REVIEW Empirical Orthogonal Functions

14 Pages·2009·2.56 MB·English

VOLUME22 JOURNAL OF CLIMATE 15DECEMBER2009
REVIEW
Empirical Orthogonal Functions: The Medium is the Message
ADAMH.MONAHAN
SchoolofEarthandOceanSciences,UniversityofVictoria,Victoria,BritishColumbia,Canada
JOHNC.FYFE
CanadianCentreforClimateModellingandAnalysis,EnvironmentCanada,UniversityofVictoria,Victoria,
BritishColumbia,Canada
MAARTENH.P.AMBAUM
DepartmentofMeteorology,UniversityofReading,Reading,UnitedKingdom
DAVIDB.STEPHENSON
ExeterClimateSystems,MathematicsResearchInstitute,UniversityofExeter,Exeter,UnitedKingdom
GERALDR.NORTH
DepartmentofAtmosphericSciences,andDepartmentofOceanography,TexasA&MUniversity,CollegeStation,Texas
(Manuscriptreceived2February2009,infinalform27May2009)
ABSTRACT
Empiricalorthogonalfunction(EOF)analysisisapowerfultoolfordatacompressionanddimensionality
reductionusedbroadlyinmeteorologyandoceanography.Oftenintheliterature,EOFmodesareinterpreted
individually,independentofothermodes.Infact,itcanbeshownthatnosuchattributioncangenerallybe
made.ThisreviewdemonstratesthatingeneralindividualEOFmodes(i)willnotcorrespondtoindividual
dynamicalmodes,(ii)willnotcorrespondtoindividualkinematicdegreesoffreedom,(iii)willnotbesta-
tisticallyindependentofotherEOFmodes,and(iv)willbestronglyinfluencedbythenonlocalrequirement
thatmodesmaximizevarianceovertheentiredomain.Thegoalofthisreviewisnottoargueagainsttheuseof
EOFanalysisinmeteorologyandoceanography;rather,itistodemonstratethecarethatmustbetakeninthe
interpretationofindividualmodesinordertodistinguishthemediumfromthemessage.
1. Introduction and dynamical mode reduction (e.g., therecent review
by Hannachi et al. 2007). In particular, it has become
Since its introduction to meteorology by Edward
common to use EOF analysis as a tool to probe the
Lorenz (Lorenz 1956), empirical orthogonal function
physics underlyingthevariabilityinageophysicalfield
(EOF) analysis—also known as principal component
of interest. There is no problem with the use of EOF
analysis (PCA), the Karhunen–Loe`ve transform, or
analysistoidentifystructuresingeophysicaldata[from
proper orthogonal decomposition—has become a sta-
observations or general circulation models (GCMs)],
tistical tool of fundamental importance in atmosphere,
whichcarryrelativelylargefractionsofvarianceinthe
ocean,andclimatescienceforexploratorydataanalysis
field under consideration. Problems can begin when
these statistical structures are interpreted as being of
individual dynamical, kinematic, or statistical meaning
Corresponding author address: Adam H. Monahan, School of
andareusedtodefinethesubsequentphysicalquestions
Earth and Ocean Sciences, University of Victoria, Victoria, BC
accordingly.Herewereviewtheconditionsunderwhich
V8W3V6,Canada.
E-mail:[email protected] EOFs are of individual physical or statistical meaning
DOI:10.1175/2009JCLI3062.1
(cid:2)2009AmericanMeteorologicalSociety 6501
6502 JOURNAL OF CLIMATE VOLUME22
andarguethattheseconditionsarenotgenerallysatis- dynamical,kinematic,orstatisticalfeaturesofthefield
fiedinrealobservationalorGCMfields. under consideration. It is in this sense that we borrow
FollowingabriefoverviewofEOFanalysisinsection2, Marshall McLuhan’s phrase that ‘‘the medium is the
theconditionsunderwhichindividualEOFmodesand message’’(McLuhan1964).Thesemathematicaldetails
dynamical modes correspond in a stable linear system are not incidental to the results of EOF analysis; in
driven by noise (a paradigmatic system for sustained fact—as the following discussion will show—they are
variability around some basic state) are computed in central.Itistothisdiscussionthatwenowturn.
section3;theseconditionsaretheexceptionratherthan
the rule for geophysical flows. Section 4 further dem- 2. EOFanalysis:Somerelevantfacts
onstrates that individual EOF modes cannot be ex-
We present a brief review of those aspects of EOF
pectedtoberelatedtoindividualkinematicdegreesof
analysis that are relevant to the present discussion;
freedom of the system, in the context of an idealized
amorecomprehensivediscussioncanbefound,forex-
modeloftheextratropicaleddy-drivenatmosphericjet.
ample,inWilks(1995),vonStorchandZwiers(1999),or
Furthermore,itisdemonstratedinsection5thatifthe
the recent review by Hannachi et al. (2007). The fol-
probability distribution of the field being analyzed is
lowing discussion also draws on basic results of linear
non-Gaussian,thenEOFmodescannotbeexpectedin
algebra, an introduction to which can be found in
general to be mutually independent (despite being un-
Arfken (1985). This discussion will not consider issues
correlatedbyconstruction).Finally,theinfluenceonthe
arising around the statistical sampling of EOFs: it will
structureofEOFsimposedbythenonlocalrequirement
beassumedthattheunderlyingprobabilitydistributions
thattheymaximizevarianceovertheentirespatialdo-
(and therefore, all statistical moments) are known ex-
mainisdiscussedinsection6.Adiscussionandconclu-
actly. Theissues consideredin this review arenot arti-
sionsfollowinsection7.
factsofsamplingvariability.
