Dec 27, 1996 - Climate Prediction Center, National Weather Service, NOAA, Washington, D.C. ... a compact methodology for assessing climate experiments with large ..... Palmer, T. N., Extended range atmospheric prediction and the Lorenz.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. D23, PAGES 29,591-29,597, DECEMBER 27, 1996
A methodologyfor assessingensembleexperiments X. L. Wang• ClimatePredictionCenter,NationalWeatherService,NOAA, Washington,D.C.
H. L. Rui 2 Environmental ModelingCenter,NationalWeatherService,NOAA, Washington, D.C.
Abstract. Climate simulationsandforecastexperimentsof increasinglylargeensemblesize arebeingperformedto assess thepredictiveskill of a dynamicmodelon seasonal and longertimescales.Especiallyin the casesof ensembleclimatesimulationor forecastforced by observedor predictedseasurfacetemperatures, themodelis expectedto maximize potentialpredictabilitydueto boundaryforcingandto minimizeinternalvariability generatedfrom dynamicinstability.In the light of smallpredictiveskill in extratropics from boundaryforcing,onemustevaluateskill of the ensemblemeanquantityagainst intersamplevariabilityor spreadof the individualensemblemember. On the otherhand, certaindominantsignalsin climatevariability,suchasE1Nifio-SouthernOscillation,have beendocumented.Predictabilityfor thesemajor signalsis the hopeof seasonaland climateforecastingusinga dynamicmodel. It may be unrealisticto anticipatea model beingable to simulateor forecastthe full spectraof climatevariability. The questionis how to evaluatea model'sperformance in capturingthe dominantclimatesignalsin ensemble experiments with increasingly largesamplesize. Theseissueshavemotivatedusto develop a compactmethodology for assessingclimateexperiments with largeensemblesize. This methodtreatsthe ensemblemeanas signaland intersamplevariabilityas spreador noisein a commonframework. Hencenot only dominantsignalsfrom boundaryforcingcan be isolated,but alsosensitivityof thesesignalsto the forcingcanbe assessed.Otherpotential applications of themethodto climatesimulationandforecasting arealsodiscussed. variabilityof the system[Madden,1976;Lau, 1985; Chervin,
1. Introduction
1986; Shea and Madden 1990].
In the caseof ensembleexperiments, it is anticipatedthatthe potentialpredictabilityresultingfrom the boundaryforcingcan be maximized while the natural variability due to internal dynamicswill be minimized. In anotherwords,the ensemble mean forecast is expected to enhance boundary-forced predictabilityby reducinginternalvariabilityor noise. In this context,predictiveskill of the ensemblemeanforecastcanbe evaluatedagainsta noiselevel reducedthroughthe ensemble averagingprocess,i.e., intersample variabilityor spread[Kumar and Hoerling, 1995]. It remainsto be determinedwhetherensemblemeansprovide usefulpredictiveskill for seasonalforecastin the extratropics. On the other hand, it is also arguedthat the impact of interannual changesin SST forcingis to createa shift in the extratropicalmean state,althoughthis shift is small and resides within the envelope of atmosphericstates attained with climatologicalSSTs [Palmer, 1993]. Given the prospectsfor small skill in the extratropicsas revealedin variancetestsof boundary-forced potentialpredictability,onequestionis how to evaluatepredictiveskill of interannualSST forcingin ensemble experiments.This paperis aimedat a compactassessment of ensembleexperimentsby evaluatingforecastskill of ensemble meanandintersamplevariabilityin a commonframework.The next section showsdetails of the technique,while the third •Alsoaffiliatedwith Research andData Systems Corporation, Greenbelt, sectiongivesa sampleapplicationof the method. Concluding Maryland.
