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Constraining climate model properties using optimal fingerprint detection methods


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MIT Joint Program on the

Science and Policy of Global Change

Constraining Climate Model Properties Using

Optimal Fingerprint Detection Methods

Chris E. Forest, Myles R. Allen, Andrei P. Sokolov and Peter H. Stone

Report No. 62

May 2000


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Constraining Climate Model Properties Using Optimal Fingerprint Detection Methods

Chris E. Forest


, Myles R. Allen


, Andrei P. Sokolov


and Peter H. Stone



We present a method for constraining key properties of the climate system that are important for climate

prediction (climate sensitivity and rate of heat penetration into the deep ocean) by comparing a modelÕs

response to known forcings over the 20th century against climate observations for that period. We use the

MIT two-dimensional (2D) climate model in conjunction with results from the UK Hadley CentreÕs coupled

atmosphere-ocean general circulation model (AOGCM) to determine these constraints. The MIT 2D model is

a zonally-averaged version of a 3D GCM which can accurately reproduce the global-mean transient response

of coupled AOGCMs through appropriate choices of the climate sensitivity and the effective rate of diffusion

of heat into the deep ocean. Vertical patterns of zonal mean temperature change through the troposphere and

lower stratosphere also compare favorably with those generated by 3D GCMs. We compare the

height-latitude pattern of temperature changes as simulated by the MIT 2D model with observed changes, using

optimal fingerprint detection statistics. Interpreted in terms of a linear regression model as in Allen and Tett

(1998), this approach yields an objective measure of model-observation goodness-of-fit (via the normalized

residual sum of squares). The MIT model permits one to systematically vary the modelÕs climate sensitivity

(by varying the strength of the cloud feedback) and rate of mixing of heat into the deep ocean and determine

how the goodness-of-fit with observations depends on these factors. This approach provides an efficient

framework for interpreting detection and attribution results in physical terms. For the aerosol forcing set in

the middle of the IPCC range, two sets of model parameters are rejected as being implausible when the model

response is compared with observations. The first set corresponds to high climate sensitivity and low heat

uptake by the deep ocean. The second set corresponds to low sensitivities for all values of heat uptake. These

results demonstrate that fingerprint patterns must be carefully chosen, if their detection is to reduce the

uncertainty of physically important model parameters which affect projections of climate change.


1. Introduction ... 2

2. Models, Methods, and Experimental Setup ... 5

2.1 MIT 2D Land-Ocean Climate Model

... 5

2.2 Optimal Fingerprint Detection Statistics

... 8

2.3 Forcings

... 13

2.4 Model Experiments

... 15

3. Results ... 16

3.1 Model Response

... 16

3.2 Filtered Response

... 18

3.3 GSO Response Ð K



... 19

3.4 GSO Response Ð Two Free Parameters

... 20

4. Discussion ... 21

4.1 Implications for Uncertainty Estimates of Future Climate

... 21

4.2 Dependency of Temperature Change Pattern on S or K


... 23

4.3 Dependency of Detection Statistics on


... 24

4.4 Remaining Issues

... 25

5. Conclusion ... 26

References ... 28





Joint Program on the Science and Policy of Global Change, Massachusetts Institute of Technology, Cambridge MA 02139 USA

Corresponding author: Chris E. Forest, Room E40-259, MIT, Cambridge MA 02139 USA (e-mail: ceforest@mit.edu)


Space Science Department, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK


Department of Physics, University of Oxford, UK


The projection of future climatechangerequires the use of complexclimatemodels

that contain uncertain representations of subgrid scale physics,and thereforedi er


uncertaintyindetermingclimatesensitivity,andtherateof penetrationofheatinto

the deepocean, whichisthe major determinantofhowrapidly warmingoccurs(see

Houghton et al., 1996)). Some information on properties of the climate system is

contained,atleast implicitly,inthe resultsof detectionand attribution studies: for

example, unambiguous detection of anthropogenic greenhouse gas in uence on


Todate, however, the focusof such studieshasbeenonsignaldetection ratherthan

the physicalimplications. Consequentlythe identi cationofanthropogenic forcings

asbeingresponsiblefor recentclimatechangeshas not yet ledtoa reductioninthe

large uncertainty in projections of future climate changes.

Inabroadsense,twotypesofuncertaintyexist: uncertaintiesassociatedwiththe


uncertainties in the forcing scenarios of future climate. In this study, we address

the formerbutrecognize thatafullassessmentof uncertainty must encompassboth


Fingerprint detection studies are conducted by running a climate modelunder a

set of prescribed forcings and using detection diagnostics todetermine whether


fromnaturalvariabilityof theclimatesystem (e.g.Hansenetal.,1997;Hasselmann,

1993, 1997; Hegerl et al., 1996, 1997; Santer et al., 1995, 1996; Tett et al., 1996).

As is wellknown, it isdiÆcult toestimate the true climatesystem variabilityfrom

observations and therefore, climate models are run with equilibriumconditions for

1000 years to provide estimates of climate variability to be used in signi cance


this is because the detection diagnostics are not directly linked to the properties

known to be important for projections of climate change. As discussed elsewhere,

the twosuch parametersare the equilibriumclimatesensitivitytochangesin

radia-tive forcings (e.g., Houghton et al., 1996) and the rate of heat uptake by the deep

ocean (Sokolov and Stone, 1998). Together, these properties describe the

decadal-to-century timescaleresponse of the climatesystem toa given forcing (e.g. via the

global meansurface temperature).

In this paper, we show how climate change detection diagnostics can be used to

quantify uncertainty in these properties and therefore quantify uncertainty in

cli-mateprojections. Wealsoexaminehow theseuncertainties a ect theattribution of

recent climate changes to speci c forcings. From recent results in climate change


emerg-ingfromthenoiseof internalvariability(Santeretal.,1996),but thissignalhasnot

yetbeenfullyexploitedforthesepurposes. Onereasonforthisisthattheenormous

resources necessary torun three-dimensionalcoupledatmosphere-oceangeneral

cir-culation models (AOGCM) limit the ability of researchers to adjust parameters

related to the climate sensitivity or adjustment timescales. In addition, the ability

toadd and subtract forcings isrestricted by the feasible numberof simulations. By

systematicallyexploringamodelparameter space,which wouldbe impossibletodo

using AOGCMs,this study providesa methodology for constraining model

proper-ties that are directly relevant to reducing uncertainty in the projection of climate

change, and thus, has direct bearing on interpreting the detection and attribution

results of 3D AOGCMs.

