## 3 Data and Methods
### Available Geotechnical Data and Assessment of the Post-Liquefaction Consolidation Subsidence
#### 3.2.2 Clay-Like Soils (\(I_{SBT}>2.6\))  
The consolidation subsidence in the cohesive upper and intermediate clays in Figure 2 is still calculated using Equation (3).  
The constrained modulus \(E_{oof}\) in Equation (3) is estimated according to the small strain shear modulus \(G_{0}\) and the Poisson ratio (\(\
u=0.3\) in Sinatra and Foti [40]) according to the elasticity theory, where the \(G_{0}\) for each layer is derived from the shear wave velocity \(V\)s as [63]:  
\[G_{0}=\frac{\gamma}{g}.V_{s}^{2}=\frac{\gamma}{g}\cdot\left(51.6\cdot\text{log}f _{s}+18.5\right)^{2}, \tag{9}\]  
where \(\gamma\) is the soil unit weight defined in Equation (7), \(g\) is the gravity force, and \(f_{s}\) is the sleeve friction.  
The excess pore water pressure (\(\Delta u\)) in Equation (3) is calculated with a simplified relationship for cohesive soils proposed by [PERSON] et al. [64] and suggested by AGI (Italian Geotechnical Association) guidelines [65]:  
\[\frac{\Delta u}{\sigma_{z}^{\prime}}=\beta\left[log\frac{\gamma_{C,max}}{ \gamma_{v}}\right], \tag{10}\]  
where \(\sigma_{z}^{\prime}\) is the initial vertical effective stress defined in Equation (6), \(\beta\) is a coefficient equal to 0.45, \(\gamma_{c,max}\) is the maximum attained shear strain during the shaking, and \(\gamma_{v}\) is the volumetric threshold strain of the soil [65]. The latter commonly derives from cyclic laboratory tests. Still, in the absence of data, it can be estimated as a function of the Over Consolidation Ratio (OCR) and two coefficients, A and B, which are functions of the plasticity index of the soils ([65], p. 373), according to:  
\[\gamma_{v}=\text{ A(OCR}-1)+\text{B}. \tag{11}\]  
For the considered clay-like soils, it is reasonable to assume OCR = 1. For a plasticity index of approximately 20% [40], a value of \(\gamma_{v}=\)0.06% is obtained as being representative of clay-like soils ([65], p. 373).  
The maximum shear strain (\(\gamma_{c,max}\)) in Equation (10) should be rigorously defined through a seismic response analysis, but in the absence of this, we used the following relationship [65]:  
\[\gamma_{c,max}=\frac{\tau_{max}}{G}. \tag{12}\]  
The maximum shear stress \(\tau_{max}\) in Equation (12) is empirically defined as:  
\[\tau_{max}=\frac{a_{max}}{g}\sigma_{v}(1-0.015z), \tag{13}\]  
where \(a_{max}\) is the peak ground acceleration (PGA) at the site, assumed as the median value of the PGAs calculated from [PERSON] et al. [42] (i.e., \(a_{max}=0.59\) g in Figure S2b), \(\sigma_{v}\) in Equation (13) is the vertical total stress, and \(z\) is the depth from the ground level [65].  
The shear modulus G in Equation (12) experimentally depends on the shear strain, as in Figure S2c. Therefore, we fitted the experimental data in Figure S2c with Equation (14) [66]:  
\[\frac{G}{G_{0}}=\frac{1}{1+\frac{\gamma}{\gamma_{r}}}\, \tag{14}\]  
where \(G_{0}\) is the shear modulus at small deformations defined by Equation (9) and \(\gamma_{r}\) is a reference shear strain used to fit the experimental data and assumed as equal to 0.085%.  
The \(\gamma_{c,max}\) value for each layer is then calculated by substituting Equation (14) into Equation (12) and solving for \(\gamma_{c,max}\).