Diagnosing Cloud Feedbacks in General Circulation Models6774/datastream/P… · Diagnosing Cloud...

Diagnosing Cloud Feedbacks in General Circulation Models

PING ZHU

Department of Earth Sciences, Florida International University, Miami, Florida

JAMES J. HACK AND JEFFREY T. KIEHL

Climate and Global Dynamics Division, National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 2 November 2005, in final form 7 August 2006)

ABSTRACT

In this study, it is shown that the NCAR and GFDL GCMs exhibit a marked difference in climatesensitivity of clouds and radiative fluxes in response to doubled CO2 and �2-K SST perturbations. TheGFDL model predicted a substantial decrease in cloud amount and an increase in cloud condensate in thewarmer climate, but produced a much weaker change in net cloud radiative forcing (CRF) than the NCARmodel. Using a multiple linear regression (MLR) method, the full-sky radiative flux change at the top of theatmosphere was successfully decomposed into individual components associated with the clear sky anddifferent types of clouds. The authors specifically examined the cloud feedbacks due to the cloud amountand cloud condensate changes involving low, mid-, and high clouds between 60°S and 60°N. It was foundthat the NCAR and GFDL models predicted the same sign of individual longwave and shortwave feedbacksresulting from the change in cloud amount and cloud condensate for all three types of clouds (low, mid, andhigh) despite the different cloud and radiation schemes used in the models. However, since the individuallongwave and shortwave feedbacks resulting from the change in cloud amount and cloud condensategenerally have the opposite signs, the net cloud feedback is a subtle residual of all. Strong cancellationsbetween individual cloud feedbacks may result in a weak net cloud feedback. This result is consistent withthe findings of the previous studies, which used different approaches to diagnose cloud feedbacks. Thisstudy indicates that the proposed MLR approach provides an easy way to efficiently expose the similarityand discrepancy of individual cloud feedback processes between GCMs, which are hidden in the total cloudfeedback measured by CRF. Most importantly, this method has the potential to be applied to satellitemeasurements. Thus, it may serve as a reliable and efficient method to investigate cloud feedback mecha-nisms on short-term scales by comparing simulations with available observations, which may provide auseful way to identify the cause for the wide spread of cloud feedbacks in GCMs.

1. Introduction

It has long been recognized that the radiative effectsof clouds play an important role in regulating theearth’s energy budget and modulating anthropogenicclimate changes. Although cloud–climate feedbackshave been intensely studied in the past decades, thesign of net cloud feedback is still a matter of uncertaintyas concluded by the Third Assessment Report (TAR)of the Intergovernmental Panel on Climate Change(IPCC; Houghton et al. 2001). Understanding the

mechanisms of cloud–climate feedbacks and reducingthe uncertainties continue to be the toughest challengesin climate research.

Most of current knowledge of cloud–climate feed-backs is from three types of climate sensitivity experi-ments. The first two are the classic �2-K sea surfacetemperature (SST) perturbation experiments (e.g.,Cess and Potter 1988; Cess et al. 1990, 1996; Zhang etal. 1994) and doubling CO2 experiments (e.g., Wether-ald and Manabe 1988; Senior and Mitchell 1993; Wil-liams et al. 2003). While the former provides the sim-plest way to measure the strength of cloud feedback,the latter further includes the complexity arising from aresponsive ocean. While these experiments allow inter-nally varied cloud fields in response to external forc-ings, the third type of experiments attempts to isolate

Corresponding author address: Ping Zhu, Department of EarthSciences, Florida International University, MARC 360, 11200 SW8th St., Miami, FL 33199.E-mail: [email protected]

2602 J O U R N A L O F C L I M A T E VOLUME 20

DOI: 10.1175/JCLI4140.1

© 2007 American Meteorological Society

JCLI4140

various feedback mechanisms by specifying cloud fieldsor cloud properties in GCM simulations (e.g., Taylorand Ghan 1992; Schneider et al. 1999).

Quantitatively climate feedback is measured by afeedback parameter defined as the ratio of the changein global mean net radiative flux at the top of the at-mosphere (TOA) R to the change in global mean sur-face temperature Ts [i.e., � � �(�R/�Ts); e.g., Wether-ald and Manabe 1988; Soden et al. 2004]. To diagnosethe feedback induced by clouds and calculate the cloudfeedback parameter, corresponding to the various cli-mate sensitivity experiments stated previously, differ-ent approaches have been developed. The partial ra-diative perturbation (PRP) approach pioneered byWetherald and Manabe (1988) allows a quantificationof all the possible feedback processes operating in asimulated climate system by explicitly computing thefeedback parameter �x � �(�R/�x)(�x/�Ts) using off-line radiative transfer calculations, where x denotes anarbitrary feedback process, such that � � �x�x repre-sents the total feedback. Although extremely powerful,this approach is computationally expensive and the re-sultant feedbacks may depend on the sampling fre-quency of the input atmospheric state and cloud fields.The biggest disadvantage of this approach is that it can-not be readily applied to observations, since to calcu-late a partial derivative only one variable is allowed tovary while the rest need to be fixed, which is easy to dofor model output but is impossible for observationaldata. Therefore, even though the PRP approach maysuccessfully identify the intermodel differences or thesource of uncertainties, it lacks the ability to furtherdetermine which is the right feedback and which mightbe the unrealistic one since the feedbacks in GCMsdiagnosed by the PRP approach cannot be verified byobservations.

Compared with the complicated PRP approach, theCess cloud radiative forcing (CRF) methodology (Cessand Potter 1988; Cess et al. 1990, 1996) provides themost straightforward measure of the contributions ofclouds to the earth’s radiation budget. Defining CRF asfcrf � Rclear � R, where Rclear is the clear-sky radiativeflux at TOA, then, the cloud feedback parameter �crf ,can be computed as �crf � (�fcrf /�Ts). In addition to itssimplicity, the CRF approach can be easily applied tosatellite measurements, so that the simulated cloudfeedbacks can be directly compared to observations.This is a huge advantage over the PRP approach.

In a recent study, by applying these two methods to�2-K SST perturbation experiments executed by theGeophysical Fluid Dynamics Laboratory (GFDL) At-mospheric Model 2 (AM2), Soden et al. (2004) foundthat the sign of global-mean net cloud feedback diag-

nosed by the two methods can be different. They ar-gued that this discrepancy stems from the effect ofclouds in masking the noncloud feedbacks since theCRF approach cannot completely separate cloud feed-backs from other feedbacks, such as water vapor andtemperature feedbacks. The same argument was alsomade by Zhang et al. (1994). Based on their calcula-tions, Soden et al. (2004) warned that there is a poten-tial for the CRF approach to misinterpret a positivecloud feedback as a negative feedback. This findingsuggests that the results from CRF analyses need to beinterpreted with caution under certain circumstances.Leaving aside the possible “cloud masking” effect ofCRF, which may be considered as a small flaw on agem, the real limitation of CRF is that it lacks the abil-ity to determine the individual contributions that re-sulted from different cloud processes to the total cloudfeedback. Thus, the CRF approach provides little helpto identify the cause for the wide spread of cloud feed-backs among GCMs.