Notethatinthisreviewwefollowconventionanduse
To begin,we consider an N-dimensional vectortime
thetermEOFmodeasshorthandforthepairofanEOF
seriesx(t),whichwemaythinkofasacontinuousfield
spatialstructureanditsassociatedtimeseries[theprin-
(such as temperature or geopotential) sampled at N
cipalcomponent(PC)].Inthepresentcontext,theword
discretepointsinspace.Withoutlossofgeneralityxcan
mode is used in the following sense (as given by the
beassumedtobeofzeromean(i.e.,anyinitiallynonzero
OxfordEnglishDictionary,http://dictionary.oed.com):
average has already been subtracted out). The co-
‘‘a particular form, manner, or variety in which some
variancematrixofxisgivenby
quality, phenomenon, or condition occurs or is man-
ifested.’’ This sense does not imply that a givenmode
C5hxxTi, (1)
isof individualsignificance(dynamically, kinematically,
orstatistically)independentofothermodes;infact,the where the angle brackets denote probabilistic expecta-
followingdiscussionisintendedpreciselytoargueagainst tion(i.e.,thepopulationmean):ifp(x)istheprobability
suchinterpretationsingeneral. densityfunctionofx,then
Empirical orthogonal function analysis is a powerful
ð
toolfordatacompressionanddimensionalityreduction
hf(x)i5 f(x)p(x)dx (2)
inatmosphere,ocean,andclimatescience:thepurpose
of this review is not to suggest that this technique is
for any function f(x). The EOFs can be defined as the
without value or to advocate its abandonment by me-
eigenvectorse ofC:
teorologistsoroceanographers.Rather,theintentisto k
bring together several strands of argument demon-
Ce 5m e , (3)
k k k
strating that some of the standard interpretations of
EOFmodesintermsoftheunderlyingphysicsthatare withcorrespondingeigenvaluesm .Conventionally,the
k
common in theliterature have a very doubtful basis in EOFsareorderedindecreasingmagnitudeofm :thatis,
k
general. That these interpretations can be problematic m $m $...$m .Itisconvenienttoassumethatthese
1 2 N
follows as a consequence of the fundamental mathe- eigenvectorsareunitnorm:
matical structure of EOF analysis. A central theme of
the following arguments is that EOF modes convolve (cid:2)(cid:2)e (cid:2)(cid:2)25e (cid:2)e 51. (4)
k k k
structure present in the data with constraints inherent
to the statistical analysis, such that the mathematical As C is a nonnegative definite square matrix, the ei-
structure of EOFs (following from basic definitions) is genvaluesm areallnonnegativeandtheeigenvectorse
k k
imprinted upon the EOF modes independent of the formacompleteorthonormalbasis.Thatis,
15DECEMBER2009 REVIEW 6503
(cid:3)1 j5k (cid:2)25hjjx(cid:3)(x(cid:2)e )e jj2i (11)
e (cid:2)e 5d 5 , (5) 1 1
j k jk 0 j6¼k
is minimized by e among all unit vectors in RN.
and any N-dimensional vector z can be expressed as 1
Having determined e , e is defined such that e is
alinearcombinationoftheeigenvectorse : 1 2 2
k orthogonaltoe andthemisfit
1
N
(cid:2)
z5 z e , (6) * 2 +
n51 n n (cid:2)25 jjx(cid:3)(cid:2)(x(cid:2)e )e jj2 (12)
k k
k51
suchthattheexpansioncoefficientsareprojectionsofz
one :
n isminimized,andsoonforincreasinge untilallN
k
z 5z(cid:2)e . (7) EOFsarefound.Thisvariationalproblemadmitsan
n n
analyticsolutionsuchthattheEOFsaregivenbythe
Inparticular,wecanwrite
eigenvectorproblem(3),whichishowtheEOFsare
N generallyfoundoperationally.Aparticularbenefitof
(cid:2)
x(t)5 an(t)en, (8) the variational formulation is that it demonstrates
n51
thatthetruncatedEOFexpansion
wheretheexpansioncoefficienttimeseriesa (t)arethe
n
principal components. It follows from the fact that the K
(cid:2)
e areorthonormaleigenvectorsofCthatthePCtime ^x5 a e , (13)
k n n
n51
seriesaremutuallyuncorrelated:
ha a i5m d (9) whereK,NistheK-dimensionalapproximationto
j k k jk
xoptimalinthesensethatthemeansquareddiffer-
andthatthevarianceofaj(t)ismj.Thus,EOFanalysis ence between x and ^x is minimal. Furthermore, by
expresses the (discretely sampled) field x as the super- Eq.(10)thefractionofvarianceaccountedforbythe
positionofNmutuallyorthogonalspatialpatternsmod- K-dimensionalapproximationis
ulatedbyNmutuallyuncorrelatedtimeseries.Thespatial
patterns and time series occur in matched pairs, which K
(cid:2)
aregenerallyreferredtoasEOFmodes. m
k
Wenotethefollowingpoints: k51 .
