Charles D. Kolstad David L. Kelly Michael E. Schlesinger Natalia G. Andronova

English

Research Papers in Economics

1
Learning About Climate Sensitivity
From the Instrumental Temperature Record
+
David L. Kelly,
* Charles D. Kolstad,
**
Michael E. Schlesinger
*** and Natalia G. Andronova
***
The debate over the magnitude of anthropogenically induced climate change has raged for
over a century (1-6).  Today considerable uncertainty remains about the magnitude of
greenhouse-gas-induced climate change, particularly the climate sensitivity – the
equilibrium change in global-mean surface temperature per unit of radiative forcing.  The
rapidity at which uncertainty in the climate sensitivity is resolved has significant policy
implications.  If resolution is expected soon, deferring action until the picture is clearer may
be prudent.  If uncertainty is likely to be resolved only slowly, then action today on the
basis of expected costs and damages may be the wisest course.  Here we use a Bayesian
learning model, the instrumental temperature record, and IPCC scenarios of future
emissions of greenhouse gases and SO2 to estimate the time required  to reduce the
uncertainty in the climate sensitivity.  We find that more than half a century is required to
be 95% confident that the true value of the climate sensitivity lies within ±20% of the
estimated value.  Further, accelerated control of greenhouse-gas emissions significantly
slows this rate of learning, while control of SO2 emissions accelerates it.
                                               
+ Work supported by US Department of Energy Grant DE-FG03-96ER62277 to the University of
California, Santa Barbara.
* Department of Economics, University of California, Santa Barbara, CA 93106 and Department
of Economics, University of Miami.
**Department of Economics, School of Environmental Science and Management, and
Environmental Studies Program, University of California, Santa Barbara, CA 93106.
*** Department of Atmospheric Sciences, University of Illinois, Urbana, IL 61801.2
It might seem that it is impossible to shed any light on future resolution of uncertainty;  after
all, who can know what future research on climate will yield?  Our approach is to assume that the
process of learning about climate sensitivity that has occurred during the last 100 years will
continue to be driven by the instrumental temperature record.  Given this assumption, we can use
standard statistical methods to quantify the current uncertainty and the rate at which it can be
expected to be resolved.
One approach to estimating the climate sensitivity, l, is to use a simple physical model to
simulate the instrumental temperature record in a best-fit, maximum-likelihood sense (7).  One
such simple physical model is (8, 9):
  
Tt = Tt-1+
1
a[ (Ft + F SO4S t )-
Tt-1- G
l ]- K(Tt-1 -Ot-1) (1a)
   Ot =Ot-1+ L(Tt-1-Ot-1) (1b)
where    Tt is the annual global temperature (￿C) difference of the upper ocean in year t from the
1961-1990 average temperature, taken to be synonymous with the surface-air temperature
difference, and    T0 = O0 = G is the initial temperature difference in some initial year;    Ot is the
corresponding temperature difference for the deep ocean;     Ft is the  radiative forcing by
greenhouse gases (GHGs), including tropospheric ozone;    St is the emission rate of sulfur dioxide
(SO2), normalized by its value in 1990 (75 TgS/yr), which is converted to sulfate aerosol in the
atmosphere;    FSO4 is the radiative forcing by sulfate aerosols in 1990; a is the heat capacity of the
upper ocean; and K and L equal the coefficient of heat transfer between the upper and deep ocean
divided by their respective heat capacities.  Equation (1) can be written as the following statistical
model (10),
  
