 days and 5 yea s, while a B ownian mo ion– ype beha io is obse ed o sho e ime scales. The mul i ac al analysis app oach unde lines a mul i ac al s uc u e whose in ensi y dec eases wi h dep h. The analysis o he shu led and su oga e e sions o he o iginal ime se ies shows ha mul i ac ali y is mainly due o long- ange co ela ions, al hough he e is a weak nonlinea con ibu ion a 1000-m dep h, which is con i med by he de ended luc ua ion analysis o ola ili y ime se ies. 1. In oduc ion A la ge po ion o he o al sola adia ion inciden on Ea h’s su ace is abso bed and s o ed in he ocean, due o i s ela i ely high hea capaci y, whe e i is edis- ibu ed h ough ocean cu en s. In addi ion o seawa e , ocean wa e masses may con ain many o he dissol ed o suspended ma e ials and li ing o ganisms. The e o e, ocean cu en s anspo ma e and ene gy om one pa o he plane o ano he , ep esen ing a key ac o in con olling Ea h’s clima e and ha ing impo an eco- logical implica ions. To an app oxima e deg ee, oceanic ci cula ion can be sepa a ed in o wo modes wi h speci ic dynamics and ime scales. In he uppe laye , wi h a hickness anging om app oxima ely 500 o 1500m ( an Aken 2007;Pine 2009), he mo e o less egula oceanic winds cons i u e he dominan p ocess d i ing ocean cu en s, al hough densi y di e ences may also play a signi ican ole. Below his laye ci cula ion is mainly de e mined by densi y di e ences, esul ing om e- gional di e ences in he exchange o hea and eshwa e be ween a mosphe e and ocean (S ewa 2009;Huang 2010). In eali y, howe e , his is no a i ial dis inc ion because bo h ci cula ion modes a e coupled. The low o wa e in he oceans is a esul o he in- e ac ion among a ious nonlinea p ocesses ha ake place ac oss a wide ange o space and ime scales. In pa icula , empo al a iabili y includes scales anging om a ew seconds ou o millions o yea s (Huybe s and Cu y 2006). The complexi y o he esul ing p ocess limi s i s p ope unde s anding and hinde s adequa e cha ac e iza ion and p edic ion o ocean cu en speeds. Wi hin his amewo k, a common and powe ul ap- p oach used o gain insigh in o he dynamics go e ning he p ocess is he analysis o ime se ies, ob ained by sampling he phenomenon unde s udy, conside ed as a ealiza ion o a andom p ocess. In his con ex , in addi ion o i s p obabilis ic s uc u e, a undamen al aspec o be conside ed o Co esponding au ho e-mail: Ge man Rod iguez, ge man. [email p o ec ed] APRIL 2017 C A B R E R A - B R I T O E T A L . 817 DOI: 10.1175/JTECH-D-16-0098.1 Ó2017 Ame ican Me eo ological Socie y. Fo in o ma ion ega ding euse o his con en and gene al copy igh in o ma ion, consul he AMS Copy igh Policy (www.ame soc.o g/PUBSReuseLicenses). cha ac e izing a andom p ocess, such as he ocean e- loci y ield, is he co ela ion deg ee be ween alues obse ed a di e en imes, also known as pe sis ence, which can be weak, s ong, o null. Fu he mo e, co - ela ion may exis be ween nea by alues in he ime se ies (sho - ange co ela ion) bu also be ween alues a away in he ime sequence (long- ange co ela ion). Long- ange dependence has been ecognized, a e he pionee ing con ibu ions o Mandelb o and co- wo ke s du ing he las pa o he 1960s (see Mandelb o and Wallis 1969 and e e ences he ein), as a cha ac e - is ic ea u e o many phenomena, which p o ides insigh s in o he unde lying mechanisms ha ule he p ocess dynamics. Since hen, he exis ence o long- ange pe - sis ence has been es ablished in a wide ange o disci- plines, including geophysical p ocesses (Kan elha d e al. 2003), ne wo k a ic modeling (Taqqu e al. 1997), economics (Man egna and S anley 1995), and physiology (Goldbe ge e al. 2002), among many o he s. The analysis o possible long- ange co ela ions, o empo al pe sis ence, in ime se ies helps o unde s and he na u al a iabili y o he p ocess unde s udy and i s dynamics. Speci ically, he exis ence o long- ange co - ela ions has impo an implica ions in e ms o o e- cas ing because scale in a iance allows he ela ionship o a iabili y be ween di e en ime scales o be quan- i ied (Bunde and Lenna z 2012). Fu he mo e, pe sis- ence implies ha emo e pa s o he ime se ies emain signi ican ly co ela ed, and hence pas e en s can ha e a no able e ec on he p esen and u u e de el- opmen o he p ocess. So, long- ange co ela ions play an impo an ole in model assessmen (Li ina e al. 2007) and in