The recentintroductionof ensembletechniqueshowsgreat prospects not onlyfor improvingon the verificationskill of the singlebestexperiment(forecastor hindcast),but alsoof dealing explicitly and quantitatively with the uncertainty of the experiment.While this is true at all forecastranges,ensemble experiments are particularlyusefulto determinethe predictive capabilityof a modelandto accessuncertaintyof forecastsover the extratropicson seasonalto climatetimescales. The climate hindcastexperimentsforced by the observed globalseasurfacetemperatures (SSTs)showpredictiveskill of extratropicalcirculationanomalieswith or without realistic initial conditions[Brankovic et al., 1994; Kumar et al., 1996; Barnett, 1995]. On the other hand, boundary-forced predictabilityin the middlelatitudesis relativelysmallasshown by observational andmodelstudies[Madden,1976;Lau, 1985; Chervin, 1986]. One explanationis that it is due to relatively large internal variability resultingfrom dynamicinstabilityin extratropics[Kumar and Hoerling, 1995]. Hence potential predictability of low-frequency atmosphericstates,such as seasonalmean average, has been traditionally assessedby comparingthe internalgeneratednaturalvariabilityto the total
2Alsoaffiliatedwith Research andGeneralServicesCorporation, Laurel, Maryland.
Copyright1996 by the AmericanGeophysicalUnion. Papernumber96JD02901. 0148-0227/96/96JD-02901
$09.00
remarks
are in section 4.
2. Method
For a generalized case of N member ensembles,the ensemble mean of a variable
29,591
Y is defined as
29,592
WANG AND RUI: METHODOLOGY FOR ASSESSING ENSEMBLE EXPERIMENTS N
N
U)
Y(x,O -
Nn=l
M
1 •'[•V(x,m)[P(t,m)-
o%,0 -
N n--lm--1
where x is index for spatialpoints(x =1.....X), t is index of temporalsamples(t =1.....T), andn is ensemble size(n =1....,N). By performinganempiricalorthogonal function(EOF) analysis, thenormalized(signifiedby an asterisk* ) ensemblemeancan Sincewe haveexpandedall of thedataontotheknownspatial be expressedas
p(t,m,n)]] 2 (8)
eigenbase {EV}, thespatial information of thespread in o2(x,t)
M
Y(x, O*=•'EV(x,m)PC(t, m)•(m)
(2)
is redundant.Thereforewe canintegratethe spreadoverthe entirespacedomainto focuson its temporalvariationsrelated to eachindividualspatialEOF modein the {EV}. That is
m=l
x
whereEV istheeigenfunction, PCisprincipal component, •, is
theassociated singular value (•2gives eigne value), and mis
1 •'•(x,O 3x(o-X x4
(9)
the number of eigenmodes(m = 1..... M). Both the eigenfunction andtheprincipalcomponent satisfythefollowing where C3x(t) represents a space integration. Bysubstituting (8) orothonormal conditions: into (9) andusingthe condition(3), (9) canbe rewrittenas
x
M
o i•j
•rX(t) --• ' oS (t,m)
•'EV(x'i)EV(x'J)--{I i-•
(10)
m=l
x--1
whereos(t,m)denotes anEOFrepresentation of thespatially integratedspread,whichcanbe shownas
•'PC(t,i)PC(t,j) : i--j
(4)
t--1
N
1 •V,[p(t,m)_p(t,m,n)]2 (11) øs(t'm)-N n--1
Both the EV andthe PC form a completeeigenfunction base, i.e., {EV} or {PC}, where {EV} is an eigenbasedefinedin s denotesan EOF representation of the spread spatialpointsand {PC} is an eigenbasedefinedin temporal The superscript using spatial eigenfunctionbase. Equations(10) and (11) points. The spreado of theN memberensembles canbe definedas definea uniquerelationshipbetweenthe spreadandthe EOF
modes of theensemble mean.Thequantity os(t,m)describes
N
1 •,[y(x,t,n)_Y(x, Oi2 øg(x't)-Nn__ 1
(5)
temporalevolutionof thespreadaffiliatedwith eachEOF mode andis a measureof predictability of climatesignalrepresented
by theEOFmodes.If oneintegrates os(t,m)overentiretime period,a spectrumof the totalspreadin theEOF space{EV} canbe obtainedfrom (10) as following
Becauseour interestis to estimatepredictabilityof the EOF modes,especiallythe leadingones,of the ensemblemean,a relationshipbetweenthe ensemblemeanEOF modesandthe spreadwill be established basedon theEOF expansions. 2.1. Spatial Expansionof the Spread
Using the spatial eigenfunctionbase {EV}, both the ensemble mean and individual members of the ensemble can be
decomposed ontothe sameEOF spaceas
M
•Yr = • oS(m) m--1
(12)
where•r isa constant anddenotes anintegration of thespread overboththespaceandthetimeandoS(m)represents a time integrationof oa(t,rn) as definedin (11). From (12) a contributionof each EOF mode to the total spreadcan be assessed. A relative measurefor reliability in a case of simulations or for predictability in a caseof predictions of EOF modesof the ensemblemeancanbe expressed as
x
P(t,m) --•'EV(x,m)Y(x, t)
(6)
x--1
O(m) -- oS(m) 2(m)
(13)
where•.(rn)is thesingular value,thesquarerootof eigenvalue, x
p(t,m,n)--•'EV(x,m)Y(x, t,n) x--1
(7)
associatedwith eigne base {EV}. For eachindividualEOF modem, thelessthequantity0(m) is, thehigherreliabilityand predictabilitythe EOF modeEV(m) has.
