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Quantifying the Role of Internal Climate Variability in Future Climate Trends
DAVID W. J. THOMPSON AND ELIZABETH A. BARNES
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

CLARA DESER
National Center for Atmospheric Research,* Boulder, Colorado

WILLIAM E. FOUST
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

ADAM S. PHILLIPS
National Center for Atmospheric Research, Boulder, Colorado(Manuscript received 4 December 2014, in final form 8 May 2015)
ABSTRACTInternal variability in the climate system gives rise to large uncertainty in projections of future climate. The
uncertainty in future climate due to internal climate variability can be estimated from large ensembles of climatechange simulations in which the experiment setup is the same from one ensemble member to the next but forsmall perturbations in the initial atmospheric state. However, large ensembles are invariably computationallyexpensive and susceptible to model bias.
Here the authors outline an alternative approach for assessing the role of internal variability in future climatebased on a simple analytic model and the statistics of the unforced climate variability. The analytic model isderived from the standard error of the regression and assumes that the statistics of the internal variability areroughly Gaussian and stationary in time. When applied to the statistics of an unforced control simulation, theanalytic model provides a remarkably robust estimate of the uncertainty in future climate indicated by a large
ensemble of climate change simulations. To the extent that observations can be used to estimate the amplitudeof internal climate variability, it is argued that the uncertainty in future climate trends due to internal variability
can be robustly estimated from the statistics of the observed climate.

1. Introduction
The signature of anthropogenic forcing in climatechange has and will be superposed on internal climate
variability resulting from a variety of physical processes(e.g., Hawkins and Sutton 2009, 2011; Deser et al. 2012a,b;Wallace et al. 2015; Kirtman et al. 2014; Collins et al.

* The National Center for Atmospheric Research is sponsoredby the National Science Foundation.

Corresponding author address: David W. J. Thompson, Department of Atmospheric Science, Campus Delivery 1782, Colorado
State University, Fort Collins, CO 80523.
E-mail: davet@atmos.colostate.edu
DOI: 10.1175/JCLI-D-14-00830.1
Ó 2015 American Meteorological Society

2013; Knutson et al. 2013; Bindoff et al. 2014). At mostterrestrial locations, a large component of the internalvariability in surface climate change arises from variations in the atmospheric circulation (Wallace et al. 1995,
2012, 2015; Deser et al. 2014). On regional spatial scales,
the internal variability can overwhelm the signature ofanthropogenic forcing not only on year-to-year timescales, but on multidecadal time scales as well (Hawkins
and Sutton 2009; Deser et al. 2012a,b; IPCC 2014). The
signal of significant warming is thus expected to‘‘emerge’’ earliest in regions with relatively small naturalvariability, such as the tropics (Christensen et al. 2007;
Mahlstein et al. 2011; Diffenbaugh and Scherer 2011;
Hawkins and Sutton 2012). Understanding and predicting the contribution of internal variability to long-term

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FIG. 1. (left) The standard deviations of the 50-yr trends in October–March mean (top) surface temperature and(bottom) precipitation based on output from the NCAR 40-member ensemble of climate change simulations. Trends
are expressed as the total change over the 50-yr period 2011–61. The trend standard deviations indicate the spread in
the trends derived from all 40 ensemble members. (right) Wintertime mean time series of (top) surface temperatureand (bottom) precipitation for grid boxes collocated with the indicated locations. The gray lines show results for all 40
ensemble members; the red and blue lines indicate the ensemble members with the largest and smallest trends overthe 2011–61 period, respectively. Tick marks at every 18C and 1 mm day21.

trends in climate is essential for both the adaptation toand the mitigation of climate change (IPCC 2014).
What is the most robust way to estimate the role ofinternal variability in future climate trends? One approach is to generate a large ensemble of climate change
simulations in which the individual ensemble members
are from the same climate model and subject to the same
external forcing but are initiated with slightly different
atmospheric initial conditions. For example, the National
Center for Atmospheric Research (NCAR) Community
Climate System Model, version 3 (CCSM3), large ensemble project includes 40 climate change simulations run
with the same coupled atmosphere–ocean–sea ice–landmodel (CCSM3) and forced with identical projectedchanges in greenhouse gases and ozone from 2000 to 2061
(the SRES A1B scenario). Since the model and forcing
are the same in all ensemble members, the differences inclimate trends from one ensemble member to the next
derive entirely from the unforced (i.e., internal) variability
in the model.
Analyses of the spread in the trends in the NCAR
40-member ensemble make clear the pronounced role ofinternal climate variability in projections of regional climate change (Deser et al. 2012a,b). For example, the left
panels in Fig. 1 show the standard deviations of the 50-yr(2011–61) trends in October–March mean near-surface

