Over a seven day period in August 2017 Hurricane Harvey brought extreme levels of rainfall to the Houston area, resulting in catastrophic flooding that caused loss of human life and damage to personal property and public infrastructure. In the wake of this event, there is growing interest in understanding the degree to which this event was unusual and estimating the probability of experiencing a similar event in other locations. Additionally, we investigate the degree to which the sea surface temperature in the Gulf of Mexico is associated with extreme precipitation in the US Gulf Coast. This talk addresses these issues through the development of an extreme value model. We assume that the annual maximum precipitation values at Gulf Coast locations approximately follow the Generalized Extreme Value (GEV) distribution. Because the observed precipitation record in this region is relatively short, we borrow strength across spatial locations to improve GEV parameter estimates. We model the GEV parameters at US Gulf Coast locations using a multivariate spatial hierarchical model; for inference, a two-stage approach is utilized. Spatial interpolation is used to estimate GEV parameters at unobserved locations, allowing us to characterize precipitation extremes throughout the region. Analysis indicates that Harvey was highly unusual as a seven -day event, and that GoM SST seems to be more strongly linked to extreme precipitation in the Western part of the region.

1. Investigating Precipitation Extremes in the US
Gulf Coast through the use of a Multivariate
Spatial Hierarchical Model
Brook T. Russell, CU Department of Mathematical Sciences
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 1 / 46

2. Outline
Introduction
Hurricane Harvey
Motivating Questions
Modeling Procedure
MV Spatial Model
Inference
Analysis
Precipitation Data and Covariate
Estimating Spatial Fields
Estimating Quantities of Interest
Discussion
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 2 / 46

3. Collaboraters
Ken Kunkel
Mark Risser
Richard Smith
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 3 / 46

4. Motivating Questions
1. How unusual was this event?
2. What is the probability of observing another event of this
magnitude in the US GC region?
3. What is the nature of the relationship between GoM SST and
precipitation extremes in the US GC region?
4. How can we account for “storm” level dependence using a
relatively simple spatial model?
NASA Richard Carson, Reuters
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 4 / 46

5. Outline
Introduction
Hurricane Harvey
Motivating Questions
Modeling Procedure
MV Spatial Model
Inference
Analysis
Precipitation Data and Covariate
Estimating Spatial Fields
Estimating Quantities of Interest
Discussion
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 5 / 46

6. Univariate Extremes: Background
Generalized Extreme Value (GEV) Distribution:
For iid X1, . . . , Xn and Mn = Max{X1, . . . , Xn}, if ∃ sequences
an > 0 and bn s.t.
a−1
n (Mn − bn)
d
→ G
for non-degenerate G, then G is GEV
GEV – three parameter family: µ ∈ R, σ > 0, ξ ∈ R (location,
scale, shape)
Importance of shape parameter
ξ < 0 ⇒ Reverse Weibull
ξ = 0 ⇒ Gumbel
ξ > 0 ⇒ Fr´echet
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 6 / 46

7. Inference in Univariate Extremes
Block Maxima Approach:
Data: independent ‘blocks’ (years, seasons, etc.)
For ‘large’ blocks use series of block maxima to estimate GEV
parameters
Characterizing Extremes via GEV Estimates:
Return level: amount that is exceeded by the block maximum
with probability p (return period is 1/p)
RLp =
µ − σ
ξ (1 − {− log(1 − p)}−ξ) for ξ = 0
µ − σ log{− log(1 − p)} for ξ = 0
Interpretations:
Avg waiting time until next event exceeding this amount is 1/p
Avg number of events exceeding this amount occurring within
return period is one
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 7 / 46

8. Exploratory Analysis: Pointwise GEV MLEs
Location MLEs:
8
10
12
14
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 8 / 46

9. MV Spatial Model
Capture spatial signal via MV spatial model
Spatially model GEV parameters
Use pointwise MLEs and covariance information as input
Use two-stage inference procedure
Approach similar to Holland et al. (2000), Tye and Cooley
(2015)
Setup:
Yt(s) – seasonal 7 day max precip at time t for s ∈ D ⊂ R2
Assume Yt(s)
·
∼ GEV (µt(s), σt(s), ξ(s))
Idea: incorporate GoM SST into location and scale parameters
Goal: estimate parameters ∀ s ∈ D, observed and unobserved
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 9 / 46