Tt =
G
al
+ b1Tt-1 + b2Ft +b3S t + KOt-1+ et , (2a)3
   Ot =Ot-1+L(Tt-1- Ot-1), (2b)
where    et  is an error term of mean zero.  Equation (2) could be estimated from observed records
of    Tt and    Ot, from which    l =b2 / (1- b1 -K) and    FS04 = b3/ b2.  However, the absence of an
observational record for the deep-ocean temperature,     Ot, makes it impossible to statistically
estimate all of the coefficients.  Because it is unlikely that    Ot has changed very much over our
sample (1856-1995), we make the assumption that    Ot is constant and thus roll    KOt-1 into the
constant term.  Equation (2a) thus reduces  to:
   Tt =b0 + b1Tt-1 +b2Ft + b3St +et = bX t +et , (3)
where    b = (b0,b1,b2,b3) and     Xt = (1,Tt-1,Ft,S t )
T.  Equation (3) can be estimated from the
historic record.  We perform ordinary least squares (OLS) estimation of  Eq. (3), using the
instrumental temperature record (1856–1995; 140 observations)  (11)  and the historical GHG
forcing and SO2 emissions (Fig. 1).  Results are summarized in Table 1.  Also shown in the table
is the implied value of l.  Since we were unable to estimate K in Eq.(2), due to an absence of
deep ocean temperature observations, we have used a value of K for computing l drawn from our
much more complex energy-balance-climate/upwelling-diffusion-ocean (EBC/UDO) model (7,12-
20).
The more familiar T2X = l F2X (with F2X = 4.39 Wm
-2) is the equilibrium temperature rise
from a doubling of GHG concentration from preindustrial levels (Table 1).  Its value of 2.3
0C with
a standard error of 0.7
0C is not inconsistent with IPCC estimates.  However, as we shall see later,
the time to reach equilibrium from a GHG shock, as implied by the coefficients in Table 1, is
unrealistically rapid.
We now define learning, the resolution of uncertainty.  We can never be perfectly certain of
our estimate of the climate sensitivity.  Statistically, at any point in time we will only have a4
certain level of confidence in our estimate.  We use the 95% confidence level from statistics as our
criterion for having learned.  We suppose uncertainty is resolved at the time in the future when,
for a standard 95% level of confidence, the true climate sensitivity l will first lie between    (1- c)ˆ  l 
and    (1+ c)ˆ  l , where    ˆ  l  is the then-estimated climate sensitivity and c defines any given confidence
interval.  Note that uncertainty in the true value of l does not mean that we are unable to reject
the hypothesis that l is zero.
To be more precise, we are interested in the first point in time at which we fail to reject the
hypothesis, H0 at the 95% level, where    H0 = {(1- c)ˆ  l < l< (1+c)ˆ  l  } .   For example, when with
95% probability will the true climate sensitivity first lie within ±20% (c = 0.2) of the estimated
sensitivity?  We use Bayes Rule (which uses information optimally and thus is the fastest way to
resolve uncertainty from the temperature record) to estimate b and then compute the median time
to achieve a particular confidence level for several scenarios of future GHG concentrations and
SO2 emissions, denoted High, Med, and Low, corresponding to their levels (Fig. 1).  Med (IS92a)
is the IPCC business-as-usual case with expansion in both    Ft and    St.  Low (IS92c) is the most
aggressive control scenario, both for GHGs and SO2.  The High (IS92e) case has the largest
increase in    Ft and    St.
We take a series of draws from the distribution of our OLS estimate of  b and the
distribution of e in the present (1995).  Each draw    (b1994,e)j together with the assumed future
   Ft and    St is sufficient to generate from Eq. (3) a sample century (1996-2095) of temperatures
  
Tt ( )j.  Letting   
ˆ  b  t be the OLS estimate of b based on data through year t > 1995 for a particular
trajectory, Bayes Rule (21) defines how   
ˆ  b  t will evolve over time as a new observation    (Yt,Xt) is
added to the data set, assuming e  is normally distributed with mean zero and variance    1/r:
  
ˆ  b  t = [ˆ  P  t-1 + r(X t ¢  X  t )]
-1[ˆ  P  t-1
ˆ  b  t-1 + rYtX t] (6a)
  
ˆ  P  t = ˆ  P  t-1+ r Xt ¢  X t ( )   . (6b)5
Here    ˆ  P  t is the precision of the estimate   
ˆ  b  t of b, and prime denotes the transpose.  For a particular
value of c and    (b1994,e), the minimum value of t for which 
  