For a computationally eÆcient model, we use the MIT 2D statistical-dynamical

climate model which was developed to examine the uncertainty in future climate

projections associated with 3D climate models (Sokolov and Stone, 1998). The

atmospheric model is coupled to a Q- ux di usive ocean model that mixes


deep ocean. The model can be forced in the same manner as 3D AOGCMs and

thus, uncertainty of the forcings can also be addressed. In particular, the

magni-tudeof theanthropogenicaerosolforcing ishighlyuncertain. Inadditiontovarying

the sensitivity and the ocean di usivity, the e ective aerosol forcing can be varied

and included as an uncertain parameter producing a three dimensional parameter

space. In this study, however, the parametrization of the aerosol forcing remains


and deep-ocean heat uptake.

The model willbe forced with changes in greenhouse gas, anthropogenic aerosol,


20th century. We will produce a large number of simulations by spanning the

pa-rameterspaceof thetwouncertain parametersandthen use theoptimal ngerprint

detectionalgorithmtodetermine the tbetween modelsimulationand

observation-s. The ngerprint diagnostics provide normalized residual sum of squares statistics

that can be used to assess the goodness-of- t as a function of the uncertain model

parameters. In this manner, a location in the model parameter space that best

ts the observational record and a con dence region surrounding this point can be

identi ed.

The method for quantifying these uncertainties is divided into two parts: the

simulationof the 20thcentury climaterecord and the comparisonof model

simula-tionswithobservationsusingoptimal ngerprintdiagnostics. We rstrequirealarge


have been systematically varied. We use the MIT 2D statistical-dynamical climate

model toprovidethe rst. Forthe second, weemployamethodof comparing

simu-lationstoobservationsthat appropriately lters\noise"fromthepatternofclimate

change. The variant of optimal ngerprinting proposed by Allen and Tett (1999)

provides this tool and yields detection diagnostics that are objective estimates of


(Section 2.1)andthe optimal ngerprintmethod(Section 2.2). Wethendiscussthe

forcings applied to simulate the 20th century climate record (Section 2.3) and the

setup of the experiments (Section 2.4). In Section 3.1, we will discuss the model

simulations and the physical implications for constructing ngerprint patterns. In

Section 3.2, we will show the e ects of ltering \noise" from the ngerprint

diag-nostics. This will be followed by a description of the distribution of the ngerprint

statistics over the model parameter space (Section 3.3) and their impacton

uncer-tainty in model forecasts (Section 4).

2 Models, Methods, and Experimental Setup

2.1 MIT 2D Land-Ocean ClimateModel


model coupled to a Q- ux di usive ocean model. The model details can be found

in Sokolov and Stone (1995, 1998). The atmospheric model is a zonally averaged

versionof the GoddardInstitutefor SpaceStudies(GISS) ModelII general

circula-tion model (Hansen et al., 1983) with parameterizations of the eddy transports of

momentum, heat, and moisture by baroclinic eddies (Stone and Yao, 1987, 1990).

Thisversionof themodel has24latitudebands(=7:826



lay-ers with 4layers abovethe tropopause. The model uses the GISSradiativetransfer

code inwhichthe e ects ofallradiativelyimportanttracegases,aswellasaerosols,

are included. The surface area of each latitude band is divided into a percentage

of land, ocean, land ice, and sea icewith the surface and radiative uxes computed

separately foreach surface type. This allows forappropriate treatmentof radiative

forcings dependent onunderlyingsurface type such asanthropogenicaerosols. The

atmospheric component of the model, therefore, simulates most of the important

nonlinear dynamicalinteractions between componentsof the atmospheric system.



though the true climate system sensitivity depends on multiple feedbacks,

atmo-spheric GCM sensitivities vary primarily from di erences in cloud feedback (Cess

et al., 1990; Houghton et al., 1996). Consequently, the MIT model's sensitivity is

controlled by altering the cloud feedback. As in most models, the MIT model

pa-rameterizes clouds based on the vertical pro le of temperature and moisture. The

computedcloudfractionismodi edinthe followingway (afterHansenetal.,1993):

C 0



)C wherek is anadjustableparameter,T


isthe changein

global-mean surface temperature, and C is the computed cloud fraction. An increased

cloud fraction (positive k) leads to a reduction of the overall warming associated

with an increased external radiativeforcing and vice versa. Because the additional

cloud fraction is proportional to the change in temperature from equilibrium, the

current mean climate remains unchanged when the climate sensitivity is changed.

As discussed in Sokolov and Stone (1998), changes in atmospheric response to a



followclosely thoseshown bythe UKMOGCM insimulationswith

di erent cloud parameterizations (Senior and Mitchell, 1993). The ability to vary

the sensitivity of the MIT climate model, along with the computational eÆciency,

providesauseful toolforexploringquestions whichwould be impracticaltoexplore

with more complex 3D AOGCMs.

Theoceancomponentofthe MITmodelisadi usiveoceaninwhichtemperature

anomalies in the ocean mixed layer are mixed vertically into the deep ocean. The

mixed-layer depths are speci ed from climatology,and vary with latitude and

sea-son. The ocean model's vertical di usion coeÆcients depend on latitude and have

been adjusted tomatchobservations of vertical mixingas determined fromtritium

observations (for a description, see Sokolov and Stone, 1998). From the tritium

measurements, the estimate of the globally averaged vertical di usion coeÆcient,

K v

, is 2.5 cm


/s (Hansenet al.,1997) with unspeci ed uncertainty. The timescale,

, for the response of the climate system to instantaneous forcings depends onthe


wellconstrainedbyobservationsanddi ers considerably fromonecoupledGCM to

another(SokolovandStone,1998). Thus, theadjustment oftheverticaldi usionof

heatprovidesaparameterforchangingthetimescaleof theclimatesystem response

independently of the sensitivity. The adjustment scales the meridional pro le of

the di usivitiessuchthat the globalmean di usivitychanges whilethe distribution

with latituderemains xed.

The ocean model includes the application of a Q- ux to account for horizontal

heat transports in the ocean. Thus, an imposed meridional heat ux is applied

in the ocean mixed layer. This Q- ux is calculated from an equilibrium climate

simulationusing present-day SST and sea-ice distributionsand itvaries seasonally.

