## 2 Materials and Methods
### Mathematical Representation of Strain and Stress Tensors  
Quantifying crustal deformation requires a robust mathematical framework to describe how external forces manifest as internal displacements within the lithosphere. In continuum mechanics, this relationship is captured through stress and strain tensors, which represent the internal force and geometric deformation states, respectively. These second-order tensors are fundamental in describing the spatial distribution and orientation of tectonic forces acting on the Earth's crust. The strain tensor encapsulates the relative displacement between infinitesimal elements of a deforming body, accounting for both extensional and shear components. Likewise, the stress tensor describes the internal traction forces per unit area acting on arbitrary planes within a material. Together, these tensors provide the theoretical basis for deriving principal directions of deformation and interpreting geodetic measurements such as GNSS displacement vectors. In geodetic applications, these tensor quantities are typically derived from differential GNSS observations using linear elasticity assumptions. The decomposition of these tensors into principal values and directions enables the identification of regional strain patterns, fault kinematics, and potential zones of seismic hazard. As such, understanding their mathematical structure is essential for the application of geophysical models to real-world tectonic systems [45; 46; 47; 48; 49; 50]. In two-dimensional Cartesian coordinates, the strain tensor \(\varepsilon\) is represented as follows:  
\[\varepsilon=\left[ \begin{array}{cc}\varepsilon_{xx}&\varepsilon_{xy};& \varepsilon_{xy}&\varepsilon_{yy}\end{array} \right] \tag{2}\]  
where \(\varepsilon_{xx}\) and \(\varepsilon_{yy}\) are the normal strain components along the x and y directions, respectively, and \(\varepsilon_{xy}\) is the shear strain component. These quantities are derived from spatial derivatives of displacement fields:  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x},\ \ \varepsilon_{yy}=\frac{ \partial v}{\partial y},\ \ \varepsilon_{xy}=\frac{1}{2}\bigg{(}\frac{ \partial u}{\partial y}+\frac{\partial v}{\partial x}\bigg{)} \tag{3}\]  
where u(x, y) and v(x, y) represent horizontal displacements in the x and y directions, typically obtained from GNSS baseline vectors. This formulation assumes infinitesimal strain, appropriate for tectonic deformation measured over yearly timescales. The corresponding stress tensor \(\sigma\) under linear elasticity is given by [PERSON]'s law for plane stress conditions:  
\[\sigma=\frac{E}{1-\
u^{2}} \begin{pmatrix}1&\
u\\ \
u&1\end{pmatrix} \Big{(}\varepsilon_{xx}\varepsilon_{yy}\Big{)}+\frac{E}{2( 1+\
u)} \begin{pmatrix}0&\varepsilon_{xy}\\ \varepsilon_{xy}&0\end{pmatrix}  \tag{4}\]  
where E is [PERSON]'s modulus and \(\
u\) is [PERSON]'s ratio. These material parameters are typically estimated from seismic velocity models or laboratory experiments on crustal rock samples. Using these values, the individual components of the stress tensor under plane stress conditions are calculated as follows:  
\[\sigma_{xx}=\frac{E}{1-\
u^{2}}\big{(}\varepsilon_{xx}+\
u\varepsilon_{yy} \big{)},\sigma_{yy}=\frac{E}{1-\
u^{2}}\big{(}\
u\varepsilon_{xx}+\varepsilon _{yy}\big{)},\sigma_{xy}=\frac{E}{2(1+\
u)}\varepsilon_{xy} \tag{5}\]  
where  
\(\sigma_{xx}\), \(\sigma_{yy}\)--normal stress components  
\(\sigma_{xy}\)--shear stress component  
\(\varepsilon_{xx}\), \(\varepsilon_{yy}\)--normal strain components  
\(\varepsilon_{xy}\)--shear strain component  
E--[PERSON]'s modulus  
\(\
u\)--[PERSON]'s ratio  
For tectonic interpretation, the principal strain components \(\varepsilon_{\mathbf{1}}\) and \(\varepsilon_{\mathbf{2}}\) are obtained as eigenvalues of the strain tensor, with corresponding eigenvectors indicating the directions of maximum and minimum strain.