Edge detection of potential field anomalies using vertical derivative of analytic signal

The analytic signal for magnetic anomalies was initially defined as a “complex field deriving from a complex potential” (Nabighian, 1972). This function can be computed easily in the frequency domain, its real part is the horizontal derivative of the field and its imaginary part is the vertical derivative. Analytic signal processing and interpretation requires few initial assumptions regarding the source body geometry and magnetization and is particularly efficient at an early stage of the interpretation even if constraints are not available.
For 2-D structures (Nabighian, 1972), the method assumes that the causative bodies have a polygonal cross-section with uniform magnetization. Such structures can also be considered as the superimposition of a finite number of magnetic steps. Narrow dikes and thin sheets can also be taken into account using a lower order of derivation; for example, the field itself instead of the horizontal derivatives. Nabighian (1972) demonstrates that the analytic signal has simple poles at each corner of the structures. The amplitude of the analytic signal is a bell-shaped symmetric function maximizing exactly over the top of each contact, with the width of the amplitude curve being related directly to the depth of the contact. This is also true for any of the derivatives of the signal (Nabighian, 1974); these properties can be used to locate the magnetic contacts and to estimate their depths.
Extension of the 2-D analytic signal to three dimensions will allow more general interpretation procedures to be developed, the two-dimensionality assumption being no longer required. The relationship between the horizontal and vertical derivatives for the 3-D case was first derived by Nabighian (1984).
As with the 2-D case, the vertical and horizontal derivatives are Hilbert transforms of each other and so the concept of the analytic signal in 2-D can be extended to three dimensions. The analytic signal amplitude can then be defined as “the square root of the squared sum of the vertical and the two horizontal first derivatives of the magnetic field” (Roest et al., 1992).
Because of interference effects, the use of the simple analytic signal in the 3-D case seems insufficient to detect geologic boundaries. Since the existing interference is usually inevitable, improving resolution becomes a requirement. In the 2-D case, Nabighian (1974) suggested using the following bell-shaped function to enhance the analytic signal from shallow sources:

(1)

where Gh and Gz are the horizontal and vertical gradients of the potential-field anomaly, respectively; h is the distance along the horizontal axis which is perpendicular to the strike of the 2-D structure; n is a positive integer; d is the depth to the top surface of the source, while the lower surface is at infinity; is the ambient parameter and is equal to when the step model of magnetic anomaly is applied; k is the susceptibility contrast of the step model; F is the earth’s magnetic field magnitude; is the dip angle of the step model; for total magnetic field anomalies; i is the inclination of the earth’s magnetic field; and is the angle between magnetic north and positive h - axis. All the above parameters are presented in Fig. 1.
In the 3-D case, the simple analytic signal is defined in Nabighian (1984) as:
(2)
and its amplitude as:
(3)
To extend equation (1) into the 3-D case, we define the nth-order enhanced analytic signal as:
(4)
and its amplitude as:
(5)
For n = 2, equation (4) corresponds to the enhanced analytic signal derived from second vertical derivative, and the amplitude of equation (5) becomes:
(6)
where G is the potential-field anomaly and ، ، .
Equation (6) is used hereafter as an example to demonstrate the improvement of the detection of geologic boundaries. In this paper the applicability of this method is demonstrated on synthetic gravity data from the vertical cylinder model. Also this method was applied on real gravity data from southwest England successfully. In this regard the granites bodies and their separating faults have been enhanced in which the results of our method have broad correlation with the geological map.