The Q- ux is imposed in all runs discussed herein. Thus, the ocean model is not

purelydi usivebutcontainsa xedtransportcomponent. Theimposedheat uxin

conjunctionwith the vertical di usionofheat representsalloceaniccirculations. In

additiontheQ- uximplicitlycompensatesforanyerrorsintheatmosphericmodel's

physics. TheQ- uxfollowsstandardpracticeintheclimatemodelingcommunityin

that it is based on current climate conditions because the appropriate equilibrium

conditionsremainunknown forinitializingsimulationsstartinginthepre-industrial

era. The modelalsocontains aninteractivethermodynamicsea-icemodel, likethat

in the GISSGCM (Hansenet al.,1984).

In summary, the MIT model has two global parameters that determine the rate

and magnitude of the decadal-to-century timescale response of the climate system

to an external forcing. These are the climate sensitivity (S) to a doubling of CO


concentrations and the global mean vertical thermal di usivity (K


) of the deep

ocean. Sokolov and Stone (1998) have shown that the global response of a given

3D AOGCM canbe duplicatedby theMIT 2Dmodelwithanappropriatechoice of

thesetwoparametersforany forcing(Figure1). Given thesematchingparameters, Fig. 1

we can then estimate the response of the 3D AOGCMs under identical forcings by


Publishedvaluesof3DAGCMmodelsensitivitiesrangefrom1.9 to5.2 C(Senior

and Mitchell, 1993). Comparisons between timeseries of transient climate changes

calculatedwiththe MITmodelandwith3D AOGCMsshowthatthe GCM's

equiv-alentvertical di usivities range from0.0to 25.0 cm


/s (see Table 1). We intend to Table 1


with observations. Provided the model response over some range of these

param-eters di ers signi cantly from the observations, we will be able to discard certain

parameter choices as being unlikely based onthe model-datagoodness-of- t.

2.2 Optimal ngerprint detection statistics

The method for comparing the climate model data to the observational record is

derived fromthe optimal ngerprint detectionalgorithm(e.g.Allenand Tett, 1999;

Hasselmann, 1979, 1993, 1997; Hegerl and North, 1997). The technique requires

choosing a patternof climatechange to be identi ed in the climate record and

de-signing anoptimal lter to maximize the signal-to-noise ratio inthe corresponding

pattern amplitude. The optimization procedure requires an a priori estimate of

noise expected inthe chosen ngerprint. Thus, weattempt tominimizethe impact

of uctuations that are not associated with the signal we are trying to detect. In

a perfect world, we would use an estimate of the noise based on observations.

Be-cause available data donot permit this, noise characteristics are estimated froman

equilibrium control run of a climate model that simulates the natural uctuations

of the climate system as represented by the model.


^ C N



forwhichextendedcontrolintegrationsare readilyavailable(Tett etal.,1997). The

internalvariabilityofthe verticalpatterns oftemperaturechange fortimescalesless

than 20 years has been examined and shown to be similar to estimates from

ra-diosonde observations(Gillettetal.,2000). Currently,noobservations areavailable


variabil-ity contributes to that on longer timescales, C N

represents the natural variability

in any predicted pattern, and is determined by the variability in the corresponding

pattern amplitude in successive segments of \pseudo-observations" extracted from

the controlrun for the climatemodel. The MIT 2D ClimateModel underestimates

the noise covariance matrix (Sokolov and Stone, 1998). Estimates of the MIT 2D

model's variability given in Sokolov and Stone (1998) were obtained from200-year

simulations. Variability estimates based on 1000-year simulations are somewhat

larger, but stillsmaller than those fromAOGCMs.

Next, we present the optimal ngerprint method followed by the modi cations

necessary for this study. We have a vector of observations, y, over a given period

and a corresponding simulation (or ensemble) by the climate model, denoted by

x. We wish to determine the amplitude of the pattern, x, that best explains the

observations, y. This isdone using a multivariate regression algorithm:

y= m X i=1 i x i +u (1) where i

isthe amplitude ofthe i


model-predictedpattern, x


,and u isthe noise

term and represents an additive internal climate variability as well as unmodeled

components of the climate change. Here, x


can represent either multiple model

responses to individual forcings or the single response to multiple forcings. The

pattern amplitudes,


, represent the amount by which we have to scale the i-th

model-predictedsignal toreproduce the observations, assumingnoprior knowledge

of the amplitude of any of these signals and taking into account the noise

charac-teristics of the internal variability.

The pattern amplitudes,


, are determined from the standard multivariate

re-gression algorithm: ~ =(x T C 1 N x) 1 x T C 1 N y (2) whereC N =E(uu T

). The(~)denotesanoptimalestimatorasopposedtoastandard


in i

are estimated by:

~ V( ~ )=F T 1 C N2 F 1 (3) whereF T 1 =(x T C 1 N 1 x) 1 x T C 1 N 1 . ^ C N 1 and ^ C N 2

are estimatedfromseparate portions

of the control run to remove \arti cial skill" in estimates of con dence intervals

for ~

as discussed in Allen and Tett (1999). The assumption that u is normally

distributedisveri edby checkingthatthe normalizedresidualsumofsquares isnot

inconsistent with a F


distribution with  m and  degrees of freedom (d.f.)

where  is the number of retained EOFs in the pseudo-inverse of C



later), m is the number of modelpatterns, and  is the degrees of freedom of

^ C N


The normalized residualsum of squares isgiven by:

r 2 = ~ u T ^ C 1 N ~ u( m)F  m; (4) where F  m;

is the F-distribution with  m and  d.f. A F


distribution is

the distribution resultingfromthe ratioof two 

2 distributions with  1 and  2 d.f.; a 2 

distributionresults fromadding the squares of  normally distributedrandom

variables with unit variance. In the limit of  ! 1, ( m)F





. As


~ V(


),wenotethatthe EOFsusedtoestimateu~and

^ C N


taken from di erent segments of the control runs to avoid \arti cial skill". (Also,

we havedropped the subscript on

^ C N

tosimplifythe notation. )


of C

1 N

. The pseudo-inverse isused and hence,a truncationof the numberof EOFs

used in the expansion must be made. (Recall that the pseudo-inverse is given by

C 1 N = E T  1

E where E is the matrix of eigenvectors of C


(aka EOFs) and 

is the associated diagonalmatrix of eigenvalues. The truncation occurs by setting

the high rank eigenvalues to zero.) Typically, this is an arbitrary task, but, by

performing an F-test on the residuals, a warning ag for over tting can be raised.

