## 1 Introduction  
Urbanization is directly linked to large migrations of masses to metropolitans resulting in land scarcity issues. It affects the daily commute from one place to another in a highly stringent manner [1]. Thus, for rapid transit, and to ensure smooth traffic flow, utilization of urban ground space became inevitable. This leads to the initiation of underground transportation in cities such as Hong Kong and Delhi [2, 3]. In Delhi itself, the metro rail network is increasing exponentially, and underground tunnels are constructed daily [4]. Tunnel construction is a challenging task due to variations in geology and topography. Modern tunnelling methods, such as the new Austrian tunnelling method (NATM) or the Norwegian tunnelling method (NTM), use sophisticated machinery such as the tunneling machine (TBM). The vibrations produced during the process develop stresses in the soil mass which are ultimately released on reaching the ground surface, causing settlement [5]. Ground settlement due to tunnelling, if not appropriately analyzed, could damage the overlying structures, and may develop tensile strains as well [6]. The risk is elevated in cities such as Delhi and Hong Kong, where population density is extremely high. Ground settlement due to tunnelling has been analyzed in the past by various researchers using stochastic cum probabilistic methods [7, 8, 9, 10], analytical methods [11, 12, 13, 14, 15, 16, 17], and empirical methods [18, 19]. Tunnelling is a 3D operation that causes settlement about both the axes (i.e., transverse and longitudinal). For simplification, researchers usually focus on steady-state ground settlements occurring transverse to the tunnel axis, though the longitudinal payment has also been analyzed by a few researchers [20; 21]. [PERSON] [18], in 1969, through various field observations, suggested that the shape of the ground settlement profile for the transverse direction can be approximated through a standard Gaussian curve, a normal-distribution curve, or a bell curve, as shown in Figure 1. The settlement at any point can be expressed using Equations (1) and (2) providing a relationship between settlement through width and tunnel radius [18]. \[S_{(x)}=S_{max}e^{(-\frac{x^{2}}{2t^{2}})} \tag{1}\]  
where, \(S_{(x)}\) is the settlement at distance \(x\) from the tunnel centre line, \(S_{max}\) is the maximum settlement, \(x\) is the transverse distance from tunnel centreline, and \(i\) is the settlement trough width from centreline up to point of inflection. \[\frac{i}{R}=\left(\frac{z_{0}}{2R}\right)^{n} \tag{2}\]  
where, \(n=0.1\) to \(0.8\), \(R\) is the radius of tunnel, \(z_{0}\) is the depth of the tunnel below ground surface, and \(i\) is the settlement trough width from centreline up to point of inflection. The normal distribution proposed by [PERSON] was simpler than the previously developed stochastic approaches and has been widely used by researchers since then, confirming its validity to most scenarios [6; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Modifications to the Gaussian curve for conditions where [PERSON]'s relations do not provide possible results were proposed [31; 32; 33; 34]. However, [PERSON]'s hypothesis is still used widely in practical applications due to its simplicity and versatility. Equation (3) was proposed by [PERSON] and [PERSON] [26] for UK tunnels based on field observations, Equations (4) and (5) were proposed for loose and dense sands, respectively, based on field observations and laboratory experiments, and Equations (6) and (7) were proposed for cohesive mediums through linear regression technique. [PERSON] [35] suggested Equation (8) based on case-history data and field observations which can apply to all types of soil media. \[\frac{i}{R}=\left(\frac{z_{0}}{2R}\right) \tag{3}\]  
\[i=0.25(1.5z_{0}+0.5R) \tag{4}\]  
\[i=0.25(z_{0}+0.5R) \tag{5}\]  
\[i=0.43z_{0}+1.1 for;3\ \leq z_{0}\leq 34 \tag{6}\]  
\[i=0.28z_{0}-0.1\ for;6\ \leq z_{0}\leq 10 \tag{7}\]  
Figure 1: Ground-settlement trough.