## 4 Methodology
### Edge Detection
#### 4.3.2 Euler Deconvolution (ED)  
The standard Euler method is based on [PERSON]'s homogeneity formula. The formula relates the (gravity) potential field and gradient elements to the location of the source, according to a certain degree of homogeneity (N), which is called the structural index [52]. Figure 5: (a) X-direction derivative of trend removed gravity data. (b) Y-direction derivative of trend removed gravity data. (c) Result of the HGM derived from third order trend removed Bouguer anomaly data (datum: WGS84 UTM Zone 36 N). This system produces results by solving the Euler equation for sub-grids (windows) within a grid using the least squares method. The most important step of the method is the choice of the correct structural index value. When choosing the window size, it is appropriate to choose a window size that includes a certain anomaly, but one that does not include other anomalies in the calculation. [PERSON] [52] provided a short explanation to describe [PERSON]'s equation. A homogeneous three-dimensional equation of degree t (as any real number) fits the following equation (Equation (6)):  
\[f(tx,ty,tz)=t^{n}f(x,y,z) \tag{6}\]  
[PERSON] [52] showed that another equation of [PERSON] is also applicable here (Equation (7)):  
\[x\frac{\partial g}{\partial x}+y\frac{\partial g}{\partial y}+z\frac{\partial g }{\partial z}=nf \tag{7}\]  
Considering the potential field data, the final Euler homogeneity relationship is shown as (Equation (8)):  
\[(x-x_{0})\frac{\partial g}{\partial x}+(y-y_{0})\frac{\partial g}{\partial y} +(z-z_{0})\frac{\partial g}{\partial z}=-N(B-T) \tag{8}\]  
In this formula, \(x_{0}\), \(y_{0}\), and \(z_{0}\) are the gravity source coordinates; \(g\) is the measured potential field density in the \(x\), \(y\), and \(z\) coordinates; \(N\) is the structural index describing the geometry of the structure; and \(T\) is the total area determined in the \(x\), \(y\), and \(z\) coordinates, that is, the regional area \(B\) is the sum of the \(\Delta T\) changes created by the source. \(T\) is also expressed as in Equation (9):  
\[T=\Delta T+B \tag{9}\]  
[PERSON] [52] and [PERSON], et al. [53] state that the choice of structural index is extremely important. Both studies defined the structural index as the rate of change of an area based on distance. They also defined the structural index according to certain structures. It has been seen that the structural index for gravity, \(N\), takes values between 0 and 1. As a result of the calculations, it was determined that \(N=0\) and \(N=1\) are suitable for fault structures and that \(N=2\) is suitable for global anomaly sources (Table 1). [PERSON]'s Deconvolution method has been shown to be a suitable and interesting method for the purpose of this study. It offers a more advanced and wider solution than other methods because the method offers the opportunity to reveal the structural geology within any zone. The success of the application of the method used in this study was thoroughly examined [54; 55; 56; 57; 58]. The various window widths and tolerance percentages were tested in the process of Euler deconvolution. Because the main purpose of the study is to reveal buried faults (step structure) in the Eskisehir Basin and we used Bouguer gravity anomaly as input data, \(N=0\) was chosen as the structural index (Table 1). Since the aim of the study was to explore \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Structural Index (N)**} \\ \cline{2-4}
**Source** & \multicolumn{3}{c}{**Data Used for ED**} \\ \cline{2-4}  & **Gravity** & \
\begin{tabular}{c} **1 st Degree Vertical** \\ **Derivative of Gravity** \\ \end{tabular} &
\begin{tabular}{c} **2 nd Degree Vertical** \\ **Derivative of Gravity** \\ \end{tabular} \\ \hline Sphere & 2 & 3 & 4 \\ Horizontal Cylinder & 1 & 2 & 3 \\ Half-Infinite Thin Layer & 0 & 1 & 2 \\ Half-Infinite Contact & \(-1\) & 0 & 1 \\ Half-Infinite Thin Dyke & 1 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table}