## 3 Environmental Contour Method
These approaches are based on the joint PDF and the failure surface \(G(y_{c},\mathbf{S})\), which are combined in order to establish a transformation into a normalized space of independent, standard Gaussian variables (which is frequently referred to as the U-space). In the FORM approach, the failure surface in normalizedspace is represented by the tangent plane at the so-called design point. The corresponding probability of failure is approximated as:  
\[p_{f}(y_{c})\approx\Phi(-\beta) \tag{5}\]  
where \(\Phi\) represents the standard normal cumulative distribution function (CDF); \(\beta\) denotes the reliability index, and this also corresponds to the distance between the design point to the origin in normalized space. It is emphasized that the relationship between the failure probability and the standard cumulative Gaussian distribution function is due to the transformation into normalized and independent variables. Furthermore, this is a first order approximation based on linearization of the mechanical limit state function. Second order corrections to this approximation can be obtained, e.g., by the so-called SORM approximation. Generally, various types of Monte Carlo simulation techniques are also applied as a supplement to (or sometimes instead of) the approximation in Equation (5). However, determination of the design load effect, \(y_{N}\), which corresponds to the capacity that is required in order to withstand the loads associated with an \(N\)-year return period generally requires iterative reliability calculations for different values of \(y_{c}\). Accordingly, it is highly relevant to simplify Step II of the long-term analysis procedure, which is precisely the objective of environmental contour methods. The design contour for the \(N\)-year return period is frequently established by means of the inverse FORM (i.e., IFORM) approach [13; 20]. The probability of failure, \(p(y_{N})\) is then first specified in order to generate the environmental contour that corresponds to that probability. An \(n\)-dimensional sphere with the radius \(\beta_{F}\), which is obtained by means of the specified failure probability as expressed by Equation (6), is then first created in the normalized Gaussian space:  
\[\beta_{F}=\Phi^{-1}(1-p_{f}(y_{N})) \tag{6}\]  
This sphere is subsequently transformed into the physical parameter space to yield the design contour. The mapping can, e.g., be based on the inverse Rosenblatt transformation for cases where the joint density function of the variables is formulated by means of the conditional modelling approach [21] or by the Nataf transformation if the marginal PDFs of variables and the correlation coefficients between these variables are given [12]. The environmental contour obtained by the IFORM is accordingly the collection of physical environmental parameters that correspond to the normalized values, which are located on the sphere with radius \(\beta_{F}\) in the U space. It is found that the main benefit of the design contour method is due to the description of the environmental parameters being uncoupled from computation of the structural response. Due to its high efficiency and satisfactory accuracy, the environmental contour method has, e.g., been widely applied for assessment of ULS criteria, and in particular during preliminary design phases. The most critical response corresponding to the environmental conditions defined by the environmental contour can then be applied for estimation of the extreme response for the specified return period. If the response process for a given ice condition is stochastic, the extreme response can be taken to be an upper fractile of the corresponding short-term extreme response distribution. This is similar to the approach applied in the case of wave-induced response [11].