A normalized statistics method in edge detection of potential field anomalies

Numerous filters are available to enhance subtle detail in potential field data, such as downward continuation, horizontal and vertical derivatives, and other forms of high-pass filters. A commonly used edge-detection filter is the total horizontal derivative (THD), which is computed as follows:
                                                                                               (1)
where f is the magnetic or gravity field.
Miller and Singh (1998) introduced a new filter based on phase variation of the data and called it "Tilt angle filter." Tilt angle is the ratio of the vertical derivative to the absolute value of the total horizontal derivative:
                                                                                       (2)
The tilt angle is positive when over the source, passes through zero when over or near the edge where the vertical derivative is zero and the horizontal derivative is maximum, and is negative outside the source region. The tilt angle has a range of -90 to +90 degrees.
Since the tilt angle is based on a ratio of derivatives, it enhances large- and small-amplitude anomalies well. However, in the cases where causative bodies are deep, the edges detected by the tilt angle are blurred as hollow. To overcome this problem, Gann et al (2004), in a new approach, suggested using the total horizontal derivative of the tilt angle as an edge detector (THDR):
                                                                                       (3)
where T is the tilt angle from Eq. 2.
The THDR successfully delineates the edges of the largest amplitude anomaly, but its results for the deeper bodies are less impressive. The theta map (Wijns et al., 2005) uses the analytic signal amplitude to normalize the total horizontal derivative. It is given by
                                                                                   (4)
where f is the potential field data and  is the theta angle filter.
The windowed computation of the standard deviation of an image is a simple measure of the local variability. It has relatively small values when the data are smooth and relatively large values when they are rough, e.g., over edges. If it is used as an edge detector, the response over large-amplitude gradients will dominate the result, similar to the results of other filters, e.g., the total horizontal derivative. We suggest using a filter based on the ratio of related normalized standard deviations (NSTD) to make large- and small-amplitude edges visible simultaneously:
                                                                                        (5)
The standard deviations  in equation 5 are computed using a moving square window of data points. The standard deviation can be computed in a given direction (to preferentially enhance edges normal to that direction).
In this paper, the filters mentioned above were applied to synthetic magnetic data from a prismatic model in both noiseless and noisy conditions. In general, presence of the disturb noises led to the detected edges being scattered, which appears as an offset between the detected edge and the actual body edge location.
Successive applications of the statistical filter on real magnetic data from the Sar-Cheshme region in Rafsanjan reveal the applicability of this filter. In this regard, we took into consideration the comparison between the main lithological units (Andezite and Trachyandezite) from a simplified geological map of the area and the results associated with the statistical filter. Application of the filter mentioned above determined the width of major geological units to be about 1520 m, which, in comparison with the measured width (1400m), produced an error of 6.67 percent, which is an admissible value.