Abstract Various methods have been proposed for defining an environmental contour, based on different concepts of exceedance probability. In the inverse first-order reliability method (IFORM) and the direct sampling (DS) method, contours are defined in terms of exceedances within a region bounded by a hyperplane in either standard normal space or the original parameter space, corresponding to marginal exceedance probabilities under rotations of the coordinate system. In contrast, the more recent inverse second-order reliability method (ISORM) and highest density (HD) contours are defined in terms of an isodensity contour of the joint density function in either standard normal space or the original parameter space, where an exceedance is defined to be anywhere outside the contour. Contours defined in terms of the total probability outside the contour are significantly more conservative than contours defined in terms of marginal exceedance probabilities. In this work we study the relationship between the marginal exceedance probability of the maximum value of each variable along an environmental contour and the total probability outside the contour. The marginal exceedance probability of the contour maximum can be orders of magnitude lower than the total exceedance probability of the contour, with the differences increasing with the number of variables. For example, a 50-year ISORM contour for two variables at 3-h time steps, passes through points with marginal return periods of 635 years, and the marginal return periods increase to 10,950 years for contours of four variables. It is shown that the ratios of marginal to total exceedance probabilities for DS contours are similar to those for IFORM contours. However, the marginal exceedance probabilities of the maximum values of each variable along an HD contour are not in fixed relation to the contour exceedance probability, but depend on the shape of the joint density function. Examples are presented to illustrate the impact of the choice of contour on simple structural reliability problems for cases where the use of contours defined in terms of either marginal or total exceedance probabilities may be appropriate. The examples highlight that to choose an appropriate contour method, some understanding about the shape of a structure's failure surface is required.

中文翻译：

---

环境等高线的边际和总超标概率

摘要 基于超越概率的不同概念，已经提出了各种定义环境等高线的方法。在逆一阶可靠性方法 (IFORM) 和直接采样 (DS) 方法中，轮廓是根据标准正态空间或原始参数空间中由超平面界定的区域内的超出来定义的，对应于边际超出概率在坐标系的旋转下。相比之下，最近的逆二阶可靠性方法 (ISORM) 和最高密度 (HD) 轮廓是根据标准正常空间或原始参数空间中联合密度函数的等密度轮廓定义的，其中超出为定义为轮廓外的任何位置。根据等高线外的总概率定义的等高线比根据边际超出概率定义的等高线更加保守。在这项工作中，我们研究了沿环境等值线的每个变量的最大值的边际超出概率与等值线外的总概率之间的关系。轮廓最大值的边际超越概率可以比轮廓的总超越概率低几个数量级，差异随着变量数量的增加而增加。例如，两个变量在 3 小时时间步长的 50 年 ISORM 等值线，通过边际回报期为 635 年的点，四个变量的等值线的边际回报期增加到 10,950 年。结果表明，DS 等高线的边际超标概率与总超标概率的比率与 IFORM 等高线的相似。但是，沿 HD 轮廓的每个变量的最大值的边际超出概率与轮廓超出概率没有固定关系，而是取决于联合密度函数的形状。举例说明了轮廓选择对简单结构可靠性问题的影响，在这种情况下，使用根据边际或总超限概率定义的等高线可能是合适的。这些示例强调，要选择合适的轮廓方法，需要对结构破坏面的形状有一定的了解。沿 HD 轮廓的每个变量最大值的边际超出概率与轮廓超出概率没有固定关系，而是取决于联合密度函数的形状。举例说明了轮廓的选择对简单结构可靠性问题的影响，在这种情况下，使用根据边际或总超限概率定义的等高线可能是合适的。这些示例强调，要选择合适的轮廓方法，需要对结构破坏面的形状有一定的了解。沿 HD 轮廓的每个变量最大值的边际超出概率与轮廓超出概率没有固定关系，而是取决于联合密度函数的形状。举例说明了轮廓选择对简单结构可靠性问题的影响，在这种情况下，使用根据边际或总超限概率定义的等高线可能是合适的。这些示例强调，要选择合适的轮廓方法，需要对结构破坏面的形状有一定的了解。举例说明了轮廓选择对简单结构可靠性问题的影响，在这种情况下，使用根据边际或总超限概率定义的等高线可能是合适的。这些示例强调，要选择合适的轮廓方法，需要对结构破坏面的形状有一定的了解。举例说明了轮廓选择对简单结构可靠性问题的影响，在这种情况下，使用根据边际或总超限概率定义的等高线可能是合适的。这些示例强调，要选择合适的轮廓方法，需要对结构破坏面的形状有一定的了解。