## 1 Introduction
### Theoretical Background for the Evaluation of the Principal Fault Contribution to PGD Hazard  
For strike-slip faults [PERSON] et al. [18] proposed for the rate \(\
u(D\geq D_{0})_{xyz}\) at which the displacement, D, on the fault exceeds a specified amount, \(D_{0}\), the following equation:  
\[\
u(D\geq D_{0})_{xyz}=\alpha(m)\int_{m,s}f_{M,S}(m,s)P[sr\
eq 0|m]\times\int_{r}P[ D\
eq 0|z,r,sr\
eq 0]\times P[D\geq D_{0}|l/L,m,D\
eq 0]f_{R}(r)drdmds \tag{1}\]  
by [PERSON] and [PERSON] [19]. Here, the \(\
u(D\geq D_{0})_{xyz}\), annual rate of exceedance \(D\geq D_{0}\); \(x,y\) are the coordinates of the site area and, \(z\) denotes the dimension of the site. The \(L\) is the rupture length, and \(l\) is the length on the fault where the closest distance from the site \(r\) is measured. The \(\alpha(m)\) is the rate of earthquake with magnitude \(m\). The \(f_{M,S}(m,s)\) is the probability density function of the earthquake with magnitude \(m\) that is due to rupture with distance \(s\) from the end of the fault. The \(P[sr\
eq 0|m]\) is the probability of non-zero surface movement happens due to earthquake with magnitude \(m\) that can be calculated as proposed by [PERSON] and Coppersmith [9]. The \(P[D\
eq 0|z,sr\
eq 0]\) is the probability for the non-zero displacement will be at the site \(z\) under the non-zero rupture condition. The \(P[D\geq D_{0}|l/L,m,D\
eq 0]\) is the conditional probability of \(D\geq D_{0}\) if the earthquake of magnitude, \(m\) happens with location \(l/L\). The \(f_{R}(r)\) is the distribution density of the possible distance of the site area to the fault. The term \(P[D\
eq 0|z,sr\
eq 0]\) is the conditional probability of non-zero surface movement at the site of area \(z\) and that is at a distance r from the fault. Interpretation of the geometrical variables in the Equation (1) is shown in Figure 3.  
Figure 3: Definition of geometrical variables ([PERSON] et al., 2011 [18]). For strike-slip faults [PERSON] et al. [18] developed bilinear, quadratic, and elliptical models for magnitude-distance relationship applicable for principal-fault displacements in the form:  
\[\ln(D)=b\cdot x^{*}+a\cdot m+c \tag{2}\]  
Here, the \(x^{*}=\sqrt{1-\frac{1}{0.5^{2}}\left[\frac{l}{L}-0.5^{2}\right]^{2}}\) depends on the location of the site with respect to the fault as it is seen in Figure 3. If the site is located at the centreline to \(L\), the \(l/L=0.5\) and \(x^{*}=1\). If the site located on the line perpendicular to the endpoint of the fault surface projection \(l/L=0.0\) and \(x^{*}=0\). The coefficients are \(b=3.3041\), \(a=1.7927\) and \(c=-11.2192\), with a standard deviation on \(\ln(D)\) of \(1.1348\). The distributions and conditional probabilities needed for Equation (1) cannot be developed for the fault zone at the study site because of lack of data as indicated in the introduction. Even the rate \(\alpha(m)\) cannot be assessed, since a single, twenty thousand years old indication of Quaternary re-activation of the fault has only be identified. Further conservative simplifications should be introduced into the Equation (1). It can be assumed, that the site is located at the centerline to the fault trace, i.e., \(l/L=0.5\) and the fault is ruptured at full length, i.e., \(s=0\), see, Figure 3. Consequently, the product of conditional probabilities \(P[D\
eq 0|z,r,sr\
eq 0]\times P[D\geq D_{0}|l/L,m,D\
eq 0]\) will reduce to \(P(D\geq D_{0}|r,m)\). Further, it is assumed that the bivariate distribution density of magnitudes and distances, \(p(r,m)\) is known for the site. Based on these assumptions, for the annual rate of exceedance, \(\
u(D\geq D_{0})\) the following equation can be written:  
\[\
u(D\geq D_{0})=\int_{m_{0}}^{m_{u}}\alpha(m)P(sr\
eq 0|m)\int_{0}^{r_{u}}P(D \geq D_{0}|r,m)p(r|m)drdm. \tag{3}\]  
Here, \(\alpha(m)\) is the annual rate of earthquake with magnitude \(m\) for the principal fault considered. \(P(D\geq D_{0}|r,m)\) is the conditional probability of the \(D\geq D_{0}\) due to the earthquake of magnitude \(m\), at the distance to the site, \(r\).