## 6 Design Contours for Gowlers, Bergy Bits, and Icebergs
### Two-Dimensional Design Contours  
The PDF for the water line length proposed in [51] is expressed as:  
\[f_{Lw}(x)=0.43\left(\frac{1}{60}e^{-\frac{x}{60}}\right)+0.57\left(\frac{1}{8} e^{-\frac{x}{8}}\right) \tag{19}\]  
This is due to the hypothesis that the number of icebergs with L greater than 20 m should approximately be equal to the number of icebergs with waterline length between 5 and 20 m. In the following, this probability model is applied for the purpose of illustrating the construction of two-dimensional environmental contours by also including the iceberg drift velocity as a random variable. Data of ice drift speed in the Pechora Sea were collected by [PERSON] and [PERSON] [48]. They found that a two-parameter Weibull distribution gave an adequate fit to the measurements:  
\[\text{F}_{\text{X}}(\text{x})=1-\exp\{-(\text{x}/\theta)^{\gamma}\} \tag{20}\]  
where the scale and shape parameters are obtained as \(\theta=0.213\) and \(\gamma=1.406\), respectively. The probability for a ship to experience an impact from an ice feature is assumed to be higher for the smallest ice features (growlers and bergy bits) versus larger icebergs (given that a geographical encounter with such an object has taken place). Denoting the occurrence rates for the two main categories of ice features by \(\rho_{1}\) and \(\rho_{2}\) (per traveled distance unit for the ship), and the corresponding conditional collision probabilities by \(p_{coll1}\) and \(p_{coll2}\), the numbers of impacts during a period of \(N_{year}\) are then obtained as follows (with subscript 1 denoting smaller ice features and subscript 2 referring to larger features):  
\[N_{1}=p_{coll1}\cdot N_{year}\cdot\rho_{1}\cdot LN_{2}=p_{coll2}\cdot N_{year} \cdot\rho_{2}\cdot L \tag{21}\]  
where \(L\) is the average distance traveled by the ship in Polar waters per year. The corresponding probabilities of exceedance and reliability indices are then obtained as:\[P_{f1}=\frac{1}{N_{1}},\beta_{F1}=\Phi^{-1}\Big{(}1-P_{f1}\Big{)}P_{f2}=\frac{1}{N_ {1}},\beta_{F2}=\Phi^{-1}\Big{(}1-P_{f2}\Big{)} \tag{22}\]  
By application of the cumulative distribution for the mass of the ice feature and the cumulative distribution for the drift speed, the corresponding two-dimensional environmental design contour can be obtained (by also assuming that mass and drift speed are uncorrelated). It is reasonable to consider collision with the two main categories of ice features as two distinctly different events, since the induced structural damage level typically is significantly different. Accordingly, the cumulative distribution function for the mass is also split in two by application of the respective terms for the density function in Equation (19). As an example, a case where the number of impact events for the growler/bergy bit category is \(N_{1}=750\) is considered, which implies that the corresponding exceedance probability is 0.00135. This gives a circle with radius equal to three in the normalized plane, and the resulting mass-velocity contour in the physical parameter plane is shown in Figure 13.  
By also including the ship speed as a deterministic parameter, a sequence of two-dimensional contours of a similar type is obtained. These are shown as a 3D surface in Figure 14a with the relative speed (ship speed plus drift speed) along the second horizontal axis, and the drift speed along the vertical axis. The corresponding 2D projections are illustrated in Figure 14b. In order to identify the relevant design scenario in terms of ice feature mass and relative velocity, the level curves corresponding to constant kinetic energy are required. The corresponding tangent point between these level curves and the mass-velocity contours then represent the most critical scenario along the environmental contour. This is illustrated in Figure 15. It is seen that for increasing ship speeds, the tangent points are shifted towards increasingly greater values for the mass of the ice feature.