1) The document analyzes the relationship between extreme precipitation events in the Gulf of Mexico region and climate change indicators like sea surface temperature and atmospheric CO2 levels. 2) It finds a statistically significant relationship, with extreme precipitation events becoming more likely as Gulf SSTs and CO2 levels increase. Using this relationship, it estimates that Hurricane Harvey in 2017 was a "very likely" 1,000-year rainfall event or rarer when accounting for climate change factors. 3) The analysis uses several statistical methods like generalized extreme value distributions and point process models to analyze extreme precipitation event data at different spatial scales and relate the frequency of extreme events to climate change indicators while accounting for seasonal and other factors.

1. The Dependence Between Extreme
Precipitation and Underlying
Indicators of Climate Change
Richard L Smith
Departments of STOR and Biostatistics, University
of North Carolina at Chapel Hill
Joint work with Ken Kunkel
North Carolina Institute for Climate Studies and Department of
Marine, Earth, Atmospheric Sciences , North Carolina State University
SAMSI Climate Extremes Workshop
May 16, 2018
1
1

2. Motivation
• 2017 featured the largest rain storm in history in the Gulf of
Mexico
• It also featured the highest mean annual (July–June) sea sur-
face temperature in history in the Gulf of Mexico (SSTGM)
• Here, we shall demonstrate a relationship among the fre-
quency of extreme storms, SSTGM and atmospheric CO2
• This enables us to answer three questions:
– How extreme was Harvey, taking SSTGM and CO2 into
account?
– To what extent can extreme events be “attributed” to
human influence?
– What are the probabilities of similar events in the future?
2

3. Other references on Hurricane Harvey:
Van Oldenborgh et al, Environmental Research Letters, 2017
Risser and Wehner, GRL, 2017
Emanuel, PNAS, 2017
Some of what I present is included in the book chapter:
Hammerling, Katzfuss and Smith, Climate Change Detection and
Attribution, to appear in Chapman and Hall Handbook of Envi-
ronmental Statistics (A. Gelfand, M. Fuentes, J. Hoeting and R.
Smith, eds)
3

4. Handbook of
Environmental and
Ecological Statistics
HandbookofEnvironmental
andEcologicalStatistics
Edited by
Alan E. Gelfand
Montse Fuentes
Jennifer A. Hoeting
Richard L. Smith
Gelfand
Fuentes
Hoeting
Smith
K27284
w w w . c r c p r e s s . c o m
copy to come
Chapman & Hall/CRC
Handbooks of Modern
Statistical Methods
Chapman & Hall/CRC
Handbooks of Modern
Statistical Methods
Statistics
K27284_cover.indd All Pages 5/11/18 11:44 AM
4

5. −100 −95 −90 −85 −80
202224262830 Region Used for Gulf of Mexico SSTs:
21 to 29 degrees N, 83 to 97 degrees W
Longitude
Latitude
5

6. 1900 1950 2000
25.526.026.5 Annual Mean SST (July−June) in Gulf of Mexico
Year
SST
6

7. To begin, I’d like to make some basic points about the use of a
Generalized Extreme Value (GEV) distribution to estimate the
probability of an extreme event such as the precipitation event
associated with Hurricane Harvey
These will be illustrated using data on a single station (instead
of spatial aggregates). Specifically, I look at the precipitation
series at Houston Hobby Airport, 1949–2017.
7

8. 8

9. 9

10. 10

11. 11

12. 12

13. 13

14. GEV Parameters Fitted to 1949–2016
1-day Maxima, MLE
Pr {Y ≤ y} = exp − 1 + ξ ·
y − µ
σ
−1/ξ
+
Parameter Estimate S.E. t-statistic p-value
µ 2.461 0.147 16.724 0.000
log σ 0.003 0.136 0.022 0.982
ξ 0.457 0.148 3.091 0.002
Probability of exceeding 2017 value: 0.0248
“40-year event“
14

15. GEV Parameters Fitted to 1949–2016
3-day Maxima, MLE
Pr {Y ≤ y} = exp − 1 + ξ ·
y − µ
σ
−1/ξ
+
Parameter Estimate S.E. t-statistic p-value
µ 3.703 0.243 15.213 0.000
log σ 0.533 0.122 4.362 0.000
ξ 0.311 0.127 2.445 0.014
Probability of exceeding 2017 value: 0.0027
“364-year event“
15

