# Numerical and Statistical Analyses of Tsunami Heights with the L-Moments Method
## 1 Introduction  
The increasing frequency of earthquakes in the regions ranging from South and Central America to North America, and the Kyushu and Honshu regions of Japan in recent years has caused a growing concern over the so-called 50-year period theory, which suggests that large earthquakes occur approximately every 50 years in the circum-Pacific seismic zone, which is referred to as the Ring of Fire [1]. Many countries close to the sea have been severely damaged by tsunamis that are caused by submarine earthquakes. For example, the impact of the 9.0-magnitude earthquake that occurred on 11 March 2011, off the east coast of Sendai, Japan, and the subsequent tsunami waves as high as 15 m, was felt worldwide. Nearly 20,000 people were killed, presumed killed or missing, and more than 150,000 people were dislocated, with the local property damage estimated at more than $220 billion USD [2]. This combined event also severely damaged the Fukushima nuclear power plant on the eastern Japan coast, the effects of which are still ongoing.  
In general, the Korean Peninsula is recognized as safe from tsunamis. However, the 1983 Central East Sea earthquake (7.7 M) and the 1993 Southwest Hokkaido earthquake (7.8 M) caused damage to the Korean Peninsula, and the probability of an undersea earthquake near the Korean Peninsula is increasing, given the 2016 Gyeongju earthquake (5.8 M) and the 2017 Pohang earthquake (5.4 M) [1]. The frequency of occurrence of submarine earthquakes in the waters around the Republic of Korea hasincreased in the 2010s compared with the 2000s. Accordingly, proper prevention measures against natural disasters have become necessary.  
There have been approximately 30 tsunamis worldwide that have caused massive damage. As huge tsunamis occur relatively rarely, observational data of tsunamis alone have limitations in predicting tsunami heights. Therefore, various methods have been proposed to predict the risks associated with tsunamis and to calculate probable tsunami heights. In general, for past tsunami events, the tsunami heights were estimated using numerical model experiments or statistical analysis methods. While the prediction of tsunami height through numerical modeling could quantify the risks associated with a tsunami with relative accuracy, it is nearly impossible to model all tsunami cases; even producing models for a small number of cases requires considerable time and cost. In contrast, statistical analysis after-the-fact is highly accurate because it uses actual tsunami measurement data. However, it is difficult to define a precise height distribution because the available number of tsunami height observation points is very limited.  
Applying statistical methods to predict tsunami events in the East Sea is difficult, given the very few past events as well as the lack of data. Therefore, in this study we predicted the maximum tsunami heights caused by the initial waveforms of virtual tsunamis and applied statistical analyses for the Imwon Port, which is the most susceptible to damage by an unexpected tsunami in the East Sea, and for the Sadong Port, which is scheduled for a new airport construction. For both cases, we combined numerical modeling with statistical analysis methodology. For the governing equations for numerical modeling, we applied linear shallow-water equations. To select the best-fit distribution of the tsunami heights obtained through such numerical modeling, we applied the L-moments method and the L-moment ratio diagram.