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air temperature and precipitation calculated over all ensemble members (i.e., the results indicate the spread in
the trends from one ensemble member to the next).
The right panels indicate time series of October–Marchmean surface air temperature and precipitation for all
40 ensemble members at two sample locations. As
noted in Deser et al. (2012b), internal variability in the
CCSM3 gives rise to temperature trend standard deviations that exceed 1 K (50 yr)21 over much of theNorthern Hemisphere and precipitation trend standard
deviations that exceed 0.5 mm day21 (50 yr)21 overmuch of the tropics. Since the spreads in the trends
indicated in Fig. 1 arise entirely from stochastic variability in the CCSM3, they may be viewed as the irreducible component of uncertainty in climate change
projections in this particular model.
The purpose of this study is to develop a simple analytic model for estimating the uncertainty in projections of future climate trends due to internal climate
variability, as exemplified in Fig. 1. The model is derived from the standard error of the regression and isbased on two statistics of the unforced climate variability: the standard deviation and autocorrelation.
The analytic model is developed in section 2. It is tested
against the NCAR 40-member ensemble in section 3 andapplied to observations in section 4. Section 5 includes a
discussion of the results. Concluding remarks are given in
section 6.

2. A simple analytic model of the role of internalvariability in future climate trends
Consider a time series x(t) with mean zero and a linear
least squares trend b. The confidence interval (CI) on
the trend in x(t) is expressed asntå (i 2 i)

(2)

2

i51

is the standard error of the trend. In Eq. (2), i denotes
time, se is the standard error of x(t) about its linear trend,
and the factor nt is included so that the standard error isgiven in the units of Dx(ntDt)21. Equations (1) and (2)are widely used to assess the significance of a trend inclimate science (Wilks 1995; von Storch and Zwiers
1999; Santer et al. 2000).
The standard deviation of the time axis [the denominator in Eq. (2)] can be expressed as a function of nt
as follows:1g(nt ) [ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5ntå (i 2 i)

2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
12,
n3t 2 nt(3)

i51

where we have used two formulas for consecutive integers
to derive the algebraic expression g(nt) (see the appendix).
Note that the units for g(nt) are per time step.
Regarding the standard error of x(t) about its linear
trend [se in Eq. (2)]: if the residuals of the regression[i.e., the detrended values of the x(t) time series] areserially correlated, then se must include a scaling factor
that accounts for the bias in the sample standard deviation introduced by persistence in the time series(Mitchell et al. 1966; Wilks 1995; von Storch and Zwiers
1999; Santer et al. 2000). As shown in the appendix, if
x(t) is well modeled as Gaussian red noise, then thestandard error of x(t) about its linear trend can be approximated as
se 5 sg(nt , r1 ) ,

(4)

where

CI 5 b 6 e ,where e is the margin of error for the trend. The trend, itsconfidence interval, and its margin of error are all expressed in units, where nt is the number of time steps and
Dt is the time step. For example, if x(t) corresponds to50 years of wintertime mean temperature data, thennt 5 50, Dt 5 1 yr, and the temperature trend in x(t) isexpressed in units of degrees Celsius per 50 yr.
If the distribution of the deviations in x(t) about its
linear trend (i.e., the residuals of the regression) is
Gaussian, then the margin of error for the trend in x(t) is
e 5 tc sb ,se
sb 5 nt sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ(1)

where the critical t statistic tc is a function of the degrees
of freedom and desired confidence interval, andsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
nt1
s[[x(i) 2 bi]2å
nt 2 2 i51(5)

is the standard deviation of the residuals,

2

31 /2

n 22
5
g(nt , r1 ) [ 4  t 
1 2 r122nt
1 1 r1(6)is the scaling factor, and r1 is the lag-one autocorrelationof the residuals. If the residuals are not serially correlated [e.g., the lag-one autocorrelation of the detrended
x(t) time series is zero], then se 5 s. Note that the denominator of Eq. (5) includes the factor nt 2 2 (rather