16. GEV Parameters Fitted to 1949–2016
7-day Maxima, MLE
Pr {Y ≤ y} = exp − 1 + ξ ·
y − µ
σ
−1/ξ
+
Parameter Estimate S.E. t-statistic p-value
µ 4.898 0.317 15.469 0.000
log σ 0.825 0.108 7.603 0.000
ξ 0.161 0.105 1.533 0.125
Probability of exceeding 2017 value: 0.00064
“1,563-year event“
16

17. GEV Parameters Fitted to 1949–2016
7-day Maxima, Bayesian
Pr {Y ≤ y} = exp − 1 + ξ ·
y − µ
σ
−1/ξ
+
Parameter Posterior Mean Posterior SD t-statistic “p-value”
µ 4.858 0.387 12.563 0.000
log σ 0.841 0.150 5.599 0.000
ξ 0.177 0.135 1.316 0.188
Posterior mean probability of exceeding 2017 value: 0.0028
“357-year event“
17

18. 18

19. Alternatively, we may represent the posterior distribution in terms
of its median (0.00077) and its 17th and 83rd percentiles (0.000036,
0.0029).
The exceedance probability for the 7-day Harvey precipitation
event is likely between 0.000036 and 0.0029.
In IPCC-speak:
• Likely means probability at least 0.66
• Very Likely means probability at least 0.9
• Virtually certain means probability at least 0.99
But it is not clear whether they intend these probabilities to be
interpreted in Bayesian fashion!
19

20. GEV Parameters With Covariates I.
We extend the preceding analyses by including two covariates:
SSTt is Gulf of Mexico annual mean SST in year t, expressed as
the deviation from 26oC
CO2t is global mean CO2 in year t, scaled as 0.01(CO2t − 350).
20

21. 21

22. 22

23. GEV Parameters With Covariates II.
Pr{Yt ≤ y} = exp

− 1 + ξ
y − µt
σt
−1/ξ
+

 ,
µt = θ1 + θ4SSTt + θ5CO2t,
log σt = θ2 + θ6SSTt,
ξ = θ3.
Fitted Parameters:
Parameter Estimate Standard error t-statistic p-value
θ1 4.70 0.29 16.22 0.00
θ2 0.56 0.13 4.25 0.00
θ3 0.15 0.09 1.64 0.10
θ4 3.06 1.49 2.06 0.04
θ5 1.95 0.82 2.36 0.018
θ6 1.24 0.50 2.48 0.013
23

24. Bayesian Estimation of Exceedance Probabilities
The preceding model was refitted using a Bayesian analysis with
a flat prior.
Exceedance probabilities for the 2017 event were calculated.
Because we are interested in long-term climatic features rather
than short-term weather events, we replaced SSTt by the fitted
curve under a 4-DF spline (same as in the previous picture)
Computed posterior median probabilities and the 17th and 83rd
percentiles of the posterior density
Result: For 2017, posterior median exceedance probability is
0.0019 (return value 525 years) with a likely range from 0.00022
to 0.00685 (return values 145 to 4472 years).
24

25. 25

26. Analysis of Extreme Gulf Storms Data
(Ken Kunkel)
1. Define a grid of 1
3-degree by 1
3-degree cells covering longi-
tudes 80–100oW and latitudes 25–35oN (large blue box on
figure). This represents an approximate area of 10,000 km2.
2. Within the grid, consider all possible 2-degree by 2-degree
boxes (all boxes like the red box in the figure).
3. Compute daily precipitation for 1949-present as a simple av-
erage of all stations in each box. All boxes that are wholly
or partly over water are not included in this analysis.
4. For each grid box, identify top 5-day precipitation totals.
5. Pool everything together and identify the top 100 events
for 1949–2017 across the entire region, ignoring those that
overlap in time or space with larger event.
6. Rank and plot these.
7. Also did same analysis on other grid sizes from 1o to 3o.
26

27. A typical grid box (red boundary)
27

28. 0 20 40 60 80 100
10152025
Plot of Ranked Precipitation Events
Index
PrecipationTotal Harvey
(Computed from 2o gridboxes)
28

29. 1950 1960 1970 1980 1990 2000 2010
10152025
Plot of Precipitation Events by Year
Year
PrecipationTotal
Harvey
(Computed from 2o gridboxes)
29

30. 0 1 2 3 4 5
05101520
Exponential QQ Plot Based on Exceedances
(Line through first n−1 points)
Reduced Value
ExcessOverSmallestValue
Harvey
(Computed from 2o gridboxes)
30