In the multivariate regression algorithm, an assumption is made that the residuals


regarding r such that we can quantify the probability of not being inconsistent

with a white-noise hypothesis. Provided we can estimate \the number of degrees

of freedom for the t" (Allen and Tett, 1999), we are able to place probability



whencompared againstanF-distribution. Wealsonote thatr

2 is

inversely relatedto typical goodness-of- t statistics,such that we are searching for

a minimum in r


toobtain a maximum inthe goodness-of- t.

The commoninterpretationof the truncation of EOFs is that weprojectthe

n-gerprint pattern onto the dominant modes of variability which contribute to the

pattern itself as determined from the external model of variability. Because C

1 N becomes E T  1 E we can substitute E T  1 2  1 2 E for C 1 N because  is adiagonal matrix. Thus, x T C 1 N


eigenvec-tor basis of C


. In addition, each new variable has been normalized to have unit

variance, which isequivalent toprewhitening the dataset. Thus, we write

x T C 1 N x=x T P T Px=(Px) T Px (5) where P =  1 2

E. If we recognize that the EOFs are the modes of variability

of the ngerprint pattern (and not the modes of climate variability from which

the EOFs were estimated), we can interpret the optimal detection procedure as

rst projecting both the ngerprint pattern and the observation pattern onto the

modesofvariabilityandretainingonlyastrictlyde nednumberofEOFs. Afterthe

projection is done, the regression coeÆcient, , isthen determined, afterwhichthe

normalizedresidualsarechecked againstawhite-noisehypothesis. Ifthe r



is found to be inconsistent with an F-distribution for +1 EOFs, but not for ,

then the truncation of EOFs is taken as .

Thescale factor, , iscalculated to provide the best t between the observations

andmodeldataaftereachdataset isrotatedontothe rstEOFs andnormalized.

In essence, 6= 1 provides an additional parameter to modify the climate model

output before the comparison to observations. Ifwe set = 1and compute r


, we


of the amplitude of the model predicted pattern. For the case of climate change

detection, the ability for to vary is essential, although = 1 implies a perfect

modelresponse. Inour case,theadditionaldegreeoffreedomisnot desiredbecause

wewish onlytoexaminethe modelresponse asafunctionof themodelparameters.

In all other aspects, the calculations remain xed. The projections onto the EOFs

andthe truncationnumberaredeterminedwhileallowing tovaryafterwhich is

set equal to1.0 and the r


is computed for the intercomparisons of modelresponse

for di erent values of model parameters (Summary shown inFigure 3). Fig. 3

Given that we can compute r


using the optimaldetection algorithm,the goalof

this work is tolocate the minimumr


asa function of the uncertain model

param-eters. Once we nd the minimum, we must ask how far can the model parameters

deviate from the locationof the minimum r


beforethe t with observations is no

longer consistent at some level of con dence. To do this, we use a property of the

estimated residuals: E(r 2 )=E(^u T ^ C 1 N ^ u)=u T C 1 N u+mF m; (6) where u and ^

u represent the true and estimated residuals, respectively. Thus, the

r 2

atanarbitrarylocationmust di erbymF


, foragivensigni cancelevel,tobe

considered di erent from the minimumr

2 value, or: r 2 mF m; (7)

This provides ameans for rejecting regionsof modelparameter space asproducing

a simulation that is inconsistent with the observational record. (Strictly, we are

rejectingmodel parameters(or simulations)that aresigni cantly di erentfromthe

best t simulation (where r


is minimized). If the best t simulation does not

providea signi cant match ( 6=0) with the observations, then the r


test would

not provide the necessary check against the observational record.)

Inallresults discussed herein,modelsimulationsare compared toupper-air


(Park-(1996). The radiosonde data are used to calculate annual mean, zonal mean

tem-perature elds on a latitude-height grid (e.g., see Figure 6 below). Because data

are unavailable for certain regions and times, a missing value mask is obtained for

both space and time. The MIT modeldata are regriddedto the observational grid,

and thensampledusingthe observationaldata masktocomputethe 1961-1980and

1986-1995 period means. Thus, the initial ngerprint patternis the latitude-height

patternof temperature changebetween the two periodmeans. The ngerprintand

observational data are identical tothose used in Tett etal. (1996) for simplicity in

comparing results. The choice of averaging periods was made to accomodate the

1958-1995 record of radiosonde data. Additionally, the averaging period excludes

the years followingmajor volcanic eruptions, 1963-4 and 1992, (see Allen and T

et-t, 1999) to reduce the e ect on the computed averages. This particular choice of

climate change diagnostic has direct bearing on the set of chosen anthropogenic

forcings and the design of the model simulationsthat are discussed next.

2.3 Forcings

For agiven set of modelparameters, we ran the model forced by changes in

green-house gas, sulfate aerosol, and ozone concentrations starting from equilibrium at

1860. Theforcings are appliedinthe same mannerasinHadCM2runs (Tett etal.,

1996) such that the radiative forcing by the combined greenhouse gases is

repre-sented as a change in an equivalent CO


concentration, the sulfate aerosols alter

the surfacealbedo,and ozonechangesare taken asdeviationsfromthe climatology.

Weuse the sameforcings as inthe GSO experimentsof the HadCM2 model

excep-t for the ozone data. We altered the ozone concentrations to match observations

(Figure 4, see Hansen et al., 1997, for description) rather than use parameterized Fig. 4

changes as done in Tett et al. (1996). Hereafter, we refer to the combined forcing

as the GSO forcing.

