## 3 Hazard-Consistent Fragility Curves of Soil Liquefaction
### Simulation of Hazard-Consistent Seismic Parameters  
To generate the pair of seismic parameters of earthquake magnitude (\(M\)) and peak ground acceleration (PGA) that match the hazard level, we first simulate the magnitude, \(M\), of a seismic event and the distance, \(R\), from the site to the source. Then we use Equation (16) to calculate the \(\mathrm{PGA_{R}}\) of the horizontal peak ground acceleration at the hard site for this given scenario of \(M\) and \(R\). In this way, we obtain the pair of (\(M\), \(\mathrm{PGA_{R}}\)) as well. Table 4 lists the random variables used in the Monte Carlo simulation. Seismic sources can be either line/area sources or point sources, depending on whether they are in the shallow zone or not. For well-defined faults (Type I sources [21]), the model uses line sources with known parameters such as the location, direction, and length of the fault line, for example, the active Type I fault in Taiwan. In this case, the random variables used for this type of source are the magnitude of \(M\), the epicentral distance of \(R_{e}\), the hypocentral depth of \(h\), and the model error term of \(E\). Future seismic events could initiate a rupture at any point along the fault line. The length of the rupture in these future events is dependent on magnitude \(M\). For area sources, we can use a line source to represent them, but we have to account for the uncertainty of the line source's location and orientation (Type III source [21]). The fault line's length depends on the magnitude of \(M\). In this scenario, the random variables include the magnitude (\(M\)), the depth of the hypocenter (\(h\)), the direction of the fault rupture (\(\theta\)), and the model error term (\(E\)). We can divide the source region into grid points and assume that each grid point can be the origin of a fault rupture. Therefore, we only use the depth of the hypocenter (\(h\)) as a random variable to capture the uncertainty of the seismic event's position. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Source Zone** & **Source Type** & **Random Variables** & **Notes** \\ \hline \multirow{4}{*}{Shallow zone} & Line source & \(M\), \(R_{e}\), \(h\), \(E\) & A line source is used to model a well-defined fault (i.e., Type I source) with known parameters such as the fault line’s location, orientation, and length, exemplified by the active Type I fault in Taiwan. \\  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \hline \multirow{2}{*}{Deep zone} & \multirow{2}{*}{Point source} & \multirow{2}{*}{\(M\), \(h\), \(E\)} & The location error of the point source might exceed the length of the fault break; the distance from the epicenter is large, so the fault break length has a relatively minor effect on the distance from the epicenter. \\ \cline{1-1}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Types of seismic sources and random variables. When seismic sources are situated in a deep zone, they can be approximated as point sources [20]. In such cases, the positional error may surpass the rupture length, and the hypocentral distance is sufficiently large to render the rupture length negligible. Therefore, the only random variables to consider are the magnitude (\(M\)) and the hypocentral depth (\(h\)) with the model error term (\(E\)). The previously mentioned random variables follow distinct distributions. The probability density function for the earthquake magnitude, \(M\), is as follows [21]:  
\[f_{M}(m)= \begin{cases}0&for\;\;m\leq M_{0}\\ \frac{2.3\;b\exp[-2.3\;b(m-M_{0})]}{1-\exp[-2.3\;b(M_{u}-M_{0})]}&for\;\;M_{0} <m\leq M_{u}\\ 0&for\;\;m>M_{u}\end{cases}  \tag{19}\]  
in which \(b\) is the seismic region coefficient, \(M_{0}\) is the smallest magnitude of engineering concern (5.0 is used in this study), and \(M_{u}\) is the upper bound magnitude, which depends on the seismic region characteristics. We assume that the model error term, \(E\), follows a log-normal distribution.