unde s anding clima e a iabili y (Ba bosa e al. 2006). Addi ionally, ime se ies wi h pe sis en beha io exhibi posi i e and nega i e de ia ions om he a e age alue o long pe iods. Hence, long- e m pe sis ence ep esen s a na u al mechanism ha leads o he clus e ing o ex eme e en s; he e o e, i has im- po an implica ions o clima e change and na u al haza ds o ecas ing (Eichne e al. 2011;Sha ma e al. 2012;Ba anowski e al. 2015). P ocesses exhibi ing an exponen ial decay o i s au- oco ela ion unc ion R( ) o la ge empo al in e als a e known as sho - ange co ela ed p ocesses, while hose wi h a powe -law decay, R( );1/ g, a e e e ed o as long- ange co ela ed, long- ange dependen , o long-memo y p ocesses. Acco dingly, he powe spec- um S( ) o a s a iona y long- ange co ela ed p ocess exhibi s a powe -law beha io o he ype S( );1/ b, wi h b512g. This powe -law scaling beha io , o scale in a iance, unde sco es he lack o a single cha ac e - is ic ime scale domina ing he dynamics o he unde lying p ocess. In o he wo ds, he exis ence o long- ange co ela ion and scale in a iance e eals simi- la s a is ical p ope ies a di e en ime scales, o equi alen ly he p ocess exhibi s s a is ical sel -simila i y o ac al s uc u e, whe eby he magni ude o sho - and long- e m luc ua ions is ela ed o each o he h ough a single scale ac o (Bassing hwaigh e e al. 1994). F ac al ime se ies can be classi ied in o wo di e en g oups. On he one hand, he scaling p ope ies o one g oup a e cha ac e ized by a single exponen , known as mono ac als; on he o he hand, hose wi h much mo e complex pa e ns, known as mul i ac als, in which a con inuous spec um o exponen s is needed o an ad- equa e cha ac e iza ion o i s scaling p ope ies (Fede 1988;Ba abasi and S anley 1995;Kan elha d e al. 2002). Mos na u al p ocesses belong o he mul i ac al class (Pa lo and Anishchenko 2007). Fu he mo e, causes leading o mul i ac ali y can be a b oad p oba- bili y dis ibu ion o he alues o he ime se ies, o a esul o di e en long- e m co ela ions o small and la ge luc ua ions (Kan elha d e al. 2002). Se e al me hods ha e been used o explo e he scaling p ope ies o na u al phenomena in e ms o mono ac al and mul i ac al beha io . Thesecan be g ouped basically in o equency domain me hods, including app oaches based on spec al and wa ele analysis (Box e al. 1970; Muzy e al. 1991;Ab y e al. 1998); and ime domain me hods, based on he andom walk heo y, such as e- scaled ange analysis (R/S; Hu s 1951) o he de ended luc ua ion analysis (DFA; Peng e al. 1994), and i s mul i ac al gene aliza ion [mul i ac al de ended luc- ua ion analysis (MFDFA); Kan elha d e al. 2002]. E en hough he au oco ela ion unc ion is a na u al es ima o o he pe sis ence, hei use o es ima ing he scaling exponen can lead o misleading esul s when dealingwi hlong imelags(Malamud and Tu co e 1999). Simila ly, he es ima ion o i s Fou ie ans o m, he powe spec um, can be limi ed by s a is ical un- ce ain ies (Talkne and Webe 2000). Acco dingly, i has been obse ed ha es ima ions o he scaling exponen s based on he powe spec um a e usually less s able and eliable han hose p o ided by he DFA me hod (Ma soukas e al. 2000). Ano he impo an ad an age o he DFA me hodology is i s capabili y o emo e poly- nomial ends o di e en o de (Hu e al. 2001). Gene ally, obse a ions o na u al sys ems exhibi nons a iona i ies, such as pe iodici ies and ends, which can lead o a alse de ec ion o long- ange co ela ions ha ha e o be emo ed om he s ochas ic componen s o es ima e i s co ec scaling beha io . Hence, i is im- po an o use me hods capable o emo ing he e ec s o possible ends in he se ies, such as he DFA and MFDFA, which a e well-es ablished me hods o de e mining he scaling o long- e m co ela ion in 818 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 34 p esence o polynomial ends (Kan elha d e al. 2002; Bashan e al. 2008;Ca aiani 2012). As p e iously men ioned, he exis ence o a ac al s uc u e has been ecognized by examining ime se ies coming om many di e en scien i ic ields. In pa ic- ula , clima ological and hyd ological se ies exhibi pe - sis ence o e a wide ange o ime scales (e.g., Pelle ie and Tu co e 1997;Ba anowski e al. 2015). Howe e , a b oad bibliog aphic e iew on he opic shows ha oceanog aphic p ocesses ha e ecei ed scan a en ion in compa ison wi h o he b anches o Ea h