2.2. Temporal Expansion of the Spread where P(t,m) and p(t,m,n) representtime coefficientsof the If one uses the temporal eigenfunctionbase {PC} to ensemble mean'sandindividualensemblemember's projection theensemblemeanandeachindividualmemberof on the spatialEOF modes. In the EOF spacethe definitionof decompose the ensemble,(6) and (7) can be modifiedto the spreadin (5) canbe rewrittenas
WANG AND RUI: METHODOLOGY
FOR ASSESSING ENSEMBLE EXPERIMENTS
29,593
3.1. Eigenbases
T
(14)
E(x,m) 5__•l PC(t,m) Y(x,t) T
e(x,m,n) =•V'PC(t,m)Y(x,t,n) t=l
(15)
whereE(x,m) and e(x,m,n)representtime coefficientsof the ensemblemean'sandindividualensemblemember'sprojection
on the temporalEOF modes.The EOF representation of the spread,i.e., (8), canbe changedto N
M
1 •V'[•'PC(t,m)[E(x,m)ø2 (x' t)- N n--lm =1
Shown in Figure 1 are the first four EOF modesof the ensemblemean500-mbargeopotentialheightanomaly. They explain about 66% of variance. The associatedprinciple componentsof the modesare shownin Figure 2. The first mode describesan atmosphericresponseof ENSO events, whichconsistsof a wavypatternemanatingsymmetricallyfrom tropicaleasternPacific into downstream extratropicsof both hemispheres(Figure l a) and a temporalevolutiondepicting ENSO characteristics(Figure 2a). The secondand the third modescapturemajorfeaturesof zonallysymmetric components of either hemisphere(Figures lb and lc), and their principle components arenotvery well associated with ENSO variations. The fourth mode shows again a wavy pattern (Figure l d). However, the wave structuresover the Pacific/America sector
By a similarargument asusedfor (8), a timeintegration of (16)
are antisymmetricwith respectto the equator,in contrastwith the symmetricpatternin the first mode. The corresponding PC (Figure 2d) showsmuch higher frequency. This mode was identified as an anomalousseasonalmodulationof global ENSO response[X. L. Wang et ai., An assessment of NCEP climatemodel:Prospects andlimitationsof climateprediction, submittedto Bulletin of theAmericanMeteorologica!Society,
yields
1996].
e(x,m,n)]] 2 (16)
M
•T(x) - •V'ot(x,m) m--1 where
N
ot(x,m)_ 1 •,[E(x,m)_e(x,m,n)]2 (18) N n--1
In orderto discussorthogonalexpansionof the spreadin an EOF space,it is helpful to acquaintfeaturesof spatiallyand temporallyaveragedquantitiesof the spreadas definedin (5). Shownin Figures3a and3b arethe time meanandglobalmean, respectively,of the spread. General featuresof time mean spread(Figure3a) are smallervalues(around5 m) over tropics andlargervalues(morethan30 m) in extratropics.In northern hemispheremidlatitudesthere are two distinctmaximaof the spreadover Pacific and Atlantic stormtracks. In termsof the globalmeanspread(Figure3b), valuesare ranged,on average, from about 24 m in summer to around 30 m in winter.