31. We get very similar plots from other
gridbox sizes between 1o and 3o
31

32. Summaries for Five Grid Sizes
Grid Size Largest Mean of next 10 Ratio
1.0 40.87 23.17 1.76
1.5 31.56 16.66 1.89
2.0 27.99 13.59 2.06
2.5 19.52 11.07 1.76
3.0 15.67 9.40 1.67
32

33. Conclusion from Initial Data Plots
1. Harvey is clearly very extreme when computed over 2o × 2o
grid boxes, but the other grid box sizes also show Harvey as
extreme
2. There is some evidence that extreme events were increasing
over time (prior to 2017) but this is hard to judge visually
33

34. Outline of Analysis Method
1. Point process approach: Smith (1989), Chapter 7 of Coles
(2001): represents exceedances over a threshold as a two-
dimensional point process, each point corresponding to the
time and level of an extreme event
2. Threshold defined by k’th largest event, for some k ≤ 100
3. Equivalent to the approach based on the Generalized Pareto
Distribution but more flexible for modeling of covariates
34

35. Point Process Approach
Assume two-dimensional plot of exceedance times and values
above a threshold u forms a nonhomogeneous Poisson process
with
Λ(A) = (t2 − t1)Ψ(y; µ, σ, ξ),
Ψ(y; µ, σ, ξ) = 1 + ξ
y − µ
σ
−1/ξ
provided 1 + ξ(y − µ)/σ > 0.
35

36. In practice:
• Group observations by month
• Threshold u taken as k’th order statistic, for any k ≤ 100.
• Assume nt values over u in month t, at yi,t, i = 1, ..., nt
• GEV parameters for month t are µt, σt, ξt
• Contribution to likelihood for month t is
exp



−
1
12
1 + ξt
u − µt
σt
−1/ξt



·
nt
i=1



1
σt
1 + ξt
yi,t − µt
σt
−1/ξt−1



.
• Parameters µt, σt, ξt may be taken as constants or functions
of time t and/or SST St
36

37. Recode:
T months, N exceedances, yi ≥ u in month ti, i = 1, ..., N.
h =
1
12
T
t=1
1 + ξt
u − µt
σt
−1/ξt
+
N
i=1
log σti +
1
ξti
+ 1 log 1 + ξti
yi − µti
σti
Minimize h based on:
• Covariance matrix X
• n1 covariates in µt, columns i1,j, j = 1, .., n1
• n2 covariates in log σt, columns i2,j, j = 1, .., n2
• µt = θ1 +
n1
j=1 θ3+jxt,i1,j
• log σt = θ2 +
n2
j=1 θ3+n1+jxt,i2,j
• ξt = θ3
37

38. Selection of covariates:
• µt includes terms cos 2πti
12 , sin 2πti
12 , where ti is month of
i’th event, to account for seasonal variation
• Either µt or log σt may include terms based on lagged SSTGM
(from HADISST) database and on annual global CO2 levels
from the RCP database (https://tntcat.iiasa.ac.at/RcpDb;
also used by Wehner and Risser in their paper)
• The SSTGM data are averaged over lagged monthly anoma-
lies from 1 to 6 months before the event in question, after
trial and error to determine the best combination of lags
• Two model selection criteria:
(a) AIC — usual definition
(b) Minimum p-value — select the model that has the
smallest p-value when tested against the model in which
SSTGM and CO2 are omitted entirely
38

39. print this first
“Best Model” Results, p-value Criterion
Size of k p-value Covariates in µt Covariates in log σt
Grid SSTGM CO2 SSTGM CO2
1o 40 0.009 Y N N N
1o 70 0.004 Y N N N
1o 98 0.001 Y N N N
1.5o 40 0.00008 N N Y Y
1.5o 70 0.001 Y Y N N
1.5o 98 0.006 Y Y N Y
2o 40 0.007 Y Y N N
2o 70 0.001 Y Y N N
2o 97 0.006 Y Y N Y
2.5o 40 0.0007 N Y N N
2.5o 70 0.013 N Y N Y
2.5o 98 0.012 N Y N Y
3o 40 0.011 N∗ Y N N
3o 70 0.013 N Y N N
3o 98 0.008 Y Y N Y
∗ only one that is different under AIC criterion
39