sciences, such as hyd ology, seismology, and me eo ology. Pa a- doxically, in an ea ly e iew on he ubiqui y o his be- ha io in as ophysics and many o he b anches o physics (P ess 1978), ocean cu en s we e iden i ied as a p ocess wi h a powe spec um exhibi ing a scaling be- ha io , hence sugges ing he p esence o long- e m de- pendence. Ne e heless, o he bes o ou knowledge, he only con ibu ion explo ing he ac al p ope ies o ocean cu en s is due o Ashkenazy and Gildo (2009). These au ho s examined he exis ence o empo al long- ange co ela ions in sea su ace cu en s’ ime se ies measu ed in he no he n Gul o Eila by using a high- equency ada sys em, du ing one yea wi h a 30-min ime esolu ion. The use o a powe spec um and DFA me hods allowed hem o conclude ha measu ed ime se ies o sea su ace cu en s exhibi a signi ican mono ac al s uc u e o ime scales anging be ween se e al hou s and less han a mon h. The aim o his s udy is o gain deepe insigh in o he ime beha io o ocean cu en s, by explo ing whe he long expe imen al eco ds o ocean cu en speed measu ed a in e media e and la ge dep hs exhibi a ac al s uc u e, using o his pu pose he DFA and MFDFA me hods. Fu he mo e, he ela i e con ibu- ion o causes leading o mul i ac ali y is examined by applying hese me hodologies o modi ied e sions o he o iginal ime se ies. The es o he pape is o ganized as ollows: The main cha ac e is ics o expe imen al da a used in he s udy, as well as he undamen als o he de ended luc ua ion analysis and mul i ac al de ended luc ua ion analysis me hods, a e ou lined in sec ion 2. The esul s de i ed om he applica ion o hese me hodologies o he measu ed speed ocean cu en da ase s, as well as o he co esponding ans o med ime se ies, a e discussed in sec ion 3, and conclusions ollow in sec ion 4. 2. Da a and me hodology a. Da a The analysis is based on ime se ies o ocean cu en speed ob ained a a single-poin moo ing on he Madei a Abyssal Plain, known as KIEL276 s a ion. I is loca ed wes o Madei a in he no he n Cana y Basin a nom- inal loca ion 338N, 228W. The measu emen s om 1980 o 2000 ha e been pe o med by he Ins i u ü Mee eskunde, Kiel, Ge many—now he GEOMAR Helmhol z-Zen um ü Ozean o schung Kiel—and since 2000 by he Leibniz Ins i u e o Bal ic Sea Re- sea ch in Wa nemünde, Ge many. All da a used in he s udy a e accessible ia he OceanSITES da abase. Analyzed ime se ies ha e been egis e ed by cu en me e s placed a app oxima ely 1000- and 3000-m dep h, al hough moo ing eplacemen o equipmen main e- nance commonly esul s in sligh changes in dep h. Time se ies consis o daily a e ages o low-pass- il e ed cu en speed measu emen s and co e a pe- iod o nea ly 30 yea s, spanning om Ap il 1980 o Decembe 2009. Gaps o weeks—and in some cases mon hs—a e due o moo ing o ba e y p oblems, in- s umen ailu es, e c.; hus, hese discon inui ies a ise om he in insic na u e o expe imen al eco dings a sea. The pe cen age o missing da a is a ound 15% and gaps a e qui e andomly dis ibu ed (see Fig. 1). A de- ailed desc ip ion o he indi idual deploymen s is gi en in Mülle and Waniek (2013), and u he de ails o- ge he wi h a comp ehensi e analysis a e gi en in F ünd e al. (2013). To deal wi h his kind o ime se ies, a common p ep ocessing me hod is o cu ou he FIG. 1. Daily cu en speed ime se ies eco ded a (a) 1000- and (b) 3000-m dep h. APRIL 2017 C A B R E R A - B R I T O E T A L . 819 un eliable agmen s and s i ch oge he he emaining pa s. This p ocedu e is pe o med be o e execu ing any s a is ical analysis and, as i has al eady been p o en, i does no ha e a signi ican impac on he scaling be- ha io o co ela ed signals—e en when up o 50% o he poin s a e emo ed (Chen e al. 2002). b. Me hodology As p e iously s a ed, a compulso y s ep p io o any p ocedu e o examining he p esence o long- ange co ela ions in a ime se ies consis s o emo ing ends and pe iodic pa e ns o a iabili y associa ed wi h ex- e nal e ec s. In pa icula , because na u al ime se ies gene ally exhibi a seasonal cycle, a deseasonaliza ion p ocedu e is pe o med by adjus ing he da a wi h he seasonal mean and s anda d de ia ion as x5(xi2xd)/sd, whe e xiis he measu ed alue o he cu en speed a day numbe i, du ing he eco ding pe iod, wi h xd and sdbeing he co esponding a e age and s anda d de ia ion o ha pa icula day o e he yea s, e- spec i ely (e.g., Li ina e al. 2011). 