N
(cid:2)
1) The EOF expansion can be interpreted geometri- m
n
cally as a change of coordinates in RN through an n51
orthogonalrotationtoabasisinwhichCisdiagonal.
Theeigenvaluesm donotincreasewithk,soasthe
k
ThisemphasizesthattheEOFexpansionisnothing
dimensionalityoftheapproximationKincreases,the
morethananotherwayofdescribingthetimeseriesx
newly included EOF modes are successively as or
intermsofanewbasissetinwhichthisdescriptionis
less important to the overall variance of the ap-
particularlysimple(fromtheperspectiveofthedis-
proximation. Truncated EOF expansions therefore
tributionofvariance).
provide a powerful tool for data compression or di-
2) The total variance of x, (cid:2)N var(x ), is simply the
n51 n mensionalityreductionwhenitisdesiredtocaptureas
sumoftheeigenvaluesm sothefractionofvariance
n much variance as possible in lower-dimensional ap-
‘‘accountedfor’’bythenthEOFmodeis
proximations. We note in passing that the fact that
var(x ) m an EOF mode accounts for little variance does not
n 5 n . (10)
N N necessarily imply that it is dynamically irrelevant
(cid:2)var(x) (cid:2)m (e.g., Crommelin and Majda 2004), so a truncated
i i
i51 i51 EOFbasismightnotbeoptimalforrepresentingthe
dynamicsofasystem.
3) EOFsofxcanbeobtainedvariationallyintermsof
a sequence of lower-dimensional linear approxima- Therearevarioustechnicalissuesassociatedwiththe
tions to x of successively decreasing (or non- computationofEOFsthatwillnotbeconsideredinthis
increasing) mean squared error. In particular, e is study,includingmixingofEOFsduetosampling-error
1
thevectorinRNsuchthatthedifferencebetweenx induceddegeneracyoftheassociatedeigenvalues(e.g.,
and the projection of x along e is minimized in Northetal.1982),theuseofdifferentnorms(suchasthe
1
amean-squaresense:thatis,themisfit total energy) to define the EOFs, or choices of spatial
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weighting (e.g., correlation-based EOFs, or latitude- As a first statement about the connection between
dependent weights accounting for sphericity of thedo- EOFs and dynamical modes, we can say immediately
main;cf.Baldwinetal.2009).Thefocusofthisreviewis that the two sets of vectors will not correspond in the
the accuracy of common interpretations of individual casethatthelinearoperatorAisnonnormal;thatis,ifit
EOF modes, which is distinct from these more opera- doesnotcommutewithitsadjointAy:AAy2AyA6¼0.In
tionalconcerns. this case, the eigenvectors of A are in general not mu-
tuallyorthogonal.AstheEOFsarenecessarilymutually
3. EOFsanddynamicalmodes orthogonalbyconstruction,theEOFsandthedynami-
calmodeswillnotgenerallybethesame.Infact,studies
Themostnaturalsysteminwhichthestatisticalmodes
suchasPenland(1996)andFarrellandIoannou(1996)
produced by EOF analysis might be expected to have
demonstrate that the EOFs and the dynamical eigen-
clearindividualdynamicalsignificanceisonegoverned
vectorsmaybeverydifferentindeed.Nonnormalityof
bylineardynamicsforwhichthenotionof‘‘dynamical
thedynamicalmatrixisthegenericcaseforthelinear-
modes’’aseigenvectorsofthelineardynamicaloperator
izeddynamicsofgeophysicalsystems,particularlyinthe
is straightforward. In fact, North (1984) demonstrated
presenceofshearorcouplingbetweensystemswithvery
thatthecorrespondencebetweenEOFsanddynamical
different time scales (e.g., Farrell and Ioannou 1996;
modes holds only in a very specialized class of linear
Kleeman2008).Itfollowsthatwecansay,asageneral
dynamicalsystemsthatareexpectedtobetheexception
rule,thatEOFsanddynamicalmodeswillnotcoincide.
ratherthantheruleinthe(linearized)dynamicsofreal
This argument does not rule out the possibility that
geophysical flows.The original North (1984)argument
EOFscorrespondtodynamicalmodesinthespecialcase
considered the statistics and dynamics of a continuous
that the linearized dynamics are governed by a normal
field with dynamics described by a noise-forced partial
operator.Inthiscase,itcanbeshown(appendixA)that
differentialequation;forthesakeofsimplicity,thefol-
the EOFs will only correspond to the dynamical eigen-
lowing discussion considers the dynamics of a finite
vectorsofAifthenoisehasnospatialstructure:thatis,if
N-dimensional vector x (which might be considered as
itisspatiallyuncorrelated.Ifthedrivingnoiseisspatially
adiscreterepresentationofthecontinuousfield).
correlated, then its structure will be imprinted on the
Many studies have shown that sustained small-
covariancematrixofthedamped,drivensystemsothat
amplitude variabilityinabroadrangeofphysicalsitu-
the EOFs of x mix the structure of the noise with the
ations in the atmosphere and ocean (e.g., Farrell and
structureofthelinearizeddynamics.Inthiscaseagain,
Ioannou 1996; Kleeman 2008; Zanna and Tziperman
theEOFsanddynamicalmodeswillnotcorrespond.
