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 based on coregionalization; 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. Nearby locations may experience extreme precipitation from the same event, resulting in dependence between annual maxima that previous spatial models of this sort have ignored. Our model incorporates dependence of this type and uses the nonparametric bootstrap to estimate its effect.

1. Employing a multivariate spatial hierarchical
model to characterize extremes with application
to US Gulf Coast precipitation
Brook T. Russell, CU Department of Mathematical Sciences
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 1 / 55

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 Climate Extremes Workshop (5/16/18) 2 / 55

3. Collaboraters
Ken Kunkel
Mark Risser
Richard Smith
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 3 / 55

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 Climate Extremes Workshop (5/16/18) 4 / 55

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 Climate Extremes Workshop (5/16/18) 5 / 55

6. 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 Climate Extremes Workshop (5/16/18) 6 / 55

7. 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 Climate Extremes Workshop (5/16/18) 7 / 55

8. 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 Climate Extremes Workshop (5/16/18) 8 / 55

9. 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 Climate Extremes Workshop (5/16/18) 9 / 55

10. 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 Climate Extremes Workshop (5/16/18) 10 / 55

11. Inference – Stage Two
Use MLEs and W as input
Estimate β, A, and ρ via sequential ML
Results in estimates ˜β, ˜A, and ˜ρ
Use fixed W :
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 11 / 55

12. Estimating W
First Approach: Wbd
Assume ε(sl ) is independent of ε(sl ) for all l = l
Results in banded W (sparse); ignores “storm” level
dependence
Could use ML based covariance estimates at each location
Second Approach: Wbs
No assumptions on W – allow for “storm” level dependence
Estimate W via NP block bootstrap; seasons are blocks
Obtain Wbs via empirical covariance matrix of BS ests
Third Approach: Regularize Wbs
Wbs is noisy, consider regularization methods
Two methods:
Sch¨afer and Strimmer (2005) method
Covariance tapering (Furrer et al., 2006; Katzfuss et al., 2016)
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 12 / 55

13. Potential Regularization Methods
1. Regularize via Sch¨afer and Strimmer (2005)
Wreg = αTtarg + (1 − α)Wbs for α ∈ [0, 1]
Ttarg is a known “target” covariance matrix
Wreg is convex combination of Ttarg and Wbs
2. Regularize via covariance tapering (Furrer et al., 2006; Katzfuss
et al., 2016)
Wtap = Wbs ◦ Ttap
◦ is Hadamard (Schur) product
Ttap is a correlation matrix based on covariance function with
property
Cov(Z(s), Z(s )) = 0 ∀ s, s such that s − s > λ > 0
Ttap is sparse ⇒ Wtap is sparse
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 13 / 55

14. Using Covariance Tapering
Covariance Functions considered in Furrer et al. (2006):
0 20 40 60 80 100 120
0.00.20.40.60.81.0
Distance
Correlation
Wendland 1
Wendland 2
Spherical
Could define
Ttap = 151T
5 ⊗ ΣW 2(λ)
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 14 / 55

15. Illustrative Example
Data for exploratory analysis
Different than data used for analysis presented later
Smaller number of stations
Different definition of season
Different criteria for excluding seasons/stations
Use model with SST in location only
Estimate fields via co-kriging
Consider Wbd , Wbs, and Wtap with several values of λ
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 15 / 55

16. First Approach: Use Wbd
ML based estimate of covariance matrices
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Log of Scale
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 16 / 55

17. Second Approach: Use Wbs
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SST Coef
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 17 / 55

18. Third Approach: Use Wtap
Using Ttap generated via Wendland 2 (λ = 300km)
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 18 / 55

19. Third Approach: Use Wtap
Using Ttap generated via Wendland 2 (λ = 150km)
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 19 / 55

20. Third Approach: Use Wtap
Using Ttap generated via Wendland 2 (λ = 75km)
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 20 / 55

21. Third Approach: Use Wtap
Using Ttap generated via Wendland 2 (λ = 35km)
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Location Const
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SST Coef
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 21 / 55

22. Thoughts Regarding W
ML based estimates of covariance matrices seem to result in
estimated fields that are too smooth
Unconstrained BS based estimate of W seems to result in
rough estimated fields
How does a non-banded estimate of W impact the spatial
model?
Could we work on correlation scale and transform back later?
Perhaps using banded W with BS based covariances is
enough...
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 22 / 55

23. 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 Climate Extremes Workshop (5/16/18) 23 / 55

24. 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 Climate Extremes Workshop (5/16/18) 24 / 55

25. 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 Climate Extremes Workshop (5/16/18) 25 / 55

26. 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 Climate Extremes Workshop (5/16/18) 26 / 55

27. 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 Climate Extremes Workshop (5/16/18) 27 / 55

28. 100 Year RL Ests
Based on “Low” SST
25
30
35
40
45
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 28 / 55

29. 100 Year RL Ests
Pointwise 90% CIs for “Low” SST (based on parametric
bootstrap)
20
25
30
35
40
90% CI (lower limits)
25
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35
40
45
50
55
90% CI (upper limits)
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 29 / 55

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

31. 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 Climate Extremes Workshop (5/16/18) 31 / 55

32. 100 Year RL Ests
Based on “2017” SST
25
30
35
40
45
50
55
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Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 32 / 55

33. 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 Climate Extremes Workshop (5/16/18) 33 / 55

34. 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 Climate Extremes Workshop (5/16/18) 34 / 55

35. 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 Climate Extremes Workshop (5/16/18) 35 / 55

36. 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 Climate Extremes Workshop (5/16/18) 36 / 55

37. 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 Climate Extremes Workshop (5/16/18) 37 / 55

38. 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 Climate Extremes Workshop (5/16/18) 38 / 55

39. 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 Climate Extremes Workshop (5/16/18) 39 / 55

40. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 40 / 55

41. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 41 / 55

42. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 42 / 55

43. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 43 / 55

44. Closer Look at Five Locations
q q
q
q
q
Houston
New Orleans
San Antonio
Tallahassee
Atlanta
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 44 / 55

45. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 45 / 55

46. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 46 / 55

47. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 47 / 55

48. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 48 / 55

49. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 49 / 55

50. 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 Climate Extremes Workshop (5/16/18) 50 / 55

51. 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)
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 51 / 55

52. 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
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 52 / 55

53. 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.
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 53 / 55

54. References II
Katzfuss, M., Stroud, J. R., and Wikle, C. K. (2016).
Understanding the ensemble kalman filter. The American
Statistician, 70(4):350–357.
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).
Sch¨afer, J. and Strimmer, K. (2005). A shrinkage approach to
large-scale covariance matrix estimation and implications for
functional genomics. Statistical applications in genetics and
molecular biology, 4(1).
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.
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 54 / 55

55. References III
Wackernagel, H. (2003). Multivariate Geostatistics. Springer
Science & Business Media.
Brook T. Russell SAMSI Climate Extremes Workshop (5/16/18) 55 / 55