1) DETRENDED FLUCTUATION ANALYSIS The DFA (Peng e al. 1994) ans o ms he au oco - ela ion unc ion decay in o an inc easing a iabili y measu e. Then, a powe law may be ob ained ha de- sc ibes he magni ude o luc ua ions as a unc ion o he empo al scale and om which i is possible o in e a scaling beha io , i any. As implied by i s name, his me hod is based on he idea o de ending local a iabili ies— ha is, ex e nal ends— ha can be sepa a ed om he s ochas ic componen s o he ime se ies. I is commonly deno ed as DFA-p, in- dica ing ha polynomial ends o o de pcan be sys em- a ically i ed and elimina ed o cha ac e ize quan i a i ely long- ange co ela ions in nons a iona y ime se ies. In his s udy, se e al es s ha e been made by applying di e en o de s o polynomial i s. Finally, i has been conside ed ha i is su icien o assume ha he unde lying p ocess is mainly a ec ed by linea ends, since he ob ained scaling exponen s did no show signi ican di e ences when e- mo ing polynomials o o de up o h ee. DFA me hodology de ails can be ound in Peng e al. (1994). In b ie , his p ocedu e can be desc ibed as ol- lows: Fi s , he o iginal ime se ies xis in eg a ed o ob ain he andom walk p o ile enhancing sel -simila i y p ope ies. I can be shown ha o iginal and in eg a ed ime se ies ha e bo h iden ical co ela ion s uc u es (Lampe i 1962;Be an 1994;Willinge e al. 1997). The in eg a ed ime se ies is gi en by y(k)5å k i51 [x(i)2x)] k51, ...,Nb, (1) whe e xis he mean o he deseasonalized ime se ies x and Nis i s leng h. Nex , he p o ile is di ided in o Nb5N/nnon- o e lapping segmen s, each con aining nda a poin s. To ensu e a eliable es ima ion o he luc ua ion unc ion F(n), nshould no be la ge han N/4 (Rybski e al. 2008). Each segmen is locally de ended using a poly- nomial unc ion yp. The luc ua ion unc ion, which can be ega ded as he a iance o he de ended ime se ies, is e alua ed in each segmen as F (n)5"1 nå n k5( 21)n11 jy(k)2yp(k)j2#1/2 51, ...,Nb. (2) Finally, by a e aging F (n) o e he Nbin e als, he mean alue o he luc ua ion unc ion is ob ained. This p ocedu e is epea ed o e di e en ime scales, ha is, n, o p o ide a ela ionship be ween F(n) and n, F(n);nH, (3) whe e H ep esen s he Hu s exponen and quan i ies he s eng h o he co ela ions. A linea ela ionship o logF(n) e sus log(n) indica es sel -simila i y, and he slope o he eg ession line is e ec i ely he scaling o Hu s exponen . Di e en ypes o beha io can be dis- inguished depending on i s alue. A slope o H50:5 co esponds o a p ocess wi h no long- e m co ela ions— ha is, whi e noise o sho - e m memo y—whe eas H51:5 is ela ed o an in eg a ed andom walk— ha is, B ownian noise—which exhibi s sel -simila i y bu no long- ange co ela ions. Fo 1 / 2,H,1 he eis pe sis ence— ha is, i in he immedia e pas he signal has a posi i e inc emen , hen on a e age an inc ease o he signal in he immedia e u u e is expec ed. On he con- a y, 0 ,H,1 / 2implies an ipe sis ence, meaning ha an inc easing alue in he immedia e pas implies a dec easing in he immedia e u u e. In he pa icula case o H51, empo al luc ua ions a e o licke -noise ype (i.e., 1/ noise), ypical o sel -o ganized c i icali y sys ems (Bak e al. 1987). Howe e , his esul could be also poin ing o a p ocess ha is in essence mul i ac al and he e o e mo e han one scaling exponen is equi ed o ully cha ac e ize i s dynamics (Be an 1994;Hausdo and Peng 1996). 2) MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS As p e iously men ioned, mul i ac als a e cha ac e - ized by high a iabili y on a wide ange o empo al scales and he desc ip ion o hei scaling p ope ies equi es many scaling exponen s. The MFDFA is a gene aliza ion o he DFA, based on he s anda d pa i ion unc ion 820 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 34 mul i ac al o malism o no malized and s a iona y measu emen s (Kan elha d e al. 2002), which is use ul o de ec ing he exis ence o dis inc scaling beha io s a di e en ime scales. By means o he MFDFA, he scaling o he q h-o de momen s o he luc ua ion unc ion a e de e mined, ins ead o ob aining only he second-o de s a is ical momen as in he case o he DFA. The p ocedu e is simila o he p e ious one, wi h he di e ence ha in his case Eq. (2) is ew i en as F ,q(n)5"1 nå n k5( 21)n11 jy(k)2yp(k)jq/2#1/2 51, ...,Nb. (4) Then, he luc ua ion unc ion o o de qis ob