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than nt 2 1, as is the case for estimating the sample
standard deviation). The factor nt 2 2 arises because
estimating the standard deviation of the residuals requires estimating not only the mean of the time series(the y intercept) but also the regression line (the slope).
In the context of climate change, s can be viewed as thestandard deviation of the internal (unforced) variability.
Substituting Eqs. (2), (3), and (4) into Eq. (1) yields
the following expression for the margin of error for atrend in x(t) in the units of Dx(ntDt)21:
e 5 tc nt sg(nt , r1 )g(nt ) .

(7)

Equation (7) provides a simple analytic model for themargin of error for a trend in a Gaussian red noise
process. It is derived from commonly used estimates oftrend significance (e.g., Lettenmaier 1976; Santer et al.
2000; Casola et al. 2009). And it makes clear that themargin of error on a trend is a function of two statisticsof the internal variability, both of which we assume arestationary in time:1) the standard deviation of the internal (unforced)
variability s and2) the lag-one autocorrelation of the internal (unforced) variability r1.
Figure 2 shows solutions for Eq. (7) as a function ofnt (trend length) and r1 (the lag-one autocorrelationof the residuals), where tc is calculated for the twotailed 95% confidence level (i.e., 2.5% of the distribution lies in each tail). The results are expressed inunits of the margin of error relative to the amplitudeof the internal variability [i.e., they show solutions to
e/s 5 tc nt g(nt , r1 )g(nt )]. As such, the results indicate
the amplitude of the trend required to exceed the
95% confidence level in units of the internal variability. For example, if e/s 5 2, then the trend must be
twice as large as the internal (unforced) variability toexceed its margin of error.
The margins of error on the trends (and thus the trend
amplitudes required for significance) increase rapidly as
the length of the trend decreases and/or the autocorrelation increases. If a time series is 40 time steps in lengthand has autocorrelation r1 ’ 0.45, then the trend in thetime series must be twice as large as the standard deviation of the internal variability to exceed the 95%
confidence level. If the autocorrelation increases to r1 ’
0.65, then the trend must be approximately 3 times aslarge as the internal variability.
As also evidenced in Fig. 2, if the autocorrelation of
the residuals r1 ’ 0 and the trend length is 50 time steps,
then the margin of error on the trend is roughly equal to
the standard deviation of the residuals; that is, inserting
r1 ’ 0 and nt 5 50 into Eq. (7) yields

FIG. 2. Analytic solutions for the uncertainty in future climate
due to internal variability. Contours indicate the trend amplitudes
required to exceed the 95% margin of error relative to the standard
deviation of the internal variability. For example, a trend of 2 indicates that the trend must be twice as large as the internal (unforced) variability to exceed the 95% margin of error. Results arederived from Eq. (7) and are shown as a function of the trend length(in time steps) and the lag-one autocorrelation r1. Contours arespaced at trend amplitudes of 0.5.

e95% ’ s(for nt 5 50

and r1 ’ 0),(8)where e95% denotes the two-tailed 95% margin of error on
the trend in the units of Dx(ntDt)21. Equation (8) provides arough rule of thumb for the uncertainty in 50-yr trends in
any Gaussian physical process that is not serially correlated
from one year to the next: the 95% margin of error on the
50-yr trends in a Gaussian process is roughly equal to its
interannual standard deviation.

3. Testing the analytic model in a large ensemble ofclimate change simulations
How well does the analytic model given by Eq. (7)predict the uncertainty in future climate trends? The
utility of the analytic model is tested by comparing
1) the margins of error in trends calculated from a large
ensemble of climate change simulations run on a coupled global climate model (the actual margins of error)
with 2) the margins of error predicted by applying theanalytic model to the statistics of the internal variability of the same coupled global climate model (thepredicted margins of error). As discussed below, theinternal variability of the coupled global climate modelis estimated from a long control simulation with fixed
anthropogenic forcing.

