## 5 Results
### Correlation Analysis and the Shape-Product Approach  
A correlation analysis is completed to better understand how the numerical results are related to the input parameters with matrices showing the resulting correlation coefficients (\(r\)) calculated by the Pearson function. The correlation matrix in Figure 6a considers all analyzed theoretical lakes. The maximum wave crest elevation does not correlate well with any input parameter, suggesting that it depends on the combination of all lake characteristics (and in a real situation also on the impacting landslide properties, which are not considered in this study). Similarly, no relevant correlations are evident for flow depth at the shoreline. In the case of the mean flow depth at the shoreline, \(r\)-values of 0.669 and 0.665 are obtained for lake width and length, respectively. Maximum and minimum flow depths yield \(r\)-values of 0.71 and 0.694 for lake width and length, respectively. Higher correlation coefficients are calculated between _WDPP_ and lake characteristics: \(r\)-values of 0.835 and 0.786 for lake width and length, respectively; and \(r\)-values of 0.830 and 0.806 for lake area and volume, respectively. By plotting _WDPP_ against lake area and water volume (Figure 7) and considering the maximum water depth as an indicator, it is found that the theoretical lake basins form two groups. Lakes with a maximum \(D_{w}\) of 20 m (small red circles in Figure 7) can be separated from the others. This separation is supported by correlation analyses provided for the two lake geometry subgroups (\(D_{w}\leq 20\) m and \(D_{w}>20\) m). In the case of lakes with maximum depths greater than 20 m, the correlations between _WDPP_ and lake area and lake volume (see Figure 6b) have \(r\) values of 0.90 and 0.88, respectively. These relationships can be approximated by hyperbolic (power-type) functions. The coefficients of determination, \(R^{2}\), which express the variation of the dependent variable predicted by the independent variable and provide a measure of how well the curves fit the data (the closer \(R^{2}\) is to 1, the better fit-Figure 7 and Table 2). (the dense fluid) enters the lake and propagates through the water body [41]. For example, the dense volume might induce an overestimation of the maximum wave crest in open water. Consequently, further analysis and discussion only consider model scenarios with maximum depths greater than 20 m. _WDPP_ is well correlated to the lake area and water volume, thus a parameter that relates different lake characteristics and _WDPP_ is formulated to improve rapid assessments of impulse waves in mountain lakes. The new parameter is the \"shape product\" (herein labeled _ShpP_), which is defined using empirical Equation (2). Results are well correlated to _WDPP_ with an _r_-value of 0.962 (Figure 6 and Table 2):  
\[ShpP=\frac{V_{w}}{A}*\left(\frac{W*L}{d_{w}^{\frac{3}{2}}}\right)^{1/3}(\text {m}^{2/3}) \tag{2}\]  
where \(V_{w}\) is lake volume, \(A\) is lake area, \(W\) and L are lake width and length, respectively, and \(d_{w}\) is mean lake depth. The shape parameter does not represent any specific physical measurement or a dimensionless parameter. On the contrary, it is the measure of a lake's geometric characteristics most closely related to the _WDPP_. A plot of _ShpP_ against _WDPP_ reveals a linear relationship with an \(R^{2}\) of 0.9257 (Figure 8a and Table 2). Linear regression of _ShpP_ against _WDPP_ verifies the reliability of this relationship (Figure 8b), as it yields the same equation and \(R^{2}\) proposed in the previous analysis (Figure 8a). After calculating _WDPP_ from a specific _ShpP_, the equations in Table 3 provide a rough estimate of the expected flow depth at the lakeshoe. Plots of _WDPP_ against maximum, mean, and minimum flow depth obtained from the numerical models are shown in Figure 9. The mean flow depth is strongly and negatively correlated to _WDPP_ (\(r=-0.849\), Table 3).