2008) can be described by linear dynamics subject to
As an illustration of the difference between normal
random(i.e.,‘‘turbulent’’)forcingrepresentingtheeffects
modes and EOFs for nonnormal systems, consider the
ofunresolvedphysicalscales:
simpletwo-dimensionalsystem[adaptedfromasimilar
dx modelconsideredinFarrellandIoannou(1996)]:
5Ax1BW_ , (14)
dt d(cid:4)x (cid:5) (cid:4)(cid:3)0.1 (cid:3)0.9cotf(cid:5)(cid:4)x (cid:5) (cid:4)0 0(cid:5) W_ !
1 5 1 1 1 .
where A is a constant N 3 N matrix, B is a constant dt x 0 (cid:3)1 x 0 s W_
N 3 M matrix, and W_ is an M-dimensional vector of 2 2 2
(16)
independentwhitenoiseprocesses:
Theanglefdeterminesthedegreeofnonnormalityof
hW_ (t)W_ (t9)i5d(t(cid:3)t9)d (15)
i j ij this system: the system is normal for f 5 908 and be-
comes increasingly nonnormal as f is increased or de-
(uncorrelated in both space and time). The matrix A
creased.Theeigenvectorsofthedynamicalmatrixare
is the linear dynamical operator governing the ‘‘de-
terministic dynamics’’ (most naturally thought of as (cid:4)1(cid:5) (cid:4)cosf(cid:5)
b 5 , b 5 , (17)
a linearization ofthenonlinear dynamicsaroundsome 1 0 2 sinf
basic state), whereas B characterizes the spatial struc-
ture of the noise forcing. The deterministic dynamics withcorrespondingeigenvaluesl 520.1andl 521.
1 2
will be stable (i.e., perturbations will all decay asymp- The eigenvaluesareboth realandnegative, andin the
totically to zero in the absence of noise forcing) if the absence of external forcing x will approach 0 as time
real parts of all eigenvalues of A are negative. In this grows.Thestochasticforcingisorthogonaltotheleast-
situation,sustainedvariabilityofxismaintainedbythe damped mode. The covariance matrix C can be com-
turbulentnoiseforcingalone. putedexplicitly[fromEq.(A27)inappendixA]:
15DECEMBER2009 REVIEW 6505
FIG.1.Dynamicaleigenvectors(bi,blackvectors)andEOFs(ei,
grayvectors)forthestochasticdynamicalsystemEq.(16)forf5
708andf51208.Forthisrepresentativenonnormalsystem,the
dynamicaleigenvectorsandEOFsdonotcoincide.Noteinpar-
ticularthatwhereastheEOFsarebyconstructionorthogonal,the
dynamicaleigenmodesarenonorthogonal. FIG.2.FirstandsecondEOFs(e1ande2respectively)ofan
ellipticalplanetaryorbit.RedraftedfollowingMoandGhil(1987).
s2(cid:4)81cot2f (cid:3)9cotf(cid:5) namical mode: in particular, strong ENSO events are
C5 . (18)
22 (cid:3)9cotf 11 notinducedbyperturbationswiththespatialpatternof
thisEOFmode.