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The actual margins of error are derived from 50-yr trends
in boreal wintertime (October–March) mean near-surface
air temperature and precipitation from the NCAR
40-member ensemble of climate change simulations. The
NCAR 40-member ensemble is described in detail in Deseret al. (2012b). Briefly, the simulations were run with a fully
coupled ocean–land–atmosphere global climate model
on a 2.88 3 2.88 latitude–longitude grid (CCSM3) andforced with the Special Report on Emissions Scenarios(SRES) A1B scenario. The ensemble members differ onlyin their initial atmospheric conditions.
The predicted margins of error are derived from a
1000-yr-long control simulation run on the NCAR CCSM3
in which greenhouse gases are held fixed at 1990 levels. In
the analyses shown here, the climate change simulations
are examined from 2011 to 2061, and the control simulation is examined for the last 500 years of the integration.
Seasonal-mean surface air temperature and precipitation
do not exhibit notable memory from one year to the next
at virtually all terrestrial locations in the control run (notshown). So, in practice, the predicted 95% margins oferror for the 50-yr trends derived from Eq. (7) are roughly
the same as the interannual standard deviations in thecontrol run [according to Eq. (8)].
Figure 3a shows the ensemble-mean 50-yr trends insurface air temperature from 2011 to 2061 averaged overall 40 members in the CCSM3 large ensemble. The
ensemble-mean trends have been discussed in previous
work (Deser et al. 2012b) and are shown here to provide
context for the amplitude of the internal variability. The
warming during the first half of the twenty-first century
is projected to be largest over the Northern Hemisphere,
where it exceeds approximately 3 K (50 yr)21 over muchof northern North America and Asia (Deser et al. 2012b;
Kirtman et al. 2014; Collins et al. 2013).
Figure 3b shows the actual two-tailed 95% margins oferror for the 50-yr trends found by 1) calculating thestandard deviations of the trends derived from all 40
ensemble members and 2) multiplying the standard
deviations by a factor of 2 (95% of the normal distribution lies within about two standard deviations of the
population mean). Note that Fig. 3b is identical to
the top-left panel of Fig. 1 multiplied by a factor of 2. The
gray bars in the surrounding panels indicate the histograms of the simulated trends at grid boxes collocatedwith the indicated cities. The actual margins of error forthe trends are due entirely to the internal variability in
the NCAR CCSM3—that is, they are not due to differences in the forcing or the model used in the simulations.
As such, they provide a quantitative estimate of the roleof internal variability in future climate trends (Deseret al. 2012a,b). By construction, the means of the histograms are equal to the trends in Fig. 3a and the

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standard deviations of the histograms are equal to
0.5 times the actual margins of errors shown in Fig. 3b.
At many terrestrial locations, the margins of error due tointernal variability are approximately 50% as large as
the forced signal (cf. Figs. 3a and 3b).
Figure 3c shows the predicted 95% margins of error forthe 50-yr trends found by applying the analytic model
given by Eq. (7) to the statistics of the control simulation.
Stippling indicates regions where the predicted margins
are not significantly different from the actual margins(see the appendix for details). The blue probability density functions in the surrounding panels show the corresponding predicted Gaussian distributions of the trends
at grid boxes collocated with the indicated cities, where
95% of the distributions lie between positive and negative e95%. Comparing Figs. 3b and 3c, it is clear that 1) themargins of error predicted by applying Eq. (7) to thestatistics of the control run provide a remarkably accurate
prediction for 2) the margin of error on the trends insurface air temperature derived from the large ensembleof climate change simulations. Over much of the globe,
the predicted margins of error are statistically indistinct
from the actual margins.
Figure 4 shows analogous results for October–Marchmean precipitation. The ensemble-mean trends (Fig. 4a)are consistent with increases in precipitation in the deep
tropics and high northern latitudes juxtaposed against
decreases in precipitation in the subtropics (Kirtman
et al. 2014; Collins et al. 2013; Held and Soden 2006).
As is the case for surface air temperature, the predictedmargins of error found by applying Eq. (7) to the statistics of the control run provide a remarkably accurate
estimate of both the spatial pattern and amplitude of
the actual margins of error throughout much of the
globe (cf. Figs. 4b and 4c). The predicted margins oferror are statistically indistinct from the actual margins
over most terrestrial locations. The apparent bimodality in the ensemble member trends at Melbourne(Fig. 4, bottom-right graph) is not reproducible inresults for adjacent grid boxes and is thus likely an
artifact of sampling variability.
We reproduced the above analyses for the boreal summer season months April–September (results not shown).
To first order, the similarities between the predicted andactual margins of error in April–September mean surface
air temperature and precipitation are comparable to thoseindicated in Figs. 3 and 4.