Over the past few decades, both the CRF and PRPapproaches have been widely applied to the analysis ofcloud feedbacks in GCMs, which helps us to recognizethe problems and gain some insight into the uncertain-ties of cloud feedbacks (e.g., Taylor and Ghan 1992;Zhang et al. 1994; Colman et al. 2001). However, theslow progress in reducing uncertainties associated withcloud feedbacks continues, which in part can be attrib-uted to the methodological difficulty in evaluatingcloud feedback processes (Bony et al. 2006), since thecurrent approaches either cannot provide detailed in-formation on various cloud feedback processes (e.g.,CRF) or generate something that cannot be directlyevaluated by observations (e.g., PRP). Over the pastcouple of decades, a great amount of satellite data ofclouds and radiation has been collected. These data canbe very useful for verifying model-simulated cloudfeedback processes if the observed change in total ra-diative fluxes at TOA can be decomposed into indi-vidual components associated with different cloud pro-cesses. Unfortunately, the current methods, either PRPor CRF, cannot fulfill this task.

The need to develop novel feedback analysis ap-proaches can be further demonstrated by the com-plexity of cloud feedbacks shown in the climate sensi-tivity tests performed by the GFDL and the NationalCenter for Atmospheric Research (NCAR) GCMs. Asshown in Fig. 1, the cloud properties and net CRFsimulated by the NCAR Community AtmosphereModel version 3 (CAM3) and the GFDL AM2 havedifferent responses to doubled CO2 and �2-K SST per-turbations at most latitudes. AM2 simulated a decreasein cloud amount and an increase in cloud condensate

1 JUNE 2007 Z H U E T A L . 2603

much stronger in magnitude than CAM3. The sub-stantial reduction of cloud amount in the AM2 simula-tions appears to be largely compensated by the markedincrease in cloud condensate, resulting in a net CRFchange much weaker than that of CAM3. To fully un-derstand the cause for this inconsistent response ofclouds and radiation to climate change forcings shownin these sensitivity experiments, it is useful to de-compose the change in total radiative fluxes into theindividual components associated with different typesof clouds (e.g., low, mid-, and high clouds) and differ-ent cloud properties (cloud amount and condensate). Inthis study, we propose a multiple linear regression(MLR) method for such a decomposition. The appro-priateness and power of MLR in exploring the relation-ship between radiative fluxes and clouds have beendemonstrated by Hartmann et al. (1992), who suc-cessfully decomposed the radiative fluxes observed bythe Earth Radiation Budget Experiment (ERBE) interms of the International Satellite Cloud ClimatologyProject (ISCCP) clouds using a linear stepwise regres-

sion. Here, the MLR approach is extended to analyzethe cloud feedbacks in the �2-K SST perturbationand doubling CO2 experiments by decomposing the to-tal cloud feedbacks into individual components. Al-though such a decomposition can also be done usingthe PRP approach, the main purpose to choose MLRis to test its performance in cloud feedback analysissince this method has a potential application to long-term satellite observations and the corresponding At-mospheric Model Intercomparison Project (AMIP)simulations by GCMs. If successful, the MLR approachcan serve as a routine powerful supplement to CRFsince it is easy to execute and yet retains the basicpower of the PRP approach to quantify individualcloud feedback processes, an ability that the CRF ap-proach lacks.

The paper is organized as follows. Section 2 describesthe procedure of using MLR to diagnose feedback pro-cesses in the climate system and briefly reviews theclimate sensitivity experiments used in this study. Sec-tion 3 presents the diagnosed individual cloud feed-

FIG. 1. Zonally averaged change in annual mean cloud amount, CWP, and net CRF in response to doubled CO2

and �2-K SST perturbations simulated by CAM3 and AM2. In the doubling CO2 experiments, the two models arecoupled to an SOM.


backs using the proposed method. A summary and dis-cussion are presented in section 4.

2. Methodology

a. Multiple linear regression

The PRP approach is based on the assumption thatthe total change in the radiative budget can be brokendown linearly into individual components (feedbacks).Considering the change in net radiative flux at TOA�R � �F � �Q due to climate change, according to thelinear assumption, �R can be expressed as

�R � �x1R � �x2

R � · · · � �xiR � · · ·

��R

�x1�x1

��R

�x2�x2

� · · · ��R

�xi�xi

� · · ·,

xi ∈ Ts, Ta, qa, c, . . ., 1

where, F and Q are the upward longwave and netdownward shortwave radiative fluxes. Here xi is a ge-neric variable representing surface temperature Ts, at-mospheric temperature Ta, specific humidity qa, cloudsc, or any other quantity that may have a feedback onthe climate system. The validity of the linear feedbackassumption at the climate scale has been supported byprevious studies (e.g., Wetherald and Manabe 1988;Zhang et al. 1994; Colman and McAvaney 1997) usingoffline radiative transfer calculations. The robustness ofthe linear relationship between total and individualfeedbacks allows us to diagnose cloud feedbacks usingthe MLR approach. Considering the partial derivativesas the coefficients of a regression equation, Eq. (1) canbe further written as

�R � �i�1

m

ci�xi � re ; ci ��R

�xi, 2

where m indicates the number of factors that may con-tribute significantly to the total feedback, ci representsthe feedback coefficient associated with a particularfeedback process, and re is the residual. The PRP ap-proach determines the partial derivatives using offlineradiative transfer calculations, whereas for MLR if onehas n sets of samples of Eq. (2), the feedback coeffi-cients C � [c1, c2, . . . , ci, . . . , cm] and the residual re canbe determined (see the appendix for details). There aretwo advantages of this method. First, it supports sepa-rating total cloud feedback in terms of cloud types (e.g.,low, mid-, and high clouds) and cloud properties (cloudamount and optical thickness), a similar feature to thePRP approach although they use different ways to ac-complish the task. Second, this method provides a self-evaluation of regression performance. The significance

of overall fitting regression and the significance of eachindependent feedback process in the linear model canbe evaluated by the multiple correlation coefficient�R, x1

, . . . , xm and t statistics ti, respectively (see theappendix for details).

