Precise edge detection of anomalies in potential field data using balanced horizontal derivative and second-order normalized total horizontal derivative

Multiple methods available to increase the resolution of the edges, which usually are the functions composed by the first-order horizontal derivative and vertical derivative of potential field data. Cooper and Cowan (2006) presented total horizontal derivative (THD) which has less performance in determining edge of deep sources. Cooper and Cowan (2006) and Wijns et al. (2005) respectively have presented TDX and the Theta which are less sensitive to depth of the anomaly, also they have greater accuracy in detecting of the edges than the THD filter. The edge detection filters are all the functions based on the horizontal derivative and vertical derivatives of the data. The maxima of total horizontal derivative and the zero of the vertical derivative are corresponding to the edges of the source.
We find that the horizontal coordinates of the maxima of total horizontal derivative and the zero of vertical derivative are both bigger than the true locations of the edges, and the coordinates of the maxima of total horizontal derivative are more close to the true edges. We prove that the ratio of the total horizontal derivative to the second-order vertical derivative can obtain more accurate edge detection results.
We use the balanced horizontal derivative (BHD) edge detection filter, which uses the ratio of the first-order horizontal derivative to the second-order horizontal derivative to recognize the edges of the source, and the recognized edges by the BHD edge detection filter are more correct and are more insensitive to noise. We also use of second-order normalized total horizontal derivative (TDX2) to detect anomaly edges, which uses the ratio of the first vertical derivative of the horizontal to the second-order horizontal derivative to recognize the edges of the source, which gives the same results as the BHD. Ma and et al. (2014) have presented the BHD and the TDX2 filters as follows:
BHD=tan^(-1)⁡(√(("∂f" /"∂x" )^2 "+" ("∂f" /"∂y" )^2 )/(k×-((∂^2 f)/(∂x^2 )+(∂^2 f)/(∂y^2 )) )) (1)
where, ∂f/∂x, ∂f/∂y and ∂2f/∂z2 are the derivatives of the data f, and, K=(mean(∂f/∂z))/(mean((∂^2 f)/〖∂z〗^2 )) and the maxima of the absolute value of the BHD are corresponding to the edges of the source.

〖TDX〗_2=tan^(-1)⁡(√(〖((∂f_z)/∂x)〗^2+〖((∂f_z)/∂y)〗^2 )/|(∂^2 f)/(∂z^2 )| ) (2)

where, fz is the first-order vertical derivative of potential field f.
The results of the modeling on synthetic and real data shows that the edges that recognized by the TDX2 and the BHD are more precise and clear, by the presented methods are consistent with the true values for the shallow anomaly .and the BHD and the TDX2 filters can display the edges of shallow and deep sources simultaneously. For the anomaly of three separated objects interfere with different domains, filters break boundary anomalies better than previous filters. Also in this study, the mentioned filters have been used on gravity data of the aqueduct in the geophysics institute. Based on the obtained results, TDX2 and BHD filters show well the general trend of the aqueduct by separating the anomaly edge of the aqueduct and other existing anomalies on the aqueduct residual map. The width of the edge obtained by applying these filters is about 1.27m whereas that is 2.7m for the TDX and Theta filters which is impossible according to the geological information of the study area. Studies show that, the obtained width for the aqueduct is determined by the BHD and TDX2 filters more accurate than others.