70

has attained equilibrium, in response to a doubling of atmospheric CO 2. In the
EM-GC framework ECS is expressed as^16 :

ECSWm

P

=

1 + -

371 2

g

l

.

(2.6)

ECS is often used to compare and evaluate climate simulations. The EM-GC run
shown in Fig. 2.5 has an ECS of 1.73 °C, which means that if CO 2 were to double
(i.e., reach 560 ppm, twice the pre-industrial value of 280 ppm) and if all other
GHGs were to remain constant at their pre-industrial level, then ΔT would rise to a
level about midway between the Paris target (1.5 °C) and upper limit (2.0 °C). As
will soon be shown, ECS is a difficult metric to use for evaluating climate models
because it depends rather sensitively on both aerosol ΔRF and ocean heat content,
both of which have considerable uncertainty.
The top rung of each EM-GC ladder plot also contains a numerical value for
reduced chi-squared (χ^2 ), a parameter that defines the goodness of fit between a
series of observed and modeled quantities. In our framework, χ^2 is defined as:

c

(s

(^212) D
1
1
1

--
́å=
()NNYEARS FITTINGPARAMETERS j )
N
j
YEARS
OBS
()TOBSjjj -DTEM-GC
2
(2.7)
where 〈ΔTOBS j 〉, 〈ΔTEM ‐ GC j〉, and 〈σOBS j〉, represent the annually averaged observed
temperature anomaly, the annually averaged modeled temperature anomaly, and the
uncertainty of the annually averaged observed temperature anomaly, respectively,
and NFITTING PAREMETERS equals 6 for the simulation shown in Fig. 2.4 (four regression
coefficients plus the two parameters γ and κ) and equals 9 for Fig. 2.5 (three addi-
tional regression coefficients). The formula for χ^2 is expressed in terms of annual
averages, rather than monthly values, due to the statistical behavior of the two time
series that appear in the formula.^17
(^16) The derivation is:
ECSRFWm
CO
P CO
CO
P
FINAL
INITIAL
P

• =


• 
  

  =
  (^11) - +
  535
  1
  2 2 2 53
  2
  g
  l
  D
  g
  l
  g
  l
  .ln. 552
  1
  Wm^22371 Wm
  P
  --ln ()= +g.
  l
  if we assume a doubling of atmospheric CO 2.
  The expression for ΔRFCO2 is from Myhre et al. ( 1998 ).
  (^17) For those familiar with statistics, the auto-correlation function of modeled ΔT is compared to the
  auto-correlation function of the measured ΔT. As shown in the supplement to Canty et al. ( 2013 ),
  these functions differ considerably for comparison of measured and modeled monthly anomalies,
  indicating either the presence of a forcing in the system not resolved by the model or else consider-
  able noise in the measurement. These auto-correlation functions are quite similar for comparison
  of measured and modeled annual anomalies, indicating proper physical structure of the modeled
  quantity and appropriate use of χ^2 , if applied to annual averages of both modeled and measured
  anomalies.
  2 Forecasting Global Warming