ained by a e aging F ,q(n) o e all he Nbsegmen s, Fq(n)51 Nb å Nb 51 F ,q(n). (5) The scaling exponen o each luc ua ion unc ion is calcula ed om log–log plo s o Fq(n) e sus n o each alue o q. Te m Fq(n) will gene ally inc ease o la ge alues o nacco ding o he ollowing powe law: Fq(n);nHq. (6) The exponen Hq, called he gene alized Hu s expo- nen , desc ibes he scaling beha io o he q h-o de luc ua ion unc ion. Fo posi i e alues o q,Hqcha - ac e izes segmen s wi h la ge luc ua ions (small alues o Hq), while nega i e alues desc ibe segmen s wi h small luc ua ions (la ge alues o Hq). Gene ally, qcan ake any eal alue. Fo q52, he s anda d DFA p o- cedu e is e ie ed and Hq5H. The gene alized Hu s exponen is only one o se e al pa ame e s used o cha ac e ize he mul i ac al s uc u e o ime se ies. The e a e o he al e na i es o desc ibe his kind o se ies, such as he singula i y spec um D(a). Commonly, Hqis ela ed o he classical mul i ac al scaling exponen —also known as he Renyi index q, whe e q5qHq21—and can be used o compu e he singula i y, o Hölde exponen a, ia a Legend e ans- o m (Fede 1988), by means o he ollowing exp ession: a5 0 q5Hq1qH0 q. (7) Then, he singula i y spec um is ob ained as D(a)5qa2 q5q(a2Hq)11. (8) The plo o D(a) e sus a, called he mul i ac al spec- um, ypically has a pa abolic conca e downwa d shape, wi h he ange o a alues inc easing wi h he p ocess complexi y. I s wid h (Da5amax 2amin) ep e- sen s de ia ions om he a e age ac al s uc u e, gi en by he Hu s exponen de i ed om he mono ac al analysis o la ge and small luc ua ions. Then, i mea- su es he deg ee o he se ies mul i ac ali y. I a spec- um has a nonsymme ic shape wi h one ail la ge han he o he one, hen he la ge ail is igno ed and he wid h is es ima ed as he sho e ail doubled (Mako iew and Fuli nski 2010). Fo q.0 he spec um will ha e a long le ail when he ime se ies has a mul i ac al be- ha io insensi i e o local luc ua ions wi h small mag- ni udes; on he o he hand, o q,0, he spec um will ha e a long igh ail when he ime se ies has a mul i- ac al beha io insensi i e o local luc ua ions wi h la ge magni udes (Mako iew e al. 2011). Gene ally, he concep o gene alized dimension D(a) co esponds o he scaling exponen o he q h momen o he measu e. Thus, D(a) a ains i s maximum alue (D(a)51) o q50—see Eq. (8). Fo a mono ac al, D(a) is a cons an unc ion o q, and no addi ional in- o ma ion is ob ained by examining highe momen s. 3) CONTRIBUTIONS TO MULTIFRACTALITY I has been p e iously commen ed ha i is possible o dis inguish be ween wo di e en con ibu ions o mul i ac ali y: on he one hand, a linea con ibu ion due o he exis ence o di e en long- e m co ela ions o small and la ge luc ua ions in he da a; on he o he hand, a nonlinea con ibu ion as a consequence o a b oad p obabili y dis ibu ion (Kan elha d e al. 2002; Sch eibe and Schmi z 2000). Bo h e ec s a e o en p esen in eal-wo ld ime se ies and can be indi idually elimina ed o quan i y he ela i e impo ance o each con ibu ion on he scaling beha io . The me hods commonly used o es he exis ence and ela i e im- po ance o each ype o mul i ac ali y a e b ie ly desc ibed below. (i) Shu ling The shu ling me hod is a use ul echnique o quan i y he s eng h o he linea con ibu ion on he scaling p ope ies o a p ocess. Shu led da a a e gene a ed by andom pe mu a ions o he o iginal ime se ies (Pe e s 1996). Random pe mu a ions gua an ee he same am- pli ude dis ibu ion han he o iginal se ies bu des oy any empo al co ela ions be ween obse a ions. Hence, shu led e sions o se ies wi h mul i ac ali y due o only long- ange co ela ions o small and la ge luc ua ions will exhibi a simple whi e noise beha io . Howe e , his p ocedu e does no a ec mul i ac al con ibu ion due o p obabili y densi y b oadness. When bo h ypes o mul i ac ali y a e p esen , APRIL 2017 C A B R E R A - B R I T O E T A L . 