Theequivalent CO



instanta-calculations, F is converted to a temperature change, T o , using T o =F( 1:2 4:3 )

where a temperature change of 1.2


C corresponds to a radiative forcing change of

4.3 W/m 2 from a doubling of CO 2 . The CO 2

concentrations are then calculated




Theanthropogenicaerosolforcing isimposed forclear sky onlyconditionsby

ad-justingthesurfacealbedousingamultiplere ectingapproximationinthe boundary

layercontainingtheaerosols(Charlsonetal.,1991;Mitchelletal.,1995). (Aversion

of the MIT model with interactive chemistry (Wang et al., 1998) is available but

wasnotusedinordertogaincomputationaleÆciency.) Thesulfateaerosolloadings

(Figure4) are converted to analbedo changevia

R s = Æ(1 R s ) 2 sec( o ) (8) where R s

isthe surface albedo, isthe upward scattering parameter, and 

o isthe

solarzenithangle(notationfromMitchelletal.(1995)). Theloadingcontributionis

Æ = B

SO 2 4

where isthemassscatteringcoeÆcient[m

2 /g]andB SO 2 4 isthesulfate loading burden [g/m 2

] (see Mitchell et al., 1995). A key point is that the albedo

changeisdependentonthe underlyingsurfacealbedosuchthatthe e ectsoverland

and sea are di erent. Because the ocean albedo issmaller,the e ect of the aerosols

ispotentiallygreater overthe sea surface. The response toaerosolforcings remains

uncertainsuchthatneither norB

SO 2 4

arewellknown (Houghton etal.,1996). To

be consistent with the HadCM2 simulations we use = 8.0 m


/g (Mitchell et al.,

1995)whichresultsinanannualmeanradiativeforcingof -0.5W/m


in1986. This



forHadCM2. AlsofollowingMitchelletal.(1995),weuse


the aerosolloadingpatterncalculated forthe 1986emissionsestimate. Wecompute

zonalaveragesoverland andsea surfacestoobtaintheaerosolloadingasafunction

of latitude and surface type. These latitude pro les are then scaled by the annual


emissionsshould be a reasonable proxy for scalingthe sulfate loadingpattern.

2.4 Model experiments

The model experiments are designed to simulate the climate response to the

pre-scribedGSO forcingwith aparticularchoice ofmodelparameterswhichare proxies

for properties of the climate system. We created a four-member ensemble

experi-ment for each parameter setting following Tett et al. (1996) to reduce noise in the

predicted response to the climate forcings resulting from internal variability of the

model. The model runs were started in 1860 with the prescribed forcing and run

to 1995. At the beginning of 1940, the model was stopped and the model state

perturbed by adding deviations from a control run to create the additional

mem-bers of the ensemble. The model perturbations were created as outputs every 10

years from a present-day climate control run. The deviations are calculated with

respect to the climatology of the control. In this manner, three perturbation runs

were started in 1940 for a total of four runs with the given forcing and parameter

settings. We recognizethat four-memberensembles are perhaps smalland that the

model's eÆciency would allowlarger ensemble sizes. However, wechose tocreate a

large ensembleof runs atdi erent parameterchoices rather than explorethe use of

a largerensemblesize at agiven choice of modelparameters.

Given the range of model parameters that correspond with 3D AOGCMs (see

Table 1 and Figure 1), we chose to vary S in the range from 0.4 to 11.0 K and K




/s(seeFigure2). WenotethattherangeofSencompassesthat

oftheIPCC(Houghtonetal.,1996)citedinKattenberg etal.(1996). Overall,these


models(seeTable1)aswellasshowingsigni cantvariationsinmodelresponse. For

example, if we examine the decadal-mean global-mean surface temperature at the

time of CO


doublingfor a simulationin which the modelis forced by a 1.0%/year

rise inCO



We ran the model to obtain the model response to the GSO forcing as it varies

within the 2D model parameter space. In all, we ran 61 experiments at di erent

modelparametervalues and 4ensemblemembers foreachparametersetting,which

corresponds to 18,544 years of simulated time. Having generated the model data,

we calculated the ngerprintdetection and the r



3 Results

This section will be split into several parts with the intent to give the reader: (1)

a sense that the model is able to produce results similar to 3D AOGCM

experi-ments,(2)an interpretationof how the optimal ngerprint ltersthe data, and (3)

an application of the method to a 35 year record of radiosonde data and model


3.1 Model Response

Given simulations of zonally averaged climate for 1860{1995, we can calculate two


result-s. The rst isthe horizontallyaveragedverticalpro leoftemperaturetrendsforthe

1979{1995period(Figure5). Theimportanceofthecorrectozoneconcentrationsfor Fig. 5

this periodisclearlyshown by the lackofa strongcoolingtrendinthe stratosphere

for runs which do not include the ozone concentration changes. The results shown

are for values of K


=2.5 cm


/s and S=3.0 K and the relevant permutations of the

combined forcings. As noted in Hansen et al. (1997) and Bengtsson et al. (1999),


in providing an explanation of the trends in satellite derived estimates of similar

trends over the 1979-1995 period. Although this 1D pattern of change is not used

in ngerprintstudies,it doesshowthe necessity fora complete set offorcings when


zonal mean temperaturedi erences for the 1961-1980to1986-1995periods(Figure

6). Forthe GSOforcing scenario, Figure6shows the MITmodelresponsefor three Fig. 6


Figure 2). Qualitatively, the pattern of zonal mean temperature di erences shows

a warmer troposphere and a cooler stratosphere overall with regions of enhanced

warming in the tropical upper troposphere and near the surface in polar regions.

These features are consistent with theoretical expectations of changes due to GHG

forcingplusfeedbacksassociatedwithwater vapor,lapserateadjustments,andhigh



simulation by the MIT model. In Figure 7, the relative amplitudes of the EOFs Fig. 7

are shown for the MIT and HadCM2 GSO simulations with equivalent changes in

ozone concentrations. The amplitudes have a correlation of 0.60 and 0.83, for the

optimizedand non-optimizedcases respectively. Thisfurther signi esthatthe MIT

modeliscapableofreproducing3DAOGCMresults. Inthenon-optimizedcase,the

amplitudes of the rst EOF in each model are roughly -1.3. We also note that the

e ect of applying the optimization increases the relative amplitudes of the EOFs

whichwillbediscussed later.

Here, wenotetwofeaturesof the GSOresponse(Figure6)thatwillbeaddressed

later. First,thestratosphericcoolingisrelativelyconstantinthethreecases shown.

We would expect this because the cooling associated with ozone losses should be

independentofboth theclimatesensitivityand thedi usivemixingofheat intothe

deep ocean. The total stratospheric cooling should be only slightly dependent on

sensitivity because the greenhouse gas response shows only a small cooling in the

upper stratosphere which is also almost independent of S and K


. This invariant

featureofthe patternhasimplicationsfordesigning ngerprintdiagnosticsthatcan

be used toconstrain modelparameters. A second notablefeature isthe qualitative


signi cantfractionofseaicemeltsandthereducedalbedoenhancesfurtherwarming.

This willbeimportantfor further discussions of the feedback mechanisms.