40. Example of Model Parameters
2o grid cells, k=70
Model:
µi = θ1 + θ4 SSTGM + θ5 CO2 + θ6 cos
2πti
12
+ θ7 sin
2πti
12
,
log σi = θ2,
ξi = θ3.
Parameter Estimate S.E. t-statistic p-value
θ1 8.39 0.23 37.10 0.00
θ2 0.55 0.17 3.25 0.00
θ3 –0.07 0.13 –0.51 0.61
θ4 1.57 0.82 1.91 0.06
θ5 1.76 0.84 2.09 0.04
θ6 –0.65 0.34 –1.94 0.05
θ7 –0.15 0.31 –0.47 0.64
40

41. Bayesian Analysis
1. Uses ”Adaptive Metropolis-Hastings” approach (Haario et
al., 2001) but with some tweaks
(a) 250,000 iterations, saved every tenth iteration
(b) Different lengths of initial warm-up period
(c) Different starting conditions for MCMC
(d) Even though AMH uses an automatic choice of step size
for the trial steps of the parameter updates, I have found
it useful to reduce this under some circumstances
2. Results:
(a) Generally good mixing assessed by eye
(b) Even a “problem case” for which ˆξ = −1 under MLE per-
formed reasonably, though the ACF of the MCMC output
decayed more slowly
41

42. Summary of Observational Data Analysis
1. “Point process approach” to modeling of extreme events —
used threshold exceedances but allows covariates to be in-
cluded in GEV parameters µi, σi, ξi.
2. Covariates include monthly SSTGM (averaged over lags 1–6)
and annual CO2, as well as seasonal terms
3. Fitted to five different grid cells and three values of the num-
ber of included order statistics k based on data from 1949–
2016
4. Maximum likelihood and Bayesian methods — convergence
fine for k = 70 or 99 but unclear for k = 40
5. Conclusion: There is a clear “climate change” signal in the
distribution of extreme events. The rest of the paper will pin
down the nature of that effect and its influence on present
and future extreme event probabilities
42

43. Climate Model Data
1. Our analyses of Gulf Storms show dependence on SSTs and
CO2
2. This raises two questions for which we would like to use
climate model data:
(a) Detection and attribution: how much has the probability
of an extreme increased as a result of global warming?
(Technique: compare predictions based on models that
do or do not include the greenhouse gas component)
(b) Projecting future extreme events: how much will the prob-
ability of an extreme event change between now and, say
2080?
(c) The latter question relies on defining a particular climate
scenario (rcp 8.5)
43

44. Outline plan
1. Access to CMIP5 dataset
(a) 41 climate models
(b) Between 1 and 10 ensemble members for each model
(c) Use three scenarios: historical, historical natural, rcp8.5
(d) For each combination of month, climate model and sce-
nario, calculate Gulf of Mexico average, defined as mean
SST over 21-28 N, 83-97 W (237,016 total observations)
2. Use model output to estimate:
(a) Differences between natural-forcings and all-forcings SSTs
for 2017
(b) Differences between SSTs for 2017 and 2080
3. These will be combined with observational data model to
estimate probabilities of extreme events
44

45. Detailed method
1. Combine historical (all forcings) and rcp 8.5 data: time frame
1949-2100
2. Same for more runs with natural forcings only, time frame
1949-2019 (but most series stop at 2005)
3. Combine monthly data into July-June annual averages
4. For replicates of a single model, average over all replicates
keeping track of number of replicates (weights for subsequent
regressions)
5. Fit linear regression with autoregression and weights
6. Use model runs to estimate (a) differences between 2017
SSTs under natural and historical models, (b) differences
between 2017 and 2080 SSTs, with standard errors for each
7. There are 19 climate models for which both sets of calcula-
tions are possible. The variability among climate models is a
first indicator of climate-model uncertainty
45

46. Summary of Climate Model Outputs
46

47. Conclusions from Climate Model Output
1. All models show positive values for both variables plotted
2. Considerable intra-model uncertainty (horizontal and vertical
widths of each rectangle represent 95% confidence intervals)
3. By the inter-model uncertainty is even greater
4. Chose four models to use as examples for ongoing analysis:
CCSM4, GISS-E2-R, HadGEM2-ES, IPSL-CM5A-LR
47

