### Abbreviations

ACF

EVA

GPD

GEV

CI

POT

MPM

MOM

MLE

PPC

*R*

IID

PACF

EVT

MEF

IQR

SE

P-P

Q-Q

LS

### 1. Introduction

### 2. Theoretical Background

### 2.1 Extreme value theory

*r*-return period value

*x*is obtained by,

_{r}*r*– return period in model scale for the

*Hr*-hour in real scale is obtained by

*D, N*, and

_{p}, dt*λ*are the test duration in hours, number of peak pressures, average time interval of impacts in seconds, and real-to-model scale ratio, respectively.

*L*) or by minimizing the negative log-likelihood function (-

*l*).

*H*is defined to obtain the

*SE*(standard error =

*stdev*of sample mean =

*CI*(confidence interval), in which

*stdev*is a standard deviation. It is a square matrix of 2nd order partial derivatives of a scalar-valued negative log-likelihood function. The estimation symbol ^ is dropped hereafter. An inverse matrix of

*H*yields the variance-covariance matrix

*V*. The

*SEs*of the distribution parameters are obtained by the square root of the diagonal terms of

*V*.

*SEs*of the

*r*-return period extreme values are more practical than the

*SEs*of the distribution, the Delta (∇) method (Xu and Long, 2005) for the GEV is used. Hence, the

*SE*is the square root of the variance of the

*r*-return period extreme value

*u*approximately yield the GPD when the IID variables follow a GEV distribution. The GPD is derived using the conditional exceedance distribution function for IID random variables.

*r*-return period value

*x*for the GPD is shown in the equation below. In this equation,

_{r}*n*is the number of measured points per unit time (1 h for the sloshing test, 1 year for the environment) and

_{y}*r*

^{′}and Δ are the number of years and measuring interval in hours, respectively.

*l*) for GPD is,

*SEs*for the

*r*-return period extreme values is shown in the equation below. Hereafter,

*u*is essential for the GPD because the POT approach provides a model for IID exceedances over a high threshold. There are various capable methods for determining

*u*, such as the MEF (mean excess function or MRL (mean residual life), most popular method (Kim and Song, 2012)) plot, parameter stability plot, dispersion index plot, rule of thumb, and multiple-threshold model method (Bommier, 2014). Because the

*MEF*is a linear function of

*u*, the starting point of linearity found in the MEF plot is to be a proper value as the threshold value.

### 2.2 Other distributions

### 2.3 Autocorrelation and partial autocorrelation

*r*(

_{x}*τ*) shows the correlation with lag

*τ*between

*x*(

*t*) and

*x*(

*t-τ*) in a given time series. The PACF

*p*(

_{x}*τ, τ*) is defined by the autocorrelation between

*x*(

*t*) and

*x*(

*t-τ*) in a given time series, with the linear dependence of

*x*(

*t*) on

*x*(

*t-1*),

*x*(

*t-2*), ...,

*x*(

*t-τ-1*) removed.

##### (15)

*CI*for sloshing impact pressures was realized in the Python programming language in this study, rather than the ‘R’ programming language and in-house code used in Choi (2016) for the extreme environmental condition.

### 2.4 Parameter estimation methods of distribution

### 3. Sloshing Impact Pressures and Conventional Fitting

*λ*= 50 for the No. 2 tank of a 160 k (160,000 m

^{3}) conventional LNG carrier was conducted by filling it to 30% with water and air under 1 year North Atlantic beam sea condition. This study dealt with the sample pressure record from a single Kistler piezo-type pressure sensor at only one location. Fig. 1 shows the extracted 764

*N*sloshing peak pressures with an initial threshold of 2.5 kPa and time window of 0.2 s, which is hereafter defined as one block for the block maxima approach in the model scale. The measured maximum pressure was 244.7 kPa in model scale. The total test duration was 2545.6 s in model scale (

_{p}*D*= 5

*h*in real scale);

*dt*was 3.33 s. Fig. 2 shows the results of the conventional procedure for the 3-parameter Weibull fitting with all 764 data points. The estimated 3 h MPM (

*Q*= 1/

*r*= 0.002) as a return level for the 3 h return period is 143.6 kPa, and it may be underestimated considering the last two large pressures, where

*r*= 3

*h*/5

*h*×764 = 458.4. A value of 143.6 kPa (75.4 bar) for water yields 69.4 kPa (35.4 bar) for LNG in real scale. The maximum value of the abscissa is obtained by multiplying the maximum measured data by 1.5.

*Q*

_{1}-

*γ*×

*IQR*to

*Q*

_{3}+

*γ*×

*IQR*are defined as the outliers, where

*Q*

_{1},

*Q*

_{3}, and

*IQR*represent the first and third quartiles and a interquartile range

*Q*

_{3}-

*Q*

_{1}of residuals, respectively.

*γ*is a constant value traditionally given as 1.5 by Tukey. However, in this paper, an appropriate value will be found, because smaller residuals yield a larger

*γ*. In the case of Fig. 2, the last two large pressures are considered as outliers with

*IQR*= 4.769 and

*γ*= 5.

*CI*, and the plots in Fig. 4 look bad (R = 0.982 in the Q-Q plot), one can say that all 764 data points are not IID. Hence, the 3 h MPM of 143.6 kPa is unconvincing.

*u*= 24.3 kPa and

*N*= 62. The estimated 3 h MPM is 164.7 kPa, which is larger than 143.6 kPa, as shown in Fig. 2. The fitting results look better than Fig. 2. In fact, considering

_{p}*IQR*= 4.172 and

*γ*= 7.57, Fig. 6 has no outlier. Furthermore, the data follows a Fréchet distribution rather than a Weibull distribution because positive shape parameter ξ= 0.434. Because this high threshold reduces the bias and the number of data points but increases the variance, it is necessary to choose a lower threshold as far as possible to increase the amount of data and decrease the variance.

*R*= 0.993 in Fig. 8.

*u*= 21.0 kPa and

*N*= 77, as shown in Fig. 10. This is slightly larger than 143.6 kPa for the 3-parameter Weibull fitting with all 764 data points in Fig. 2. The last two large pressures are considered the outliers, with

_{p}*IRQ*= 14.738 and

*γ*= 3.58. Considering ξ > 0 in Fig. 6, the Weibull with MOM is not an appropriate distribution in this case.

### 4. GEV and GPD for IID Variables

*CI*,. Hence, all of the cases pass the IID checks. The 11% case has the largest number of data points (

*N*= 85) among them, and data larger than 11% cannot pass the IID check. Therefore,

_{p}*u*= 19.8 kPa for the 11% case is finally chosen for the GPD threshold.

*N*=85). The estimated 3 h MPM for GPD is 167.14 kPa. This figure also has no outlier with

_{p}*IRQ*= 4.504 and

*γ*= 5.85.

*R*= 0.995. Hence, the final block size of this study is 10, with a time window of 2.0 s.

*IRQ*= 0.087 and

*γ*= 200.