821 mul i ac ali y o he o iginal se ies will be s onge han ha o i s shu led e sions. (ii) Su oga e da a Gene a ion o su oga e da a has been sugges ed as a p ocedu e o es ing he exis ence o nonlinea i ies in ime se ies (Theile e al. 1992). The es s a e based on he gene a ion o syn he ic ime se ies, deno ed as su - oga e se ies, which p ese e some s a is ical cha ac- e is ics o he o iginal, o es ed, ime se ies, bu andomize he Fou ie phases and hence emo e non- linea i ies. The e a e se e al algo i hms o gene a ing su oga e ime se ies. The p ocedu e used in his s udy is known as i e a i e ampli ude-adjus ed Fou ie ans- o m (IAAFT) and enables syn hesizing ime se ies in which nonlinea i ies p esen in he o iginal ime se ies a e emo ed while p ese ing i s powe spec um and p obabili y dis ibu ion (Sch eibe and Schmi z 1996). (iii) Vola ili y ime se ies An addi ional echnique o e alua e he nonlinea i y deg ee on he scaling p ope ies o a p ocess consis s o s udying he co ela ions o he ola ili y ime se ies (Kalisky e al. 2005). Vola ili y is de ined as he abso- lu e alues o he inc emen o he o iginal da a Dxi5jxi112xij(Liu e al. 1999). I has al eady been shown ha long- ange co ela ions in ola ili y ime se ies a e ela ed o mul i ac ali y and in pa icula o he nonlinea i y cha ac e is ics o a p ocess (Ashkenazy e al. 2003). Thus, ola ili y ime se ies de i ed om nonlinea and long co ela ed eco ds exhibi long co ela ions oo, acco ding o a mono- ac al analysis. On he con a y, i he o iginal da a a e comple ely linea and p esen long- ange co ela ions, hen hei ola ili y ime se ies will no show long- ange co ela ions. 3. Resul s and discussion Time se ies o daily cu en speed a 1000- and 3000-m dep h a e shown in Figs. 1a and 1b, espec i ely. As expec ed (e.g., Talley e al. 2011), he s eng h and ange o a iabili y o cu en speed a e signi ican ly smalle a 3000-m dep h. Thus, a simple s a is ical da a analysis e eals a clea dec ease wi h dep h o he mean and s anda d de ia ion o cu en speed—4.23 63.73 cm s 21 (1000 m) and 2.06 61.28 cm s 21 (3000 m)—as well as a d as ic educ ion o he ange o a iabili y, om 0 o nea ly 34 cm s 21 a 3000 m and be ween 0 and 9 cms 21 a 1000 m, app oxima ely. These di e ences a e mainly due o s ong episodic cu en e en s de ec ed a 1000-m dep h, wi h speeds exceeding 15 and e en 30 cm s 21 . a. Mono ac al analysis The log–log plo s o he DFA unc ion e sus ime scale in days, o he o iginal and shu led cu en speed eco ds a 1000- and 3000-m dep h, a e shown in Figs. 2a and 2b, espec i ely. I is qui e clea om hese igu es ha a bo h dep hs he e exis s a c osso e poin ha sepa a es he ime-scale ange in o wo subbands wi h di e en powe -law beha io s. The scaling exponen s associa ed wi h each band ha e been es ima ed as he slope o he s aigh line i ed o he da a by means o he leas squa es me hod a each side o he c osso e poin . I can be obse ed ha o ime scales la ge han app oxima ely 5 yea s, bo h i s de ia e om he scaling beha io , exhibi ing an inc easing dispe sion. In his sense, i is impo an o ake in o accoun ha he numbe o segmen s used o ob ain he a e age alue o F(n) dec eases as ninc eases and, consequen ly, he s a is ical s abili y o hese es ima ions educes. Ac- co dingly, he discussion on he scaling beha io is FIG. 2. Log–log plo s o he luc ua ion unc ion s he scale (days) o he o iginal and shu led ime se ies a (a) 1000- and (b) 3000-m dep h. Solid lines ep esen leas squa es i ed lines. 822 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 34 limi ed o ime scales below 5 yea s, whe e he da a adequa ely con o m o a s aigh line. C osso e s a e loca ed a ime poin s close o 50 days a bo h dep hs. Thus, luc ua ions o cu en speed a ime scales be ween app oxima ely 50 days and 5 yea s exhibi long- ange co ela ions wi h scaling exponen s equal o 0.85 and 0.79 o 1000 and 3000 m, espec i ely. The co esponding scaling exponen s o sho e ime scales a e 1.54 and 1.39, implying a B ownian mo ion– ype beha io (Peng e al. 1994). The appea ance o c osso e s in he cu es de i ed h ough DFA seems o be a common ea u e o geo- physical ime se ies. In pa icula , Mone i e al. (2003) s udied he pe sis ence o he sea su ace empe a u e and ound wo di e en scaling beha io s o ime scales below and abo e 10 mon hs. Mo eo e , Li ina e al. (2011) obse ed he exis ence o a c osso e a a ew mon hs’ scale when explo ing long- e m co ela ions in i e lux da a, a a iable closely ela ed o cu en speed. Ne e heless, ca e ul conside a ion should be gi en o he p esence o c osso e s, since hey may no be e- lec ing di e en beha io s a di e en empo al scales, bu an e ec o ends embedded in he ime se ies (Hu e al. 2001). O he nons a iona i ies can also cause c osso e s i hey a e no p ope ly aken in o accoun , such as seasonal a ia ions (Li ina e al. 2011). Mo eo e , i has o