48. Analysis
For each size of cell (1, 1.5, 2, 2.5, 3):
• Compute probability of exceeding August 2017 level during
2017, based on data 1949-2016
• Three scenarios:
– 2017 calculation (“climate” not “weather” — used long-
term trends to establish SST and CO2 values for 2017)
– Compare exceedance probabilities for 2017 under anthro-
pogenic forcings v. 2017 under natural forcings; compute
relative risk or FAR
– Compare exceedance probabilities for 2080 under rcp8.5
v. 2017 under same; compute relative risk or FAR
• Express all results in terms of posterior probabilities and cred-
ible intervals to capture uncertainty
48

49. Results
49

50. Probability of a “Harvey” in 2017
All forcings
Grid size k Posterior Very likely Virtally certain
(degrees) Mean less than less than
1 40 0.00032 0.00039 0.0073
1 70 0.00059 0.0015 0.0092
1 100 0.00025 5e-04 0.0047
1.5 40 0.0015 9.5e-05 0.042
1 70 9.7e-05 4.3e-06 0.0025
1.5 100 0.00098 0.002 0.018
2 40 7e-04 0.0013 0.015
2 70 0.00024 5e-04 0.0045
2 100 0.00099 0.0021 0.018
2.5 40 0.0017 0.0046 0.027
2.5 70 0.004 0.011 0.06
2.5 100 0.0026 0.0073 0.034
3 40 0.0012 0.0029 0.025
3 70 0.0023 0.007 0.025
3 100 0.009 0.026 0.075
50

51. Probability of a “Harvey” in 2017
Natural forcings
Grid size k Posterior Very likely Virtally certain
(degrees) Mean less than less than
1 40 0.00014 4.5e-05 0.0039
1 70 0.00033 0.00066 0.0061
1 100 0.00012 0.00013 0.0026
1.5 40 3.2e-06 0 0
1 70 3.8e-05 0 0.00098
1.5 100 4.4e-05 4.1e-06 0.00095
2 40 0.00034 0.00029 0.0089
2 70 0.00011 0.00015 0.0024
2 100 1.4e-05 1e-07 0.00024
2.5 40 0.001 0.002 0.02
2.5 70 5.6e-05 1e-05 0.0015
2.5 100 8.5e-05 6.3e-05 0.0021
3 40 0.00067 0.00064 0.017
3 70 0.0018 0.0054 0.022
3 100 1e-04 0.00014 0.0022
51

52. Probability of a “Harvey” in 2080
rcp 8.5
Grid size k Posterior Likely Very likely
(degrees) Mean greater than greater than
1 40 0.029 0.00099 0
1 70 0.01 0.00068 1e-07
1 100 0.0075 0.00036 0
1.5 40 0.56 0.56 0.49
1 70 0.58 0.32 0.012
1.5 100 0.41 0.35 0.042
2 40 0.47 0.099 0.0042
2 70 0.16 0.007 0.00028
2 100 0.44 0.43 0.1
2.5 40 0.54 0.23 0.016
2.5 70 0.45 0.43 0.078
2.5 100 0.4 0.34 0.03
3 40 0.48 0.18 0.0083
3 70 0.061 0.0049 4e-04
3 100 0.51 0.52 0.24
52

53. Relative Risks: “All forcings” versus
“Natural forcings” in 2017
Grid size k Ratio of Very likely Virtually certain
(degrees) posterior means greater than greater than
1 40 2.2 2 1.1
1 70 1.8 1.7 1.2
1 100 2.1 2.2 1.3
1.5 40 470 Inf 2.8
1 70 2.5 2.7 1.4
1.5 100 22 19 0.46
2 40 2 1.8 1.1
2 70 2.1 2.1 1.4
2 100 72 340 3.2
2.5 40 1.7 1.4 1.1
2.5 70 72 52 0.2
2.5 100 30 13 0.19
3 40 1.8 1.4 0.96
3 70 1.2 1.1 1
3 100 87 52 3.5
53

54. Relative Risks: 2080 versus 2017 (rcp 8.5)
Grid size k Ratio of Very likely Virtually certain
(degrees) posterior means greater than greater than
1 40 91 16 1.8
1 70 18 3.8 1.5
1 100 30 7.4 2
1.5 40 370 5700 14
1 70 6000 18000 8.5
1.5 100 420 140 6.1
2 40 670 32 1.6
2 70 660 23 3.1
2 100 450 220 23
2.5 40 320 16 1.6
2.5 70 110 42 5.3
2.5 100 150 49 0.41
3 40 390 32 1.3
3 70 27 1.9 1
3 100 56 22 7.1
54

