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Safety and Reliability

# A Spatiodirectional Model for Extreme Waves in the Gulf of Mexico

[+] Author and Article Information
Philip Jonathan

Shell Technology Centre Thornton, P.O. Box 1, Chester CH1 3SH, UKphilip.jonathan@shell.com

Kevin Ewans

Shell International Exploration and Production, P.O. Box 60, 2280 AB Rijswijk, The Netherlandskevin.ewans@shell.com

J. Offshore Mech. Arct. Eng 133(1), 011601 (Nov 04, 2010) (9 pages) doi:10.1115/1.4001949 History: Received February 26, 2009; Revised June 11, 2010; Published November 04, 2010; Online November 04, 2010

## Abstract

The characteristics of extreme waves in hurricane dominated regions vary systematically with a number of covariates, including location and storm direction. Reliable estimation of design criteria requires incorporation of covariate effects within extreme value models. We present a spatiodirectional model for extreme waves in the Gulf of Mexico motivated by the nonhomogeneous Poisson model for peaks over threshold. The model is applied to storm peak significant wave height $HS$ for arbitrary geographic areas from the proprietary Gulf of Mexico Oceanographic Study (GOMOS) hindcast for the US region of the Gulf of Mexico for the period of 1900–2005. At each location, directional variability is modeled using a nonparametric directional location and scale; data are standardized (or “whitened”) with respect to local directional location and scale to remove directional effects. For a suitable choice of threshold, the rate of occurrence of threshold exceedences of whitened storm peak $HS$ with direction per location is modeled as a Poisson process. The size of threshold exceedences is modeled using a generalized Pareto form, the parameters of which vary smoothly in space, and are estimated within a roughness-penalized likelihood framework using natural thin plate spline forms in two spatial dimensions. By reparameterizing the generalized Pareto model in terms of asymptotically independent parameters, an efficient back-fitting algorithm to estimate the natural thin plate spline model is achieved. The algorithm is motivated in an appendix. Design criteria, estimated by simulation, are illustrated for a typical neighborhood of $17×17$ grid locations. Applications to large areas consisting of more than 2500 grid locations are outlined.

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## Figures

Figure 1

Contours of the maximum value of rescaled storm peak HS∗ for a typical neighborhood of locations N. The background gray-scale is graduated from the lowest (darkest) to the highest (lightest) values per location.

Figure 2

Rescaled storm peak HS∗ data with storm peak direction for a typical 5×5 grid of neighboring locations with estimates of the local median (solid) and 0.99 quantile with respect to direction. There is considerable variation in the values of rescaled storm peak HS∗ with direction. Storm peak direction gives the direction from which storms emanate, measured clockwise, with North at 0 deg.

Figure 3

Whitened storm peak HS data with storm peak direction for the 5×5 grid of neighboring locations in Fig. 2 with estimates of local median (solid) and 0.99 quantile with respect to direction. There is considerably less variation in the values of whitened storm peak HS with direction with respect to the two quantiles concerned.

Figure 4

Estimated value of generalized Pareto shape parameter using spatial model, based on whitened data with whitening band [qL=0.5, qU=0.99], incorporating moderate directional smoothing, and extreme value threshold set at the 0.75 quantile per location

Figure 5

Estimated value of generalized Pareto scale parameter using spatial model, based on whitened data with whitening band [qL=0.5, qU=0.99], incorporating moderate directional smoothing, and extreme value threshold set at the 0.75 quantile per location

Figure 6

Variation in generalized Pareto shape parameter with threshold for six locations selected from different regions of the GoM. Selection of a suitable threshold for extreme value analysis is problematic.

Figure 7

Annual occurrence rate for a typical location estimated using the Poisson model

Figure 8

Contours of median rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.99] quantile whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation.

Figure 9

Contours of the first quartile (0.25 quantile) of rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.99] quantile whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation.

Figure 10

Contours of the third quartile (0.75 quantile) of rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.99] quantile whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation.

Figure 11

Contours of median rescaled HS100∗, estimated using independent GP fits per location on whitened data. A [0.5,0.99] quantile whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation.

Figure 12

Contours of median rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.9] whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation. There is reasonable correspondence with Fig. 8.

Figure 13

Contours of median rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.99] whitening band was used with heavy directional smoothing with a 0.75 quantile threshold for model estimation. There is reasonable correspondence with Fig. 8.

Figure 14

Contours of median rescaled HS100∗, estimated using the NTPS model on original (unwhitened) data. A [0.5,0.99] quantile whitening band was used with moderate directional smoothing with a 0.75 quantile threshold for model estimation.

Figure 15

Contours of median rescaled HS100∗, estimated using independent GP fits per location on original (unwhitened) data. A 0.75 quantile threshold for model estimation per location.

Figure 16

Rescaled storm peak HS100∗ data with storm peak direction for location L with relative longitude=0.5 and latitude −0.75 within neighborhood N. Also shown is a 0.25 quantile directional threshold used to fit a Fourier directional GP model.

Figure 17

Estimates for generalized Pareto shape (gray) and scale (black) at location L

Figure 18

Contours of median rescaled HS100∗, estimated using the NTPS model on whitened data. A [0.5,0.99] whitening band was used with considerable directional smoothing with a 0.90 quantile threshold for model estimation. There is reasonable correspondence with Fig. 8.

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