3.2 Filtered Response

The optimal ngerprint algorithm as described earlier is equivalent to deriving an

optimal lter to compare model simulations and observations. To illustrate the

lteringprocess, weshowthe observationaldata afteritis lteredby the optimized

and non-optimized lters (Figure8). (The non-optimal lter is that which projects Fig. 8

the data onto the rst  EOFs but does not weight the EOF amplitudes by the

inverse of the noise, i.e. the singular values.) The optimal lteremphasizes climate

changes associated with what appears as shifts in the Hadley Circulation. The

non-optimal lter emphasizes a broad region of warming over most of the earth

with two main areas in the upper troposphere in northern mid-latitudes and in

the tropics. The e ect of weighting the EOF pattern by its associated variance

gives moreweight tothe high rankEOFs with lowvariance. When weconsider the

patterns andvarianceofthehighrankEOFs(Figure9and10), thebanded features Fig. 9

Fig. 10

of EOFs 11{14 dominate the optimized ngerprint pattern (Figure8).

As others have noted (Allen and Tett, 1999; Hegerl and North, 1997), the use of

highrankEOFsishighlysuspect ifthe climatenoisedata isinsuÆcienttocorrectly

samplethese modes of variability. Whilewe donot showthe climatenoise variance

explainedby EOFs, the cumulativevariance of theobservationaldata explainedby

the EOFs (Figure 10) shows that the rst two EOFs explain 77% of the spatial

variance in the observational data and the remaining12EOFs explain another6%

ofthevariance. Clearly,arelativelysmallamountofinformationiscontainedinthe



Wenow present results for comparing the modelresponse toGSO forcing with the

radiosonde observations. In this rst case, we compare the modelresponse to GSO

forcingsfordi erentclimatesensitivitiesandholdK



2 /s

(Figure11). Alsoweremindthereaderthatwecomparethemodelresponsedirectly Fig. 11

with observations without additional scaling by (i.e., = 1 ). In these gures,

two types of comparisons are made against the (1) observations and (2)

pseudo-observations. We use pseudo-observations asgiven by the modelresponse ata best

t location. (For each value of K


, a best t location is chosen by locating the

minimum r


when a comparison is made against observations. In this case, the

comparison of the model response with itself willyield r


=0. Thus, the r



of the \perfect model"must bein atedsuch that the minimum value matches that

fromthecomparison withobservations.) InFigure11,we comparetwo methodsfor

estimating con dence intervals for climate sensitivity, S. The rst is based on the

likelihood of r


being di erent from the minimum value owing to random chance

(Eq. 7) indicated by the dashed horizontal lines (80th and 95th percentiles are

shown). Provided r


is less than this value, the modelresponse cannot be rejected

asbeingdi erentfromtheobservations. Asecondmethodforestimatingcon dence

intervals is to use Equation (3). Given the estimates of

~ and ~ V( ~ ), we apply this

scale-factortothe model's sensitivity and use the associatedvariancetoprovidean

optimal estimateof the climate sensitivity, (

~ S = ~ S ~

S, where S is the model's

climatesensitivity). Thecon denceintervalsfromthis secondmethodareshownby

ahorizontalbarthrough the minimum r


and represent a2-limit(95%con dence

interval). In the case of a 3D model where the sensitivity is xed, this is the only

available estimate for the uncertainty in the climate sensitivity based on available

statistics. As expected, r


at the minimum matches the value for the comparison

againstobservationsbecausewecomparethe modelpatternwithitselfandobtaina


this method suÆces. ] For K v

= 2.5 cm


/s, the 80% con dence interval for S is

1.7 to3.3K. The 95%con dence intervalis 1.2 to3.8K.ForK


= 40.0 cm



80% con dence interval for S has a lower bound at 1.5 K but no upper bound for

the region spanned by our simulations. These estimates can be compared with the

95% con dence intervals based on 


which are 0.6 to 3.8 K and 0.2 to 3.4 K for

the K v =2.5 cm 2 /sand 40cm 2 /s, respectively. Using ~ V( ~

) as indicated by the horizontalbar through the minimum r


-value in

Figure11,we ndabiasintheuncertaintyregiontowardslowersensitivitiesbutthis

is partially a result of a model's over or under prediction of the observations. The

comparison of uncertainty bounds alsoindicates that the statistics for the variance

of S may not have a normal distribution which results from the transfer function

from to S not having a linear form. The direct application of V(


) to estimate

the V(S)doesnotappear tobevalid. This pointhas been examinedbyAllenetal.

(1999)via anenergy balancemodelcalculationfor globalmean temperature. Here,

we make use of the full detection methodology to make a similar point. We also

showaperfectmodelcomparison(Figure11)inwhichthe observationsarereplaced

with the modelresponse atthe minimum locationfor agiven value of K


. We note

thatwhenthe r


valuesforthemodeltopseudo-observation comparisonaresmaller

than forthe model toobservation comparison,this impliesthat the modelresponse


of S.

3.4 GSO Response - Two Free Parameters

We further examine the set of GSO simulations by varying K


, in addition to S,

and create distributions of r


and in two dimensions for the comparison of the

modelresponse withthe pseudo-observations (Figure12) and the trueobservations Fig. 12

(Figure13). From the twith pseudo-observations(Figure12), the modelresponse

Fig. 13


parameters move into the \low" or \high response" regions the r values correctly

showthatthe modelresponsesare signi cantlydi erentasFigure6suggests. Inthe

model-observation t (Figure 13), we see a similar distribution of r


but multiple

minima indicate that di erences in the predicted response contributes to noisiness

in the distribution of r


. We expect that larger ensemble sizes would reduce this




distribution. Thethicklines andshadingrepresentthe boundariesforthe80-th,

95-th,and99-thpercentilecon denceregionsandindicatethatonlythe\highresponse"

region can be rejected at the 1% level of signi cance. A second region of \low

response" can be rejected at the 5% level of signi cance. We note that using this

pattern of temperature change, only the NCAR W&M model is rejected by this

method. All other 3D AOGCMs (see Table 1 and Figure 2) are within the 95%

con dence region of acceptable parameters. We note that the response to future

climate forcings of these remaining AOGCMs does not represent the full range of

possible predictions which are consistent with the observations.

The distribution of (Figure 13) indicates the relative magnitude of the model

response as compared against observations. The contour for = 1indicates where

themagnitudeofthe lteredmodelresponsematchesthatofthe observations. This

impliesthat variationsof r


along this lineshouldrepresent di erences inthe

mod-eled pattern of change. The lack of signi cant di erences in r


along the = 1

curvesuggests that di erences inthemodelresponse are smallforabroad regionof

modelparameters based onthis temperaturechange diagnostic.