The core of the MLR approach is to obtain thesamples of Eq. (2). Traditionally, a climate state is de-fined by the statistical mean over a sufficiently longperiod, say, 10 or 20 yr depending on the specific cli-mate sensitivity experiment. However, any shortertime-scale mean (e.g., monthly or yearly mean) withinthe time series, although containing noise, includes theinformation of feedbacks. Thus, each monthly or yearlymean of model output can be considered as a sample ofEq. (2). Assuming that one has 20-yr data of two dif-ferent climates for example, the present and doubledCO2 climates, if the yearly mean is used, then the dif-ference between the two climates yields 20 sets ofsamples of Eq. (2); if monthly mean is used, then thenumber of samples increases to 240. Note that in thelatter case, the samples may contain seasonal signals,but they have been much reduced since only the differ-ence, not the absolute value of each month, is consid-ered. We have tested the regression using both yearlyand monthly mean data. They basically produce similarresults, but the estimation error is smaller when themonthly mean is used, apparently due to more sets ofsamples. For this reason, the diagnosed feedbacksshown in the following sections are computed using themonthly mean data obtained from the climate sensitiv-ity experiments.

b. Climate sensitivity experiments

In this study, the proposed MLR methodology isused to analyze cloud feedbacks in the climate sensitiv-ity experiments performed by NCAR CAM3 andGFDL AM2. The detailed description of the relevantmodel physics and the simulated climatology in CAM3and AM2 are referred to Collins et al. (2004) and theGFDL Global Atmospheric Development Team(2004). Two types of experiments are analyzed in thisstudy. The first is the �2-K SST perturbation experi-ments. Unlike the classic Cess experiments, which arethe perpetual July simulations, here both models wereforced by the observed monthly mean climatologicalSST with seasonal variation uniformly added by a per-turbation of �2 K. AM2 was integrated for 9 yr for eachrun and the last 8-yr monthly mean data are used forthe analysis, which gives 96 sets of samples of Eq. (2),whereas CAM3 was executed for 10 yr, thus, 108 sets(last 9 yr) of samples are available for the analysis.These two experiments hereafter are named as CAM3–CESS and AM2–CESS, respectively. We would like to

1 JUNE 2007 Z H U E T A L . 2605

note that �2-K SST perturbation forcing is highly un-realistic. The main purpose of including this type ofexperiments in this study is to show that the differentcloud feedbacks between CAM3 and AM2 in coupledsimulations (see below) also occur in the fixed SST per-turbation experiments. This has an important implica-tion that the cloud feedback issue can be properly ac-cessible in an uncoupled atmospheric modeling system.More specifically, it means that we may use the AMIPsimulations to identify the possible cause of the differ-ent cloud feedbacks among GCMs if the diagnosedcloud feedbacks from simulations can be compared di-rectly with satellite observations.

The second type of climate sensitivity tests is the dou-bling CO2 experiment. In this experiment, the atmo-spheric model is coupled to a slab ocean model (SOM).In the control simulation, the CO2 concentration wasfixed to the present climate value. This value wasdoubled at the beginning of integration for the sensi-tivity simulations. Since the ocean mixed layer tem-perature is a prognostic variable when the SOM is en-abled, the model generally takes a much longer time toreach equilibrium than when SSTs are prescribed, so

that both models were integrated for 50 yr, hereafternamed as CAM3–2CO2 and AM2–2CO2, respectively.The last 20-yr monthly mean data are used for theanalysis. Thus, 240 sets of samples are available for theregression analysis. Figure 2 compares the ISCCP an-nual mean total cloud amount and vertically integratedcondensed water path [(CWP) i.e., liquid water path(LWP) � ice water path (IWP)] and the ERBE annualmean net CRF with those from the CAM3 and AM2control simulations. Overall, there is fairly good agree-ment in cloud amount and CWP between observationsand simulations in the Tropics and subtropics. How-ever, large model biases are shown in the mid- and highlatitudes. Both models significantly overestimate cloudamount beyond 70°. Serious positive biases in CWP arearound 60° in both hemispheres. Despite the large bi-ases in cloud properties, the simulated net CRF at TOAmatches the ERBE observations unexpectedly well atmost latitudes, especially for CAM3. The most likelyexplanation for this agreement is that the radiative bi-ases generated at different levels due to incorrect cloudproperties cancel each other to result in a radiativeforcing superficially right at TOA although fundamen-

FIG. 2. Zonally averaged annual mean total cloud amount, CWP, and net CRF. Dark lines indicate the valuesfrom the ISCCP and ERBE observations. Light and dashed lines are the results from the CAM3 and AM2 controlruns for both coupled and SST forced simulations.


tally wrong. We also note that the good agreement incloud properties and CRF in the Tropics does not nec-essarily ensure a consistent cloud feedback there. Asshown in Fig. 1, the two models disagree substantiallyon the change in cloud properties and CRF in responseto climate change forcings in the Tropics. Therefore, anappropriate decomposition of the change in clouds andradiation may help to diagnose the cause for the modelbiases, the inconsistency, and the superficial agreement.

3. Regression results

Cloud–climate feedback is determined by many pro-cesses. Theoretically, any possible feedback process canbe explicitly analyzed using Eq. (2) as long as enoughsets of samples are available. However, in this study weonly focus on cloud feedbacks. All the other noncloudfeedbacks are grouped into the residual term in Eq. (2).The net cloud feedback is further divided into the con-tributions from cloud amount and cloud condensate(taken as CWP in this study) of low (P � 680 hPa), mid-(440 hPa P 680 hPa), and high (P 440 hPa)clouds in a total of six categories. In CAM3, the cloudvertical overlap follows Collins (2001), while AM2 usesa resolution-invariant overlap scheme (Pincus et al.2003).

a. Regression significance and the change inclear-sky radiative flux and CRF

If Eq. (2) does represent the feedback processes inthe climate system in response to climate change forc-ings, then it can be used to deduce the change in clear-sky radiative flux and CRF from the change in full-skyradiative fluxes. Considering the previously classifiedthree cloud types (low, mid, and high) and two cloudproperties (amount and CWP), applying Eq. (2) tolongwave and shortwave radiative fluxes, respectively,yields

�F � ��i�1

6

cFi�xi�cloud � rFe, 3

�Q � ��i�1

6

cQi�xi�cloud � rQe. 4

Physically, rFe and rQe should represent the change inclear-sky longwave and shortwave radiative flux, re-spectively, and rFe � rQe is the net clear-sky flux changeif MLR forms a good approximation of the effect ofclouds on the radiation balance (Hartmann et al. 1992).Following Ramanathan (1987), the longwave, short-wave, and net CRF are defined flcrf � Fclr � F, fscrf ��(Qclr � Q), and fcrf � (Fclr � F) � (Qclr � Q), re-

spectively, where Fclr and Qclr are the clear-sky upwardlongwave and net downward shortwave radiativefluxes. Thus, the change in CRF due to external forcingcan be represented by