whered(x,m) is an EOF representation of thetimeintegration
of thespread andGr(x)isthetime-integrated spread.A similar
3.2. Spatial Expansion
Usingthe spatialEOF modesshownin Figure 1, the spread defined in (5) can be expressedin termsof the eigenbaseas seenin (8). Shownin Figure4 are the first four modes(m=l-4) M of spatiallyintegratedspreadcy•(t,m)as definedin (11). The spreadassociated with thefirsttwo EOF modesappearsto have (19) large seasonal cycle, with wintermaxima and summerminima. m=l The magnitudeof the spreadis muchlargerfor the secondEOF whereor(m)refersto a spaceintegration of ot(x,m)asdefined mode. Consideringthe fact that the first EOF modeexplains in (18). A relative measurefor predictabilityof the ensemble about39% of variancewhile the secondexplainslessthan 13%, onecan expectthat the spatialstructuresof the first EOF mode meanEOFs in termsof temporalexpansionis expressedas aremuchmorereproducibleamongthe ensembles andtherefore are of largerpredictability.The spreadassociated with the third EOF is as large as the first one, while the fourth EOF shows ,•(m) much smallerspread.
spectrum of the total spreadin theEOF space{PC } is defined as
•T = •V'or(m)
y(m) - øt(m)
(20)
By combining the information obtained from •(t,m) and
ot(x,m),onecanquantifybothspatialandtemporalvariations of thespreadassociated with eachindividualEOF modeof the ensemble mean.
3. Results On the basis of the method described in section 2, we have
analyzeda 13-memberensembleof climatesimulationdoneby Climate Modeling Branch of National Centers for EnvironmentalPrediction(NCEP). The simulationwas forced by observedSST from 1950 to 1994 (a total of 45 years). The model
used is the NCEP
T40
climate
model.
From
the
experiment the 500-mbar geopotentialheights from each ensemblehindcastwere extractedandusedin this study.
3.3. Temporal Expansion
Shownin Figure 5 are the first four modes(m=l-4) of time
integratedspreadot(x,m),following(17) and (18). General featuresof Figure 5 are smaller spreadin tropicsand large values over midlatitudes to high latitudes. The spread associatedwith the temporal structureof the first mode is overallthe largest,and amplitudesof the spreaddecreasewith modes. Over the North Pacific-Americasectorthe largest spreadappearsin the secondmode. 3.4. Predictability
As mentionedpreviously,reproducibilityof the ensemble mean EOF modes are intimately linked to their associated spread. Hence a predictabilityassessment for individualEOF
29,594
Mode1 (59.0%)
_ •••_
60N
.•.(•
•
a
,
Mode2 (12.8%)
b
- --.... ---- ' •--•__::: • I-:•_-•-,',-••:::-[ .......... •-.,•__--::•_ :•---
•••'-'••••••:::::•••_.•-
o.o.•_-••:::
60N
30N
EQ
30S
"'......
•....
.0••' EQ .•o•
60S
0
60E
120E
180
120W
60W
0
Mode5 (8.7%)
0
60E
120E
180
120W
0
60E
120E
c
60W
180
120W
60W
0
Mode4 (5.0%)
0
0
60E
120E
180
120W
d
60W
0
Figure1. The firstfourempiricalorthogonal function(EOF)modes,(a) mode1, (b) mode2, ¸ mode3, and (d) mode4, of theensemble mean500-mbargeopotential heightanomaly.Valuesareorthonormalized, and negativevaluesaredashed.Intervalsare0.005. a 0.5
O
-0.5 -1 1
b
19'55
19'60
19'65
19'70
19'75
19'80
19'85
19'90
1995
0.5
0
-0.5 -1
,
,
,
,
,
,
,
,
1955
1960
1965
1970
1975
1980
1985
1990
1995
0.5
-1
o
19'55
•., •_w•v•/,v•.,•v.•• •. V•,...• ,v• 19'60
19'65
19'70
19'75
19'80
19'85
19'90
0
1995
0.5
-0.5 -1
V
19'55
19'60
'vvv,v,vvvr 'v 19'65
19'70
19'75
19'80
0
19'85
19'90
Figure 2. Principalcomponents of theEOFmodes(a) 1,(b) 2, (c) 3, and(d) 4 asshownin Figure1.
