## 4 Results
### The Probabilistic Analysis
#### 4.3.1 Statistics of Ground Surface Settlement  
As depicted in Figure 11a, the ground surface settlement curve without considering the spatial variability of parameters is regular grooved, as shown by the red solid line in Figure 11a. While if the spatial variability is taken into account, each settlement curve computed from a certain realization is basically groove-shaped but bends and curls in an unique irregular way, as presents in the thin gray line in Figure 11a, which is utterly different from the former deterministic curve. These squiggly settlement curves computed from all the realizations cluster together and form a collection of settlement curves. It can be observed that the settlement curves of the two sides are not symmetrical, since the parameters of either side of the pit are typically non-symmetrical at all. The thick black dashed line herein envelopes the settlement curve collections and provides a full field-scale variation range that includes all possible deformations. Consequently, the variation of settlement can be implied that the spatial variability of the parameters is associated directly with the surface settlement at a certain location outside the seepage curtain. While in practical engineering, such variability has been verified and demonstrated by massive studies [1; 29; 30; 33]. Therefore, it is required to be accounted for when performing settlement analysis. For each \(D\), it has a variation range of settlement corresponding to the curve collection, denoted by \(\Delta H\). The correlation between \(D\) and \(\Delta H\) is illustrated in Figure 11b. It is squiggly downward groove, which is literally the reverse of the settlement curve. Overall, the particular location with a greater settlement is generally prone to having a larger \(\Delta H\). A closer inspection of \(\Delta H\) shows it has a maximum value of 15.32 mm, which is 38.4% of the maximum settlement of determined analysis. In addition, the \(\Delta H\) is quite insignificant for a great \(D\), which is an unexpected outcome. It implies that the variation range of settlement is independent of variable parameters and is comparable with the deterministic results. To identify the overall distribution of settlement data at different locations outside the curtain, six monitoring points (\(D_{1}\)\(\sim\)\(D_{6}\)) were set in the model. The settlements obtained by all realizations at each monitoring point are represented by a violin plot, as shown in Figure 12. The settlements for each point have a bell-shaped distribution with only one peak. It depicted an overall trend that the variability of settlement decreases with increasing distance from the curtain (\(D^{\prime}\)). A comparison of the median of \(H\) at each location outside  
Figure 10: The convergence of the central moments of the performance function with the number of MCS runs: (**a**) mean and standard deviation; (**b**) third-order dimensionless central moment and fourth-order dimensionless central moment. the curtain (in the black dashed line) with the deterministic settlement curve (in the red dashed line) indicated that it has an insignificant difference outside \(D_{2}\), but a significant difference within the range. The maximum settlement (\(H_{m}\)) and the distance of the maximum settlement location from the curtain (\(D^{\prime}{}_{m}\)) for all 1000 realizations were statistically analyzed. Their histograms and probability density functions (PDFs) are shown in Figure 11. Following the Kolmogorov-Smirnov (K-S) test, \(H_{m}\) in Figure 13a was subjected to a log-normal distribution with a most frequent value within the interval of 40-42 mm, which was a bit larger than the deterministic analysis. It has a COV of 0.15, which was the same as the COV of the input random parameters \(\lambda\) and \(\kappa\) (COV\({}_{\lambda,\kappa}\)). Meanwhile, \(D^{\prime}{}_{m}\) in Figure 13b was distributed 5-110 m from the curtain and was subjected to a log-normal distribution.