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Apr 10, 2014 - NAV-Edge: Edge detection of potential-field sources using normalized anisotropy variance. Heng Lei Zhang1, Dhananjay Ravat2, Yára R.
GEOPHYSICS, VOL. 79, NO. 3 (MAY-JUNE 2014); P. J43–J53, 12 FIGS., 1 TABLE. 10.1190/GEO2013-0218.1

NAV-Edge: Edge detection of potential-field sources using normalized anisotropy variance

Heng Lei Zhang1, Dhananjay Ravat2, Yára R. Marangoni3, and Xiang Yun Hu4

Cordell, 1987; Fedi and Florio, 2001), vertical derivatives (Cooper and Cowan, 2003; Cowan and Cooper, 2005), and the tilt angle (Miller and Singh, 1994). Recently, a large number of new methods have been developed for the problem of edge detection. To map shallow basement structures and mineral exploration targets, Verduzco et al. (2004) suggest using the total-horizontal derivative of the tilt angle (THDT). Wijns et al. (2005) propose the theta map method that uses total-gradient amplitude to normalize the total-horizontal derivative. For detecting lineaments, Hansen and de Ridder (2006) present a new approach based on analysis of the curvature of the total-horizontal gradient of the total-magnetic field reduced to the pole. Cooper and Cowan (2006) use the hyperbolic tangent function in the tilt angle calculation to achieve better delineation of the edges with a method called hyperbolic tilt angle (HTA). Phillips et al. (2007) discuss the curvature based on HGA (C-HGA) as well as two other special functions of potential field to detect source edges and to estimate their depth. Salem et al. (2008) apply the tilt angle to derive a simple linear equation, similar to the 3D Euler equation. They used the new method to detect horizontal location and the depth of magnetic bodies. Cooper and Cowan (2008) present normalized standard deviation (NSTD) based on the ratios of the windowed standard deviation of derivatives of the field. Cooper and Cowan (2009) apply profile curvature instead of the Laplacian derivative operator to improve the edge detection. Later, they introduced the generalized derivative operator that was a linear combination of the horizontal and vertical field derivatives, normalized by the total-gradient amplitude (Cooper and Cowan, 2011). Oruç et al. (2013) describe an edge-detection method based on the eigenvalues and determinant obtained from the curvature gravity gradient tensor (CGGT), and the contours of the smaller of the two eigenvalues from CGGT (SECGGT) are used in their approach to identify source edges. Although the above methods have been used extensively to detect source edges (Salem et al., 2008; Cameron and Goussev, 2010;

ABSTRACT Most existing edge-detection algorithms are based on the derivatives of potential-field data, and thus, enhance high wavenumber information and are sensitive to noise. The normalized anisotropy variance method (NAV-Edge) was proposed for detecting edges of potential-field anomaly sources based on the idea of normalized standard deviation (NSTD). The main improvement over the balanced, windowed normalized variance method (i.e., NSTD) used for similar purposes was the application of an anisotropic Gaussian function designed to detect directional edges and reduce sensitivity to noise. NAV-Edge did not directly use higher-order derivatives and was less sensitive to noise than the traditional methods that use derivatives in their calculation. The utility of NAVEdge was demonstrated using synthetic potential-field data and real magnetic data. Compared with several existing methods (i.e., the curvature of horizontal gradient amplitude, tilt angle and its total-horizontal derivative, theta map, and NSTD), NAV-Edge produced superior results by locating edges closer to the true edges, resulting in better interpretive images.

INTRODUCTION An important goal in the interpretation of potential-field data is to determine the location of the source, which helps in its geological interpretation. To date, a variety of methods have been focused on edge detection to facilitate source interpretations of gravity or magnetic data, using simplified assumptions with data at high latitudes, reduced-to-pole (RTP), nondipping sources, etc. These methods include the horizontal gradient amplitude (HGA) (Grauch and