��flcrf � Fclr � F2 � Fclr � F1

�fscrf � �Qclr � Q2 � Qclr � Q1

�fcrf � �flcrf � �fscrf

, 5

where subscript 1 and 2 indicate two different climates.In the regression formulation, terms (�6

i�1cFi�xi)cloud

and (�6i�1cQi�xi)cloud in Eqs. (3) and (4) represent the

change in longwave and shortwave radiative fluxes dueto clouds. In accordance with the definition of CRF, thevalues of �(�6

i�1cFi�xi)cloud, (�6i�1cQi�xi)cloud, and their

difference in the regression relations should be close tothe change in longwave, shortwave, and net CRF. Butthey are not exactly the same because of the slight dif-ference in their definitions. In the MLR approach, thechange in longwave and shortwave radiative fluxes dueto the cloud change is directly regressed from thechange in full-sky radiative fluxes. In some sense, theregressed radiative flux change associated with thecloud change is similar to that derived from the PRPapproach. However, if the change in noncloud proper-ties such as temperature and moisture is strongly cor-related to the cloud change, then the MLR approachwill not separate them as cleanly as the PRP approach,an effect similar to cloud masking of the CRF approach.In the worst situation, the cloud masking of MLR maybe comparable to that of CRF, but should be no morethan that. Thus, like the CFR approach, the resultsfrom the MLR analysis should be also interpreted bear-ing the possible cloud masking effect in mind. We shallfurther investigate this issue in our future studies whenthe MLR approach is applied to long-term satellite ob-servations and the corresponding AMIP simulations.

Although the MLR approach has been applied to theentire globe, in this paper we only show the resultsbetween 60°S and 60°N. One reason is that the clear-sky shortwave radiative fluxes in a narrow band aroundAntarctica simulated by CAM3–2CO2 show a signifi-cant bias compared with the ERBE satellite observa-tions (Fig. 3). No substantial bias, however, is found inthe full-sky shortwave radiative fluxes. Thus, the short-wave CRF in this band simulated by CAM3–2CO2 musthave an equivalent bias with an opposite sign in orderto offset the bias in clear-sky shortwave flux. The rea-son for this bias may have to do with the incorrect seaice cover prediction in this latitude band. Thus, we dis-card the regression analysis beyond 60°S and 60°N, andfocus on cloud feedbacks in the low and midlatitudes.Another reason for limiting our analyses to 60°S–60°N

1 JUNE 2007 Z H U E T A L . 2607

is that the multiple correlation coefficient (Fig. 5) goesdown beyond there. Details of regression significancewill be discussed shortly.

To the first-order approximation, the MLR-deducedclear-sky radiative flux change and the cloud radia-tive flux change should be close to those directly simu-lated by the models; thus, the comparison betweenthem provides a way to measure the fidelity of MLR indecomposing the total cloud feedback. Figure 4 com-pares the zonally averaged change in annual mean netCRF and clear-sky radiative flux in response to doubledCO2 and �2-K SST perturbations simulated by CAM3and AM2 with the corresponding change deduced fromMLR. Note that the radiative flux change has beennormalized by the corresponding surface temperaturechange. First, this figure shows that the clear-sky radia-tive feedback behaves quite differently in the doublingCO2 experiment and the �2-K SST perturbation ex-periment. There is a nearly 2 W m�2 K�1 difference inclear-sky radiative flux change between the two experi-ments at most latitudes between 60°S and 60°N. Alikely explanation for this difference is that the directradiative forcing that resulted from doubled CO2 hasnot been subtracted from the change in radiative fluxes

in the CO2 experiment. But we will not rule out thepossibility that this difference reflects the differentclear-sky feedback that resulted from the doubled CO2

and the �2-K SST forcings since the latter is muchstronger and unrealistic.

Second, the most promising result shown in the fig-ure is that CAM3 and AM2 predicted a fairly consistentchange in clear-sky radiative fluxes in both the doublingCO2 and the �2-K SST perturbation experiments, in-dicating the robustness of clear-sky feedback simulatedby the two models. This result is consistent with previ-ous studies (e.g., Cess et al. 1990).

Third, in contrast to the consistent change in clear-sky radiative fluxes, the two models predicted a ratherdifferent change in cloud radiative fluxes in both typesof sensitivity experiments. CAM3 predicted a decreasein net CRF in the Tropics and subtropics but an in-crease in the midlatitudes, which is almost opposite tothe change simulated by AM2. Such an inconsistentchange in CRF reflects the uncertainty and complexityof cloud feedbacks simulated by the current GCMs. Inthe following sections, we will break down the net CRFchange in terms of different cloud types and cloudproperties.

FIG. 3. (a) Zonally averaged difference of annual mean clear-sky shortwave radiative fluxbetween the CAM3–2CO2 simulations and the ERBE observations (solid lines: present cli-mate minus ERBE; dashed lines: doubled CO2 climate minus ERBE). (b) Same as in (a), butfor full-sky shortwave radiative flux.


Fourth, the MLR-deduced change in net cloud andclear-sky radiative fluxes is in fairly good agreementwith the corresponding simulated change in all the ex-periments performed by CAM3 and AM2, indicatingthat this method performs sufficiently well in analyzingand interpreting cloud feedback processes. There aresome noticeable differences here and there between thesimulations and the estimations, which in part can beattributed to the slightly different definition betweenthe change in CRF [Eq. (5)] and the cloud radiative fluxchange deduced from MLR [Eqs. (3) and (4)], and inpart reflect the estimation error of MLR. For example,in our MLR analysis we did not consider the effect ofcloud height and cloud phase on the radiative fluxes atTOA, which was shown to be important especially forthe longwave radiation according to Colman et al.(2001). However, these effects do not seem to be sig-nificant enough to affect the performance of MLR con-sidering the general agreement between the simulatedand regressed change in cloud radiative fluxes (Fig. 4).The agreement, in a way, suggests that the effect ofcloud height and cloud phase might not be as importantas it was expected to be. But there is also a possibilitythat the errors generated in different layers have been

cancelled out. We note that the regression analysisdone by Hartmann et al. (1992) also neglected the pos-sible effect from cloud height and cloud phase. Thisissue will be further investigated using the long-termsatellite observations and the AMIP simulations in ourfuture research.

To give an evaluation of the overall performance ofthe MLR analyses, Fig. 5 shows the multiple correlationcoefficients (similar to the correlation coefficient in thesimple linear regression with one independent variable)of the four climate sensitivity experiments. The MLRanalysis works well in most places, especially at the lowlatitudes. This is different from the analysis done byHartmann et al. (1992), who showed that their regres-sion analysis fails to represent the ERBE clear-sky ra-diative fluxes using the ISCCP clouds in the tropicalconvection regime where the sky is overcast since theregression is overwhelmed by the cloudy radiances andit receives little information on what clear-sky radi-ances are. But this problem no longer exists in ouranalysis: first, the radiances in our analysis were notmeasured but explicitly calculated by models and, sec-ond, our regression is applied to the difference betweenthe two climates. For overcast conditions, even though

FIG. 4. (a) Zonally averaged change in annual mean net CRF at TOA in response to (left) doubled CO2 and(right) �2-K SST perturbations simulated by CAM3 and AM2 compared with the corresponding change in netcloud radiative fluxes deduced from MLR (dashed lines). (b) Same as in (a), but for the change in clear-skyradiative flux. Note that the radiative flux change has been normalized by the corresponding surface temperaturechange.

