## 1 Introduction
### The Proposed Methodology for On-Fault Displacement Evaluation  
The study site is located within the of mid-Pannonian area with typical diffuse seismicity. Although the pre-Quaternary faults are mapped, the data are insufficient for the parametrization of the recent activity of the presumably reactivated in the Quaternary fault in the site area. The probabilistic seismic hazard analysis exists for the site is based on comprehensive seismotectonic modelling of source areas (see [14]). Therefore, the methodology for PGD hazard evaluation developed below is based on probabilistic seismic hazard analysis available for the site. This method is a specific realization of \"earthquake-based approach\" [5,6] for probabilistic permanent displacement hazard analysis  
As part of the PSHA, the seismic hazard can be disaggregated into magnitude-distance bins for every selected hazard level \(\lambda\) [1/a] (see [15]). The probability weight associated to magnitude \(Mw\) and distance \(R_{JB}\) pairs characterize the contribution of the magnitude distance pairs/bins to the hazard. The disaggregation of hazard at given level \(\lambda\) can be transformed into a bivariate probability density function \(\wp(Mw,R_{JB})\). In discrete case, the joint probability mass function is given for magnitude and distance bins (\(Mw_{i}\div R_{JB}\)) with n \(\times\) m size weightsmatrix, \(\left[W_{ij}\right]\). The marginal densities \(p_{m}(Mw)\) and \(p_{m}\big{(}R_{JB}\big{)}\) are represented by \((w_{i};Mw_{i})\) and \(\Big{(}w_{j};R_{JB}\big{)}\), while:  
\[\sum_{i=0}^{i=i_{u}}\big{[}W_{i,j}\big{]}=\big{[}w_{i}\big{]}\text{ and }\sum_{i}w_{i}=1 \tag{4}\]  
where the upper index \(i_{u}\) corresponds to the maximum magnitude up-to those the disaggregation is performed. The marginal density of the distances, \(r\) can be calculated similarly to the Equation (4). Further, the relation between distance between the fault trace and the site, \(r\) and \(R_{JB}\) should be established. The \(R_{JB}\) is the Joyner-Boore distance (see [15]), the closest distance to the projection of the fault plane on the surface, see Figure 4. As it is shown in Figure 4, the \(R_{JB}\) can be assessed as a conservative estimation for \(r\) since it corresponds to the situation when \(r\) is minimum. Further, for the sake of simplicity, the variables \(Mw_{i}\) will be replaced by \(m_{i}\) and the \(R_{JB}\) by \(r_{j}\) (or by \(m\) and \(r\) in continuous form of equations). For the hazard level \(\lambda\), the Equation (3) can be modified as follows:  
\[P(D\geq D_{0})\big{|}_{\lambda}=\left.\left[\int_{m_{0}}^{m_{u}}P(sr\
eq 0|m) \int_{0}^{r_{u}}P(D\geq D_{0}|r,m)\times p_{\lambda}(m,r)drdm\right]\right|_{ \lambda}. \tag{5}\]  
Integrating over \(r\), the Equation (5) takes the following form:  
\[P(D\geq D_{0})\big{|}_{\lambda}=\left.\left[\int_{m_{0}}^{m_{u}}P(sr\
eq 0|m) P(D\geq D_{0}|m)p_{m,\lambda}(m)dm\right]\right|_{\lambda}. \tag{6}\]  
Here the \(p_{\lambda}(m,r)\) is the bivariate density and \(p_{m,\lambda}(m)\) is the marginal distribution density of magnitudes at hazard level \(\lambda\). Equations (5) and (6) give the probability of \(D\geq D_{0}\) at annual level \(\lambda\). Note, that the disaggregation of seismic hazard at hazard level \(\lambda\) is nothing else as the discrete representation of the bivariate density \(p_{\lambda}(m,r)\) of magnitude and distances. Thus, Equations (5) and (6) can be written in discrete form as follows:  
\[P(D\geq D_{0})\big{|}_{\lambda}=\left.\left[\sum_{i=0}^{i=i_{u}}\sum_{j=0}^{i =i_{u}}P(sr\
eq 0|m_{i})P\big{(}D\geq D_{0}|r_{j},m_{j}\big{)}W_{i,j}\right] \right|_{\lambda}. \tag{7}\]  
Figure 4: Definition of geometrical variables (\(r\) is distance from the site to the mapped fault trace on the surface and \(R_{JB}\) the Joyner-Boore distance).