# Improved Simplified Engineering Fault Displacement Hazard Evaluation Method for On-Fault Sites
## 2 Development of the Improved, Simplified Hazard Evaluation Method  
### Sources of Conservatism of Simplified Fault Displacement Hazard Evaluation Methods  
The fault displacement hazard for strike-slip faults can be evaluated as proposed in [26; 34]. The geometry and the geometrical variables are shown in Figure 1a. The evaluation results in rate \(\
u(D\geq D_{0})_{xyz}\) at which displacement \(D\) on the fault exceeds a specified amount \(D_{0}\) at the site with coordinates \(x\) and \(y\) and size \(z\) shown as follows:  
\[ \begin{split}\
u(D\geq D_{0})_{xyz}=\alpha(m_{0})\int_{m,s}f_{m,s }(m,s)P[sr\
eq 0|m]\times\int_{r}P[D\
eq 0|z,sr\
eq 0]\times\\ P[D\geq D_{0}|l/L,m,D\
eq 0|f_{R}(r)drdmds,\end{split}  \tag{1}\]  
where \(\
u(D\geq D_{0})_{xyz}\) is the annual rate of exceedance; \(x,y\) are the coordinates of the site area, and \(z\) denotes the site dimension; The \(\alpha(m_{0})\times f_{m,s}(m,s)\) is the rate of the earthquake with magnitude \(m\), where \(\alpha(m_{0})\) is the rate of events with magnitude \(m\geq m_{0}\) and \(f_{m,s}(m,s)\) is the bivariate probability density function for magnitude \(m\) and distance from the end of the fault \(s\); \(L\) is the rupture length, and \(l\) is the length on the fault where the closest distance from the site \(r\) is measured; \(P[sr\
eq 0|m]\) is the probability of non-zero surface movement due to the earthquake with magnitude \(m\) that can be calculated by the correlations of [35; 36; 37]; the term \(P[D\
eq 0|z,sr\
eq 0]\) is the conditional probability of non-zero surface movement at the site of area \(z^{2}\) at distance \(r\) from the fault; \(P[D\geq D_{0}|l/L,m,D\
eq 0]\) is the probability of \(D\geq D_{0}\) if the rupture caused by the earthquake with magnitude \(m\) causes a rupture with \(l/L\) regarding the site. The geometry is simplified, when the site is on the mapped fault line, as shown in Figure 1b. Since the site is on the fault line, \(f_{R}(r)\) reduces to the delta function, \(f_{R}(r)=\delta(r=0)\). The simplified engineering hazard evaluation methods, published in [4; 5] and also several other studies, e.g., [2], assume that \(f_{m,s}(m,s)=f(m)\times f(s)\). \(\alpha(m_{0})\) and \(f(m)\) are obtained from the Gutenberg\(-\)Richter relation. The following total probabilities are calculated:  
* The total probability of non-zero surface displacement \(P[sr\
eq 0]\), which depends on the magnitude distribution: \[P[sr\
eq 0]=\int_{m_{0}}^{m_{u}}f(m)\times P[sr\
eq 0|m]dm,\] (2)where \(m_{u}\) is the maximum probable magnitude for the source;
* The total probability that the rupture with length \(L\) caused by an earthquake with magnitude \(m\) and initiating at coordinate \(x_{i}=s\) crosses the site; In [5] and [31], simple geometrical considerations are made for the distribution of the fault ruptures along the total fault line and the probability of crossing the site. For example, in [5], the probability of crossing the site is proportional to the rupture length relative to the fault length;
* The most important simplification assigning the average displacement along the entire rupture length \(L\). Thus, the exact position of the rupture relative to the site, \(l/L\), can be neglected. Thus, the total probability of the average displacement \(D_{ave}\geq D_{0}\) is calculated via:  
\[P(D\geq D_{0})=\int_{m_{0}}^{m_{u}}f(m)P[D_{ave}\geq D_{0}|m,D\
eq 0]dm \tag{3}\]  
using empirical magnitude scaling relations \(D_{ave}=D_{ave}(m)\) and its inverse \(m=D_{ave}^{-1}(D_{ave})\) and the density function \(f(m)\). The methods following the concept above are over-conservative, since they do not consider the points as follows:  
1. The position of the site on the fault line;
2. The position of the starting point of the rupture concerning the site;
3. Whether all rupture lengths cross the site;
4. The displacement distribution along the fault. Remarks 1 to 3 relate to the site and faults' geometrical configuration.