10. MV Spatial Model
At location s and time t, define the GEV parameters via
µt(s) = θ1(s) + SSTtθ2(s)
log σt(s) = θ3(s) + SSTtθ4(s)
ξ(s) = θ5(s)
For θ(s) = (θ1(s), θ2(s), θ3(s), θ4(s), θ5(s))T at location s,
assume
θ(s) = β + η(s)
Mean parameter values over region:
β = (β1, β2, β3, β4, β5)T
Spatially correlated random effects:
η(s) = (η1(s), η2(s), η3(s), η4(s), η5(s))T
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 10 / 46

11. Spatial Model
Use coregionalization model (Wackernagel, 2003)
η(s) = A δ(s)
for δ(s) = (δ1(s), δ2(s), δ3(s), δ4(s), δ5(s))T
A: lower triangular matrix (Finley et al., 2008)
δi : indep. second-order stationary GPs with mean 0 and
covariance function
Cov(δi (s), δi (s )) = exp − s − s /ρi
Assumes stationary and isotropic
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 11 / 46

12. Inference – Stage One
First stage of inference:
Obtain MLEs ˆθ(sl ) = (ˆθ1(sl ), ˆθ2(sl ), ˆθ3(sl ), ˆθ4(sl ), ˆθ5(sl ))T at
station l ∈ {1, . . . , L}
Assume
ˆθ(sl ) = θ(sl ) + ε(sl )
Estimation error (indep. of η):
ε(sl ) = (ε1(sl ), ε2(sl ), ε3(sl ), ε4(sl ), ε5(sl ))T
Further assume
(ε1(s1), . . . , ε1(sL), . . . , ε5(s1), . . . , ε5(sL))T
∼ N(0, W )
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 12 / 46

13. Resulting Hierarchical Model
Define:
Θ = (θ1(s1), . . . , θ1(sL), . . . , θ5(s1), . . . , θ5(sL))T
ˆΘ = (ˆθ1(s1), . . . , ˆθ1(sL), . . . , ˆθ5(s1), . . . , ˆθ5(sL))T
Hierarchical model
ˆΘ|Θ ∼ N(Θ, W ) and Θ ∼ N(β ⊗ 1L, ΩA,ρ)
Marginal model
ˆΘ ∼ N(β ⊗ 1L, ΩA,ρ + W )
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 13 / 46

14. Inference – Stage Two
Use MLEs and W as input
Estimate β, A, and ρ via sequential ML
Results in estimates ˜β, ˜A, and ˜ρ
Selecting W :
Use NP block BS to capture “storm” level dependence
(seasons are blocks)
Obtain Wbs via empirical covariance matrix of BS ests
Wbs is noisy, regularize via covariance tapering (Furrer et al.,
2006)
Wtap = Wbs ◦ Ttap
Ttap: generated using using covariance function s.t.
Cov(Z(s), Z(s )) = 0 ∀ s − s > λ
Use Wendland 2 covariance function with λ = 150km
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 14 / 46

15. Outline
Introduction
Hurricane Harvey
Motivating Questions
Modeling Procedure
MV Spatial Model
Inference
Analysis
Precipitation Data and Covariate
Estimating Spatial Fields
Estimating Quantities of Interest
Discussion
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 15 / 46

16. Precipitation Data
Obtain seasonal (June – Nov) daily precip totals (1949–2016)
from GHCN
Omit seasons with more than 5 missing values during
hurricane season
Exclude stations with at least 10 years omitted (n = 326)
Extract series of 7 day seasonal maxima for each station
2017 data held out
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Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 16 / 46

17. Incorporating SST
Monthly TS of avg GoM SST via Hadley Centre Sea Ice and
Sea Surface Temperature data set (Rayner et al., 2003)
between 83◦ − 97◦W and 21◦ − 29◦N
Exploratory analysis at each station suggests using avg SST
March–June
Centered and scaled SST covariate:
1950 1970 1990 2010
−2−1012
Year
GoMSST(centeredandscaled)
GoM SST (centered and scaled)
Frequency
−2 −1 0 1 2
0246810
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 17 / 46

18. Inference and Spatial Interpolation
Two-step Inference Procedure:
1. Assume
Yt(s)
·
∼ GEV (θ1(s) + SSTtθ2(s), θ3(s) + SSTtθ4(s), θ5(s));
use precip data and SST series to get station MLEs ˆΘ
2. Assume marginal model
ˆΘ ∼ N(β ⊗ 1L, ΩA,ρ + W );
use ˆΘ and W to get estimates ˜β, ˜A, and ˜ρ (via seq likelihood)
Spatial Interpolation:
Goal: At s0 ∈ D (observed or unobserved), estimate θ(s0)
Use co-kriging and model output ( ˜β, ˜A, and ˜ρ) to obtain
estimate ˜θ(s0)
Estimate spatial fields by interpolating over a grid of points
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 18 / 46

