# Improved Simplified Engineering Fault Displacement Hazard Evaluation Method for On-Fault Sites
## 2 Development of the Improved, Simplified Hazard Evaluation Method
### Geometrical Conditions for the Sites on the Mapped Fault Line  
The displacement at the site should be calculated by accounting for the probability that the rupture crosses the site, depending on two random variables \(x_{i}\) and \(L\) (Figure 1a). The simplest assumption is the uniform distribution of the initiation coordinate of the ruptures along the fault length \(x_{i}\) with uniform probability density function \(f(x_{i})=1/FL\). However, apparent geometrical limitations should be considered, since \(x_{i}\) and \(L\) are not independent. At any site position \(x_{s}\), these are limitations of the range of random variables \(x_{i}\) and \(L\), which are mutually dependent within the fault length \(FL\). This is illustrated in Figure 2.  
Figure 1: Geometrical arrangement for off-fault sites (**a**) and on-fault sites (**b**) cases. Definitions of variables were used for characterising the rupture, position of the rupture along the mapped length of the fault, position of the site and the distance from the end of the fault to the initiating point of the rupture. The explanations are given on the drawings. Let us fix the mapped rupture length \(FL\) and the site's coordinate \(x_{s}\). In Figure 2a, \(x_{s}=1/3 FL\), and in Figure 2b, \(x_{s}=1/2 FL\). The starting point, \(x_{i}\), and rupture length, \(L\), vary in Figure 2. For illustration purposes, elliptical displacement distribution is assumed along the rupture length; however, the values are used only for illustrating the physical picture. The lengths and coordinates in Figure 2 are normalised to the fault length, \(FL\). The displacements are normalised to the maximum displacement at \(l=0.5\times FL\). Consequently, while evaluating the surface displacement at the site, those ruptures should be accounted for, for which the following geometrical conditions hold:  
1. The rupture should start below the site's coordinate, \(x_{i}<x_{s}\) (ruptures illustrated by red dotted curve do not cause displacements at the site; Figure 2a);
2. Those ruptures contribute to the site displacement that crosses the site, that is, \(x_{i}+L>x_{s}\) or \(L>(x_{s}-x_{i})\equiv l\) (see the solid lines in Figure 2a,b). The ruptures, illustrated by the dotted lines and shorter ones, do not cause displacements at the site;
3. Ruptures with \(x_{i}+L\leq FL\) are realistic only. The actual rupture with length \(L\) should fit inside the fault length \(FL\). Figure 3 displays the geometrical conditions listed above. Those ruptures contribute to the site hazard, which complies with the geometrical conditions. The number of ruptures contributing to the site hazard is zero if the \(x_{i}\geq x_{s}\). The number contributing to the site hazard is the largest if the site is in the middle of the fault since \(x_{i}\), which in this case can vary in \(0\leq x_{i}\leq x_{s}=0.5\times FL\). In this case, 50% of all possible ruptures should be considered, the length of which is \(L>1/2\times FL\). The ratio of the area of the parallelogram to the total area of the triangle 1-0-1 is 0.5. If the site is at \(x_{s}/FL=1/6\), the maximum number of the ruptures to be considered is initiated at \(0\leq x_{i}\leq x_{s}=0.2\times FL\) and the maximum number of ruptures to be considered is -32% of all possibles. Figure 2: Ruptures that contribute to the fault displacement hazard at a site located \(x_{s}=1/3 FL\) (**a**) and at \(x_{s}=1/2 FL\) (**b**). Table 1 shows the case where the site is in the middle of the mapped fault length, \(x_{s}/FL=0.5\) and \(s=x_{i}=0\). The last column of Table 1 shows the result of the screening if \(x_{i}/FL=0.25\). Table 2 illustrates the case where \(x_{s}/FL=0.25\times FL\) and \(x_{i}=0\), and the last column shows the result of the screening for \(x_{i}=0.1\times FL\). The first two columns in Tables 1 and 2 are data taken from the case study for the Paks site in Hungary [5]. As seen in Tables 1 and 2, the dependence of whether a rupture contributes to the hazard is a complex function of \(x_{s}\), \(x_{i}\), \(L\) and \(FL\).