Thenormalmodesb andtheEOFse areillustratedin The correspondence between EOFs and dynamical
i i
Fig.1forf5708andf51208.Evidently,thedynamical modesisevenlessclearinthecaseofsystemsgoverned
modes and EOFs of this nonnormal system are quite bynonlineardynamics,inwhichtheconceptofthedy-
different. namicalmodemustbegeneralizedtothemoreabstract
The relevance of individual EOF modes to the pre- notion of dynamically invariant subspaces (which the
dictabilityofalinearforcedsystemisdeterminedbythe system will not leave once having entered). Such sub-
degree of nonnormality of the dynamics. For a system spaceswillnotingeneralevenbeplanar,incontrastto
governedbynormaldynamics,thestrongestresponseto the case of subspaces spanned by EOFs or linear dy-
a perturbation occurs when the spatial pattern of the namical modes. For such systems the EOFs will be of
forcingisthesameasthatoftheleast-dampeddynam- course determined by—but on an individual basis can-
icaleigenmode(i.e.,thetwovectorsareparallel);inthis notbeexpectedtobearanysimplerelationshipto—the
case, the response and the (leading) EOF will be col- dynamics.ConsiderfirstanexampledrawnfromMoand
linearwiththeforcing(e.g.,FarrellandIoannou1996). Ghil (1987),in which thevector x(t) describestheper-
Incontrast,constructive/destructiveinterferenceamong fectly elliptical orbit of a planetary body around a star
nonorthogonaleigenmodesofanonnormalsystemre- (Fig. 2). In the absence of external perturbations, this
sults in a more complex response to a perturbation, elliptical orbit will be ‘‘dynamically invariant.’’ By
suchthattheperturbationthatgeneratesthemaximum symmetry,theleadingEOFe willbealignedalongthe
1
response will in general project on many dynamical semimajoraxisoftheellipse(whichislongerandthere-
eigenmodes—as will the EOFs of the system. These fore carries the most variance) and the second EOF e
2
facts are particularly pertinent in the context of pre- along the semiminor axis. Both of these directions are
dictionsofElNin˜o/SouthernOscillation(ENSO).Inthe dynamically significant, but neither is a ‘‘dynamically
tropical Pacific, the leading EOF of sea surface tem- invariant subspace’’ on its own. In particular, a self-
perature(SST)projectsstronglyonthecompositemaps containeddynamicalmodelrestrictedtothesemimajor
ofSSTanomaliesformatureElNin˜oorLaNin˜aevents. axisoftheellipsewouldnotmakeagoodtheoryofor-
However,thoseperturbations(orinitialconditions)that bitaldynamics.Anotherillustrativeexample,alsodrawn
induce rapid growth into mature episodes bear little from Mo and Ghil (1987), is that of the EOFs of the
resemblance to this EOF mode (e.g., Penland 1996; Lorenz(1963)modelwiththestandardparametervalues
Kleeman 2008). The leading EOF mode carries a sub- for which the system’s trajectory settles down onto a
stantialfractionofthetotalvariance,butitisnotady- strangeattractor(Fig.3).Onceagain,theEOFsidentify
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FIG.3.Lorenz(1963)attractorx(t)forstandardparametersproducingastrangeattractor.
Thecoloredlinesaretheprojectionsofx(t)ontothethreeEOFmodes:p(t)5[x(t)(cid:2)e]e.
j j j
RedraftedfollowingMoandGhil(1987).
the orthogonal directions carrying the most variance ( [x(cid:3)x (t)]2)
but donot cleanly correspond toany invariant aspect u(x,t)5U(t)exp (cid:3) c , (19)
2s2
of the dynamics. The EOFs of this highly nonlinear 0
dynamical system do not have individual dynamical
significance. wherexisameridionalcoordinate.Inthismodel,thejet
strengthU(t)andpositionx (t)arethenaturalkinematic
c
variables of the jet—what we will call the kinematic
4. EOFsandkinematicdegreesoffreedom
degrees of freedom. For convenience, we will assume
We have seen that EOFs will not generally be of that fluctuations in U(t) and x (t) are independent and
c
individual dynamical significance. Because statistical Gaussian. Based on observations of the extratropical
propertiesofasystemaredescriptionsofvariabilityand zonal-mean eddy-driven jet (in either hemisphere), we
thus inherently kinematic, we might ask if individual will assume that both of l 5 std(U)/mean(U) and h 5
EOFmodeswillbesimplyrelatedtonaturalkinematic std(x )/s are (cid:4)1. With these assumptions, the co-
c 0
descriptors of variability. That this cannot be expected variancematrixofu(x,t)canbecomputedanalytically
tobethecaseingeneralwillbeillustratedbytheexample andexpandedasaTaylorseriesinthesmallparameters
ofasimplemodelofafluctuatingjetinzonal-meanzonal l, h [details of these computations are presented in
wind,forwhichtheEOFproblemisanalyticallysolvable. MonahanandFyfe(2006)].Fromtheseexpansionsthe
In both observations and atmospheric GCMs, the leading EOFs can be determined in terms of the nor-
leadingEOFofextratropicalzonal-meanzonalwindis malized basis vectors f (x), f (x), and f (x) (Fig. 4),
1 2 3
a dipole with a central zero-crossing at approximately correspondingrespectivelytoamonopole,adipole,and
themeanlatitudeoftheeddy-drivenjet(e.g.,Hartmann atripole.Bysymmetry,thedipoleisorthogonaltothe
and Lo 1998; Codron 2005; Fyfe and Lorenz 2005; monopole and tripole, but the monopole and tripole
Eichelberger and Hartmann 2007). To address the ki- themselvesarenotmutuallyorthogonal(andtherefore
nematicsignificanceofthisEOFmode,weconsiderajet cannotsimultaneouslybeEOFs).
in zonal-mean zonal wind with Gaussian profile and In the case of pure fluctuations in jet strength, the
fluctuatinginstrengthandposition: only EOF with nonzero variance is the monopole. For
15DECEMBER2009 REVIEW 6507
FIG.4.Normalizedbasisvectorsf1(x),f2(x),f3(x)(respectivelythemonopole,dipole,andtripole)from
whichareconstructedtheEOFsoftheidealizedjetinzonal-meanzonalwind.