19. Characterizing Precipitation Extremes
Using Parameter Estimates:
˜θ(s0) can be used to estimate quantities of interest at s0
Return levels – consider 100 yr. RLs
Exceedance probabilities
Observed return periods
Consider three SST scenarios
“Low” SST = −1
“High” SST = 1
“2017” SST ≈ 1.71
Methods for quantifying uncertainty
Delta method – simple but has known problems
Profile likelihood – challenges using at unobserved locations
Bootstrapping – parametric vs NP
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 19 / 46

20. 100 Year RL Ests
Based on “Low” SST
25
30
35
40
45
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Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 20 / 46

21. 100 Year RL Ests
Pointwise 90% CIs for “Low” SST (based on parametric
bootstrap)
20
25
30
35
40
90% CI (lower limits)
25
30
35
40
45
50
55
90% CI (upper limits)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 21 / 46

22. 100 Year RL Ests
Based on “High” SST
25
30
35
40
45
50
55
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Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 22 / 46

23. 100 Year RL Ests
Pointwise 90% CIs for “High” SST (based on parametric
bootstrap)
20
25
30
35
40
45
90% CI (lower limits)
30
40
50
60
90% CI (upper limits)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 23 / 46

24. 100 Year RL Ests
Based on “2017” SST
25
30
35
40
45
50
55
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Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 24 / 46

25. 100 Year RL Ests
Pointwise 90% CIs for “2017” SST (based on parametric
bootstrap)
20
25
30
35
40
45
50
90% CI (lower limits)
30
40
50
60
70
80
90% CI (upper limits)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 25 / 46

26. Comparing 100 Year RL Ests
Ratio: “High” SST versus “Low” SST
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Ratio of 100 Yr RLs (High SST vs Low SST)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 26 / 46

27. Comparing 100 Year RL Ests
Pointwise 90% CIs for ratio of 100 yr RLs (“High” SST versus
“Low” SST)
0.7
0.8
0.9
1.0
1.1
1.2
1.3
90% CI (lower limits)
0.7
0.8
0.9
1.0
1.1
1.2
1.3
90% CI (upper limits)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 27 / 46

28. 100 Year RL Ests
Ratio: “2017” SST versus “Low” SST
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Ratio of 100 Yr RLs (2017 SST vs Low SST)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 28 / 46

29. Comparing 100 Year RL Ests
Pointwise 90% CIs for ratio of 100 yr RLs (“2017” SST versus
“Low” SST)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
90% CI (lower limits)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
90% CI (upper limits)
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 29 / 46

30. 2017 Event in Houston Area
Approximately 100cm in Houston area
Observed return periods for this amount in downtown
Houston based on spatial model
SST Obs Ret Per CI Lwr CI Upr
Low 8512.24 2079.44 70123.68
High 3471.28 1059.97 20659.49
2017 2549.66 404.20 6435.60
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 30 / 46

31. 2017 Event in Houston Area
Approximately 100cm in Houston area
Estimate probability of exceeding this amount in downtown
Houston based on spatial model
SST Est Exc Pr RR vs “Low” RR CI Lwr RR CI Upr
Low 0.000117 – – –
High 0.000288 2.45 1.47 4.66
2017 0.000392 3.34 2.30 24.61
Brook T. Russell SAMSI CLIM Transition Workshop (5/16/18) 31 / 46

