## 2 Materials and Methods  
### Data Preparation  
As mentioned above, the focus was specifically on surface displacements recorded by benchmarks. Simultaneously, this study aimed to investigate the potential relationship between surface displacement and rapid underground displacement, also known as shear zones. To compare these two types of displacement, data standardization was necessary. Shear displacements typically occur in the direction perpendicular to the dam and are measured by inclinometers. This direction of displacement is the most critical. Therefore, this study focused on analyzing the resultant direction of displacement. The data obtained from benchmark displacement measurements provide information on the displacement in three directions relative to the initial state. The measuring points are placed at different distances from each other around the reservoir, with more benchmarks often placed in critical areas to obtain more precise measurements. Additional measurement points are added to new levels of the dam, resulting in different measurement histories for each point. Measurements are performed at least once a year at each point. The critical value when analyzing the dam stability of the tailings storage facility is the displacement perpendicular to the primary dam, which is indicated by the measurement points around the TSF that show gradual outward displacements. To standardize the data and facilitate analysis of correlation with inclinometer test data, the resultant value of the displacement perpendicular to the primary dam is determined and analyzed consistently. Often, the resultant vector of displacement is directed precisely in this direction or with a slight difference in angle. In practice, this requires a geometric transformation using vector projection. Basic dam location points have to be used \(\{(X_{i},Y_{i}):i\in(0,N)\}\). Determining the value of the displacement perpendicular to the dam at each point can be performed according to the following procedure where \((U_{x},U_{y})\) is the displacement in the horizontal directions of the selected benchmark at the location \((x,y)\):  
1. Find the closest point \((X_{k},Y_{k})\) on the basic dam for a given benchmark \[\left\{(X_{k},Y_{k}),k\in(0,N):\underset{i\in(0,N)}{min}\left(\sqrt{\left(X_{ i}-x\right)^{2}+\left(Y_{i}-y\right)^{2}}\right)\right\}\] (1)
2. Determine the normal vector at this point of the basic dam: 1. Determine the tangent vector to the dam \[X_{\mathcal{T}k}=\left\{ \begin{array}{c}\frac{X_{k+1}-X_{k-1}}{Z},\text{ for }k\in(1,N-1)\\ X_{k+1}-X_{k},\text{ for }k=0\\ X_{k}-X_{k-1},\text{ for }k=N\end{array} \right.,\] (2)\[Y_{Tk}= \begin{cases}\frac{Y_{k+1}-Y_{k-1}}{2},\text{ for }k\in(1,N-1)\\ \quad Y_{k+1}-Y_{k},\text{ for }k=0\\ \quad Y_{k}-Y_{k-1},\text{ for }k=N\end{cases} .\] (3) b.  Rotate the vector 90 degrees \[X_{Nk}=-Y_{Tk}Y_{Nk}=X_{Tk}. \tag{4}\]
3. Project the displacement vector \(\overset{\rightarrow}{U}=\left(U_{x},U_{y}\right)\) to the normal vector to the \(\text{dam}\overset{\rightarrow}{W}=\left(X_{Nk},Y_{Nk}\right)\). \[\overset{\rightarrow}{U}_{N}=\frac{\overset{\rightarrow}{U}^{\circ}\overset{ \rightarrow}{W}}{|\overset{\rightarrow}{W}|}\cdot\overset{\rightarrow}{W}\] (5)
4. Calculate of the vector length, i.e., the value of the displacement in the direction perpendicular to the dam. \[U_{Nt}=|\overset{\rightarrow}{U}_{N}|\] (6)  
The displacements calculated in this way in relation to the initial state make it possible to determine the annual increments, which we can treat as the velocity expressed in the units mm/year. And also similarly, accelerations, which will be an increase in velocity. These parameters were used for further analysis. The most critical displacements are those that result from soil shearing, which can be recorded using inclinometers that measure displacement at various depths from the initial state. The displacement recorded by the inclinometer has a direction perpendicular to the primary dam, which is the same as that obtained after transforming the displacement of benchmarks.