4 Discussion

4.1 Implicationsfor uncertainty estimates of future climate

The ability to estimate uncertainty in predictions of future climate is strongly

de-pendent onbeing able to vary uncertain physical parameters that a ect the future



ty distribution of the parameters. This identi es a signi cant limitation of 3D

AOGCMs in which the model structure remains xed. Thus, in order to reduce

uncertainty in either S or K


, more exible 3D climate models would be desirable.

Unfortunately, the majority, if not all, of climate models are structurally xed and

their climate change properties are xed as well. To adjust for this shortcoming,

s-tudiessuchas thisone must beused toassess theprobability distributionfunctions

(pdf) of the response. We note that the GISS model, parent of the MIT model,

did have a similar adjustment for the cloud feedback and therefore can produce

simulationswith di erent climatesensitivities(Hansen et al.,1993).

As discussed in Forest et al. (2000), these results provide uncertainty estimates

by placingboundsonthejointprobabilitiesforSand K


whichcanthen beusedto

estimate limits on future climate change when used in conjunction with estimates

of forcing scenarios asgiven by changesin greenhousegas concentrations oraerosol

loadings. As an example of such bounds we refer back to Figure 2 and compare

it with Figure 13. These results imply that a 5% chance exists that the warming

under a 1%/year compounded increase in CO


concentrations will will lie outside

0.5to2.9K.These resultscan becompared forthis particularscenariowith results

from the second phase of the Coupled Model Intercomparison Project (CMIP2)

(Meehl et al., 2000) in which the identical simulation is being carried out for a

suite of 3D AOGCMs. These joint probability distributions provide a context for

intercomparing models and also allows one to calculate uncertainty estimates for

simulationsby any single AOGCM.

Among many possible uses of these pdfs, we can assign a quantitive probability

statementto theoft quoted uncertainty inclimatesensitivity,1.5{4.5K, (Houghton

etal.,1996). Asbrie ydiscussedinForestetal.(2000),theIPCCrangecorresponds

toroughlyan80%con denceintervalwhichisobtainedfromthepdfforSevaluated

at K v = 5.0 cm 2 /s, the value of K v

that matches the upwelling di usion/energy


the non-uniform likelihoodsfor being outside this interval. There is a 13% chance

that Swill beabove 4.5K and a7% chancethat S willbe less than 1.5K.Wealso

note thatthe upperboundonSincreases rapidly asK


isincreased and thus, these

estimates of uncertainty for S are less robust than thosefor the climate predictions

where the isolines of model response are roughly parallel to the isolines of r


. This

particular example highlights the need for exploring joint probability distributions

aswellasidentifying,through further use ofobservations,wherethe \true"climate

system liesin such aparameter space. Doublingor halving the value of K


used in

this exercise willcertainly alter the implications.

4.2 Dependency of temperaturechange pattern onS orK


One goal of this work is to provide quantitative probabilistic bounds on S and

K v

. We also desire to identify means to further tighten these constraints. At rst

glance, we can either use additional datasets or improve the use of existing data.

To further exploit existing data, we must focus on the patterns of climate change

that are most dependent on climate sensitivity or ocean heat uptake. A simple

methodforidentifyingsuchfeaturesistotaketheset ofsimulations(orsubset)and

estimate two patterns of temperature change, one that is invariant to changes in

S or K


and a second that varies linearly with S or K


. We present the invariant

pattern and the linearly varying pattern for model simulations with K


= 2.5and

40.0 cm


/s (Figure 14) and with S = 1.6, 3.0 and 4.5 K (Figure 15). Comparing Fig. 14

Fig. 15

the invariantpatterns withthe fullresponse(Figure6),wesee that certainfeatures

of the modelresponse are not useful for constraining S. For example, temperature

changes in the stratosphere are invariant with S. The linearly varying component

indicatesthatchangesinthe tropicaluppertroposphere andatthe surfacenearthe

sea-ice edges are most dependent on climate sensitivity. Thus, a ngerprint that

focuses on these regions would best discriminate between models with di erent S.


change must be observed to be signi cant. In all, this exercise demonstrates the

need for a better understanding of how the ngerprint pattern depends on climate

systempropertiesinadditiontoitsdependenceonthe variousforcings. Thistypeof

analysis wouldlead toalternativechoices of ngerprintpatterns which aredesigned

toreduce uncertainty in model parameters.

Aseparate issueishowthese resultsa ect theassumption oflinearsuperposition

of ngerprint patterns in the detection and attribution algorithm. We note that

the variations of T associated with variations in S or K


can be of the same

magnitude as the overall temperature changes. These variations combined with

the threshhold type behavior noted earlier suggests that simple combinations of

xed patterns may be inappropriate. Therefore, perhaps we should examine more

closely the assumption that one can scale patterns of climate change to predict

future climate change and also determine the extent to which uncertainty in these

predictions can be estimated fromdetection diagnostics.

4.3 Dependency of detection statistics on

In addition to exploring the dependency of the diagnostic on properties of the

cli-mate system, wecan examinethe dependency of the detection statisticson

trunca-tion number. As we have already shown in Figure13, the distribution of



our expectations by following a typicalresponse surface asshown inFigure 2. The

regionsof high (low) response require

~ <1(


>1) toreduce (increase)the model

predictedresponseand matchthe observations. The wellde ned



pass-ing through the acceptable region of parameter space adds a consistency check for

the methodology. Hence, the distributionof


isconsistent withour understanding

of the climate change detection methodology. As part of our analysis, we varied

the truncation number, , toexplore how r


varied. Because the lowerorder EOFs

represent the largest spatial scales, we wished to explore the possibility that small



distribution. Infact,ther



robust to changes in  (Figure 16). We also explored the distribution of in the Fig. 16

same fashion and found two strangee ects. First, the positionof the


=1 isoline

shifts dramaticallyas  varies. Second, the values of


for S <1 begin todecrease

for all<13. For 13,the distribution of


remainsfairly robust.

Thisbehaviorhasbearingontwoissuesofthedetectionproblem. First,thechoice

of remainsatopicof debatebecausethe levelof signi canceforthe detectionand


determinedfromthecontrolrun (HadCM2,inthiscase.) Ifamodelunderestimates

the noise ina particular EOF, the possibility of a false detection of climate change

increases. Ifeithertheobservations orthe modelprojectontothispattern, thenthe

increased weighting (through

^ C

1 N




uncertainty in

~ .