Prob (1- c)ˆ  l < l < (1+ c)ˆ  l  { }‡ 0.95,
   tj, is then determined numerically from the sampling distribution for    ˆ  l  which is generated by
Monte Carlo method (1000 draws) from the distribution of the OLS estimate of b.
Let    t
+(c) be the median value of    tj taken over 1000 trajectories.     t
+(c) is the time at
which we can expect the true climate sensitivity to be within a prespecified confidence interval
(–c%).  Figure 2a shows    t
+(c) for each of the three scenarios, that is, the median year when we
are 95% sure that the true value of climate sensitivity lies within a specified range (–c%) of the
estimated value.  The results are striking: reducing climate sensitivity from its current uncertainty
of approximately –50% to –20% will take a long time.
We see from Figure 2 that learning that the value of climate sensitivity lies within –20% of
the estimated value takes nearly a century with the High or Med scenarios.  This time is cut to
approximately 30 years for the low scenario.  The High scenario has high levels of GHGs which
increase the temperature signal, but also high SO2 emissions which reduce the signal.  The net
effect is that the signal is weaker for “High” than for “Low.”
Figure 2b decouples the effects of GHG concentrations from sulfate aerosols by fixing SO2
emissions at the levels associated with Med, the mid-range emission scenario.  Figure 2b shows
that controlling GHG emissions slows down the rate of learning about climate sensitivity.  The
most rapid learning occurs when we have the largest signal from GHGs: over 50 years to achieve
a –20% confidence band.  Aggressive control makes the climate change trend more difficult to see
within the noisy temperature record.  The learning model has difficulty discerning between
random warm and cold years and the warm years due to emission of GHGs.  Figure 2c shows the
result for GHG concentrations set at their level in the Med scenario, and varying SO2 emission
rates.  Here aggressive control of SO2 increases the upward trend in temperature and thus makes
the climate change more visible within the noisy temperature record.6
As mentioned earlier, one troubling feature of Eq. (3) as estimated with the instrumental
record is that the response from an instantaneous and permanent increase in GHG forcing is
surprisingly, and unrealistically, rapid.  Since the effect on the temperature n years after a
hypothetical shock occurs is     b1
nT0, where T0 is the temperature at the time the shock occurred,
the closer b1 is to unity, the slower the response to a forcing shock.  Because of this, we have also
estimated Eq. (3) assuming a value of b1 that is consistent with the physical model; in fact, we use
a value obtained from our more detailed EBC/UDO model.  We obtain b1 estimates by simulating
the effects of a CO2 doubling for 245 years (equal in time to 1856-2100), using the EBC/UDO
model, for three different assumed temperature sensitivities, T2x, and then use the generated data
to estimate the appropriate coefficient values in Eq. (1) with    Ot neglected.  For T2x = 1.5
oC,
2.5
oC, and 4.5
oC, the results imply  b1 = 0.87941, 0.92277 and 0.95167, respectively.
Consequently, we have also estimated Eq. (3) assuming a fixed value for b1 of 0.9, and then
estimated the equation again assuming b1 = 0.94.
Table 2 shows how the learning times vary with the assumed value of b1.  The table clearly
demonstrates that increasing the value of b1 only serves to slow down learning about climate
sensitivity.  This is as would be expected since fixing b1 leads to a poorer fit of Eq. (3) to the
instrumental record, and thus more error.  This logically increases the amount of time necessary to
reduce the error in the estimate of climate sensitivity.
In conclusion, using a simplified model of learning we have shown that achieving some
confidence regarding the value of climate sensitivity may take many decades.  Although additional
factors could be included in our model (such as volcanoes, the sun, and regional temperature
variations), the results suggest a very slow resolution to the question of the magnitude of climate
sensitivity.  This notwithstanding, we have shown that accelerated control of GHGs and SO2
emissions will have opposing effects on the rate at which we learn about climate sensitivity, the
former slowing learning and the latter accelerating it.  The policy implication is that if we wait7
until uncertainty is resolved before controlling emissions of greenhouse gases, we may wait a very
long time.
Acknowledgments.  We thank T. Wigley for providing the GHG forcing and SO2 emissions used
in the 1995 IPCC report, and D. Parker and M. O'Donnell the temperature observations also used
in the 1995 IPCC report.  We are also grateful to D. Steigerwald for comments on the paper.
This work was supported by the U.S. Department of Energy under Grant DE-FG03-96ER62277,
and the U.S. National Science Foundation and the Office of Biological and Environmental
Research of the U.S. Department of Energy under grant ATM-9522681.8
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Figure Captions
Figure 1. (a) Global average temperature relative to 1961-1990 (
0C); (b) Radiative forcing due to
greenhouse gases (GHGs) (22), and (c) SO2 emission (22) normalized by its 1990 value of 75
TgS/yr.
Figure 2 (a) Median year    t
+(c) at which we can be 95% sure that the true climate sensitivity is
within ±c% of the estimated climate sensitivity, under emissions assumptions associated with the
Low, Med and High scenarios. (b) as in (a), except with the SO2 emission scenario fixed at the
Med(IS92a) scenario; (c) as in (a), except with the CO2 emission scenario fixed at the  Med
(IS92a) scenario. Parameter estimates as in Table 1.11
 Table 1.  Ordinary Least Squares Estimates of Parameters in Equation (3)
Quantity Estimate Standard Error
   b0 -0.2209 0.0428
   b1 0.5589 0.0712
   b2 0.2295 0.0751
   b3 -0.2517 0.1455
e NA 0.1008
l 0.5350 0.1592
   T2x 2.3486 0.6988
NB: 1.     l ”b2 / (1- b1 -K)  computed using K = 0.012 from complex energy-balance-
climate/upwelling-diffusion-ocean (EBC/UDO) model (7,12-20).
2.    T2x ” lF2x, with    F2x = 4.39Wm
-2.
Table 2. Estimated Learning Times for Three Models.
c    b1 = 0.5589*    b1 = 0.9    b1 = 0.94
% IS92e IS92a IS92c IS92e IS92a IS92c IS92e IS92a IS92c
5 >2100 >2100 >2100 >2100 >2100 >2100 >2100 >2100 >2100
10 >2100 >2100 2077 >2100 >2100 >2100 >2100 >2100 >2100
15 2096 2085 2056 >2100 >2100 >2100 >2100 >2100 >2100
20 2089 2073 2033 >2100 >2100 >2100 >2100 >2100 >2100
30 2067 2039 2011 >2100 >2100 2081 >2100 >2100 >2100
50 2001 2001 2000 >2100 2081 2044 >2100 >2100 >2100
*Results from Eq. (3) with all parameters estimated.  Other cases involve b1 fixed at indicated
value during estimation of Eq. (3).
NB:
>2100 states that learning is resolved at some time after the year 2100, due to a lack of estimates
of greenhouse gas and sulfate forcing post-2100.
Learning time is defined as the first year in which we expect to be 95% confident that the true
climate sensitivity is within ±c% of the estimated climate sensitivity.12
Figure 1:13
Figure 2:

Learning About Climate Sensitivity from the Instrumental Record

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