Abstract Originally discussed in the context of ocean statistics for oil rig platform decks and satellite data, spatiotemporal exceeding probability distributions have recently been thought to explain the failure of second and third-order nonlinear models of distributions at a fixed point in space. To investigate this possibility, we derive a simplified model for the (2+1) exceeding probability distribution of ocean waves based on the Gaussian random field approach, with a particular interest in rogue waves. Using standard statistical tools, we show that (2+1)) Longuet–Higgins’ distribution does not depend on the spatiotemporal scale of the ocean surface, rather on the directional spectrum and the dimensionless wave height. Moreover, we use extreme value theory Gumbel (1958)[24] to find the expected maximum wave height in (2+1) dimensions, showing that Piterbarg’s framework provides a smaller expected maximum than Adler’s approach. Unfortunately, both are known for assigning maximum heights that do not saturate when the surface area is very large. Taking into account the mean rogue wave profile, we derive a new expression for the maximum wave height that provides realistic estimates. Finally, comparing the theory for the exceeding probability and expected maximum dimensionless height with the North Alwyn data set of Stansell (2004)[47] we conclude that both (2+1) versions of Longuet-Higgins (1952, 1980)[31,34] distributions fail to provide an accurate description of both return period and maximum observed wave heights, thus undermining the argument that the error in ocean wave statistics can be cured by extending point statistics to higher-dimensions.

中文翻译：

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(2+1) 高斯海中的流氓波统计 I：窄带分布

摘要 最初是在石油钻井平台甲板和卫星数据的海洋统计背景下讨论的，最近人​​们认为时空超出概率分布可以解释空间中固定点分布的二阶和三阶非线性模型的失败。为了研究这种可能性，我们基于高斯随机场方法推导出了海浪的 (2+1) 超出概率分布的简化模型，特别关注流氓海浪。使用标准统计工具，我们表明 (2+1)) Longuet-Higgins 的分布不依赖于海面的时空尺度，而是依赖于方向谱和无量纲波高。此外，我们使用极值理论 Gumbel (1958)[24] 来找到 (2+1) 维的预期最大波高，表明 Piterbarg 的框架提供了比 Adler 的方法更小的预期最大值。不幸的是，两者都以分配最大高度而闻名，当表面积非常大时，该高度不会饱和。考虑到平均流氓波剖面，我们推导出了最大波高的新表达式，提供了现实的估计。最后，将超过概率和预期最大无量纲高度的理论与 Stansell (2004)[47] 的 North Alwyn 数据集进行比较，我们得出结论，Longuet-Higgins (1952, 1980)[31, 34] 分布未能提供对重现期和最大观测波高的准确描述，从而削弱了通过将点统计扩展到更高维度可以消除海浪统计中的错误的论点。