1 JUNE 2007 Z H U E T A L . 2609

the cloudy radiances dominate the clear-sky radiances,their difference between the two climates may not. Thelarge value of multiple correlation coefficient suggeststhat the MLR analysis receives sufficiently strong sig-nals in both clear-sky and cloudy radiances.

The significance of individual terms in the regressionequation (2) can be evaluated by t statistics (see theappendix). A term in Eq. (2) is considered to be sig-nificant in the regression when the following conditionis met:

ti2 � F�, 6

where t2i is defined by Eq. (A9), � is the confidence

level (in some textbooks, 1 � � is used and refers tosignificance level), and F� satisfies

�F�

�

F �, � dx � �, 7

where F(�, �) is the F distribution defined by Eq.(A11). The t statistics was performed during the MLRcalculation at confidence levels � � 0.9 and � � 0.7.Figure 6 shows the values of t2

i and F� of six cloudproperties associated with the regression analysis of thenet radiative fluxes. In most cases, t2

i is greater than F0.9,indicating that the change in six cloud properties hassignificant contributions to the change in net radiativefluxes at a confidence level of � � 0.9. Some variablessuch as the midcloud amount change appear to be lesssignificant than other variables at some latitudes, buttheir significance is still above the confidence level of� � 0.7. There is a good agreement on the t2

i value ofhigh clouds between CAM3 and AM2, but substantial

differences are shown in low and midclouds. In theCAM3 simulations, the change in CWP appears to bemore significant than the cloud amount change, whilethe opposite is true for the AM2 simulations. The exactreason is not clear, but since all these cloud propertiesshow their significance above the confidence level of� � 0.9 at most latitudes, none of them can be ne-glected in the regression analysis.

In short, the significance tests and the general agree-ment between the regressed and simulated change incloud and clear-sky radiative fluxes at TOA suggestthat the MLR approach performs sufficiently well andprovides an appropriate way to interpret the cloudfeedbacks in the climate system. In the following sec-tions, the regression analysis on cloud feedbacks will bediscussed in detailed.

b. Total cloud radiative feedbacks

In this section, we examine the MLR-deduced radia-tive flux change at TOA associated with the change intotal cloud amount and vertically integrated CWP. Fig-ure 7 shows the zonally averaged change in longwave,shortwave, and net radiative fluxes at TOA due to thechange in total cloud amount and CWP in response todoubled CO2 and �2-K SST perturbations, where theradiative flux change has been normalized by the sur-face temperature change. Significant differences be-tween the two models are shown at most latitudes. Thecloud amount change in the AM2 simulations causes alarger change in longwave and shortwave radiativefluxes than that in the CAM3 simulations. This appearsto be consistent with the larger cloud amount change inAM2 shown in Fig. 1. But since the change in longwaveand shortwave radiative fluxes due to cloud amountchange has a different sign, the cancellation results in anet radiative flux change equivalent to that of CAM3.In contrast, the large CWP change in AM2 does notresult in a large flux change as expected. In particular,the AM2–CESS experiment simulates a much smallerchange in shortwave fluxes than that of CAM3–CESSat the low latitudes. This inconsistency may be relatedto the cancellation between different cloud feedbackprocesses, which we will show in the following sections.

The averaged (60°S–60°N) longwave, shortwave, andnet cloud feedback parameters associated with thechange in total cloud amount and CWP in response todoubling CO2 and �2-K SST perturbations are shownin Fig. 8, where the total longwave and shortwave cloudfeedback parameters are defined as �LW � �x � (�F/�x)(�x/�Ts) and �SW � �x(�Q/�x)(�x/�Ts), respectively,where x denotes individual cloud property. Despite thedifferences in many aspects, CAM3 and AM2 do pro-duce the same sign of individual longwave and short-

FIG. 5. The multiple correlation coefficient of the MLRanalyses.


wave feedbacks associated with the change in cloudamount and CWP. For example, they both agree on anegative longwave feedback and a positive shortwavefeedback due to the cloud amount change, and a posi-tive longwave feedback and a negative shortwave feed-back due to the CWP change. The cancellation betweenthem determines the sign and magnitude of net cloudfeedback. CAM3 produced a negative net cloud feed-back mostly due to the strong negative shortwave feed-

back caused by the CWP change, whereas the net posi-tive cloud feedback in AM2 results mainly from thestronger positive shortwave feedback associated withthe cloud amount change.

c. Low cloud radiative feedbacks

To identify the cause for the different cloud feed-backs shown in the CAM3 and AM2 simulations, wefurther examine the cloud feedbacks associated with

FIG. 6. Values of t2i of six cloud properties associated with the MLR analysis on the change in net radiative fluxes. The t statistics of

cloud amount and CWP are represented by the solid and dashed lines, respectively. Light dotted and light dash–dotted horizontal linesindicate the values of F� with the confidence levels of � � 0.9 and � � 0.7.

1 JUNE 2007 Z H U E T A L . 2611

individual cloud types. This section focuses on lowclouds. Figure 9 shows the change in low-cloud amountand CWP in response to doubled CO2 and �2-K SSTperturbations. AM2 predicted an overall low-cloud

amount decrease with the maximum reduction roughlyat 40°S and 40°N. The reduction of low-cloud amountbetween 35° and 60° in the two hemispheres is also seenin the CAM3 simulations, but the reduction in the

FIG. 7. Zonally averaged change in longwave, shortwave, and net radiative fluxes at TOA due to thechanges in (left) total cloud amount and (right) CWP in response to (a) doubled CO2 and (b) �2-K SSTperturbations. Note that the radiative flux change has been normalized by the surface temperature change.


Northern Hemisphere is much smaller than that ofAM2. Large increase of low-cloud amount in the Trop-ics and subtropics is shown in the CAM3 simulations.The increase and decrease of low-cloud amount at dif-ferent latitudes tend to cancel each other. As a result,CAM3 predicted a weak overall low-cloud amount in-crease between 60°S and 60°N.

Both CAM3 and AM2 predicted a small change ofCWP in the Tropics and subtropics. But beyond 35°Sand 35°N, CAM3 simulated a CWP decrease, which isapproximately in phase with the cloud amount change.In contrast, AM2 produced a substantial increase ofCWP in this region, which is almost off-phase with thecloud amount change, indicating clouds shrink andthicken in the warmer climate. We will show later thatsuch an opposite change in cloud properties (i.e., de-crease in cloud amount and increase in CWP) also oc-

curs in mid- and high clouds. We shall discuss this issuelater. Overall, we see an opposite sign in low-cloudchange between 60°S and 60°N in the CAM3 and AM2simulations, and the AM2 predicted low-cloud changeis much stronger in magnitude than that of CAM3.