4. Application to observations
The results in the previous section indicate that therole of internal variability in a large ensemble of climatechange simulations can be quantified to a high degree of6448

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FIG. 3. Using the control run to estimate the 95% margins of error on 50-yr trends in October–March mean surface temperature. (a) The
forced response defined as the linear trends in October–March mean surface temperature averaged over all 40 ensemble members inkelvin per 50 yr. (b) The actual 95% margins of error on the 50-yr trends derived from all 40 ensemble members. (c) The predicted 95%
margins of error on the 50-yr trends derived by applying Eq. (7) to the statistics of the control run. Stippling indicates regions where thepredicted margins are not significantly different from the actual margins (see appendix for details). (left),(right) The probability distribution functions of the 50-yr trends at grid boxes collocated with the indicated cities. The gray bars denote the histograms derived from all
40 ensemble members. The standard deviations of the blue curves correspond to the predicted margins of error on the trends found by
applying Eq. (7) to the statistics of the control run; the means of the blue curves correspond to the ensemble-mean trends. The areas underthe blue curves are normalized so that they match the areas under the attendant gray bars.

accuracy from the statistics of the variability in an unforced control simulation. The results highlight the importance of simulating correctly the internal variability
in a control simulation: if the standard deviation and/or

autocorrelation of the simulated internal variability arebiased relative to the observations, then those biases
will project directly onto the uncertainty in simulationsof climate change. Since model simulations inevitably

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FIG. 4. As in Fig. 3, except for October–March mean precipitation. Trends are expressed in millimeters per day per 50 yr.

contain biases, the internal variability of the real world isarguably best estimated from the real world itself—that is,from observations.
In this section, we apply the analytic model to estimatesof internal variability derived from two observational
data sources: 1) precipitation data from the Global
Precipitation Climatology Project (GPCP), version 2.2,
combined precipitation dataset (Adler et al. 2003) and2) surface air temperature data from the HadCRUT4
dataset (Kennedy et al. 2011; Osborn and Jones 2014).

The precipitation data are analyzed on a 2.58 3 2.58 meshand were obtained from the NOAA/ESRL Physical Sciences Division; the surface air temperature data are analyzed on a 58 3 58 mesh and were obtained from the
Climatic Research Unit at the University of East Anglia.
The observed internal climate variability is assumed to
be closely approximated by the statistics of the detrended, seasonal-mean gridpoint values over the period
1979–2013. In principle, 1) the anthropogenic forcing of
the past few decades is not perfectly linear, and 2) the6450

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FIG. 5. Using observations to estimate the 95% margins of error on 50-yr trends in October–March mean surface temperature. (a) The
predicted 95% margins of error on the 50-yr trends derived by applying Eq. (7) to the statistics of the observations. (b) The predicted 95%
margins of error derived by applying Eq. (7) to the statistics of the control run. The results in (b) are as those shown in Fig. 3c, except that 1)
the stippling indicates regions where the modeled and observed margins of error are significantly different from each other at the 95%
confidence level (ratios .1.5:1 or ,1:1.5 exceed the 95% confidence level based on a test of the F statistic assuming one degree of freedom
per year) and 2) the model output has been interpolated to the same mesh as the observations. (left),(right) The probability distributionfunctions of the 50-yr trends for grid boxes collocated with the indicated cities. Solid and dashed curves denote the distribution functions
predicted by applying Eq. (7) to the statistics of the observations and the interpolated control simulation output, respectively. Distributions are normalized so that they have the same area.

