# Prediction of Surface Deformation Induced by Ultra-Shallow-Buried Pilot Tunnel Construction
## 1 Introduction
### Calculation of Surface Deformation Induced by Unit Excavation  
First, a vertical section of the geotechnical soil mass is studied. Take the ground surface as the \(X\)-axis and the \(Z\)-axis with downward as the positive direction, establishing a Cartesian coordinate system as shown in Figure 1. Consider a micro soil element \(\mathrm{d}\zeta\mathrm{d}\eta\) at a depth of \(H\). The excavation of this micro-element will induce ground settlement, and tunnel-excavation-induced ground settlement can be regarded as the superposition of settlements induced by each micro-element. Statistically, the settlement induced by excavating a micro-element at any point A(X,Z) above it is a random event with a certain probability [27]. The probability of this event is \(f(X^{2})\mathrm{d}X\), where \(f\) denotes the probability density function. Figure 1: Schematic diagram of unit excavation impacts. Based on the above coordinate system, assuming the positive Y-axis direction is perpendicular to the page inward, the probability \(P(\mathrm{d}S)\) that the excavation of a micro soil element \(\mathrm{d}\xi\mathrm{d}\zeta\mathrm{d}\eta\) induces settlement at point A(\(X\),\(Y\),\(Z\)) is expressed as  
\[P(\mathrm{d}S)\ =\ f\Big{(}X^{2}\Big{)}f\Big{(}Y^{2}\Big{)}\mathrm{d}X\mathrm{d}Y \tag{1}\]  
The magnitude of this probability is independent of the coordinate system selected, thus establishing an equivalence relation. Considering the boundary conditions, where the probability approaches 0 at both positive and negative infinity, solving the equation yields  
\[f\Big{(}X^{2}\Big{)}\ =\ q(Z)\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}X^{2}\biggr{]} \tag{2}\]  
\[f\Big{(}Y^{2}\Big{)}\ =\ q(Z)\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}Y^{2}\biggr{]} \tag{3}\]  
where \(q(Z)\) and \(r(Z)\) are functions of the coordinate \(Z\). Substituting Equations (2) and (3) into Equation (1), we obtain  
\[P(\mathrm{d}S)\ =\ q^{2}(Z)\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}(X^{2}+Y^{2}) \biggr{]}\mathrm{d}X\mathrm{d}Y \tag{4}\]  
Therefore, in three-dimensional space, the probability density function of the settlement at point A induced by the excavation of a micro soil element is  
\[f(X,Y,Z)\ =\ q^{2}(Z)\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}(X^{2}+Y^{2})\biggr{]} \tag{5}\]  
It is generally accepted that the volume of stratum ground loss due to tunnel excavation equals the volume of the final surface settlement basin. Similarly, the volume of excavated micro soil elements equals the volume of the subsidence basin at point A. The subsidence basin at point A is composed of smaller \"unit subsidence\" (hereafter referred to as \"unit subsidence\"), defined as the product of the micro soil element volume and the probability density function after the basin stabilizes. After excavating a micro soil element at time t, the unit subsidence _We_ at point A is  
\[f(X,Y,Z)\ =\ q^{2}(Z)\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}(X^{2}+Y^{2})\biggr{]} \tag{6}\]  
Under the soil incompressibility assumption, we first derive \(q^{2}(Z)\) and substitute it into Equation (6) for simplification. The resulting expression is integrated over the infinite Y-axis. By imposing convergence as \(t\)\(\rightarrow\)\(\infty\) (i.e., subsidence stabilization) and treating \(r^{2}(Z)\) as a depth-dependent constant, the final unit settlement expression is obtained:  
\[W_{e}(X)\ =\ \frac{1}{r(Z)}\exp\biggl{[}-\frac{\pi}{r^{2}(Z)}X^{2}\biggr{]} \mathrm{d}\xi\mathrm{d}\eta \tag{7}\]  
Equation (7) is the basic formula for investigating subsidence at any point within the formation under the influence of arbitrary excavation under plane strain conditions. Now consider a plane strain problem: an arbitrary cross-sectional shape tunnel is located underground, with the center of its cross-section at a burial depth \(H\) from the surface, as shown in Figure 2. The surface uses an XOY coordinate system, and the excavated micro soil elements use a \(\xi\mathrm{O}\eta\) coordinate system.