## Abstract

Physical considerations and previous studies suggest that extremal dependence between ocean storm severity at two locations exhibits near asymptotic dependence at short inter-location distances, leading to asymptotic independence and perfect independence with increasing distance. We present a spatial conditional extremes (SCE) model for storm severity, characterising extremal spatial dependence of severe storms by distance and direction. The model is an extension of Shooter et al. 2019 (Environmetrics **30**, e2562, 2019) and Wadsworth and Tawn (2019), incorporating piecewise linear representations for SCE model parameters with distance and direction; model variants including parametric representations of some SCE model parameters are also considered. The SCE residual process is assumed to follow the delta-Laplace form marginally, with distance-dependent parameter. Residual dependence of remote locations given conditioning location is characterised by a conditional Gaussian covariance dependent on the distances between remote locations, and distances of remote locations to the conditioning location. We apply the model using Bayesian inference to estimates extremal spatial dependence of storm peak significant wave height on a neighbourhood of 150 locations covering over 200,000 km^{2} in the North Sea.

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## Acknowledgements

The authors would like to thank Jenny Wadsworth (Lancaster University) and Matthew Jones, David Randell and Ross Towe (Shell) for helpful discussions. We also thank two anonymous reviewers and an associate editor for comments on an earlier draft of the manuscript.

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R. Shooter would like to acknowledge financial support from Engineering and Physical Sciences Research Council grant EP/L015692/1 (STOR-i Centre for Doctoral Training) and Shell Research Ltd.

## Appendices

### Appendix A: Grouped adaptive MCMC

Under the ABN model (with piecewise linear distance-direction representations for *α* and *β*) the parameter set *Ω* to be estimated is \(\{\{\{\alpha _{k\ell },\beta _{k\ell }\}_{\ell =1}^{n_{\theta }},\mu _{k},\sigma _{k},\delta _{k}\}_{k}^{n_{d}},R_{1},R_{2}\}\) with *n*_{P} = 2*n*_{d}*n*_{θ} + 3*n*_{d} + 2 parameters.

### Prior specification

Impose uniform prior distributions for each parameter. Explicitly, \(\alpha _{k\ell }\sim \text {Unif}(0,1)\), \(\beta _{k\ell }\sim \text {Unif}(0,1)\), \(\mu _{k}\sim \text {Unif}(-1,1)\), \(\sigma _{k}\sim \text {Unif}(0,\sqrt {2}+0.1)\) and \(\delta _{k}\sim \text {Unif}(0.5,2.5)\) for all *k* = 1, 2, … , *n*_{d}, *ℓ* = 1, 2, … , *n*_{θ}, and \(R_{1}\sim \text {Unif}(0,100)\), \(R_{2}\sim \text {Unif}(0,2)\),

### Starting solution

We obtain a random starting solution *Ω*^{(0)} by sampling the elements of *Ω* from their prior distributions, verifying that the starting solution has a valid likelihood (satisfying the conditional quantile constraints if applied).

### Metropolis-within-Gibbs for *n*
_{MiG} iterations

Writing \(\varOmega _{k}^{(i)}\) as the value of the *k* th parameter of *Ω* at the *i* th iteration, use adaptive random walk Metropolis-within-Gibbs scheme for *n*_{MiG} (> 2*n*_{P}) iterations. That is, for *i* = 2, … , *n*_{MiG}, update each \(\varOmega _{k}^{(i)}\) in turn, proposing candidate value \(\varOmega _{k}^{(i)c}\) from distribution

### Grouped adaptive Metropolis-within-Gibbs for *n*
_{GA} iterations

For *i* > *n*_{MiG}, use a grouped adaptive random walk Metropolis-within-Gibbs scheme, updating groups \(\boldsymbol {\varOmega }_{G_{k}}^{(i)}=\{\{\alpha _{k\ell },\beta _{k\ell }\}_{\ell =1}^{n_{\theta }},\mu _{k},\sigma _{k}\}\) jointly for *k* = 1, 2, … , *n*_{d} in turn, before updating \(\{\delta _{k}\}_{k=1}^{n_{d}}\), *R*_{1} and *R*_{2} separately as before. Propose candidate \(\boldsymbol {\varOmega }_{G_{k}}^{(i)c}\) from distribution

where *β* = 0.05, as suggested by Roberts and Rosenthal (2009), and **C**_{i} is the empirical variance-covariance matrix of the parameters \(\boldsymbol {\varOmega }_{G_{k}}^{(i)}\) from the previous *i* iterations.

### Iteration to convergence

Throughout, a candidate state is accepted using the standard Metropolis-Hastings acceptance criterion. Since prior distributions for parameters are uniform, and proposals symmetric, this is just a likelihood ratio. Candidates lying outside their prior domains, or violating the conditional quantile constraints if applied, are rejected.