amplitude of the internal variability on decadal timescales may be underestimated in the relatively short
1979–2013 period (the GPCP precipitation data are
available only after 1979). However, in practice, 1) thestatistics of the gridpoint surface air temperature andprecipitation observations are effectively identicalwhether the anthropogenic signal is modeled as a firstorder (linear trend) or second-order polynomial fit,and 2) variations on decadal time scales account for a
relatively small fraction of the total variance in surface
air temperature and precipitation on regional scales(not shown).
Figures 5 and 6 compare the 95% margins of error forthe 50-yr October–March mean surface air temperatureand precipitation trends derived from the observations(top-center panels and solid distributions in left andright panels) and the CCSM3 control simulation (bottomcenter panels and dashed distributions in left andright panels). As is the case for the control simulation
output, observed October–March mean surface airtemperature and precipitation do not exhibit statisticallysignificant memory from one year to the next at virtually
all terrestrial locations (not shown). Hence, in practice1) the 95% margins of error on the 50-yr trends predicted by applying Eq. (7) to the observations are effectively equal to 2) the standard deviations of the(detrended) October–March mean observations, according to Eq. (8). Note that the results in Figs. 5b and6b are identical to those shown in Figs. 3c and 4c, respectively, except that 1) the stippling in Figs. 5b and 6b
indicates regions where the modeled and observed interannual variances are significantly different from each

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FIG. 6. As in Fig. 5, but for October–March mean precipitation. The observations have been interpolated to the same mesh as themodel output.

other at the 95% confidence level (see Fig. 5 caption)and 2) the control simulation output used in Figs. 5 and6 has been interpolated to the same mesh as the observations before calculating the interannual standard
deviations (i.e., the temporal variance of area-meansurface air temperature and precipitation generally
decreases when averaged over successively larger
spatial regions).
The margins of error predicted by the CCSM3 control
simulation and observations exhibit similar spatial patterns but have significantly different amplitudes overlarge regions of the globe (stippling). For example, thecontrol simulation exhibits significantly different margins of error in surface air temperature over much ofwestern North America, southern Asia, and tropicalSouth America and Africa (Fig. 5). It also exhibits significantly different margins of error in precipitation overmuch of North America, South America, and eastern
Asia (Fig. 6). The differences between the margins oferror predicted by the observed and control interannual
standard deviations are visually apparent at several of

the indicated cities (probability distribution functions).
Comparable differences are found during the April–
September season in both surface air temperature andprecipitation (results not shown).

5. Discussion
The analytic model developed in section 2 is based on
three primary assumptions. First, it assumes that theinternal variability is roughly Gaussian and is not dominated by, say, bimodal or oscillatory behavior. The climate system exhibits various forms of quasi-periodic
variability other than the seasonal cycle [e.g., theMadden–Julian oscillation (Zhang 2005) and El Niño–
Southern Oscillation]. But a substantial fraction of climate variability is well modeled as a Gaussian process,particularly at extratropical locations (Hartmann and Lo
1998; Feldstein 2000; Newman et al. 2003) and on interannual time scales. The strong similarities between
the actual and predicted margins of errors in Fig. 4
suggest that even precipitation is sufficiently Gaussian

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on seasonal-mean time scales to justify the assumptions
that underlie the analytic model.
Second, the analytic model assumes that the signatureof climate change in surface climate can be estimatedfrom linear trends. In principle, the analytic model could
be generalized to a higher-order polynomial fit than alinear trend. But to zeroth order, the predicted changes
in surface climate over the next 50 yr project strongly
onto linear trends. Linear trends are widely used to estimate the amplitude of climate change in the climate
literature, including all of the IPCC reports.
Third, it assumes that the standard deviation and autocorrelation of the internal variability are stationary intime. There is evidence that the standard deviation of
surface air temperature will change over select locations
in response to climate change, with decreases in temperature variance over the high latitudes of the Northern
Hemisphere during winter (e.g., Gregory and Mitchell
1995; Screen 2014; Schneider et al. 2015) and increases
over various terrestrial regions in summer (Fischer and
Schär 2009). The most noticeable changes in interannual
variance in the NCAR CCSM3 large ensemble of climatechange simulations are found over eastern Europe/
western Russia, where the standard deviations of surface
air temperature decrease during the forcing period (seeFig. A1 in the appendix). But even in this region, thedifferences between the predicted and actual margins oferror are not statistically significant (Fig. 3c). As demonstrated in Figs. 3 and 4, the analytic model provides aremarkably robust estimate of the uncertainty in simulated climate change due to internal variability over thevast majority of the globe.
In fact, the analytic model given by Eq. (7) may beviewed as a ‘‘null hypothesis’’ for the role of internalvariability in future climate on any time scale in any
physical field that is well modeled as a Gaussian process.
The results in Figs. 7a and 7b show trends in temperatureand precipitation as a function of trend length at themodel grid point collocated with Los Angeles (results forthe same grid point are highlighted in Figs. 1, 3, and 4).
Red dots indicate trends derived from all 40 ensemble
members for the period starting 2011 and ending on theyear indicated on the abscissa. For example, the red dots
at 2061 indicate trends derived from all 40 ensemble
members for the period 2011–61 (in units per 50 yr) andare identical to the results used to generate the grayhistograms in the top-left panels of Figs. 3 and 4. The
dashed lines indicate the ranges of the trends given by
bnt 6 e, where b is the ensemble-mean trend and e isfound by applying Eq. (7) to the statistics of the control
run. Figure 7c indicates analogous results for October–March mean precipitation averaged over grid points
that lie within the Colorado River watershed. Fig. 7d

