## 4 Statistical Models of Freeze Length Characteristics
\]  
Again, the largest variation of the common ratio among the quadrants  
campaign in [20], i.e., \(q=0.73\) for \(d=10\,\mathrm{m}\) and \(q=0.77\) for \(d=15\,\mathrm{m}\) (results for \(d=12.5\,\mathrm{m}\) were not reported). The result that most of the scale parameters \(a_{\mathrm{s}}\) of the models for the short freeze lengths are larger than 1 indicates a shift of the fitting functions toward larger \(x\) values, crossing the point \((x=1+x_{0},y=1)\). In contrast, a fit of a non-shifted exponential or geometric CCH would cross the point \((x=1,y=1)\) representing the discrete freeze length and corresponding maximum CCH value. The actual shift \(x_{0}\) of a CCH can be obtained as the solution to the following condition:  
\[y_{\mathrm{s}}(1+x_{0})=a_{\mathrm{s}}\,\exp[b_{\mathrm{s}}(x_{0})]=1 \tag{6}\]  
which yields  
\[x_{0}=\frac{1}{b_{\mathrm{s}}}\ln\biggl{(}\frac{1}{a_{\mathrm{s}}}\biggr{)} \tag{7}\]  
Table 3 shows the shifts \(x_{0}\) for the different quadrants and distances which assume positive non-integer values \(0<x_{0}<1\) with the exception of Q1 for \(d=15\,\mathrm{m}\). To comply with the property of a CCH \(G(x)\), i.e., \(0\leq G(x)\leq 1\), the shift \(x_{0}\) inherent in the model \(y_{\mathrm{s}}(x)\) in (5), causing the crossing of the point \((x=1,y=1)\), is accounted for by defining  
\[G(x)=\min\{y_{\mathrm{s}}(x),\;1\} \tag{8}\]  
Second, we consider the domain of long freeze lengths \(x\in\mathcal{X}_{\mathrm{t}}=\{10,\ldots,30\}\) constituting the tails of the freeze length progressions. In particular, tails with slower decays captured within the domain \(\mathcal{X}_{\mathrm{t}}\) compared to domain \(\mathcal{X}_{\mathrm{s}}\) appear for the following cases (see also Figures 10 and 11):  
\[ \begin{array}{ll}d=12.5\,\mathrm{m}:&\text{all, Q3, Q4}\\ d=15\,\mathrm{m}:&\text{all, Q1, Q3, Q4}\end{array} \]  
Table 4 shows the coefficients and common ratio of the exponential and geometric progressions, respective, for the tails of domain \(\mathcal{X}_{\mathrm{t}}\) along with the goodness of fit performance for the following fitting functions:  
\[y_{\mathrm{t}}(x)=a_{\mathrm{t}}\,\exp[b_{\mathrm{t}}(x-1)]=a_{\mathrm{t}}\, q_{\mathrm{t}}^{(x-1)} \tag{9}\]  
We observe adjusted coefficients of determination in the range \(0.89<\mathcal{R}_{\mathrm{a}}^{2}<1.00\) which again relates to very good fitting performance. However, in contrast to the short freeze length domain \(\mathcal{X}_{\mathrm{s}}\), the values of the scale and decay parameters vary widely in the ranges \(0.08<a_{\mathrm{t}}<0.71\) and \(-0.15<b_{\mathrm{t}}<-0.03\), respectively. On the other hand, as \(a_{\mathrm{t}}<1\), the concept of a shift \(x_{0}\) is not needed for the modeling of the tails. To reveal additional insights into the freeze length characteristics associated with the tails, the fitting function (9) may be formulated as compound probability  
\[y_{\mathrm{t}}(x)=p_{\mathrm{t}}\cdot\Pr(X>x\,|\,\mathrm{LF}) \tag{10}\]  
where the probability \(p_{\mathrm{t}}\) that a long freeze (LF) occurs is given by  
\[p_{\mathrm{t}}=a_{\mathrm{t}} \tag{11}\]  
Further, the probability \(\Pr(X>x\,|\,\mathrm{LF})\) that a freeze \(X>x\) occurs conditioned on such a freeze being long, is given by  
\[\Pr(X>x\,|\,\mathrm{LF})=\exp[b_{\mathrm{t}}(x-1)]=q_{\mathrm{t}}^{(x-1)} \tag{12}\]  
Let us consider quadrant Q3 for \(d=15\,\mathrm{m}\) that shows a particularly flat and low tail. The probability that the tail occurs is rather small, i.e., \(p_{\mathrm{t}}=a_{\mathrm{t}}\simeq 0.088\). However, the corresponding tail decays very slowly: \(b_{\rm t}\simeq-0.039\Leftrightarrow q_{\rm t}\simeq 0.962\), indicating a risk for extraordinary large freeze lengths. Finally, Figure 12 shows the fittings over all quadrants according to the models formulated in (4) and (9) for freeze lengths in the domains \(\mathcal{X}_{\rm s}\) and \(\mathcal{X}_{\rm t}\), respectively.