be s essed ha c osso e s mus no be con used wi h mul i ac ali y, since mul i ac ali y is cha ac e ized by di e en scaling beha io s o di e en momen s o e he ull ange o ime scales (Bouchaud e al. 2000). Ano he poin o men ion is he dec ease o Has he dis ance om he sea su ace inc eases. This is p obably ela ed o he change in he numbe o in ol ed physical p ocesses ope a ing a di e en ime scales, which is highe a 1000-m dep h han a 3000-m dep h. In his ega d, i is in e es ing o no e ha , al hough he dynamics o su ace ocean cu en s a e subs an ially di e en om ha o deep ocean cu en s, Ashkenazy and Gildo (2009), when analyzing long- ange co ela- ions o ocean su ace cu en s in a semienclosed sea, ob ain simila scaling exponen s as he ones ob ained in he p esen s udy in he scale ange abo e app oxima ely 50 days. Ne e heless, he empo al scale ange exam- ined by hese au ho s is be ween a ew hou s and less han one mon h, which is well below he ime scales conside ed in his s udy. Applica ion o DFA o he co esponding shu led ime se ies a 1000- and 3000-m dep h esul s in a sole scaling exponen close o 0.5 in bo h cases, as depic ed in Fig. 2. This e eals ha by andomizing he o iginal ime se ies, long- ange dependence disappea s, implying ha he scaling beha io o cu en speed luc ua ions is mainly domina ed by empo al co ela ions. The p edominance o long- ange co ela ions in he p ocess and he educ ion o hei s eng h wi h dep h ag ees wi h he exis ence o e en s du ing which cu en speed eaches ex eme alues somewha clus e ed in ime, as seen in Fig. 1, especially a 1000-m dep h (Fig. 1a). b. Mul i ac al analysis As discussed in he p e ious sec ion, mos na u al eco ds do no exhibi a simple mono ac al scaling be- ha io . The e o e, a mul i ac al analysis is pe o med o de e mine whe he he cu en speed ime se ies e- qui es mo e han one exponen o a ull desc ip ion o i s scaling beha io in he same ange o ime scales. The dependence o Hqwi h qis shown in Fig. 3a. I can be obse ed ha Hqdec eases mono onically wi h he inc ease o q, e ealing a di e en scaling beha io o small and la ge luc ua ions o cu en speeds, and sug- ges ing ha he deep cu en speed ime se ies exhibi s mul i ac al cha ac e is ics a bo h dep hs. Fo a FIG. 3. Gene alized Hu s exponen as a unc ion o q o (a) cu en speed ime se ies a 1000- (solid line) and 3000-m dep h (dashed line), and (b) mul i ac al spec a o 1000- (solid line) and 3000-m dep h (dashed line). APRIL 2017 C A B R E R A - B R I T O E T A L . 823 mono ac al ime se ies, Hqis independen o q. Hence, he a e o change o Hqcan be used as a measu e o he s eng h o he mul i ac al cha ac e o a p ocess. Ac- co dingly, he mul i ac ali y deg ee dec eases wi h dep h. Fu he mo e, he dec easing pa e n o Hqwi h q is mo e nonlinea o cu en s a 1000 han a 3000 m, indica ing a la ge di e ence in he scaling o small (q,0) and la ge luc ua ions (q.0) in he uppe poin o measu e. Ano he use ul way o cha ac e izing he mul i ac al beha io o a p ocess is by means o D(a), de i ed om Hq[Eq. (8)]. This quan i ies in de ail he long- ange co ela ion p ope ies o a ime se ies (Ashkenazy e al. 2003). In pa icula , he Dao mul i ac al spec a cons i u es a measu e o he mul i ac ali y deg ee. Theo e ically, o a mono ac al ime se ies he mul i- ac al spec um educes o a single poin . Thus, he wide he Da, he s onge he mul i ac ali y will be, and he mo e complex he p ocess om which he ime se ies de i e. Mul i ac al spec a o cu en speed ime se ies a 1000- and 3000-m dep h a e shown in Fig. 3b. I is ema kable ha in bo h cases he spec um exhibi s a ypical pa abolic conca e downwa d shape, indica ing a mul i ac al s uc- u e a bo h dep hs. The loca ion o he maxima a ies wi h dep h, peaking a ound a;1:14 and a;1:08 a 1000 and 3000 m, espec i ely. These alues a e close o he mean alue o he scaling exponen s ob ained in he DFA, sugges ing ha he maximum o he mul i ac al spec um gi es insigh in o he dominan scaling beha io . Conce ning he wid h o D(a), i is obse ed how i dec eases as he dep h inc eases. Thus, he complexi y deg ee is highe a 1000 m, wi h Da;0:66, han a 3000 m, whe e Da;0:4. This ac also mani es s i sel in he gene alized Hu s exponen a ia ion ange (Fig. 3a), which is clea ly la ge o 1000 m, a ying om a ound 1.35 o 0.78. I can also be no iced ha a 1000-m dep h, he spec um is sligh ly skewed o he