Manuscript received by the Editor 12 June 2013; revised manuscript received 21 November 2013; published online 10 April 2014. 1 China University of Geosciences, Institute of Geophysics and Geomatics, Wuhan, China and São Paulo University, Geophysics Department, São Paulo, Brazil. E-mail: [email protected]. 2 University of Kentucky, Earth and Environmental Sciences, Lexington, Kentucky, USA. E-mail: [email protected]. 3 São Paulo University, Geophysics Department, São Paulo, Brazil. E-mail: [email protected]. 4 China University of Geosciences, Institute of Geophysics and Geomatics, Wuhan, China. E-mail: [email protected]. © 2014 Society of Exploration Geophysicists. All rights reserved. J43

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Al-Garni, 2010; Beamish and White, 2011; Beamish, 2012), there are some limitations to these methods. For example, Li (2006) points out that in the theta map the positive-to-negative range of the tilt angle is renormalized between zero and one and hence the information on the relative magnetization variation is lost (also confirmed by Wijns et al., 2006, in their reply). Zhou et al. (2013) suggest that the HTA method may produce false maxima not corresponding to edges in some situations. Cooper (2013) thought that thresholding used in the HTA could remove valid edges; he also proposed using a positive constant to stabilize the results of the HTA method when the denominator of the HTA equation is small. Santos et al. (2012) point out that when the amplitude of the analytical signal, HGA, tilt angle, and THDT is applied to offshore anomalies, the enhancement or the continuity of small geological features becomes inadequate for interpretation because of the variation in amplitude of the anomalies due to varying source to observation distances. In this paper, we propose a new edge-detection approach that is also useful for analyzing elongated sources; we call it NAV-Edge or the normalized anisotropy variance method for edge detection. We compare this method with some popular existing edge-detection methods (C-HGA, tilt angle, THDT, theta map, NSTD, and SECGGT).

BACKGROUND The tilt angle (Miller and Singh, 1994) is given by

  9 ð∂f∕∂zÞ > > β ¼ arctan > = HGA sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 ; ∂f ∂f > > > HGA ¼ þ ; ∂x ∂y

(1)

where β is the tilt angle, f is the potential field, ð∂f∕∂xÞ, ð∂f∕∂yÞ, and ð∂f∕∂zÞ are the first-order derivatives of the potential field in the x-, y-, and z-directions, respectively. The zero value of the tilt angle delineates the source edges. Several applications of the tilt angle are discussed by Cooper and Cowan (2008, 2009, 2011), Salem et al. (2008), Lahti and Karinen (2010), and Beamish and White (2011), and elsewhere. Verduzco et al. (2004) suggest using the THDT that delineates the edges by its maxima. The THDT is given by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ∂Tilt 2 ∂Tilt 2 : THDT ¼ þ ∂x ∂y

(2)

The theta map proposed by Wijns et al. (2005) uses total-gradient amplitude to normalize the HGA, which is given by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð∂f∕∂xÞ2 þ ð∂f∕∂yÞ2 cos θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð∂f∕∂xÞ2 þ ð∂f∕∂yÞ2 þ ð∂f∕∂zÞ2

A þ Bx þ Cy þ Dx2 þ Exy þ Fy2 ≈ gðx; yÞ;

where A, B, C, D, E, and F are coefficients of the quadratic surface approximating the HGA and gðx; yÞ is the gridded function of HGA. The curvature matrix is constructed from the coefficients of the second-order terms:

0

∂2 g B 2 B ∂x B 2 @ ∂ g ∂x∂y

1 ∂2 g  C  2D E ∂x∂y C ¼ : C E 2F ∂2 g A ∂y2

(5)

The eigenvalues of the curvature matrix are given by



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD − FÞ2 þ E2 λ1 ¼ ðD þ FÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : λ2 ¼ ðD þ FÞ − ðD − FÞ2 þ E2

(6)

If both eigenvalues are negative, the extremum is a maximum corresponding to the edge, and its location ðx0 ; y0 Þ is given by

8 > > > < x0 ¼ −

Be2x þ Cex ey 2ðDe2x þ Eex ey þ Fe2y Þ ; Ce2y þ Bex ey > > > y ¼ − : 0 2ðDe2x þ Eex ey þ Fe2y Þ

(7)

where ðex ; ey Þ is the eigenvector associated with the largest of the two eigenvalues. The NSTD method (Cooper and Cowan, 2008), which delineates the edges with maxima is based on the ratios of the windowed standard deviation of derivatives of the field and is defined as

