## 2 Materials and Methods
### Excitation Modelling
#### 2.1.2 Conditional Random Field Simulation of Ground Motions for Multiple-Support Structures  
In this paper, the method of conditional stochastic simulation of ground motions using the spatiotemporal correlation function proposed in [27; 28; 29] was applied to generate the unknown acceleration time histories at consecutive supports of a multi-support structure. The above-mentioned papers present an effective method of conditional simulation of the space-variation of ground motion. This approach is formulated under the assumption of linear soil and structural behavior. The frequency dependence of the spatial correlation function is simplified so that only the correlation of the predominant frequency of the quake is accounted for and that the simulation can be easily performed. The algorithm, given in [30], allows for the generation of a signal at a point located a certain distance from the registration place, based on the actual event recording and the adopted velocity and correlation function. The spatial correlation function of the field, defined as the correlation between the values of the field at different points (\(i\), \(j\)), takes the form:  
\[K\big{(}D_{ij}\big{)}=\sigma^{2}e^{(\frac{-\mu_{d}D_{ij}}{2m\alpha})} \tag{6}\]  
where:  
\(D_{ij}\)--separation distance between two field points (\(i,j\)),  
\(\omega_{d}\)--predominant frequency of the shock,  
\(v\)--wave velocity,  
\(\alpha\)--space scale parameter (\(\alpha>0\)), depending on local geological conditions,  
\(\sigma\)--standard deviation of the recorded shock. For two points the spatial correlation function can be written in a matrix form:  
\[\mathbf{K}= \begin{bmatrix}\mathbf{K}_{11}&\mathbf{K}_{12}\\ \mathbf{K}_{21}&\mathbf{K}_{22}\end{bmatrix}  \tag{7}\]  
In order to create a random signal at point \(j\) (\(\mathbf{x}_{u}\)), based on the registered (original) signal at point \(i\) (\(\mathbf{x}_{k}\)), the following procedure was adopted:  
\[x_{us}=-a_{max}+2a_{max}\cdot r_{d}\;,\quad s=1,\cdots,n \tag{8}\]  
where:\(a_{max}\)--maximum ground acceleration of the original record,  
\(r_{d}\) --uniform random variable from a range <0, 1>. Under the assumption that the shock records satisfy a zero-mean Gaussian distribution, the formula for the conditional probability density of symmetrically truncated Gaussian distribution of the generated signal could be formulated as:  
\[f(\mathbf{x}_{u}|\mathbf{x}_{k})=(1-l)^{-\frac{1}{2}}(det\mathbf{K}_{c})^{- \frac{1}{2}}\cdot(2\pi)^{-\frac{1}{2}}\cdot exp(-\frac{1}{2(1-l)}(\mathbf{x}_{ u}-\mathbf{m}_{c})^{T}\mathbf{K}_{c}^{-1}(\mathbf{x}_{u}-\mathbf{m}_{c})) \tag{9}\]  
where:  
\(\mathbf{x}_{u}\)--vector of unknown values (generated signal),  
\(\mathbf{x}_{k}\)--vector of known values (registered signal),  
\(\mathbf{K}_{c}=\mathbf{K}_{11}-\mathbf{K}_{12}\mathbf{K}_{22}^{-1}\cdot \mathbf{K}_{21}\)--conditional covariance matrix of the random field,  
\(\mathbf{m}_{c}=\mathbf{K}_{12}\mathbf{K}_{22}^{-1}\mathbf{x}_{k}\)--vector of conditional mean values. \(l\)--truncation parameter, depending on the ratio \(\frac{a_{max}}{\sigma}\). If this ratio reaches 4.0 for a given signal, the truncation can be disregarded (\(l=0\)). This condition is usually met in case of mining tremors, where the intensive shock phase is relatively short. The newly created vector (\(\mathbf{x}_{u}\)) can be accepted only if it satisfies the condition of von Neumann elimination:  
\[R\leq f(\mathbf{x}_{u}|\mathbf{x}_{k}) \tag{10}\]  
where:  
\[R=(det\mathbf{K}_{c})^{-\frac{1}{2}}\cdot(2\pi)^{-\frac{1}{2}}\cdot r_{d} \tag{11}\]  
If the condition was not met, another vector of unknown values (8) was generated, and the steps described by Formulas (9)-(11) were repeated. Applying this procedure requires the assumption of two parameters characterizing the geological and soil conditions: wave velocity and space scale parameter.