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indicates results for the model North Atlantic Oscillation (i.e., northern annular mode), calculated here as thestandardized difference between October–March meansea level pressure anomalies collocated with Iceland andthe Azores.
Figure 7 highlights two main points: 1) it makes clearthat the analytic model given by Eq. (7) provides a remarkably accurate estimate of the uncertainty in climatechange projections on an array of time scales, not justthe 50-yr time scale exemplified in Figs. 3 and 4, and 2) ithighlights the utility of the analytic model for estimating
the uncertainty in surface climate trends averaged over a
relatively large spatial region and in the large-scale atmospheric circulation. The uncertainties in projectedchanges in precipitation averaged over the Colorado
River watershed (Fig. 5c) and in the North Atlantic
Oscillation (Fig. 5d) indicated by the large ensemble areclosely approximated by applying Eq. (7) to the statisticsof the control simulation.

6. Concluding remarks
Uncertainty in projections of future climate change
can arise from three different factors (e.g., Hawkins andSutton 2009): 1) model uncertainty—the uncertainty
that arises from differences between climate models, 2)
scenario uncertainty—the uncertainty that arises fromdifferences in forcing scenarios, and 3) uncertainty due
to internal variability—the uncertainty that arises from
unpredictable internal variability in the climate system.
Model and scenario uncertainty can be reduced asmodels and our understanding of future forcing scenarios improve. The uncertainty due to internal climate
variability arises from the chaotic nature of the climate
system and—to the extent that it is unpredictable—ispresumably irreducible.
The analytic model given by Eq. (7) provides a simple basis for estimating the uncertainty in climatechange projections due to internal variability using thestatistics of the unforced climate. It provides a zerothorder estimate of the uncertainty in future trends due
to internal variability in a range of physical fields including, for example, precipitation averaged over awatershed, surface air temperature averaged over anagricultural region, and the atmospheric circulation at
middle latitudes.
The analytic model is based on three assumptions:
1) the internal variability is well modeled as Gaussian,2) the signature of anthropogenic forcing in surface climate
can be modeled as a linear trend, and 3) the standard
deviation and/or autocorrelation of the internal climate
variability do not change in response to anthropogenic forcing. The robustness of the model to all three15 AUGUST 2015

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FIG. 7. Actual trends (red) and the predicted ranges of the trends (black dashed lines) infields indicated. Trend periods start in 2011 and end on the year indicated on the abscissa. Units
are change over the length of the trend period. Red dots indicate the actual trends derived fromall 40 ensemble members. Black dashed lines indicate the predicted ranges of the trends foundby applying Eq. (7) to the statistics of the control run as a function of trend length. The margins
of error on the trends indicated by the black dashed lines [e in Eq. (1)] are given by the statisticsof the control run; the amplitudes of the forced signal are given by the ensemble-mean trends.
The units on the North Atlantic Oscillation index are standard deviation of the interannual
variability.

assumptions is strongly supported by the close similarities between 1) the uncertainties in climate trends estimated from the statistics of an unforced control
simulation and 2) the uncertainties found in a large
ensemble of climate change simulations. To the extent
that the assumptions of the analytic model hold, theresults suggest that large ensembles provide little information on the role of internal variability in futureclimate that cannot be inferred from the statistics of an
unforced control simulation.
The analytic model also makes clear the direct relationship between 1) biases in a model control simulationand 2) the uncertainty in projections of future climatechange. If the amplitude of internal climate variability isbiased relative to observations in the control simulation of agiven climate model, then so is the uncertainty in futureclimate change simulated by the same model.

