Benefits of mitigation for future heat extremes under RCP4.5
compared to RCP8.5
Claudia Tebaldi1, Climate and Global Dynamics Division, National Center for
Atmospheric Research,
Michael F. Wehner, Lawrence Berkeley National Laboratory,
Supplementary Material
1
PO BOX 3000, Boulder, CO, 80307 USA tebaldi@ucar.edu 1-303-497-1710
Figure S1: Like Figure 2, but for extremes computed as annual maxima of 1-day
minimum temperature.
Figure S2: Like Figure 3, but for extremes computed as annual maxima of 3-day
average minimum temperatures.
Figure S3: Like Figure 4, but for extremes computed as annual maxima of 1-day
minimum temperature.
Figure S4: Like Figure 5, but for extremes computed as annual maxima of 3-day
average minimum temperatures.
2025
2050
2075
RCP4.5
0.6
1.3
1.9
RCP8.5
0.7
1.9
3.3
Table S1. Projected 21st century changes in global mean surface air temperature (oC)
by CESM1-CAM5 relative to the baseline period 1986-2005.
One-day
Minimum
Temperature
One-day
Maximum
Temperature
RCP RCP
4.5 8.5
2025
2050
2075
Three-Day
Minimum
Temperature
RCP RCP
4.5 8.5
Three-Day
Maximum
Temperature
RCP RCP
4.5 8.5
RCP RCP
4.5 8.5
1
18
47
2
31
50
1
67
85
2
80
90
2
2
0
1
0
2
5
0
10
21
0
22
32
0
3
0
0
0
0
0
0
1
1
0
4
7
0
4
0
0
0
0
0
0
0
0
0
0
1
0
1
87
98
46
93
96
42
99 100
47
98
99
42
2
15
73
3
31
78
3
54
93
3
72
93
3
3
1
25
0
6
37
0
10
52
0
25
60
0
4
0
3
0
1
10
0
1
15
0
6
30
0
5
0
0
0
0
3
0
0
2
0
1
10
0
6
0
0
0
0
1
0
0
1
0
0
3
0
7
0
0
0
0
0
0
0
0
0
0
1
0
1
97 100
94
96 100
94 100 100
96 100 100
94
2
60
98
50
76
96
56
82 100
49
99
98
53
3
11
87
6
28
91
13
40
95
7
92
95
11
4
2
57
1
7
67
1
8
72
1
55
81
1
5
0
26
0
2
39
0
2
46
0
24
54
0
6
0
6
0
0
20
0
1
18
0
8
34
0
7
0
2
0
0
8
0
0
5
0
2
17
0
8
0
0
0
0
3
0
0
1
0
1
7
0
9
0
0
0
0
1
0
0
0
0
0
3
0
Table S2: Percentage of land affected by changes in the magnitude of the 20-year
event by thresholds from 1° to 9°C. Differences in the changes at a given time between
the two scenarios are summarized in the shaded column. All fractions are computed by
considering only those grid cells with statistically significant changes (and dividing by
the total extent of the global land area). The differences between scenarios are not
simply the result of subtracting the numbers to the left of them, since the significance
of that difference is taken into account.
Statistical analysis
The GEV distribution has the following form:
𝐺−𝐺
𝐺(𝐺) = exp {− [1 + 𝐺 (
𝐺
)]
−1/𝐺
},
where G(z) indicates P(X<=z), the cumulative distribution function of the variable X,
𝐺(𝐺−𝐺)
> 0} and the parameters 𝐺, 𝐺 and 𝐺
satisfy the conditions −∞ < 𝐺 < ∞, 𝐺 > 0 and −∞ < 𝐺 < ∞.
defined on the domain {𝐺: 1 +
𝐺
The three GEV parameters have a one-to-one relation to 20-year return values
(which we will often abbreviate as RVs) or to the new return periods (RPs) for
previously estimated 20-year return values.
This is readily seen by considering that the inverse of a return period coincides with
a quantile of the distribution. For example, the 20-year return value coincides with
the 95th percentile of the distribution. According to the GEV form, for a generic p
(0.95 in the above example)
𝐺
𝐺𝐺 = {
𝐺 − [1 − {−𝐺𝐺𝐺(1 − 𝐺)}−𝐺 ],
𝐺
𝐺 − 𝐺𝐺𝐺𝐺{−𝐺𝐺𝐺(1 − 𝐺)},
𝐺≠0
𝐺 = 0.
Therefore, estimates and uncertainties derived for the three parameters translate to
estimates and uncertainties for zp for a given p (or for p given z). In this study,
uncertainty in return values and their changes are estimated by a parametric
bootstrap technique where the 3 GEV parameters are first estimated for the actual
samples. This is followed by the construction of 100 synthetic samples of either 150
(RCP4.5) or 300 (RCP8.5) values using a random number generator distributed by
the fitted GEV distribution of the actual sample. Each of these synthetic samples is
then fit to GEV distributions and the uncertainty in the actual return values
estimated by the standard errors of the bootstrap distributions of synthetic return
values (Hosking and Wallis 1997).
