Edge Detection and Depth Estimation Using a Tilt Angle Map from Gravity Gradient Data of the Kozaklı-Central Anatolia Region, Turkey

Abstract

In this paper the application of an edge detection technique to gravity data is described. The technique is based on the tilt angle map (TAM) obtained from the first vertical gradient of a gravity anomaly. The zero contours of the tilt angle correspond to the boundaries of geologic discontinuities and are used to detect the linear features in gravity data. I also present that the distance between zero and $\pm {\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}$ pairs obtained from the TAM corresponds to the depth to the top of the vertical contact model. Alternatively, the half distance between $- {\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}$ and $+ {\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}$ radians is equal to the depth to the same model. I illustrate the applicability of the present method by gravity data due to buried vertical prisms, imaging the positions of the edges of the prisms. The results obtained from the theoretical data, with and without random noise, have been discussed. The analysis of the TAM has been demonstrated on a field example from the Kozaklı-Central Anatolian region, Turkey, and the location and depth of the edges of the structural uplifts of the Kozaklı graben are imaged. The results indicated that depth values from these sources have ranged between 0.2 and 0.6 km. I have also compared the Euler deconvolution technique with the TAM images obtained from the first vertical gradient of residual gravity anomaly. Both techniques have agreed closely in detecting the horizontal location and depth of the uplift edges in the subsurface with good precision.

Appendix A: Derivation of structural index from the second vertical gravity gradient of semi-infinite contact

For 2D sources, where ${{\partial^{2} g_{z} } \mathord{\left/ {\vphantom {{\partial^{2} g_{z} } {\partial y\partial z}}} \right. \kern-\nulldelimiterspace} {\partial y\partial z}} = 0$, Eq. (12) is written as

$$ (x - x_{0} ){\frac{{\partial^{3} g_{z} }}{{\partial x\partial z^{2} }}} + (z - z_{0} ){\frac{{\partial^{3} g_{z} }}{{\partial z^{3} }}} = - N{\frac{{\partial^{2} g_{z} }}{{\partial z^{2} }}} . $$

(A1)

The derivatives of Eq. (3) with respect to x and z are defined, respectively;

$$ {\frac{{\partial^{3} g_{z} }}{{\partial x\partial z^{2} }}} = 2G\rho \sin d{\frac{{\sin d[(z - z_{0} )^{2} - (x - x_{0} )^{2} ] - 2(x - x_{0} )(z - z_{0} )\cos d}}{{[(x - x_{0} )^{2} + (z - z_{0} )^{2} ]^{2} }}} $$

(A2)

and

$$ {\frac{{\partial^{3} g_{z} }}{{\partial z^{3} }}} = 2G\rho \sin d{\frac{{\cos d[(x - x_{0} )^{2} - (z - z_{0} )^{2} ] - 2(x - x_{0} )(z - z_{0} )\sin d}}{{[(x - x_{0} )^{2} + (z - z_{0} )^{2} ]^{2} }}} . $$

(A3)

Substitution into the left-hand side of Euler’s Eq. (A1) yields

$$ \left( {x - x_{0} } \right){\frac{{\partial^{3} g_{z} }}{{\partial x\partial z^{2} }}} + (z - z_{0} ){\frac{{\partial^{3} g_{z} }}{{\partial z^{3} }}} = {\frac{1}{{r^{4} }}}\left\{ {(x - x_{0} )\sin d[ - (x - x_{0} )^{2} - (z - z_{0} )^{2} ] + (z - z_{0} )\cos d[ - (x - x_{0} )^{2} - (z - z_{0} )^{2} ]} \right\} = - {\frac{1}{{r^{2} }}}[(x - x_{0} )\sin d + (z - z_{0} )\cos d] $$

(A4)

If Eq. (A4) is compared with the right-hand side of Eq. (A1), the structural index N leads to 1.0.