NSTD ¼

σð∂f∕∂zÞ ; σð∂f∕∂xÞ þ σð∂f∕∂yÞ þ σð∂f∕∂zÞ

(8)

where σð∂f∕∂aÞ is the standard deviation of the derivative of the field f with respect to a computed using a moving window of data points. Here, a represents the x-, y-, and z-directions. The SECGGT method described by Oruç et al. (2013) uses the smaller of the two eigenvalues from the CGGT and is defined as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2ffi   1 ∂f x ∂f y ∂f x ∂f y 2 ∂f x þ − − ; SECGGT ¼ þ4 2 ∂x ∂y ∂x ∂y ∂y (9)

(3)

Equation 3 is a ratio of the magnitude of the HGA to the totalgradient amplitude, thus the theta map can be thought of as a normalization of the HGA, and it also delineates the edges with its maximum values. The C-HGA method (Phillips et al., 2007) locates the edges based on a search for the maximum of the HGA using curvature. The first step is to use a linear least-squares approximation in each three by three points window of HGA

(4)

where f x and f y are the horizontal vectors of the potential field f. The zero value of the SECGGT delineates the source edges.

THE NAV-EDGE METHOD Based on the 2D Gaussian function, the anisotropic Gaussian function with a direction θ and a window size of M points by M points is constructed as

Edge detection using NAV-Edge

  1 1 2w Þ ; exp − ðw þ cof 1 2 2πσ 2 2σ 2     2 Mþ1 Mþ1 where w1 ¼ i − cos θ þ j − sin θ ;w2 2 2     2 Mþ1 Mþ1 − i sin θ þ j − ¼ cos θ ; 2 2 Qði;jÞ ¼

(10) where i ¼ 1; : : : ; M, j ¼ 1; : : : ; M, and σ is the standard deviation of the Gaussian function; cof expresses the anisotropy scale, which we will discuss later. For map data, the NAV-Edge of potential-field data f is defined as

8 Pi;j¼M > > i;j¼1 f½fðxi ;yj Þ − fðx;yÞQði;jÞg > < NAVðx;yÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pi;j¼M i;j¼M 2 2; ½fðx ;y Þ − fðx;yÞ i j i;j¼1 i;j¼1 Qði;jÞ > > P > : fðx;yÞ ¼ 12 i;j¼M fðxi ;yj Þ i;j¼1 M (11) where the index (i or j) indicates the observation point (x ¼ xi ; y ¼ yj ) inside the data window. For profile data, the NAV-Edge is defined as

8 Pi¼M > > > i¼1 f½fðxi Þ − fðxÞQðiÞg < NAVðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pi¼M 2 Pi¼M 2; ½fðx Þ − fðxÞ i i¼1 i¼1 QðiÞ > > P > 1 i¼M fðx Þ : fðxÞ ¼ i i¼1 M (12) where Q is the 1D Gaussian function, and i ¼ 1; : : : ; M such that

    1 1 Mþ1 2 QðiÞ ¼ exp − 2 i − : 2 2πσ 2 2σ

(13)

In equation 11, the numerator of the NAV function is similar to the calculation of a variance and it is anisotropic in the sense that it has a direction; and hence the name normalized anisotropy variance (NAV). The NAV function can be computed using a moving window of data points.

Applicability of NAV-Edge It is shown in Appendix A that the sign of the NAV function is controlled by the horizontal derivative of the potential-field data, and the zero crossing value of NAV shows the same location as the maxima of the horizontal derivative of the potential-field data. Thus, NAV-Edge works for detecting edges in situations, in which other methods based on the derivatives of the potential-field anomaly data can work: i.e., the magnetic anomaly should be reduced to the pole. Through examples, we show that the performance of NAV-Edge is better than the other methods in this category. The zero crossing value of NAV is used for detecting edges. Recalling equation 11, the Gaussian function has two roles. First, it makes the numerator less than zero when the derivative of f is increasing and

