## 4 First-Year Ice Ridges
### Statistical Models for Key Parameters Associated with Ice Ridges  
The description of the ship-ridge interaction process given above gives reason to select four variables for characterization of ice ridges, and these also form the basis for establishment of environmental design contours. These four variables are the thickness of the consolidated layer, its crushing strength, its flexural strength, and the draft of the keel. Statistical models are introduced for each of these variables. The thickness of the consolidated layer depends on geographical area, its environmental conditions, and the time of the season. Here, the consolidated layer thickness of ridges located in the Barents Sea is considered. The data sets for the thickness of the consolidated layer were collected from field measurements that were carried out from 2002 to 2011 [24]. Figure 4: Illustration of a ship and ice ridge interaction process. Based on a statistical analysis, the Gamma distribution is found to provide a satisfactory description of the experimental data. The Gamma probability density function is given by:  
\[f(h_{cl})=\frac{1}{\Gamma(k)\theta^{k}}h_{cl}^{k-1}exp(-\frac{h_{cl}}{\theta}) \tag{7}\]  
where \(\Gamma(\cdot)\) represents the gamma function, and \(\theta\) and \(k\) are the scale and the shape parameters, respectively. The resulting values of the two parameters are found to be 2.97 (shape) and 0.54 (scale) based on application of the moment estimators. Fitting of a Gamma distribution to the data is illustrated in Figure 5 [39]. The distribution of the consolidated layer thickness will be different for different Polar regions, and also associated with large uncertainties. The data measured in the Barents Sea were obtained based on mechanical or thermal drilling, and these provide at least a certain amount of information. For other Polar regions, where only limited or lacking data coverage exists, utilization of data related to the thickness of the surrounding level ice can possibly provide an estimate of the distribution of the consolidated layer thickness by application of a representative \"enhancement factor\". Because there are very limited experimental data for the mechanical properties of the consolidated layer, these values are assumed to be close to those of the surrounding level ice due to limited experimental data for the consolidated layer of ice ridges. For the flexural strength of first-year sea ice, a large number of measurements have been performed in different Arctic regions [22]. This strength parameter depends basically on the ice temperature, salinity, and inherent brine volume. The data of flexural strength in the present study are based on those provided by Timco [40]. These comprise in situ experimental data obtained from different Arctic regions, such as Baffin Island, Greenland, and the Gulf of Bothnia, and the Canadian and Alaskan Beaufort Sea. The two-parameter Weibull distribution can typically be applied to describe probabilistic distribution of the flexural strength \(\sigma_{f}\), which is expressed as:  
\[f\left(\sigma_{f}\right)=\frac{\beta}{\alpha}\left(\frac{\sigma_{f}}{\alpha} \right)^{\beta-1}exp\left(-\left(\frac{\sigma_{f}}{\alpha}\right)^{\beta}\right) \tag{8}\]  
where the scale parameter \(\alpha\) and the shape parameter \(\beta\) are determined as 0.582 and 2.090, respectively, by means of regression estimators as implemented in a probability paper. A fitted Weibull density function (PDF) for the flexural strength of first-year sea ice in the Arctic is shown in Figure 6 [39]. Figure 5: Density function for the thickness of the consolidated layer \(h_{cl}\) fitted to the measurements. The level ice crushing strength depends on various parameters such as loading direction, porosity, salinity, temperature, failure mechanism, etc. For ship-ridge interaction, samples that are loaded in the horizontal direction should be considered. However, only limited data are published for such samples in relation to Arctic regions.