Deriving environmental contours from highest density regions

Environmental contours are an established method in probabilistic engineering design, especially in ocean engineering. The contours help engineers to select the environmental states which are appropriate for structural design calculations. Defining an environmental contour means enclosing a region in the variable space which corresponds to a certain return period. However, there are multiple definitions of environmental contours for a given return period as well as different methods to compute a contour. Here, we analyze the established approaches and present a new concept which we call highest density contour (HDC). We define this environmental contour to enclose the highest density region (HDR) of a given probability density. This region occupies the smallest possible volume in the variable space among all regions with the same included probability, which is advantageous for engineering design. We perform the calculations using a numerical grid to discretize the original variable space into a finite number of grid cells. Each cell's probability is estimated and used for numerical integration. The proposed method can be applied to any number of dimensions, i.e. number of different variables in the joint probability model. To put the highest density contour method in context, we compare it to the established inverse first-order reliability method (IFORM) and show that for common probability distributions the two methods yield similarly shaped contours. In multimodal probability distributions, however, where IFORM leads to contours which are difficult to interpret, the presented method still generates clearly defined contours.