Both observational and climate model estimates ofinternal climate variability have shortcomings. The
observational record provides robust estimates of internal climate variability on intraseasonal and interannual time scales. However, the record is relatively
short and thus provides limited insight into internal
climate variability on multidecadal and longer timescales. In contrast, numerical models provide long records for estimating the amplitude of internal climate
variability. However, models frequently exhibit systematic biases in their representation of internal climate variability not only on decadal time scales (e.g.,
Laepple and Huybers 2014) but on interannual timescales as well (e.g., Figs. 5 and 6). To the extent thatimperfect observational records provide a more realistic representation of the real world than a climate
model, it follows that the role of internal variability in6454

JOURNAL OF CLIMATEVOLUME 28

FIG. A1. Ratio of variances between the periods 2051–60 and 2011–20 from the NCAR 40-member ensemble ofclimate change simulations. The variances are calculated from the pooled detrended seasonal-mean data from all
ensemble members. Ratios greater than ;1.4 and less than ;0.71 (indicated by stippling) are significant at the 95%
level based on the F statistic.

future climate trends is arguably best estimated notfrom a long control simulation or a large ensemble ofclimate change simulations, but from observational
estimates of internal climate variability.

it follows that

Acknowledgments. We thank John M. Wallace, Susan
Solomon, and an anonymous reviewer for insightfulcomments on the study. DWJT, EAB, and WEF aresupported by the National Science Foundation Climate
Dynamics program.

and thus

APPENDIX
Analysis Details
a. Expanding the standard deviation of the time axis
The standard deviation of the time axis [the denominator in Eq. (2)] can be expanded as a function of nt,
since the time axis corresponds to a series of consecutive
integers. Using two formulas for consecutive integersnt

n (n 1 1)
å i 5 t t2
i51

nt

and

n (n 1 1)(2n 1 1),
åi 5 t t 6 t
i51
2

n

(n 1 1)1 t,
i5 å i5 t2nt i51

nt

nt

nt

nt

å (i 2 i) 5 å i 2 å 2i  i 1 å i 2
2

i51

i51

2

i51

i51

nt

nt

i51

i51



nt 1 1 2
2i51
nt

5 å i2 2 (nt 1 1) å i 1 å

nt (nt 1 1)(2nt 1 1) nt (nt 1 1)2 nt (nt 1 1)2
2
1
6
2
4
3
n 2 nt.
5 t
12

5b. Accounting for autocorrelation in the standard
error of the residuals
A simple and commonly used method for accounting
for the bias in the sample standard deviation is to substitute an effective sample size neff for the sample size nt

15 AUGUST 2015

THOMPSON ET AL.

in the calculation of the standard deviation [i.e., Eq. (5)].
If x(t) is well modeled as Gaussian red noise (and thus its
autocorrelation function decays exponentially with lag),then neff can be approximated as (Mitchell et al. 1966;
Santer et al. 2000)

1 2 r1
neff ; nt.
1 1 r1
Note that the time between independent samples in a
red noise time series is equal to approximately 2 times
the e-folding time of the autocorrelation function(Leith 1973).
Substituting neff for nt in the denominator of Eq. (5)yields the following relationship between the standard
error se and the standard deviation s of the residuals:
se 5 sg(nt , r1 ) ,
where

2

31/2

n 22
5
g(nt , r1 ) [ 4  t 
1 2 r122nt
1 1 r1.

c. Estimating statistical significance in Figs. 3c and 4c
The stippling in Fig. 3c denotes regions where
the predicted margins shown in Fig. 3c fall within the
95% confidence intervals of the actual margins shownin Fig. 3b. The 95% confidence intervals of the actual
margins are found by 1) calculating the 95% confidence
intervals of the interannual standard deviations fromthe control simulation using the chi-squared distribution assuming 40 degrees of freedom and 2) multiplying
the resulting confidence intervals by a factor of 2 toconvert them to confidence intervals of the predictedmargins of error. The stippling in Fig. 4c is found in an
analogous manner.

d. Changes in interannual variance over the 2011–60
periodFigure A1 indicates the ratio of variances between
the periods 2051–60 and 2011–20 from the NCAR
40-member ensemble of climate change simulations. See
caption for details.
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