The second issue is that the value of


guides the climate modeling community

astowhether themodelunder-orover-predictsthe responsetothe appliedforcing.

Whenthe diagnosticissimplyglobalmeantemperaturechange,thishardlymatters

because the one can easily view whether the model reproduces the observational

record. When the complexity of the optimal lter is considered, the comparison

becomes less intuitive. In fact, it appears likely that particular regions, for

ex-ample, high latitudes near the surface, could be over- or under-weighted and thus

inappropriatelycontribute to

~ .

4.4 Remainingissues

This paper wasintended to discuss issues that were left unexplored in Forest etal.

(2000) due to limited space. Of those not discussed previously, two issues remain

thata ectthediscussionoftheuncertaintiesandtheirimplications: (1)dependency

of the climate variability on the climate model properties and (2) the uncertainty

in the applied radiative forcing. In Forest et al. (2000), we acknowledge that we

assume independence of the noise model, C


, and the properties of the climate

system being varied (S and K



sensitivity and variabilityshould alsoexist (e.g., Hall and Manabe, 1999; Heldand

Soden, 2000). Additionally,theoceanisknownto a ectlong-termclimate

variabil-ity through various mechanisms such as ENSO, NAO, or AO. This suggests that

estimates of variability based on the HadCM2 control run may not be appropriate

for all values of S and K


. Preliminary results using the MIT model indicate that

interannual variability depends on the model parameters. To apply these results

to the ngerprint statistics which use the HadCM2 control run, we require a

com-pleteanalysis of theMIT model's modesof variabilityand theirrelationtothose of

HadCM2. Ifthese prove tobe reasonablysimilar,wewillbeable to scale the noise

estimatefrom the HadCM2 model.

Thesecondissueistheuncertaintyintheradiativeforcingsspeci edforthe

twen-tieth century. In these results, wehave kept the aerosol forcing xed at-0.5 W/m


forthe1986 distributionofaerosols. However, theuncertainty inthe aerosolforcing

ranges from 0 to -2.5 W/m


if the possible indirect e ects are included (Houghton

et al., 1996). We note that the forcing, as modeled by a change in surface albedo,

representsthenet aerosolforcing andincludesthee ectsof allprocessesthatwould

have aradiative forcing with a spatial distribution similarto that produced by the

sulfate aerosols. We have not considered the anthropogenicforcing by tropospheric

ozone which may o set the cooling by sulfate aerosols. We also neglect natural

forcings suchas variationsin solar and volcanic activity. However, asnoted inTett

etal.(1996),the data averagingprocedurewaschosen tominimizethe e ect ofthe

volcanicforcingsontheclimatechange signalbyremovingdatainyears 1963-4and

1992 whichcorrespond toeruptions of Mt. Agung and Mt. Pinatubo,respectively.

5 Conclusions

We have presented a method for constraining model properties that is based on

the distribution of the r



physical processes that control the climate system's response to external forcings.

Using patterns of climatechange estimated from the record of radiosonde

observa-tions, we can place constraints on the model properties in a rigorousfashion. Two

types of constraintscan be estimated. A 1D constraint can be estimated assuming

the remaininguncertain parametersare known a priori. Ifthis isnot possible, then

constraintsontwo ormoreparameters may be estimated,however, the uncertainty

regionnecessarilyincreases aswell. Asappliedinthis work,ifthe \true"K



climatesystemis known, thencon denceintervalswithassociatedprobabilitiescan

be placedonS.If neitherSnor K


are known a priori, then the con denceinterval

is transformed into an uncertainty region and two regions of parameter space can

be rejected with agiven levelof signi cance.

Theseresultsindicatethat thechoiceofdiagnostichas importantimplicationson

which model parameters can be constrained well. The use of surface or upper air

data alone does not provide suÆcient information to place strong constraints. An

approachusingacombinationofeachdatasetmayhelp. Alternatively,comparisons


statisticaltechniquestoproducemaximumlikelihoodconstraints. Estimatesofprior



wouldbetaken fromstudiessuchasWebsterandSokolov


Asdiscussed earlier,the transient response ofthe MIT modeltoincreased

radia-tive forcings can match the response of 3D AOGCMs for a unique S and K

v for

any forcing. Thus, quantitativeboundson Sand K


providea means for

quantita-tively comparing the predictive capabilities of 3D AOGCMs under similar forcings

(see Figure 2). As emphasized in Forest et al. (2000), these results only reject the

NCAR W&Mmodelas being inconsistent withthe radiosonde observations forthe

period examined. However, these results also fail to make signi cant

distinction-s among the remaining models and therefore suggest that results from individual


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Acknowledgements. We wishto thankC.Wangforincorporatingtheozoneconcentration

changes into the model;M. Webster fordata in Figure 2; the UK Natural Environment

Research Council (MRA) for support; C. Senior, S. Tett, T. Johns, and J. Mitchell for

providingdataandinformationregardingtheHadCM2 model;J.HansenandM.Satofor

providingtheGISSozoneconcentrationdata. ThecontrolrunofHadCM2wasfundedby


Table 1. Model Parameters fortheMIT2DClimateModelwhichcorrespondto various coupledAOGCMs. Model S K v GFDL 3.7 5.0 ECHAM1/LSG 2.6 25.0 NCAR W&M 4.6 0.0 HadCM2 2.5  7.5 NCAR CSM 2.1  7.5 

These sensitivities were obtained through personal communications with theindicated








Years from start of simulation






Degrees Centigrade





Fig.1. Annual-meanglobal-mean surfacetemperatures forvariouscoupledclimate

mod-els (solid) and the matching MITmodel (dashed) under identical forcings. The forcings


0.00 1.00 4.00 9.00 16.0 25.0 36.0 49.0 64.0












Climate Sensitivity [K]




]) for 1%/yr CO




















Fig. 2. Responseintheglobalmeansurfacetemperatureat thetimeof doublingofCO




concentration. ThecorrespondingSandK



Fig. 1. Annual-mean global-mean surface temperatures for various coupled climate mod-
Fig. 2. Response in the global mean surface temperature at the time of doubling of CO
Fig. 3. Summary of optimal detection algorithm (left) and contrast with direct estimation
Fig. 4. Summary of applied model forcings in the MIT - GSO experiments. (A) E-


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