fluctuationsinpositionalone,theleadingtwoEOFsare toincludefluctuationsinjetwidth(MonahanandFyfe
the dipole and the tripole, respectively. When the jet 2006), to relax the assumptions of Gaussian jet pro-
fluctuates in both strength and position, if fluctuations fileandkinematicparameterprobabilitydistributions
in position are relatively large compared to those of (Monahan and Fyfe 2009), and to consider the geo-
strength(asisthecaseinobservations)thentheleading potential EOFs associated with the fluctuating jet
EOF is the dipole f (x). The leading PC time series is (Monahan and Fyfe 2008). The central conclusion re-
1
givenby(toleadingorderinthesmallparameterh) mains unchanged: despite the fact that they directly re-
flectthekinematicsofvariability,thedefiningconstraints
p
(cid:4) ffipffiffiffi(cid:5)1/2 onEOFs(orthogonalstraight-lineaxeswithuncorrelated
a(t)5 U(t)x (t). (20)
1 2s c timeseries)preventthemingeneralfrombeinginsimple
0
one-to-one correspondence with kinematic descriptors
That is, while the dipole pattern arises because of the ofthefield.
presence of fluctuations in position, the fluctuations in
jetstrengthalsoprojectuponitandarethereforemixed
5. EOFsofnon-Gaussianfields
intotheassociatedPCtimeseries.ThefirstEOFmode
bundles together both kinematic degrees of freedom We have seen that individual EOF modes cannot in
andcannotbeuniquelyassociatedwithpositionfluctu- generalbeexpectedtocorrespondtoindividualdynam-
ations. Furthermore, the spatial pattern of the second icalorkinematicmodes.Thatis,EOFswillnotgenerally
EOFisamonopole/tripolehybridwherethedegreeof partition the state space into dynamically or kinemati-
hybridizationisdeterminedbythequantity cally independent structures. One might at least hope
that the EOF decomposition will produce statistically
3h4 independentstructures:afterall,thePCtimeseriesare
d5 . (21)
8l2 mutually uncorrelated. However, decorrelation does
notimplyindependence:twouncorrelatedvariablescan
When d (cid:4) 1, e is a monopole and when d (cid:5) 1 it is still be nonlinearly related (appendix B). If the distri-
2
a tripole: in between, it is a linear combination of the butionofxismultivariateGaussian,thenthePCmodes
two.Themonopolecomesinfromstrengthfluctuations will in fact be both mutually uncorrelated and indepen-
and the tripole from position fluctuations, but because dent. If x is non-Gaussian, however, PC modes will not
these are not orthogonal they cannot both simulta- generallybemutuallyindependent.
neouslybeEOFs.SpatialstructuresthatareEOFsinthe Theidealizedzonaljetmodelwithfluctuationsinboth
case of fluctuations in a single kinematic degree of strengthandpositionconsideredinsection4providesan
freedomonitsownwillnotnecessarilybeEOFsinthe exampleofasysteminwhichthePCmodesarenotin-
presence of multiple fluctuating degrees of freedom as dependent.Whenthefluctuationsinpositionaretaken
a consequence of the requirement that the EOFs be to be Gaussian, the resulting field is non-Gaussian [as
mutually orthogonal. Not surprisingly, the time series u(x, t) is a nonlinear function of x (t)]. A series of nu-
c
associated with the second EOF, a (t), also mixes to- mericallysimulatedscatterplotsofa (t)againsta (t)are
2 1 2
gethervariabilityinbothstrengthandposition. presented in Fig. 5 for values of the parameter d [Eq.
Theobservedextratropicaleddy-drivenjetfluctuates (21)]between0.1and10.Acleardependencebetween
instrength,position,andwidth(withthefirstandthird a anda isevidentwhendisnotverysmall(i.e.,when
1 2
of these correlated as a consequence of momentum positionfluctuationsaresufficientlystrong).Inparticu-
conservation).Theaboveargumentscanbegeneralized lar, strong negative and positive excursions of a are
1
6508 JOURNAL OF CLIMATE VOLUME22
FIG.5.Scatterplotsofnumericallycalculateda1(t)/U0vsa2(t)/U0forh50.26andd50.1,0.25,1,2.5,and10.Blue
dotsdenotethosepointsforwhichU,U andreddotsdenotethosepointsforwhichU.U.
0 0
both associated with strong positive excursions of a . deviationfield.Alsonotableisthesimilaritybetweene
2 2
ThecolorsinFig.5correspondtothosepointsforwhich and the skewness field, the reason for which becomes
thejetisanomalouslystrong[U.mean(U),reddots]or evidentthroughaninspectionofascatterplotofa with
1
anomalouslyweak[U,mean(U),bluedots].Thecur- a (Fig.8).Fromthisplotitisevidentthatstrongposi-
2
vature of the interface between the two regions is an- tive and negative anomalies of a (corresponding re-
1
other measure of the coupling between these two PC spectively to El Nin˜o and La Nin˜a events) are both
time series. That this coupling of EOF modes is not associatedwithstrongpositiveanomaliesofa .Inother
2
simply an artifact of this idealized system is illustrated words, the second EOF mode makes a positive contri-
byscatterplotsofa againsta for500-hPazonal-mean bution to the SST field during both extreme phases of
1 2
zonal winds from the Southern Hemisphere summer- ENSO, so onaverage the strongest positive SST anom-
timeinreanalysisdataandfromadryprimitiveequation alies during El Nin˜o are located farther east than the
GCM (Fig. 6). The idealized GCM is characterized by strongestnegativeSSTanomaliesduringLaNin˜a.This
stronger fluctuations of the jet position than are the asymmetryintheSSTfieldbetweentheopposingphases
observations, and the two scatterplots in Fig. 6 corre- of ENSO is then manifest in the skewness field. Again
spondinglyresemblescatterplotsinFig.5forlargerand we see a relationship between non-Gaussianity of the
smallervaluesofd,respectively. fieldandstatisticaldependenceoftheEOFmodes.