1995
WANG AND RUI: METHODOLOGY FOR ASSESSING ENSEMBLE EXPERIMENTS
Total Mean
a
GlobalMean
45
60N
29,595
b
40
30N
35
30S
60S
60E
120E
180
120W
60W
0
15
1955
1950
1955
197;
1975
1980
1985
1990
1995
Figure3. (a) Time meanspreadof 500-mbargeopotential heightanomalyfrom 13-member ensemble.Intervals are 10 m. (b) The globalaveragedspreadof 500-mbargeopotential heightanomalyfrom the same13-member ensemble.
Unit is meters.
modes can be made by estimatingrelative importanceof variancecontribution(expressed by eigenvalues)of the mode againstits uncertainty (associated spread),following(13) and (20). Shownin Figure6 are 0 (equation(13)) and¾(equation (20)) valuesfor the first eight eigenmodes. Smallervalues indicatelargerpredictability. In general,both0 and¾valuesincreasewith modenumber, indicatingthat the first EOF modeis mostpredictableand
highermodeshavelowerpredictability.It is very interestingto notethe differencebetweenthe spatialexpansionand temporal expansionof spread. In the caseof spatialexpansion,relative predictabilities for the secondandthirdmodesare substantially less than the first mode and are even less than the fourth mode.
Recall that the spatialstructuresfor the first and the fourth modesare zonallyasymmetricwavy patterns,while the second andthird modesdescribezonallysymmetricfeatures(Figure 1).
a
30
25
o
5
0b
30
19'55 19'60 19'65 19'70 19'75 19•0
19•5
19'90 1995
1955
1985
1990
25 20 15 10 5 0
,
C
3O 25
1960
1965
1970
1975
1980
1995
_
20
15 o
10 5 0
30
d
v•jwv•w V'-w W'w•''v• N'••• '"vv '•'k/ VV •vv v.'v•'•y•,vv' V¾•• uw' 1955
1960
1965
1970
1975
1980
1985
1990 ,
1995
25-
20 15 o
10 V'V' 0
"-,,Jv'-'•-
19'55
-v-
,vVv,,-v--
19'60
v ,,
U•.,,V,,-vvw-
19'65
,,vv
•/'• "•.•"v"'•/¾'v"VV'-'V'•
19'70
19'75
'""- v
•
vv•,m-
19'80
v- ",/•""•,r'v'V"
19'85
'¾v
'""
19'90
Figure 4. The squarerootvaluesof o' (t,m)for m=(a) l, (b) 2, (c) 3, and(d) 4. Referto (l 0) and(l l) for details.Unit is meters.Solidlinesrepresenttimevalue.
1995
29,596
WANG AND RUI: METHODOLOGY FOR ASSESSINGENSEMBLE EXPERIMENTS
M ode
o
120E
60E
1
180
120W
Mode ß
o
6oE
60W
3
0
0
60E
120E
c
18o
b
120w
60w
Mode 4
0
d
..
180
• 20E
Mode 2
a
120W
60W
0
0
60E
120E
180
120W
60W
0
Figure 5. Thedistributions ofsquarerootofot (x,m)form=(a)l, (b) 2, (c) 3, and(d) 4. Referto (17) and(18) for details.Intervalsare I m, andvaluesgreaterthan3 m areshaded. Therefore the feature shownin Figure 6a suggeststhat zonal mean responsesof ENSO are less predictable,while wavy responses are muchmorerobustin the ensemble[Wanget al., 1996]. In the caseof temporalexpansion,the ¾valuesare increasing monotonically,suggestinga linear decreaseof reproducibilityof time-dependent variationof ensemblemean
responsein stationarywaves,which is highlyreproduciblein the model; (3) zonal mean responsesof ENSO are more difficult
to simulate.