Under the ABP model (with parametric forms for *α* and *β* with distance only), the estimation scheme is simplified so that the full set of parameters is updated at the grouped adaptive stage. The resulting procedure is the same as the original scheme of Roberts and Rosenthal (2009). Uniform priors \(A_{1\ell }\sim \text {Unif}(1,20)\), \(A_{2\ell }\sim \text {Unif}(0.1,5)\), \(B_{1\ell }\sim \text {Unif}(0.1,1)\), \(B_{2\ell }\sim \text {Unif}(1,5)\) and \(B_{3\ell }\sim \text {Unif}(0,20)\), *ℓ* = 1, 2, … , *n*_{θ} were applied. The prior distributions for all other parameters under the ABN and ABP models are the same.

### Appendix B: Conditional quantile constraints

We optionally restrict the space of feasible combinations of *α* and *β* to ensure that conditional quantiles from asymptotic independent models do not exceed those from asymptotic dependent models, as proposed by Keef et al. (2013). For any pair *α* and *β* corresponding to distance *d*_{k} and direction *θ*_{ℓ}, *k* = 1, 2, … , *n*_{d}, *ℓ* = 1, 2, … , *n*_{θ} on the distance-direction lattice, we require either

or

In the above, *ν* is a value of the conditioning variate (on standard Laplace scale) above the threshold level at which the SCE model is applied. Further *z*^{+}(*q*) is the quantile of the distribution of standardised residuals from the conditional extremes model with non-exceedance probability *q*, and *z*^{+}(*q*) is quantile of the distribution of standardised residuals from the conditional extremes model assuming asymptotic positive dependence (i.e. by imposing *α* = 1, *β* = 0) with non-exceedance probability *q*. In practice, as suggested by Keef et al. (2013), it is sufficient to satisfy the constraints above for *q* = 1 and *ν* equal to the maximum observed value of the conditioning variate.

### Appendix C: Supporting diagnostic plots

Figure 10 supports the discussion in Section 4 around Fig. 8 regarding agreement between simulated spatial trajectories under the fitted SCE model, and observed trajectories. The figure shows 100 simulated trajectories for transects emanating to the North-East and to the North-West, from the central conditioning location, in black. The trajectories all correspond to a conditioning value *x*_{0} with non-exceedance probability in the interval (0.95, 0.96). Also shown are observed trajectories available (22 and 20 in number respectively for the left and right hand cases), for the same conditioning, in red.

Figure 11 illustrates the fitted conditional Gaussian correlation function of Eq. 6. Consider two remote locations, equidistant from a third conditioning location. For each of the curves in the figure, the x-coordinate of the black disc indicates the distance between the remote locations. The curve passing through each disc gives the value of conditional residual correlation as a function of the (common) distance of the remote locations from the conditioning site. Thus, the leftmost curve pertains to two remote locations approximately 30 km apart. When the distance of these locations to the conditioning site is large, the remote locations have large residual correlation since conditioning has effectively no effect: the residual process is essentially unconstrained, and the value of correlation determined by the powered exponential *ρ*. However, as distance to the conditioning site is reduced, residual correlation decreases, since more sample variation is explained by the SCE model (as opposed to the SCE conditional residual process). The minimum value of distance to conditioning site occurs when the three locations are collinear, with the conditioning site mid-way between remote locations. For this arrangement, conditioning induces a small negative residual correlation between remote locations. As the distance between remote locations increases, for different curves left to right, the maximum conditional residual correlation is reduced.

### Deviance and DIC

The table below provides MCMC deviance statistics and DIC values to support the discussion in Section 4.

Model |
| Constraints | \(\overline {D(\varOmega )}\) | sd( | \(D(\bar {\varOmega })\) | DIC |
---|---|---|---|---|---|---|

ABN | 1 | Y | 1770 | 8 | 1740 | 1800 |

ABN | 6 | Y | 1550 | 11 | 1490 | 1620 |

ABP | 6 | Y | 1580 | 3 | 1570 | 1580 |

ABN | 1 | N | 1730 | 7 | 1760 | 1760 |

ABN | 6 | N | 1470 | 10 | 1422 | 1520 |

ABP | 6 | N | 1510 | 4 | 1500 | 1510 |

Columns in the table are as follows: *n*_{θ} is the number of directional parameters in the model, ‘Constraints’ indicates whether the conditional quantile constraints of Keef et al. (2013) were imposed, \(\overline {D(\varOmega )}\) is the posterior mean deviance, sd(*D*) is its standard deviation and \(D(\bar {\varOmega })\) the deviance calculated using posterior mean parameters. In summary, there is evidence in favour of a directional model, since lower values of deviance and DIC are obtained for *n*_{θ} = 6 compared with *n*_{θ} = 1 both with and without application of conditional quantile constraints. DIC values for ABP are somewhat lower than for the corresponding ABN, but differences are small.

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Shooter, R., Tawn, J., Ross, E. *et al.* Basin-wide spatial conditional extremes for severe ocean storms.
*Extremes* **24**, 241–265 (2021). https://doi.org/10.1007/s10687-020-00389-w

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DOI: https://doi.org/10.1007/s10687-020-00389-w