32. Chances of observing another event of this magnitude
Idea: use annual avg precip as baseline
Houston: 70cm is 53% of annual avg., 48cm is 36.5% of
annual avg.
PRISM annual avg. precip map1
50
100
150
200
1
PRISM Climate Group, Oregon State University,
http://prism.oregonstate.edu
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33. Estimated Exceedance Probabilities
Based on “Low” SST
0e+00
2e−04
4e−04
6e−04
8e−04
1e−03
Est Prob of Exceeding 53% of Annual Avg
0.000
0.005
0.010
0.015
Est Prob of Exceeding 36.5% of Annual Avg
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34. Estimated Exceedance Probabilities
Based on “High” SST
0e+00
2e−04
4e−04
6e−04
8e−04
1e−03
Est Prob of Exceeding 53% of Annual Avg
0.000
0.005
0.010
0.015
Est Prob of Exceeding 36.5% of Annual Avg
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35. Estimated Exceedance Probabilities
Based on “2017” SST
0e+00
2e−04
4e−04
6e−04
8e−04
1e−03
Est Prob of Exceeding 53% of Annual Avg
0.000
0.005
0.010
0.015
Est Prob of Exceeding 36.5% of Annual Avg
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36. Closer Look at Five Locations
q q
q
q
q
Houston
New Orleans
San Antonio
Tallahassee
Atlanta
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37. Houston, TX
0.00.20.40.60.81.0
Proportion of Average Total Annual Precipitation
7DayEstimatedExceedanceProbability
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
3.29 6.57 9.86 13.15 16.43 19.72 23.01 26.3 29.58
Precipitation (cm)
Low SST
High SST
2017 SST
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38. New Orleans, LA
0.00.20.40.60.81.0
Proportion of Average Total Annual Precipitation
7DayEstimatedExceedanceProbability
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
4.01 8.02 12.03 16.05 20.06 24.07 28.08 32.09 36.1
Precipitation (cm)
Low SST
High SST
2017 SST
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39. San Antonio, TX
0.00.20.40.60.81.0
Proportion of Average Total Annual Precipitation
7DayEstimatedExceedanceProbability
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
1.9 3.8 5.7 7.6 9.5 11.4 13.29 15.19 17.09
Precipitation (cm)
Low SST
High SST
2017 SST
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40. Tallahassee, FL
0.00.20.40.60.81.0
Proportion of Average Total Annual Precipitation
7DayEstimatedExceedanceProbability
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
3.54 7.08 10.62 14.16 17.69 21.23 24.77 28.31 31.85
Precipitation (cm)
Low SST
High SST
2017 SST
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41. Atlanta, GA
0.00.20.40.60.81.0
Proportion of Average Total Annual Precipitation
7DayEstimatedExceedanceProbability
0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
3.27 6.55 9.82 13.1 16.37 19.65 22.92 26.19 29.47
Precipitation (cm)
Low SST
High SST
2017 SST
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42. Outline
Introduction
Hurricane Harvey
Motivating Questions
Modeling Procedure
MV Spatial Model
Inference
Analysis
Precipitation Data and Covariate
Estimating Spatial Fields
Estimating Quantities of Interest
Discussion
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43. Incorporating GoM SST Projections
SST projections from Alexander et al. (2018):
GoM: 0.2 − 0.4◦
C decade−1
(1976 – 2099)
Below 60◦
Lat.: Little change in year to year variability
“The shift in the mean was so large in many regions that SSTs
during the last 30 years of the 21st century will always be
warmer than the warmest year in the historical period.” (1976
– 2005)
1950 1960 1970 1980 1990 2000 2010
25.025.526.0
Gulf of Mexico Mean SST (Mar−Jun)
Year
Temperature(Celcius)
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44. Additional Thoughts and Future Work
Quantifying the degree to which Harvey was unusual:
accounting spatial extent
Impact of accounting for storm level dependence in W
Regularization methods for W – choice of λ
M. Yam, LA Times, Getty J. Raedle, Getty Images
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45. References I
Alexander, M. A., Scott, J. D., Friedland, K. D., Mills, K. E., Nye,
J. A., Pershing, A. J., and Thomas, A. C. (2018). Projected sea
surface temperatures over the 21st century: Changes in the
mean, variability and extremes for large marine ecosystem
regions of Northern Oceans. Elem Sci Anth, 6(1).
Finley, A. O., Banerjee, S., Ek, A. R., and McRoberts, R. E.
(2008). Bayesian multivariate process modeling for prediction of
forest attributes. Journal of Agricultural, Biological, and
Environmental Statistics, 13(1):60–83.
Furrer, R., Genton, M. G., and Nychka, D. (2006). Covariance
tapering for interpolation of large spatial datasets. Journal of
Computational and Graphical Statistics, 15(3):502–523.
Holland, D. M., De, O. V., Cox, L. H., and Smith, R. L. (2000).
Estimation of regional trends in sulfur dioxide over the eastern
united states. Environmetrics, 11(4):373–393.
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46. References II
Rayner, N., Parker, D., Horton, E., Folland, C., Alexander, L.,
Rowell, D., Kent, E., and Kaplan, A. (2003). Global analyses of
sea surface temperature, sea ice, and night marine air
temperature since the late nineteenth century. Journal of
Geophysical Research: Atmospheres, 108(D14).
Tye, M. R. and Cooley, D. (2015). A spatial model to examine
rainfall extremes in Colorado’s Front Range. Journal of
Hydrology, 530(Supplement C):15 – 23.
Wackernagel, H. (2003). Multivariate Geostatistics. Springer
Science & Business Media.
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