Changes in return times are calculated in a similar fashion where the future extreme
value samples are fit to GEV distributions by the L-moments method. Then the
return times for each future sample are estimated for the present day 20-year
return values. In the absence of climate change, future return times for
temperatures of these magnitudes would remain to be 20 years. However, in the
warmer worlds considered here, future return times of present day temperature
extremes are always more frequent.
Uncertainty in future return times is also estimated by the parametric bootstrap
method described above. In this case, the uncertainty metric is the standard error of
the return times of the synthetic extreme value samples. A full characterization of
the future return time uncertainties would also combine an estimate of the present
day return value, which we have neglected in this part of our calculations. However,
we find that for 20-year return values of these large ensembles, the uncertainties
are small enough and the changes large enough, that our conclusions would remain
highly statistically significant with the full inclusion of this latter type of uncertainty.
We note here that other studies of extremes have adopted the GEV distribution,
among them Kharin and Zwiers, 2005, and Wehner, 2005).
Validation
We use HadEX data for two ETCCDI indices (see main text), TXx and TNx for the
period 1951-2003 (downloadable from
http://www.metoffice.gov.uk/hadobs/hadex/) to validate CESM output.
The two indices represent yearly maxima of, respectively, maximum and minimum
temperature. Thus we compute yearly maxima from the same period of the
historical simulations of the Large Ensemble, using the same mask as the observed
values after interpolating the CESM grid to the coarser scale of the HadEX dataset
(spaced by 3.75 degrees in longitude and 2.5 degrees in latitude) gathering 53 years’
time series for each of the 30 ensemble members. Ideally we would like both
observed absolute values (i.e. climatological means) and trends over the 50 years
(or multidecadal changes between the two twent year periods of 1951-1970 and
1984-2003) to fall within the envelope of the large ensemble values. The
climatological means (computed either over the entire period or over the last 30
years of the observational record) show large biases when comparing observed to
simulated values, with observations falling for most of the covered areas outside of
the ensemble envelope. Figures S5 and S6 show such results and Table S3 lists the
quantiles of the bias values obtained by computing them for each ensemble member
and considering the distribution of values across ensemble members and gridpoints. In addition, Figure S7 shows the ensemble median values (grid-point by gridpoint) of such biases.
However, when we look at description of change in these two indices in
observations and the model, validation appears more successful: for trends and
multidecadal changes most grid-point values fall indeed within the ensemble range
as Figures S8 and S9 show, and quantile histograms are flatter (we would ideally
want them uniform). Based on these results we consider our analysis of changes
within scenario and comparative changes between scenario on solid grounds, since
it is the behavior of these indices over time and thus the reactivity of the quantities
to forcing and internal variability rather than the absolute values of these quantities
that we want to quantify and characterize.
Percentile
0th
10th
20th
30th
40th
50th
60th
70th
80th
90th
100th
TXx
TNx
TXx
TNx
climatology
climatology
climatology
climatology
(1951-2003)
(1951-2003)
(1974-2003)
(1974-2003)
-27.5
-20.9
-27.5
-20.9
-7.2
-5.7
-7.2
-5.8
-4.4
-3.0
-4.3
-3.1
-3.0
-1.9
-3.0
-2.0
-2.0
-1.2
-1.9
-1.3
-1.1
-0.7
-1.1
-0.7
-0.4
-0.1
-0.2
-0.2
0.5
0.5
0.6
0.4
1.3
1.3
1.5
1.3
2.5
2.7
2.7
2.6
16.1
13.6
14.3
13.8
Table S3: Distribution of bias values across the ensemble members and grid-points for
the two indices TXx and TNx, computed by comparing values of climatological
averages. Two periods are considered (but the results are not significantly different):
1951-2003 and 1974-2003.
Figure S5: results from the comparison of climatological values of TXx (annual
maximum of daily maximum temperatures) from observations (top left panel) and the
large ensemble (whose point-wise minimum, median and maximum are shown in the
three bottom panels). Values are averaged over 1951-2003. The top right panel show
the quantile where observational values fall, point-by-point, within the ensemble, while
the histograms show a distribution of the quantiles in the map (weighted by cosine of
latitude on the left and unweighted on the right).
Figure S6: Like Figure S5, for TNx (annual maximum of daily minimum temperatures).
Figure S7: Ensemble median values (grid-point by grid-point) of biases for the
climatological averages over 1951-2003 of TXx (top) and TNx (bottom).
Figure S8: Like figure S5, but for the comparison of observed and simulated trends in
TXx values. Trends are linear fits over the period 1951-2003, grid-point by grid-point.
Figure S9: Like figure S7, for trends in TNx.
Additional References
Kharin VV, Zwiers FW (2005) Estimating extremes in transient climate change
simulations.
J Climate 18:1156–1173
Wehner, MF (2005) Changes in daily precipitation and surface air temperature
extremes in the IPCC AR4 models. US CLIVAR Variations, 3, pp 5-9. LBNL-61594