Another field in which non-Gaussianity is manifest We therefore see that individual EOFs will not cor-
throughstatisticaldependenceofEOFmodesistropical respond in general even to statistically independent
Pacific sea surface temperature, as was discussed in structures.Thisfactismitigatedbytheobservationthat
Monahan and Dai (2004). Maps of the leading EOF formanygeophysicalfieldsdeviationsfromGaussianity
pattersofSSTascomputedfromtheHadleyCentreSea arenotstrong.Nevertheless,inconsideringtheresultsof
IceandSSTdataset(Rayneretal.2003)arepresentedin an EOF analysis, it is worth bearing in mind that as
Fig.7.Alsopresentedaremapsoftheestimatedstandard ageneralruleanEOFdecompositionwillnotgenerally
deviation and skewness fields; the latter corresponds partitionstatespaceintodirectionsthatareindependent
to the normalized third-order moment (measuring the dynamically,kinematically,orstatistically.
asymmetryofaprobabilitydistributionarounditsmean)
6. NonlocalityofEOFs
*(cid:4)a(cid:3)hai(cid:5)3+
skew(a)5 (22) We haveseen that individual EOFmodes arenot in
std(a) general mutually independent either dynamically, ki-
nematically,orstatistically:asanevenweakerresult,we
and vanishes if the distribution is Gaussian. Nonzero mighthopethatwhenafieldcontainsstructuresthatare
values of this statistic are therefore a measure of non- truly statistically independent, then these will be sepa-
Gaussianity. The skewness fieldillustrated in Fig. 7in- ratedbytheEOFs.Infact,thiswillnotbethecaseifthe
dicates that the SST probability density tilts toward structuresarelocalizedornotorthogonal.Theseresults
positiveanomaliesintheeasternequatorialPacificand followfromthenonlocalityofEOFs,whichbydefinition
towardnegativeanomaliesinahorseshoe-shapedband maximizevarianceovertheentireanalysisdomain,and
from the central subtropical South Pacific through the from the requirement that by construction EOFs be
westernequatorialPacificbackuptothenorthernsub- mutuallyorthogonal.
tropics. The leading SST EOF, which carries the most Asimpleillustrativeexampleofthegeneralinability
variance, bears a strong resemblance to the standard ofEOFstoseparatelocalizedstructureswaspresented
15DECEMBER2009 REVIEW 6509
FIG. 6. As in Fig. 5, but for 500-hPa zonal-mean zonal wind data (left) from Southern
Hemispheresummertimereanalysesand(right)fromadryprimitiveequationGCM.Adapted
fromFyfeandLorenz(2005(cid:2)AmericanMeteorologicalSociety,reprintedwithpermission.).
in Ambaum et al. (2001). This example considers the dichotomy.Thehemispheric-scaleNAMappearsasthe
three-dimensionalvectorx5(x,y,z)suchthatxandz leadingEOFcomputedovertheentireNorthernHemi-
areofunitvarianceandaremutuallyindependent,and sphere extratropics whereas the regional NAO is ob-
y52x2z;eachofthesevariablesmightbethoughtof tainedwhentheanalysisislimitedtotheEuro-Atlantic
ascorrespondingto,forexample,thetimeseriesofre- sector, as is illustrated in Deser (2000). Noting that the
gionalgeopotentialheightanomalies.Itfollowsthatthe correlationbetweencirculationanomaliesintheAtlantic
covariancematrixofthissystemis and Pacific sectors is weak, Deser (2000) questioned
the extent to which the NAM can be interpreted as
0 1 (cid:3)1 0 1 a coherent hemispheric-scale phenomenon. Interpret-
ing the variables x, y, and z in their simple statistical
C5@(cid:3)1 2 (cid:3)1A. (23)
model as corresponding respectively to geopotential
0 (cid:3)1 1
heightanomaliesoverthe NorthAtlantic,Arctic,and
North Pacific Oceans, Ambaum et al. (2001) demon-
OnlytwoEOFsareofnonzerovariance:
stratedthatindependentvariabilityintheAtlanticand
0 1 1 0(cid:3)11 Pacific sectors can be combined into a leading EOF of
hemispheric scale because of their joint covariability
e 5@(cid:3)2A, e 5@ 0 A, (24)
1 2 with the Arctic sector. While the debate regarding the
1 1
degreeofzonallocalizationofextratropicalatmospheric
variability remains open (e.g., Wallace 2000; Vallis and
carryingrespectively75%and25%ofthetotalvariance.
