## 2 Deformation of Retaining Structures  
The excavation of deep foundation pits in urban areas will inevitably result in the deformation of the soil and the retaining structures (e.g., retaining walls). Excessive deformation will trigger significant damage to adjacent buildings and infrastructure facilities. During the excavation process, the soil in the pit is removed and the earth pressure at the two sides of the retaining wall would be changed. The earth pressure on the outer side of the wall changes from static earth pressure to active earth pressure, and thus the difference in earth pressure between the inner and outer sides will lead to the horizontal deformation of the wall [34; 35; 36]. When the wall is supported at the top, as is often the case in engineering practice, the overall deformation of the wall along the vertical direction would behave in a \"concave\" way [37], as illustrated in Figure 1a (adopted from [38]).  
A threshold for the lateral displacement is often used to determine the behaviour of the deformed retaining wall, which would be dependent on the excavation depth of the pit. When the deformation of the wall is monitored at some selected depths, the wall is deemed as _satisfactory_ if all the monitored deformations do not exceed the predefined threshold, and vice versa. This method is relatively straightforward and reflects the current engineering practice for the performance evaluation of retaining structures in foundation pits, providing a simple yet useful tool for decision-makers. This approach only concerns the deformation at some certain depths (where monitored). In fact,the failure of retaining walls (i.e., the displacement exceeds the threshold value) could occur at any depth. Illustratively, as shown in Figure 1b, even if all the displacements at the monitoring points are smaller than the limit, the actual displacement curve may still exceed the threshold, leading to the failure of retaining wall. Thus, a reliability assessment approach is essentially needed moving from \"point control\" to \"surface control\". Conceptually, increasing the number of monitoring points and inclinometers will increase the accuracy of the traditional point control-based method. While this idea could improve the monitoring practice in engineering, it is often difficult to implement due to the significant constraints of costs and monitoring equipment. This fact explains the motivation of the proposed \"surface control\" method.  
In this paper, the deformation reliability of retaining structures are considered, which is defined as the probability that the maximum displacement of the retaining structure within the monitored area does not exceed the displacement limit. Suppose that there are \(n\) pieces of monitored deformation data at different depths \(d_{1},d_{2},\ldots d_{n}\); namely, \(y_{1},y_{2},\ldots y_{n}\) respectively. Let \(y_{\text{lim}}\) denote the predefined threshold for each \(y_{i}\). The existing approach (\"point control\") simply concludes that the structural behavior is satisfactory if \(y_{\text{lim}}\geq y_{i}\) for \(\forall i=1,2,\ldots n\) and unsatisfactory otherwise. Clearly, the possible occurrence of _failure_ (displacement exceeding limit) at locations rather than \(\{d_{1},d_{2},\ldots d_{n}\}\) has been neglected. Also, the point control fails to take into account the ME associated with the monitored data. These disadvantages are addressed in this paper.