55. Effect of Varying the Climate Model
• Previous results were based on CCSM4 — low projections of
future SSTs compared with other models
• I also tried the same calculations using the HadGEM2-ES
which modeled larger trends
• As might be expected, those analyses that used SSTGM as
a covariate showed even larger increases in projected proba-
bilities through 2080
• A problem with all of these analyses: none of the climate
models shows very good agreement between the observed
and simulated SSTGMs.
55

56. Time Series of SSTGM – Three models and Observations
56

57. For the last part of the talk I return to my opening example
(Hammerling et al. 2018) where we took a different approach
to reconciling the model-based and observational constructions
of SSTGM.
Future plan is to do this for Ken Kunkel’s data as well.
57

58. Climate Model Data I.
Data from CMIP5: used to calculate annual SST means over
the Gulf of Mexico
• Historical all-forcings data up to 2005 or 2012
• Historical natural-forcings data up to 2005 or 2012
• Future forcings data under the RCP 8.5 scenario
• All model runs have been converted to anomalies
• Four climate models; also computed average over the four
models
58

59. Climate Model Data II.
The model Gulf of Mexico SSTs do not follow the observa-
tional data very closely so, in order to use the regression model
fitted previously to observational SSTs, we proceed as follows.
The observational SSTs for 1949–2017 are regressed on two
covariates: first, the difference between historical-forcings and
natural-forcings climate model runs, and, second, the natural-
forcings climate model runs on their own. The two components
together are then used to define the “all forcings” signal and
the second component on its own is used to define the “natural
forcings” signal. Both components are represented via smooth-
ing splines to give a smooth signal. This exercise is repeated for
each of the four climate models and also with all four models
averaged to give the curves in the next figure.
59

60. 60

61. Climate Model Data III.
This exercise was repeated to obtain future projections of Gulf
of Mexico SST up to 2080; see Figure. Since there are no
natural-forcings projections over this time period, only the RCP
8.5 values are shown.
61

62. 62

63. Calculations of Exceedance Probabilities I.
We now repeat the calculation of the probability of a Harvey-sized
event under the circumstances, (a) for 2017 under all forcings,
(b) for 2017 under natural forcings, (c) for 2080 under RCP 8.5.
The calculation is repeated for all four climate models and for the
average over the four models; we used the same posterior density
output as before to obtain Bayesian posterior curves. Finally, we
took the ratio of (a) to (b) (relative risk for 2017 under the
all-forcungs and natural-forcings scenario), and the ratio of (c)
to (a) (relative risk for a Harvey-sized event in 2080 compared
with 2017). The results are in Table:
63

64. Calculations of Exceedance Probabilities II.
Model Present Future
Lower Mid Upper Lower Mid Upper
CCSM4 1.5 2.0 3.2 9.0 26.2 133
GISS-E2-R 1.8 2.5 4.8 13.5 43.5 244
HadGEM2-ES 1.6 2.1 3.5 23.6 73.3 415
IPSL-CM5A-LR 1.5 2.0 3.3 10.8 33.8 186
Combined 1.7 2.4 4.4 14.3 46.0 254
Relative risks. The columns labelled “Present” refer to relative
risks for the 2017 event under an all-forcings scenario versus
a natural-forcings scenario, computed under four climate mod-
els and with all four models combined. Lower, mid and upper
bounds correspond to the 17th, 50th and 83rd percentiles of the
posterior distribution. The columns labelled “Future” are relative
risks for such an event in 2080 against 2017; same conventions
regarding climate models and percentiles.
64

65. Summary
For the combined-model results, the relative risk of the Harvey
precipitation under all-forcings versus natural-forcings scenarios
is estimated as 2.4, “likely” between 1.7 and 4.4. For all five
sets of model results, the lower bound exceeds 1, proving that
it’s “likely” that anthropogenic conditions affected Harvey. This
is consistent with earlier results reported by van Oldenburgh et
al. (2017), Risser and Wehner (2017) and Emanuel (2017).
For the relative risks of a Harvey-sized event in 2080 against
2017, the posterior means range from 26 to 73, with “likely”
bounds ranging from 9 to 415. Evidently, the uncertainty range
for future projections is very wide. Given that Emanuel (2017)
obtained an estimated relative risk of 18 by completely different
methods, there seems to be some agreement that a drastic rise
in the frequency of this type of event is to be expected.
65