As we previously showed in the total cloud feedback,a large change in cloud properties may not lead to alarge change in radiative fluxes because of the possiblecancellation between individual cloud feedbacks. Tosee if this is the case for low clouds, we plotted theregression-deduced low-cloud longwave, shortwave,and net feedback parameters in Fig. 10. AM2 predicts astrong positive shortwave feedback due to the largelow-cloud amount decrease and a strong negative short-wave feedback due to the low-cloud CWP increase. Butthey tend to cancel each other out and result in aweaker positive net shortwave feedback. In the mean-

FIG. 8. The averaged (60°S–60°N) longwave, shortwave, and net feedback parameters associated with the change in (left) total cloudamount and (middle) CWP in response to doubled CO2 and �2-K SST perturbations simulated in CAM3 and AM2. (right) The totalcloud feedbacks.

1 JUNE 2007 Z H U E T A L . 2613

time, AM2 simulated a negative longwave feedback,but it is too weak to offset the positive net shortwavefeedback. As a result, the residual gives a weak netpositive feedback of low clouds in the AM2 simula-tions. On the other hand, cancellation is not a big factorin the CAM3 simulations. CAM3 simulated a negativeshortwave feedback due to the CWP change, whichseems to dominate other feedbacks. Thus, the resultantnegative net feedback of low clouds in CAM3 is equiva-lent in magnitude to the positive net low-cloud feed-back in AM2.

d. Midcloud radiative feedbacks

Figure 11 shows the change in cloud amount andCWP of midclouds in response to doubled CO2 and�2-� SST perturbations simulated by CAM3 andAM2. Again, marked differences are shown in thesimulations. AM2 predicted a substantial decrease incloud amount and an increase in CWP. The maximumreduction of cloud amount occurs approximately at40°S and 40°N. But there is only a weak change in cloudamount in the Tropics. The change in CWP does notappear to be correlated to the cloud amount change.The increase of CWP is fairly uniform in the warmerclimate at most latitudes.

In the CAM3 simulations, the cloud amount increase

is seen in the Tropics, while the cloud amount decreaseoccurs in the wide range of subtropics and midlatitudes.The CWP change appears to be weakly correlated tothe cloud amount change. Since the increase and de-crease of cloud amount and CWP at different latitudestend to cancel each other out, the resultant mean cloudchange between 60°S and 60°N in CAM3 is muchweaker than that in AM2. In particular, the overallCWP change in CAM3 is nearly tenfold weaker thanthat in AM2.

Figure 12 shows the longwave, shortwave, and netfeedback parameters associated with the midclouds.First, individually the longwave and shortwave feed-backs associated with the cloud amount change arestronger than those associated with the CWP change.However, since there is a strong cancellation betweenthe longwave and shortwave cloud amount feedbacks,the resultant net cloud amount feedback is equivalentto that of the CWP feedback. Second, the strong indi-vidual feedbacks in AM2 seem to be consistent withtheir large change in cloud properties (Fig. 11). How-ever, the large cancellation between them causes AM2to produce a weaker net midcloud feedback than thatof CAM3. This is similar to what we have seen in low-cloud feedback. Since the net feedback is usually muchweaker than individual feedbacks because of the strong

FIG. 9. Changes in low-cloud amount and CWP in response to doubled CO2 and �2-K SST perturbationssimulated by CAM3 and AM2.


cancellation between them, the sign of net feedbackmay strongly depend on the estimation error even if theerror is proven to be negligible to the individual feed-back. Third, in the doubling CO2 experiment, bothCAM3 and AM2 predict a positive net midcloud feed-back; while in the CESS experiment a negative netfeedback is produced by CAM3, indicating that netcloud feedback may also depend on the types of exter-nal forcings.

e. High cloud radiative feedbacks

Figure 13 shows the change in high-cloud amountand CWP in response to doubled CO2 and �2-� SSTperturbations. AM2 predicted a decrease in cloudamount at most latitudes between 55°S and 55°N excepta slight increase in cloud amount just north of the equa-

tor shown in the doubling CO2 experiment. Cloudamount increase is seen at high latitudes beyond 55° inboth hemispheres. CAM3, on the other hand, predicteda much complicated high-cloud amount change. Gen-erally, high-cloud amount increases in the Tropics andhigh latitudes but decreases in the subtropics. There isa large reduction of cloud amount at 15°N in theCAM3–2CO2 experiment. The cancellation gives asmall total high-cloud amount increase between 60°Sand 60°N in the two CAM3 experiments.

It is interesting to see that both CAM3 and AM2predicted an increase of high-cloud CWP at all latitudesin both doubling CO2 and �2-� SST perturbation ex-periments. This appears to be the only change in cloudproperties that is agreed by both models, although thedetailed variation at a specific latitude is different. The

FIG. 10. Same as in Fig. 8, but for low-cloud amounts and feedbacks.

1 JUNE 2007 Z H U E T A L . 2615

optically thick high clouds formed in a warmer climateshown in the CAM3 and AM2 simulations may be con-sidered to be a robust phenomenon between the twomodels since they are different in every aspect in thecloud parameterizations. In AM2, large-scale cloud wa-ter is parameterized based on the prognostically deter-mined specific humidity of liquid and ice. Cloud micro-physics are parameterized with an updated treatment ofmixed-phase clouds (Rotstayn et al. 2000). Cloudamount is also treated as a prognostic variable follow-ing the parameterization of Tiedtke (1993). While inCAM3, the parameterization of large-scale cloud pro-cesses follows Rasch and Kristjánsson (1998) andZhang et al. (2003) with a prognostic cloud condensatescheme combining a bulk microphysical representationof condensation and evaporation. Cloud amount is di-agnostically determined based on relative humidity, at-mospheric stability, and convective mass fluxes. Withall these differences, both models are still able to pre-dict the same trend of an increase in optical depth ofhigh clouds in a warmer climate. From the water budgetpoint of view, the potential condensate available in highclouds is determined by the thermodynamic propertiesof the boundary layer air, which contains more mois-ture in a warmer climate. Ramanathan and Collins(1991) argued that the enhanced moisture in the bound-ary layer air over warm oceans may enhance the con-

vective system due to the increased parcel buoyancyassociated with the increased contribution of water va-por to virtual temperature, and thus lead to thicker andmore extensive anvil clouds. However, we point outthat the total high-cloud amount may not increase sincethe thicker anvil clouds have a higher precipitation ef-ficiency that is unfavorable to the formation of highthin cirrus. A negative correlation between thick cirro-stratus and thin cirrus is often seen in the ISCCP ob-servations (J. Lin 2005, personal communication). Thisphenomenon is also shown in the CAM3 simulations.In the CAM3–CESS experiment, the ISCCP simulator(Klein and Jakob 1999; Webb et al. 2001) was activated,which allows us to examine the climate sensitivity of aspecific type of clouds. Figure 14 shows the change incirrus and cirrostratus amount in response to �2-� SSTperturbations simulated by CAM3, where following theISCCP convention, cirrus and cirrostratus are definedas the clouds within the range: P 440 hPa, � 3.6,and P 440 hPa, 3.6 � 23, respectively. Appar-ently, the increase of high-cloud CWP shown in Fig. 13is caused by the increase of thicker cirrostratus. How-ever, the net change in total high-cloud amount is de-termined by the change of both cirrus and cirrostratus.In this case, the increase of cirrostratus amount isnearly canceled by the decrease of cirrus amount, whichleads to a barely changed total high-cloud amount. If

