# Numerical and Statistical Analyses of Tsunami Heights with the L-Moments Method
## 4 Results
### Results of Statistical Analysis and the L-Moments Method  
We used the method of L-moments and the maximum tsunami heights to create virtual tsunami scenarios, and then we drew the results into L-moment ratio diagrams to analyze the probability distribution of the tsunami heights at the Imwon Port and the Sadong Port. Various distributions, including generalized logistic (GL), generalized extreme value (GEV), generalized Pareto (GP), 3-parameter log normal (LN3), and Pearson type 3 (PA3), were candidates for the best-fit distribution.  
Figures 6 and 7 show the L-moment ratio diagrams used to identify the best-fit distribution type for representing the tsunami heights in this study. The candidate model probability distributions are plotted as curvilinear lines, and pairs of the sample third and fourth L-moment ratio values are numerically represented. The point numbers represent the undersea earthquake identification numbers.  
Figure 4: Maximum tsunami heights simulated for the representative cases at the Imwon Port.  
Figure 5: Maximum tsunami heights simulated for the representative cases at the Sadong Port.  
Figure 6: L-moment diagram along with the numbered points representing the third and fourth L-moment ratios of the simulated tsunami heights at the Imwon Port.  
By analyzing the tsunami heights through the L-moment diagram, we confirmed that the probability distribution of the tsunami heights could be changed for each tsunami case or location area. This indicates that previous studies of tsunami heights in limited areas that followed the log-normal distribution were not always accurate (e.g., [10]). Therefore, caution must be taken when applying a log-normal distribution to tsunami heights, regardless of the spatial width.  
From the perspective of disaster prevention, it is reasonable to select a probability distribution out of the many possibilities because it is impossible to predict the distribution of all the cases. By integrating all the cases, the results of applying the L-moment ratio are shown in Figures 8 and 9. Figure 8 shows that the log-normal distribution is appropriate for the Imwon Port, whereas Figure 9 shows the generalized Pareto distribution for the Sadong Port.  
Figure 8: L-moment ratio diagram based on integrating all of the cases for the Imwon Port.  
Figure 7: L-moment diagram along with the numbered points representing the third and fourth L-moment ratios of the simulated tsunami heights at the Sadong Port.  
Additionally, we compared three different goodness-of-fit measures, namely, negative log-likelihood (NLogL), Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC). When fitting models, it is possible to increase the likelihood by adding parameters, but doing so may result in overfitting. Both BIC and AIC attempt to resolve this problem by introducing a penalty term for the number of parameters in the model. Lower BIC (or NlogL and AIC) is better for the model, as summarized in Table 11. We thus confirmed that the best-fit distributions of the tsunami heights for the Imwon and Sadong Ports were the log-normal distribution and generalized Pareto distribution, respectively. Based on these results, Figures 10 and 11 show the cumulative distribution function (CDF) of the distributions of the two ports. Through CDF, the tsunami heights could be quantitatively predicted by calculating the probability of the tsunami height caused by an earthquake that might occur in the East Sea. Based on this quantitative prediction methodology, realistic and economical responses and measures can be presented for preparing hazard maps or designing sea-facing facilities.  
Figure 10: Cumulative distribution function of the log-normal probability of tsunami heights at the Imwon Port.  
Figure 9: L-moment ratio diagram based on integrating all of the cases for the Sadong Port.