igh , which indica es ela i ely s ongly weigh ed high ac al ex- ponen s (Shimizu e al. 2002), and ega ding i s ails, he cu en speed ime se ies seems o ha e a mul i ac al s uc u e mainly domina ed by long- ange luc ua ions. The a ia ion o mul i ac ali y wi h dep h sugges s a di e en speci ic in luence o he wo ac o s con ibu - ing o he mul i ac al s uc u e o he obse ed ime se ies. To co obo a e his poin , he DFA and MFDFA me hods ha e been applied o shu led, su oga e, and ola ili y ime se ies associa ed wi h he o iginal one. I he o de o obse a ions in a mul i ac al ime se ies is andomly modi ied, hen i s memo y componen is e- mo ed and some pa o i s mul i ac ali y should dis- appea , esul ing in a na owe mul i ac al spec um. On he o he hand, mos o he emaining mul i ac ali y should disappea i nonlinea i ies p esen in he ime se ies a e elimina ed by andomizing he co esponding Fou ie phases. Thus, he ela i e educ ion in wid h o he mul i ac al spec um gene a ed by he shu ling and phase- andomiza ion p ocedu es p o ides in o ma ion on he speci ic impo ance o he linea and nonlinea con ibu ions o he obse ed deg ee o mul i ac ali y. The a ia ions o he gene alized Hu s exponen wi h qa e shown in Fig. 4a o he cu en speed ime se ies a 1000 m and in Fig. 4b o 3000-m dep h. No e ha he ange o Hq alues educes d as ically o he shu led ime se ies, being almos independen o qand close o 0.5 a bo h dep hs. The slope o Hqa ound q50 educes o he su oga e ime se ies and eaches e en lowe alues o shu led e sions. This ac is pa icula ly ue o cu en s a 3000-m dep h. These esul s indica e ha mul i ac ali y is mainly caused by di e en long- ange empo al co ela ions o small and la ge luc ua ions, while he e ec o he a - ailed dis ibu ion is almos FIG. 4. Gene alized Hu s exponen as a unc ion o q o he o iginal (do ed line), shu led (c ossed line), and su oga ed (dashed line) cu en speed ime se ies a (a) 1000- and (b) 3000-m dep h. 824 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 34 negligible a he lowe dep h and weak a he mo e su- pe icial poin o measu emen . Fu he mo e, some nonnegligible di e ences can be obse ed in he e- la ionship o Hqwi h q, pa icula ly o q.0, o he o iginal and su oga ed ime se ies a 1000-m dep h, which hin s a he exis ence o nonlinea con ibu ions a ec ing he scaling beha io o la ge luc ua ions. Addi ional suppo o hese indings can be ob ained by explo ing changes in he s uc u e and loca ion o he mul i ac al spec um o he o iginal and modi ied ime se ies. Fo his pu pose, mul i ac al spec a associa ed wi h o iginal, shu led, and su oga e ime se ies a e shown o cu en speeds eco ded a 1000- and 3000-m dep h in Figs. 5a and 5b, espec i ely. I can be obse ed ha he wid h o he shu led ime se ies spec a de- c eases signi ican ly and ha he loca ion o hei max- ima shi s owa d 0.5 (app oaching he beha io o a whi e noise), wi h ega d o ha o he o iginal se ies. Howe e , hese e ec s a e almos negligible o he su oga ed ime se ies mul i ac al spec a, mainly a 3000-m dep h, indica ing possible nonlinea con ibu- ions in he obse a ions eco ded a 1000-m dep h. To es and p o ide ex a suppo o his ac , he scaling p ope ies o he ola ili y ime se ies de i ed om each cu en speed eco d a e examined. A com- pa ison o DFA esul s o ola ili y ime se ies and hei co esponding su oga e e sions shows a sligh dec ease o he scaling exponen o he su oga e ola ili y se ies (Hy50:82 and Hy2s50:76, espec i ely) a 1000-m dep h (Fig. 6a), while he exponen emains cons an , Hy5Hy2s’0:74, a 3000-m dep h (Fig. 6a), endo sing he idea o some weak nonlinea con ibu ion o he ac al cha ac e is ics o he speed ime se ies measu ed a 1000-m dep h. In his con ex , i is in e es ing o no e ha Ashkenazy and Gildo (2009) pe o med a simila analysis o ola ili y sea su ace cu en speed ime se- ies and iden i ied a signi ican nonlinea con ibu ion o he ac al s uc u e o he p ocess. On he basis o he indings abo e, i seems easonable o admi a dec ease in he mul i ac ali y, and hence he FIG. 5. Mul i ac al spec a o cu en speed a (a) 1000- and (b) 3000-m dep h. Do ed, c ossed, and dashed lines co espond o spec a o he o iginal, shu led, and su oga ed ime se ies, espec i ely. FIG. 6. DFA o ola ili y and su oga ed ola ili y cu en speed ime se ies a (a) 1000- and (b) 3000-m dep h. APRIL 2017 C A B R E R A - B R I T O E T A L . 825