## 3 Parametric Study for Determining the Optimal Sampling Strategy  
This paper examines the case of a wall retaining a fully drained cohesionless soil against active failure in plane strain conditions. Different sampling strategies are examined, i.e., sampling from a single point and sampling from a domain, through a parametric analysis for defining the strategy that minimizes the probability of failure and thus the statistical error (i.e., the optimal sampling strategy). The soil mass is discretised into a 48 \(\times\) 34 mesh (number of elements at the horizontal and vertical direction respectively) consisting of eight-noded square elements with side lengths equal to 0.1 (Figure 1). Various wall heights are considered ranging from \(H\) = 1.4 m to 2.9 m (meaning that the wall extends to a depth ranging from 14 to 29 elements), while the mesh geometry has been kept the same for all cases. A 24-element wall will be generally considered in the analysis (hereafter called a \"reference wall\"); the other wall heights will be used for the investigation of the effect of wall height on the optimal sampling strategy. The 48-element mesh in the horizontal direction was chosen so that the failure mechanism in the RFEM analysis will not be affected by the proximity of the right boundary. In this respect, for the highest wall considered (29-element-high wall), as the \(\phi^{\prime}\) value approaches zero (extreme case), according to [PERSON]'s theory, the failure mechanism in the active state will occupy a horizontal distance from the wall face equal to one wall height (that is, 29 elements; value 40% smaller as compared to the 48 horizontal elements of the geometry). In this study, both \(\phi^{\prime}\) and \(\gamma\) are treated as log-normal random fields. The soil is assumed cohesionless with \(\mu_{\phi^{\prime}}\) = 30\({}^{\circ}\) and \(\mu_{\gamma}\) = 20 kN/m\({}^{3}\), while various standard deviation and \(\theta\) values are examined. Moreover, since \(K_{o}\) depends on \(\phi^{\prime}\) (the initial horizontal stresses are defined by [PERSON]'s [22]\(K_{o}=1-\sin\phi^{\prime}\) equation), this is also treated as a random field. In addition, it is mentioned that although the elastic parameters of soil (\(\
u\) and \(E\)) affect the required wall movement in order for the active state to be fully reached, preliminary parametric analysis carried out by the authors showed that they have no influence on the optimal sampling strategy. Thus, these values have been kept constant and equal to 0.3 and 10\({}^{5}\) kN/m\({}^{2}\) respectively throughout the entire analysis presented herein. Furthermore, the random fields are assumed to have the same spatial correlation length (\(\theta\)) and an exponentially decaying (Markovian) correlation structure (see [18]):  
\[\rho(\tau)=\exp[-2\tau/\theta] \tag{3}\]  
Figure 1: Active earth failure of the “reference” wall. Graphical representation of a random field of \(\phi^{\prime}\) (this is a typical random finite element method (RFEM) realization); light areas correspond to lower friction angles and vice versa. For the soil shown, \(\theta/H=8.3\) and \(COV_{\phi^{\prime}}=0.3\). Finally, a safety factor \(FS\) equal to 1.3 is generally assumed in the analysis; the effect of \(FS\) on the sampling strategy is also investigated later. As samples are taken from a material field (i.e., the ground), which simultaneously is a stress field (stresses caused by the self-weight of the soil and also any external load), the location of the optimal sampling points is affected by the coexistence of these two fields.