## 2 Random Spatial FE Models of the Subsoil
### The Subsoil Material Parameters Described by Three-Dimensional Random Fields  
The geotechnical parameter uncertainty of soils may be relevantly reflected by three-dimensional random fields. This action requires advanced numerical tools and theoretical background in advanced probabilistic methods. Although random field generation is an advanced issue, implemented in relevant software [11; 37], the random field methodology is still rare in real-life engineering applications. The only software of wide present use includes simple generators of random fields to be directly entered as input data into standard FEM software. In this paper, such a simple computational path is also executed. Dedicated software is used to generate the FE input-ready Gaussian two-dimensional and three-dimensional random fields, and the software is described in detail in [38; 39; 40]. The software conducts random variable generation employing a conditional acceptance and rejection concept. A conditional function \(f_{s}(\mathbf{X}_{u}/\mathbf{X}_{k})\) is applied to directly generate an unknown vector \(\mathbf{X}_{u}\), assuming that the field part defined by a vector \(\mathbf{X}_{k}\) is already generated and is known:  
\[f_{s}(\mathbf{X}_{u}/\mathbf{X}_{k})=(\det\mathbf{K}_{c})^{-1/2}(2\pi)^{-m/2} \exp\left(-0.5\bigg{(}\mathbf{X}_{u}-\overset{-}{\mathbf{X}_{c}}\bigg{)}^{ \mathrm{T}}\mathbf{K}_{c}^{-1}\bigg{(}\mathbf{X}_{u}-\overset{-}{\mathbf{X}_{ c}}\bigg{)}\right) \tag{1}\]  
where \(\mathbf{K}_{c}\) is a conditional correlation matrix, \(\mathbf{X}_{c}\) is a conditional vector of mean values, and \(m\) stands for the dimension of the correlation matrix \(\mathbf{K}\). The most important aspect of random field modelling is the correlation function adoption, dedicated to a given problem. Various correlation function definitions are found in the literature; usually, the function is assumed a priori. Comprehensive studies on parameter determination based on targeted research results can be found (see, e.g., [21]). Random fields of standard engineering materials (e.g., soils, concretes, composites) make use of the exponential correlation function related to the first-order autoregressive function or a Markov process model. In a three-dimensional problem case, it has the following definition [41]:  
\[\rho(x,y)=\rho(x)\rho(y)\rho(z)=\mathrm{e}^{-d_{x}|x|-d_{y}|y|-d_{z}|z|} \tag{2}\]  
where \(d_{x}\), \(d_{y}\), and \(d_{z}\) are damping parameters describing the correlation decay, and \(|x|\), \(|y|\), and \(|z|\) are the distances between the points of the random field, along respective axes. Due to simple engineering interpretation, the correlation function (2) has been used in many cases (see, e.g., [17; 18; 19; 42]). In the calculations, diverse random field types were generated, differing in the value of the adopted damping parameter \(d\equiv d_{x}=d_{y}=d_{z}\)\(\mathrm{\;\;[m^{-1}]}\). The parameters vary from \(d=0.01\)\(\mathrm{\;\;m^{-1}}\) (a relatively large correlation range) through \(d=0.1\)\(\mathrm{\;\;m^{-1}}\), \(d=0.2\)\(\mathrm{\;\;m^{-1}}\), and \(d=1.0\)\(\mathrm{\;\;m^{-1}}\) to \(d=5.0\)\(\mathrm{\;\;m^{-1}}\) (a relatively low correlation range). Figure 2 illustrates the impact of the assumed damping parameter \(d_{x}\) on the correlation between the points (finite elements of the subsoil block). The random field applied in the FE model of the subsoil block is based on the previous truncated Gaussian variable of [PERSON]'s modulus \(E\) (\(\mu_{E}=28,000\) kPa, \(\sigma_{E}=7000\) kPa, \(t=4\)). A number of 200 three-dimensional Gaussian fields of \(n_{re}=1000\) points are generated, corresponding to each of the five correlation decay parameter values, directly implemented in the FE model. The number is limited to 200 samples only, as the time required to generate the fields, to create FE models incorporating these data, to conduct the numerical calculations, and to analyse the results is extremely significant.