## 4 Methodology
### Edge Detection
#### 4.3.1 Horizontal Gradient Magnitude (HGM)  
The method based on the magnitude of horizontal derivatives of gravity anomalies (HGM) was first proposed by [PERSON] [49]. The principal advantages of the method are that it does not require vertical derivatives, and that the horizontal derivatives can be easily calculated in the spatial domain. HGM data are shown as the square root of the sum of the squares of the first horizontal derivatives in the \(x\) and \(y\) directions of gravity anomaly maps (Equation (5)):  
\[HGM(x,y)=\sqrt{\left(\frac{\partial g(x,y)}{\partial x}\right)^{2}+\left(\frac {\partial g(x,y)}{\partial y}\right)^{2}} \tag{5}\]  
The method is based on determining only the regions in which the HGM is maximum along the grid data of the gravity anomaly map. The maxima of the HGM are particularly effective in determining vertical or approximately vertical density boundaries, faults, and geological contacts. Another advantage of the method is that the HGM maxima can be calculated with better sensitivity to noise effects. The HGM method has been discussed and used in various ways by numerous researchers [15; 49; 50; 51]. Solutions can be produced for both shallow and deep boundaries using the HGM method. The disadvantage of the method is that if the borders diverge from the vertical and are very close to each other, the HGM maxima do not occur on those limits.  
First, the Bouguer gravity data had to be filtered, and then, the first, second, and third-order trend data were removed, respectively. The focal depths of the earthquakes caused by the earthquake-forming faults in the study area vary between five and ten km when earthquake distribution is considered. To eliminate the effect of deep structures and to reveal the effects of shallow structures with possible lineaments, a low-pass filter was first applied to the Bouguer anomaly data. The HGM method applied to each trend removed items of data one by one. On the basis of the observations made, it was decided that the third-order trend-removed solution was the preferred result to continue interpretation. The HGM map was prepared by taking the square root of the sum of squares of the x-directional  
Figure 4: The radially averaged amplitude spectrum of the Bouguer anomalies. The line segments with three different slopes (a, b, and c) were fitted in each radial wavenumber section. The critical wave numbers of the line segment giving the basement rock (2.6 km) are marked as k1 and k2. The depths from the a and b segments correspond to the mean depth of 0.8 km of engineering bedrock and mean depth of 0.2 km soft sedimentary layer, respectively.  
first derivative (Figure 5a) and the y-directional first derivative (Figure 5b) of the filtered gravity anomaly. The HGM results show shallow structures in the study area that display linearity. The maximums of the HGM indicate possible lineaments. Fault-type structures and mass boundaries presenting density differences in certain parts of the study area can also be seen on the HGM maps. Upon examining the HGM map, it can be seen that there are maximum amplitude areas in the southern and northern regions of the study area (Figure 5c).  
The maximum amplitude values of the HGM were determined to be compatible with the characteristics of the tectonic discontinuities of the Eskisehir Fault Zone in the north and northwest regions, and with the form of short, far from systematic, scattered maxima in the north (Figure 5c).