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greater than zero when the derivative of f is decreasing (see equations A-1 and A-7). So NAV value varies from negative to positive values, and the method NAV-Edge uses the zero crossing of NAV to identify the edges. Second, the weighted computation of the Gaussian function in a window can suppress noise. The NAV function, being a dimensionless ratio, responds equally well to shallow as well as deep sources (as indicated by upward continuation); it also represents well sources having a large dynamic range of amplitudes. In Figure 1, using theoretical (a) and noise (b) contaminated gravity examples, we show how different sources and edge-detector methods (including NAV-Edge) delineate source edges. We use two homogeneous simulated sources with density contrasts of 0.01 and 0.05 g∕cm3 for Figure 1a and 1b, respectively (gray prisms in the bottom). We illustrate this in a profile sense for the ease of comparison. The gravity field yielded by the two sources is shown in Figure 1. Tilt angle and NAV-Edge use the crossover zero amplitude to detect edges, and the other methods use the maximum amplitude. The zero values of NAV function show the same locations with the maxima values of the horizontal derivative of the gravity field. The central value of the NAV-Edge response is of the same sign as the original residual anomaly and it changes to the opposite sign outside the source region. In Figure 1b, we present anomalies and results after contamination of the theoretical anomaly with noise. Because NAV-Edge does not use of any higher-order derivatives directly and uses Gaussian function to smooth the noise, it is less sensitive to noise than the other methods.

Parameters of NAV-Edge Equation 10 or 11 has four parameters: standard deviation (σ), direction (θ), window size (M), and anisotropy scale (cof). For the Gaussian function, if the distance of the data point in a window to the central point is larger than three σ, we discard it. This results in the following simple relationship for the selection of σ: σ ¼ ðM − 1Þ∕6. For the direction (θ) of the anisotropic Gaussian function in equation 10, we compute

  f θ ¼ arctan x ; fy

(14)

where f x and f y are the first-order horizontal derivatives of the potential-field data f along the x- and y-directions, respectively. For noisy data, we use

9 > −i > f x ¼ f × Gx ¼ f × ·G > = 2 σ  ; −j > > f y ¼ f × Gy ¼ f × ·G > ; σ2 

(15)

where Gx and Gy are the first derivatives of the Gaussian function Mþ1 2 2 G ¼ expð− 2·σ1 2 ½ði − Mþ1 2 Þ þ ðj − 2 Þ Þ along the x- and y-directions, respectively. As described above, the practical procedure of calculating NAVEdge requires selection of two parameters by the user: one is the window size (M) and the other is the anisotropy scale (cof). The window size M is associated with data quality; for noise-free data, M ¼ 3 is acceptable. Appropriately larger windows are necessary in the presence of increasing noise. However, a large window also smears out the edges due to averaging. In this paper, we use the

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L2 -norm misfit function used by Pašteka et al. (2012) to optimally define the window size of the Gaussian filters. The L2 -norm is the root mean square value of the elements of a vector; the vector in this case is the difference between each subsequent window’s data point by data point NAV-Edge result:

L2 ‐norm ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mean½ðNAVM − NAVM−1 Þ2 ;

TEST CASES OF MODEL DATA (16)

where NAVM denotes the NAV value obtained from equation 11 using a window size of M. When the L2 -norm is plotted against the value of the window size (M), the minimum value and/or the knee of the L2 -norm misfit function yields the optimum value of M. The application of the directional strategy for the anisotropy scale can make the filtering approach more flexible and the edge-detection precise. The anisotropy scale (cof) chosen for a given noisy data set depends on the separation of two juxtaposed sources in the long dimension of the largest source. Generally, the closer the two sources, the larger the chosen anisotropy scale. This allows Figure 1. Comparison of several edge-detecting techniques. (a) Noise-free results and (b) noisecontaminated results (10% Gaussian random noise with zero mean). The open circles indicate the edge locations detected by the C-HGA method based on the HGA. For NSTD and NAV-Edge, the window sizes are 3 and 25 points for the noise-free data and noisy data, respectively. The lines with the arrowheads show the edges interpreted by the respective methods.

Figure 2. Magnetic response of the synthetic test model. (a) Theoretical anomaly and (b) data contaminated with 5% Gaussian random noise with mean zero. The widths of the sources A and B are 10 m. See other parameters in the text.

in smoothing of noise along the direction perpendicular to the gradient direction, and that results in better detection of edge details along the gradient direction. In this paper, we have taken an empirical approach to select this parameter (cof).