FIG. 11. Same as in Fig. 9, but for midclouds.


such a cancellation mechanism also holds for the otherexperiments, it is most likely that the decrease of cirrusdominates the increase of cirrostratus in the AM2 simu-lations. Unfortunately, the ISCCP simulator was notactivated in the AM2 simulations and the CAM3 dou-bling CO2 experiment. Thus, we are unable to examinethe detailed high-cloud change in these experiments.

Figure 15 shows the computed feedback parametersassociated with high clouds. Compared with low- andmidcloud feedbacks, a relatively consistent high-cloudfeedback is shown in the simulations. Both CAM3 andAM2 predict the same sign of individual cloud amountfeedback and cloud condensate feedback although thefeedback strength in AM2 is much stronger. Both mod-els agree on a positive net longwave feedback and anegative net shortwave feedback. In CAM3, the posi-tive net longwave feedback is dominated by the nega-

tive net shortwave feedback, which results in a negativenet feedback of high clouds. However, in AM2 the op-posite net longwave and shortwave feedbacks havenearly the same magnitude, which leads to a very weaknet high-cloud feedback. From a different angle, theweak net high-cloud feedback in AM2 also resultedfrom a strong cancellation between the negative cloudamount feedback and the positive cloud condensatefeedback.

To examine whether the SOM-type coupled simula-tion produces similar cloud feedbacks to those gener-ated by a fully coupled system, the regression analysis isalso applied to the IPCC 1% increase CO2 experiment(to doubling CO2) performed by the fully coupledNCAR Community Climate System Model version 3(CCSM3), which uses the same atmospheric model asCAM3. Note that this IPCC 1% increase CO2 experi-

FIG. 12. Same as in Fig. 10, but for midclouds.

1 JUNE 2007 Z H U E T A L . 2617

ment was performed at a high resolution of T85, whichis different from T42 resolution used by CAM3–2CO2

simulations. Thus, it allows us to further examine thesensitivity of simulated cloud feedbacks to model reso-lution. The regression diagnosed cloud feedback inCCSM3 does not show substantial difference from thatin CAM3–2CO2. Thus, no further results are presentedhere.

4. Discussion and summary

Cloud–climate feedback simulated by GCMsstrongly depends on the treatment of clouds and radia-tion in models. In this study, we have shown that theNCAR CAM3 and GFDL AM2 exhibit a wide range ofsensitivity to doubled CO2 and �2-� SST perturbationsalthough they produce a similar cloud climatology intheir control runs. CAM3 predicts a weak increase inglobal-mean cloud amount and cloud condensate in thewarmer climate. In contrast, a substantial decrease incloud amount and an increase in cloud condensate aresimulated by AM2. The substantial reduction of cloudamount in AM2 appears to be largely compensated bythe increase in cloud condensate to result in a net CRFchange much weaker than that of CAM3. To identifythe cause for this important inconsistent change in

clouds and radiation and explore the complicatedcloud–radiation feedback shown in the climate sensitiv-ity tests, it is useful to decompose the total cloud feed-back into individual components associated with differ-ent cloud processes. To do so, we have developed a new

FIG. 14. Change in cirrus and cirrostratus amount (diagnosed bythe ISCCP simulator) in response to �2-K SST perturbationsimulated by CAM3.

FIG. 13. Same as in Fig. 9, but for high clouds.


approach based on the MLR technique to diagnosevarious cloud feedback processes. It is showed thatthere is a general agreement between the regressedchange in clear-sky and cloud radiative fluxes and thechange directly simulated by the models. The statisticsignificance analysis also indicates the appropriatenessof the MLR approach for diagnosing cloud feedbacks inthe climate sensitivity experiments. Using this ap-proach, this study specifically examines the cloud feed-back process associated with the cloud amount changeand the cloud condensate change involving low, mid-,and high clouds between 60°S and 60°N. The analysissuccessfully reveals the important similarities and dis-crepancies of individual cloud feedback processes simu-lated by CAM3 and AM2, which are hidden in the totalcloud feedback measured by the CRF approach. Thefollowing presents the major regression results and fur-ther discussions on some important issues.

First, the decomposed cloud feedbacks clearly indi-

cate that both CAM3 and AM2 predict the same sign ofindividual longwave and shortwave feedback associatedwith the change in cloud amount and condensate for allthree types of clouds (Figs. 10, 12, and 15) except thelow-cloud and midcloud condensate longwave feed-backs, but they are much weaker than the other indi-vidual feedbacks. This consistency has important impli-cations considering the different cloud and radiationparameterizations used by the two models. Our regres-sion analysis supports previous studies (e.g., Colman etal. 2001) that the longwave and shortwave feedbacks orcloud amount and cloud condensate feedbacks gener-ally have opposite signs. Thus, as a subtle residual of all,the net cloud feedback is very sensitive to a smallchange in any part of clouds, external forcings, or evena specific radiative transfer calculation. For example,the regression analysis yields strong individual feed-backs in the AM2 simulations, which appears to beconsistent with its predicted large change in cloud prop-

FIG. 15. Same as in Fig. 10, but for high clouds.

1 JUNE 2007 Z H U E T A L . 2619

erties. But the strong cancellation between them resultsin a very weak net cloud feedback. Based on this, weargue that until cloud and radiation parameterizationscan sufficiently generate accurate individual cloud feed-backs, it is impossible for climate models to predict aconsistent net cloud feedback since it is an extremelydelicate process resulted from a large cancellation.

Second, despite the same sign of individual cloudfeedback predicted by the two models, CAM3 andAM2 disagree on the relative importance of the threetypes of clouds in determining the total cloud feedback.In the AM2 simulations, the net feedbacks of mid- andhigh clouds are small; thus, the net low-cloud feedbackdetermines the sign and strength of the total cloud feed-back. The MLR analysis further indicates that thestrong positive shortwave feedback from low-cloudamount dominates the other feedback processes. This isconsistent with the large decrease in low-cloud amountsimulated by AM2. The large increase in low-cloudcondensate does induce a negative shortwave feedback,but it is not strong enough to offset the positive short-wave feedback due to the reduction of low-cloudamount. On the other hand, in the CAM3 simulations,the total cloud feedback appears to be largely deter-mined by the low- and high-cloud condensate feed-backs. The different importance of cloud amount andcloud condensate feedbacks involving low, mid-, andhigh clouds simulated by CAM3 and AM2 raises aquestion for the radiative transfer calculation in mod-els. With the same cloud loading, do the radiative trans-fer codes used in CAM3 and AM2 produce the sameclimate sensitivity of the radiative flux? This questioncould be answered by using the offline calculation pro-vided that the detailed cloud information is available.