Gerber 2008), this example clearly illustrates the fact
Despitethefactthatxandzareindependent,theyare
thatthepresenceoflarge-scalestructuresinaparticular
mixed together in the leading EOF because of their
EOFpatternisnotsufficientevidencefortheexistence
mutualrelationshipswiththethirdvariabley.Because
ofsuchcoherentstructuresintheactualflow.
theleadingEOFbyconstructionmaximizestheexplained
In the case of independent but nonorthogonal pat-
variance overthe entire domain,the localizedfeatures
terns, the argument for a lack of correspondence be-
arecombined.Asageneralrule,independentlocalized
tween EOFs and statistically independent modes is
featuresthatsharesomevarianceincommonwithother
straightforward.Supposeforexamplethat
partsofthefieldwillnotbeseparatedbyEOFanalysis.
ThisexamplewasintroducedbyAmbaumetal.(2001)
x(t)5a (t)u 1a (t)u , (25)
inthecontextofadebateregardingtheextenttowhich 1 1 2 2
low-frequency variability in the Northern Hemisphere
extratropical atmospheric circulation is hemispheric or whereu (cid:2)u 6¼0andthetimeseriesa anda aretruly
1 2 1 2
regionalinscale,asexpressedbytheNorthernAnnular independent. The expansion (25) has the form of a
Mode(NAM)versusNorthAtlanticOscillation(NAO) (truncated)EOFexpansion,butasthevectorsu andu
1 2
6510 JOURNAL OF CLIMATE VOLUME22
FIG.7.MapsoftheleadingtwoEOFpatternsoftropicalPacificSST,alongwiththestandarddeviationandskewness
[Eq.(22)]fields.RedrawnfromMonahanandDai(2004).
arenotorthogonal,thesecannotbetheEOFs.Amore bethegravestmodewithawavelengthdeterminednot
detaileddiscussionofthispointanditsconsequencesis bytheproperties ofthefieldbut bythesizeof thedo-
presentedinDommengetandLatif(2002). main. If interpreted as being of individualsignificance,
Intheend,canweatleasthopethatthestructureof theseEOFmodeswillimposespatialstructurewherein
individual EOFs will be entirely determined by the fact there is none. Dommenget (2007) uses this fact to
statistical features of the field under consideration? In suggesta‘‘stochasticnullhypothesis’’fordeterminingif
fact,theanswerisno—thiswillnotalwaysbethecase. EOFstructuresmorereflectthevariabilityofthefieldor
Whenafieldischaracterizedbyspatiallyhomogeneous the geometry of the domain. Furthermore, it has been
statistics(i.e.,invariantfromplacetoplace),theEOFs notedthathemispheric-scaleEOFstructuressuchasthe
will be strongly influenced by the size and shape of NorthernandSouthernAnnularMode(SAM)canarise
the domain (e.g., Buell 1975, 1979; Richman 1986; from variability that is zonally localized (e.g., mid-
Dommenget 2007). For example, if x represents the latitude eddy activity) but with statistics that display
valuesofafieldobservedatpointsalongalinesuchthat ahighdegreeofzonalsymmetry(e.g.,Cashetal.2005;
neighboringpointsarerelatedbyanAR(1)process— KushnerandLee2007;VallisandGerber2008).Thatis,
thatis, the (hemispheric) zonal scale of the SAM and NAM
maybesetbythezonalsymmetryofmidlatitudedynam-
1 icalprocessesratherthanbytheexistenceofhemispheric-
x (t)5 x (t)1a(cid:2) (t), (26)
j d j(cid:3)1 j scalecoherentstructures.
where (cid:2) are independent and identically distributed
j 7. Conclusions
random variables—then the EOFs of x are sinusoids
regardlessofthemagnitudesofdora(AllenandSmith The study of atmospheric and oceanic fields (either
1994). This result can be shown to follow from the observed or simulated by a GCM) requires consider-
Wiener–Kinchintheorem,fromwhichweknowthatthe ation of datasets of very high dimensionality (not un-
Fouriercoefficientsofastationaryfieldareuncorrelated commonlyoforder103).Suchdatasetsobviouslycannot
random variables [e.g., Gardiner (1997) and Yaglom be visualized in their raw form: some sort of statistical
(1961),inwhichitisalsoshownthattheEOFsofaho- processing is required to reduce the dimensionality of
mogeneous random field on a sphere will be spherical the data to a size accessible to the human mind. This
harmonics]; the spatial orthogonality of the Fourier statisticalprocessingimposesafilteronthedata,sothat
modesthenensuresthattheseareEOFs.Thescalesof thestructuresthatemergeareaconvolutionofboththe
theEOFspatialstructureswillbedeterminedbythesize data itself and the statistical technique used for the
ofthedomain.Inparticular,theleadingEOFmodewill analysis.Thatis,thestatisticsusedleavetheirimprinton