Figure 2 shows the magnetic response from three vertical-sided prisms (A, B, and C) with 200-m thickness, and their edges are indicated by the white lines. The top of the three prisms (A, B, and C) are located at depths of 10, 50, and 10 m, respectively. The magnetization contrasts of the three prisms (A, B, and C) are 1.0, 0.5, and 1.0 A/m, respectively. The inducing field has an inclination of 90° and a declination of 0°. The prisms have no remnant magnetization. The data set was computed on plane z ¼ 0 m at the nodes of a 21 × 61 grid with a grid spacing of 10 m in the north–south and east–west directions. Figure 2a and 2b shows the noise-free and noise-corrupted total-field anomalies produced by the simulated sources (A–C). We corrupt the theoretical anomaly (Figure 2a) with additive Gaussian random noise with zero mean and standard

Edge detection using NAV-Edge deviation of 5% of the maximum amplitude of the magnetic anomaly. For comparison, we present in Figure 3a–3f (left panel) the results of filtering the noise-free total-field anomaly (Figure 2a) with C-HGA, tilt angle, THDT, theta map, NSTD, and SECGGT, respectively. Notice that the outlines of sources A and B are correctly delineated with zero contour of the tilt angle, whereas some edges of source A are not identified in the SECGGT result. The zero crossings of the tilt angle and the SECGGT can distinguish anomaly B from anomaly A, whereas the other methods map the edges outside of the sources boundaries. On the other hand, the C-HGA, NSTD, and SECGGT methods provide more accurate edges of the source C than the other methods. The edge of the source C is weaker than others, and indeterminate on the theta map. We note that the C-HGA method searches for maximum values (the open circles shown in Figure 3a) on the ridge of the HGA function; it is able to detect small values of HGA, but that is difficult for the method in the pres-

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ence of noise (Figure 3a, right panel). We applied the above methods to the noise-corrupted total-field anomaly (Figure 2b). The enhanced anomalies of the SECGGT method (Figure 3f, right panel) are better than the others (Figure 3a–3e, right panel), but its results are difficult to interpret as edges of the sources. Notice that the enhancements produced by applying the NSTD filter (Figure 3e, right panel) do not delineate the outlines of the sources A and B, even when using a larger window size (seven by seven points) that suppresses the noise. In contrast, the application of NAV-Edge to the noise-free totalfield data provides a good estimate of outlines of all of the sources (black dotted lines and also color change from white to gray in Figure 4a). Notice that the zero crossings of NAV-Edge (Figure 4a) are very close to the edges of all of the sources. For the noisy data, we used increasingly higher anisotropy scale parameters (cof in equation 10) of one, two, and five for the NAV-Edge method, and the results are shown in Figure 4b, 4c, and 4d, respectively. Figure 3. Edge detection by existing methods on anomalies in Figure 2. Each method detects edges using a particular characteristic of its function (maxima, zero crossings, etc.) and that characteristic is given in the text. The left and right panels are results generated from Figure 2a and 2b, respectively. The open circles shown in Figure 3a indicate the edge locations detected by the C-HGA method. For the NSTD method shown in Figure 3e, the window sizes are three by three, and seven by seven grid points for the left and right panels, respectively.

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We use the L2 -norm misfit function (equation 16) to define the window size M. As shown in Figure 5, the optimal values are picked at the knee of the curve. We choose the window size of seven for the