Third, the detailed examination indicates that the cli-mate sensitivity of cloud amount simulated by CAM3and AM2 at least shares certain similarities. For ex-ample, our analysis indicates that both models pre-dicted a low-cloud increase in the subtropical subsid-ence regimes where SST is less increased in the dou-bling CO2 experiment (not shown here). However,almost no similarity in the condensate change of lowand midclouds is seen in the two models. We have ex-ecuted a new experiment to force CAM3 using the cli-matological SST generated by the AM2 doubling CO2

experiment. The basic feature of low-cloud amountchange in response to doubled CO2 simulated by AM2is somewhat reproduced by CAM3 with such an SSTforcing although with biases. Again, no common fea-ture in cloud condensate change is seen in the two simu-lations. Thus, as an important step to understand theimpact of clouds on climate sensitivity and reduce theuncertainty in cloud–climate feedbacks, more attention

should be paid to improve the treatment of cloud mi-crophysics that determines the amount and phase ofcloud condensate. It is also important to develop a self-consistent cloud parameterization that can provide aninternal mechanism to constrain the change in cloudamount and cloud condensate.

Fourth, the large cancellation between individualfeedback processes suggests that quantifying the subtleglobal-mean cloud feedback may depend strongly on aparticular analysis approach and the estimation errorassociated with the approach. This is because even ifthe estimation error is proven to be negligible to theindividual cloud feedback, the globally accumulated es-timation error is not necessarily negligible to the globalnet cloud feedback. In other words, an accurate mea-sure of the global-mean net cloud feedback is difficultsince the real signal of the weak net cloud feedbackmay be overwhelmed, or affected by the estimation er-ror and “cloud masking” effect suggested by Soden etal. (2004). Thus, the results from feedback analysesshould be interpreted with caution.

Finally, as indicated by this work and many previousstudies, at the current stage diagnosing individual cloudfeedback processes may be more useful than just ex-amining the total cloud feedback since the former pro-vides an efficient way to isolate differences and identifythe cause for the widespread of cloud–climate sensitivi-ties among GCMs. Fortunately, the signals of individualcloud feedback processes are usually strong; thus, theestimation error may not be as critical as it is to thetotal cloud feedback. In this study, we successfully de-compose the total cloud feedback simulated by CAM3and AM2 into individual components associated withlow, mid-, and high clouds using the MLR approach.Such a decomposition can also be done by the PRPapproach. The real problem of this kind of studies isthat we have generated something that cannot be evalu-ated by observations. Thus, although the differentcloud feedback processes between models have beenclearly illustrated using the MLR approach or the PRPapproach, we are unable to determine which is the rightfeedback and which might be the unrealistic one. Itwould be a great help if the diagnosed cloud feedbackcould be directly compared with observations. One ad-vantage of the proposed MLR approach is that it can beapplied to observations as illustrated by Hartmann etal. (1992) who have successfully regressed 1-yr ISCCPcloud data to the ERBE radiative flux data. Over thepast few decades, a great amount of satellite data ofclouds and radiation has been collected. With the MLRapproach, these satellite data can be used to diagnosethe individual cloud feedback processes associated withshort-term scale climate anomalies, such as ENSO. The


derived result from observations can be used to evalu-ate the individual cloud feedback processes diagnosedfrom the corresponding AMIP simulation by GCMs.We believe that such a comparison between observa-tions and simulations can shed some new light on cloudfeedback issues and provide a useful guide to improvecloud and radiation parameterizations. This will be oneof the aspects we focus on in our future research.

Acknowledgments. The authors wish to acknowledgethe support for this work from the Geophysical FluidDynamics Laboratory (GFDL). We also wish to thankDrs. Ming Zhao and Cecile Hannay for providing theCAM3 and AM2 output. We are very grateful to thetwo anonymous reviewers for their constructive com-ments, which led to an improved paper.

APPENDIX

Multiple Linear Regression

The following briefly summarizes the basic proce-dures for the MLR calculations. For more details, read-ers may refer to statistical analysis textbooks (e.g., Afifiand Azen 1979). Assuming one has n sets of samples ofEq. (2), �Rk, �x1k, �x2k, . . . , �xmk, (k � 1, 2, . . . , n),then the regression coefficient C � [c1, c2, . . . , cm] canbe determined as

C � L�1l0, A1

where L�1 is the reciprocal matrix of L represented by

L ��l11 l12 ... l1m

l21 l22 ... l2m

... ... ... ...

lm1 lm2 ... lmm

� ;

L�1 ��f11 f12 ... f1m

f21 f22 ... f2m

... ... ... ...

fm1 fm2 ... fmm

� . A2

The matrix elements of L can be calculated by

lij � �k�1

n

�xik � �xi�xjk � �xj, i, j � 1, 2, . . . , m,

A3

where the overbar denotes the mean of a variable overn sets of samples, for example,

�xi �1n �

k�1

n

�xik, i, � 1, 2, . . . , m. A4

The same definition also holds for other variables. Vec-tor l0 � [l01, l02, . . . , l0m] in Eq. (A1) is calculated by

l0i � �k�1

n

�xik � �xi�Rk � �R, i, � 1, 2, . . . , m.

A5

Once the coefficient C is determined, the regressionresidue re can be calculated as

re � �R � �k�1

m

ci�xi. A6

The overall performance of regression determined bythe previous procedures can be evaluated by the mul-tiple correlation coefficient �R,x1

, . . . , xm represented by

�R,x1, . . . , xm� � 1

l00�i�1

m

l0ici�12

, i, � 1, 2, . . . , m,

A7

where

l00 � �k�1

n

�Rk � �R2. A8

The significance of each independent feedback processin the linear regression equation (2) can be evaluatedby t statistics ti represented as

ti �ci

2fii12

s, i, � 1, 2, . . . , m, A9

where fii is the diagonal element of matrix L�1 in Eq.(A2). Here s2 is the residual mean square represented by

s2 �

l00 � �i�1

m

cil0i

n � m � 1, i, � 1, 2, . . . , m. A10

The t statistics defined by Eq. (A9) follows the F dis-tribution represented by

F �, �x�2�1

B�2, 2 ��

��2�1 ��x

��2

,

A11

with the degree of freedom of � � 1 and � � n � m �1. Here B(� /2, � /2) is the � function.

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