## 2 Materials and Methods
A paramount research work in this field was conducted by [PERSON] (1890-1968), who identified 378 tidal harmonics that were collected in a celebrated catalog in 1921 [24]. The largest tidal harmonics [25] are shown in \[|\vec{a}_{\mathcal{M}}(R)|=\frac{GM}{(R-r)^{2}}-\frac{GM}{R^{2}}\simeq\frac{2 MGr }{R^{3}} \tag{2}\]4 The elastic deformation of the Earth has two modes: spheroidal and toroidal, but tidal forces excite only the spheroidal modes [26]. For the sake of simplicity, we assume that the Earth is spherically symmetric, nonrotating, elastic, and isotropic. Nevertheless, the effect of sphericity of the Earth's layering cannot be neglected if one wishes to calculate surface waves of long wavelength. Spheroidal deformations of the SNREI model as a function of depth, [PERSON]'s coefficients and the gravitational acceleration can be evaluated by using a set of functions \(y_{i}\) with \(i=1,\,\ldots,\,6\) which satisfy a set of six ordinary differential equations [27]  
\[|\vec{a}_{\mathcal{M}}(R)|=\frac{GM}{(R-r)^{2}}-\frac{GM}{R^{2}}\simeq\frac{2 MGr }{R^{3}} \tag{2}\]5  
where \(n\) is the \(n\)-th mode. For our purpose, it is enough to consider \(n=2\). Displacements can be written in spherical coordinates as follows:  
\[\left\{ \begin{aligned} u_{r}&=\frac{h_{2}(r)}{g(r)}W(r,\theta,\phi)\\ u_{\theta}&=\frac{l_{2}(r)}{g(r)}\frac{\partial W(r,\theta,\phi)}{\partial\theta}\\ u_{\phi}&=\frac{l_{2}(r)}{g(r)\sin\theta}\frac{ \partial W(r,\theta,\phi)}{\partial\phi}\end{aligned} \right. \tag{15}\]  
The value of \(g(r)\) rises from the surface up to a depth of about 700 km to a maximum of 9.99 m/s\({}^{2}\). In the lower mantle, \(g(r)\) layers stable and increases abruptly near the Gutenberg discontinuity, reaching 10.16 m/s\({}^{2}\). Gravity continuously falls in the core with a rate that depends on density till it reaches a zero-value at the center of the Earth. In this work, tidal perturbations are considered with respect to seismicity and, above all, crustal seismicity (up to \(\sim\)100 km at depth), so that if we assume \(g(r)\simeq g(r=6371\) km) an error \(\sim\)0.1% is introduced, which is negligible. The strain components in spherical coordinates are obtained from (15) by derivation  
\[\left\{ \begin{aligned} \varepsilon_{rr}&=\frac{ \partial u_{r}}{\partial r}\\ \varepsilon_{\theta\theta}&=\frac{1}{r}\bigg{(} \frac{\partial u_{\theta}}{\partial\theta}+u_{r}\bigg{)}\\ \varepsilon_{\phi\phi}&=\frac{1}{r\sin\theta} \bigg{(}\frac{\partial u_{\phi}}{\partial\phi}+u_{r}\sin\theta+u_{\theta}\cos \theta\bigg{)}\\ \varepsilon_{\theta\phi}&=\frac{1}{r}\bigg{(} \frac{1}{\sin\theta}\frac{\partial u_{\phi}}{\partial\phi}+\frac{\partial u_{ \phi}}{\partial\theta}-u_{\phi}\cot\theta\bigg{)}\end{aligned} \right. \tag{16}\]  
For computational simplicity, the radial component of strain can also be evaluated by using  
\[\varepsilon_{rr}=-\frac{\
u}{1-\
u}\big{(}\varepsilon_{\theta\theta}+ \varepsilon_{\phi\phi}\big{)} \tag{17}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Doodson Number** & **Period (Days)** & **Amplitude (m)** \\ \hline \(M_{2}\) & 2 5 5 5 5 5 & 0.518 & 0.6322 \\ \(S_{2}\) & 2 7 3 5 5 5 & 0.500 & 0.2941 \\ \(K_{1}\) & 1 6 5 5 5 5 & 0.997 & 0.3686 \\ \(M_{f}\) & 0 7 5 5 5 5 & 13.661 & \(-\) 0.0666 \\ \(M_{m}\) & 0 6 5 4 5 5 & 27.555 & \(-\) 0.0352 \\ SSA & 0 5 7 5 5 5 & 182.622 & \(-\) 0.0310 \\ h & 0 5 6 5 5 4 & 365.264 & \(-\) 0.0049 \\ P & 0 5 5 6 5 & 3232.605 & 0.0002 \\ N/2 & 0 5 5 5 7 5 & 3399.048 & \(-\) 0.0003 \\ N & 0 5 5 5 6 5 & 6798.097 & 0.0279 \\ \hline \end{tabular}
\end{table} 
Table 1: Largest tidal frequencies. where \(\
u\) is the [PERSON]'s coefficient that can be computed at depth with  
\[\
u(r)=\frac{\lambda(r)}{2(\lambda(r)+\mu(r))} \tag{18}\]  
in turn, the [PERSON]'s coefficients can be obtained starting from the speed of the seismic P and S waves as a function of depth (PREM, [28]). \[ \begin{cases}\lambda(r)=\rho(r)\big{(}\
u_{P}^{2}(r)-\
u_{S}^{2}(r)\big{)}\\ \mu(r)=\rho(r)\
u_{S}^{2}(r)\end{cases}