4. Concluding Remarks
EOFs
in individual ensemble member. These differences This paper presents a compact procedure to access indicate(1) the modelis able to discriminatespatialresponses multimemberensembleexperimentsin termsof analyzingthe betweenspreadandensemblemeanin EOF space. of ENSO, while temporalvariationsof ENSO cycleare more relationship A fundamental characteristic used in the derivation in section difficult to be differentiated; (2) there is a canonicalENSO
sigma-s mean
a
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
sigma-t mean
o. 1 o
•,
:•
•.
•
EOF MODES
•
3
8 EOF MODES
Figure 6. (a) The nondimensional valuesof 0(m) from (13) associated with the first eightEOF modes. (b) The nondimensionalvaluesof ¾(m)from (20), corresponding to the first eightEOF modes.
b
WANG AND RUI: METHODOLOGY FOR ASSESSING ENSEMBLE EXPERIMENTS
2 is the orthogonalityof eigenfunctionbaseformedby EOF analysis. Thereforeone may performsimilar analysisusing differenttypeof orthogonal eigenfunction base,suchasnormal
29,597
areappreciated. Thisworkis partiallysupported by NationalOceanic andAtmospheric Administration (NOAA) Officeof GlobalPrograms (OGP).
modes.
With a few modificationsthismethodcanbe easilyusedto study different aspectsof ensembleexperiments. Those alternative approachesinclude but are not limited to the following. (1) EOFs can be computedin a regionaldomain, such as over the Pacific-North America (PNA) area. The reliability and predictabilityof regionallydominatedfeatures from ensembleexperimentscanthenbe assessed, (2) EOFs can be calculatedfor different seasons,such as winter only or summer only, which may enable one to study seasonal dependencyof the forecast, (3) EOFs can alsobe calculated from observational data, so error characteristics of ensemble
experimentscan be analyzed, (4) In the case of forecasts, reliabilityof forecasteddominantmodescan be evaluatedby combiningapproaches1, 2, and3. Branstatoret al. [ 1993] studiedpredictabilityof extendedrangeforecastusingEOF technique.They concludedthat they foundanEOF decomposition of the500-mbarheightfield to be an effectivemeansof distinguishing the forecastabilityof flow elements, withthosecomponents thatprojectontoleadingEOFs tendingto be betterforecastthancomponents thatprojectonto trailingEOFs. In the case of ensemble forecasts, the method described in
this paper will be able to distinguishwell forecastedfrom poorly forecastedmodes or flow elementsby using the approaches3 and 4. To benefit from such assessments of climateensembleforecasts,onemustbe adaptedto the notion that certain large-scale,low-frequencymodes can not be accuratelyforecastedwith specifiedboundaryconditions. There is an importantissuethat hasnot beenraisedin this paper. That is, how do thosestatistics computedherechange withensemble size?A moregeneralized question is,howmany members areenoughin anensemble experiment?Thisquestion can be addressed, at leastpartially,by repeatingcalculations with continuously changingensemblesizein section3. Acknowledgments. Discussionwith C.F. Ropelewski,V.E. Kousky, and P.-T. Penghelpedclarify severalaspectsof our work. Theconstructive comments fromthereviewerstrengthen thepaperand
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Wang, X. L., H. L. Rui, and A. Leetmaa,The dynamicsof ENSO anomalyin NCEPensemble climatesimulations - Impactof mean stationary wave,Adv.Atmos.Sci.,in press,1996. H. L. Rui, EnvironmentalModelingCenter,NationalCentersfor Environmental Prediction, National Weather Service, NOAA, W/NP24, Washington, D.C. 20233. (e-mail: wd0lhr@ sun3.wwb.noaa.gov) X. L. Wang, Climate PredictionCenter, National Centersfor Environmental Prediction, National Weather Service, NOAA, W/NP24, Washington, D.C. 20233. (e-mail:
wd52xw@sgil 4.wwb.noaa.gov) (ReceivedApril 9, 1996;revisedSeptember 17, 1996; accepted September 20, 1996.)