anisotropy scale parameters of one, two, and five based on Figure 5. In fact, when the anisotropy scale parameter is equal to one, the NAV-Edge method degenerates into an isotropic method, which makes resolving sources with small separation across their long dimension less precise (compare Figure 4b, 4c, and 4d). Using a large anisotropy scale parameter (e.g., five), one can see that the source B can be resolved from source A very well (Figure 4d), whereas smaller anisotropy scale parameter of one (Figure 4b) cannot resolve these sources adequately. In addition, to examine the variation of the NAV-Edge results as a function of different window sizes, we computed the results based on the noisy data (Figure 2b) using an anisotropy scale parameter of five and different window sizes of three, five, nine, and 11 (these results are not shown). As expected, the results indicate that a smaller window size (e.g., three) leads to somewhat noisy result but the NAV-Edge image is sharper, whereas a larger window size (e.g., 11) smears Figure 4. Edge detection by the NAV-Edge method. Figure 4a is generated from Figout the edges of sources A and B. It is obvious ure 2a using a window size of three by three grid points, and an anisotropy scale paramthat the result of applying the NAV-Edge techeter of five. (b), (c), and (d) are generated from Figure 2b using the window size of seven nique is better than the previous methods by seven grid points, and the anisotropy scale parameters (cof) of one, two, and five, (Figure 3) in terms of the source features and respectively. sharpness of their edges; and especially for elongated sources, and it can also obtain useful results from noisy data, which indicates that NAV-Edge is more robust than the existing methods. In this model case, sources A and B are designed such that their north and south edges give rise to strong interfering anomalies. This is a nearly impossible situation for resolving edges for any edge-detection method. However, using these examples, we have demonstrated that NAV-Edge is better at locating edges in comparison to other methods. The NAV-Edge also detects the source C’s edges well (whose width is large compared to elevation) despite the higher values of cof. The user can select cof by observing significant change in the NAV function with respect the value of cof.

APPLICATION TO A NONVERTICAL CONTACT MODEL Figure 5. The L2 -norm misfit function from the NAV-Edge method with respect to window size for different anisotropy scales (cof). The optimum value is at the knee of the curves.

For a nonvertical contact case, we set up a model similar to that published in Santos et al. (2012). We computed the total-field anomaly produced by two sources, in which the edge locations are indicated by the black lines (Figure 6a). The inducing field has an inclination of 90° and a declination of 0°. Figure 6b shows the wedge-shaped vertical cross section along profile AA′. The theoretical anomaly in Figure 6a was corrupted with additive Gaussian random noise with zero mean and standard deviation of 1% of the maximum amplitude of the magnetic anomaly. Note that the anomaly produced by source 1 is very large except for the portion located at the north edge. In contrast, source 2 produces no noticeable response because of its greater depth and weak magnetism. Figure 6. (a) Total-field anomalies produced by two 3D sources with the edges shown For comparison, we present the edge detecby black lines and (b) vertical cross section along profile AA′ shown in (a) by the white tions based on Figure 6a by NSTD, tilt angle, line.

Edge detection using NAV-Edge SECGGT, and NAV-Edge methods shown in Figure 7. The NSTD, tilt angle, and NAV-Edge methods detected all of the edges of source 1 well, whereas SECGGT could detect only the southern edge of source 1 and not the north edge. On the other hand, NAV-Edge could detect the edges of source 2 clearly, whereas the other methods could not.

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where f denotes the magnetic anomaly, f k1 and f k1 −Δk denote the noise reductions achieved by cutting high wavenumbers greater than the wavenumber k1 and k1 − Δk, respectively. The Δk denotes a small change of the wavenumber. Then the percentage of low-end wavenumbers retained is defined as

k% ¼ APPLICATION TO FIELD DATA We also compare the performance of the NAV-Edge filter to six commonly used filters applied to a magnetic anomaly associated with a magnetite ore body from Qinghai Province in northwest China. Magnetite has a very high magnetic susceptibility, whereas the host rocks, granodiorite and dacite in this case, have weak magnetic susceptibilities (Zhang and Liu, 2011). Hence, the magnetic anomaly is caused mainly by the magnetite ore that is located below the thick Quaternary overburden in this area with a thickness of ∼150 m (Zhang et al., 2013). Figure 8 depicts the reduced to pole magnetic anomaly within an inducing magnetic field that has an inclination of 56.5° and a declination of 0.2°. The magnetic anomaly is broad, whose precise edges are indistinguishable. The data were gridded from ground observations with an interval of 5 m and the data set is 339 by 360 points in size. To aid mining operations, the methods discussed and developed in this study are intended to define more realistic boundaries of the ore deposit than the general areas of occurrences of the ore body that can be inferred from drillholes. In the previous synthetic tests, we have shown that the C-HGA, tilt angle, THDT, and theta map are very sensitive to noise; all of them produce inaccurate responses for the field in Figure 8 because of the random noise in the actual field data (these results are not shown), although the data are smooth in appearance. To reduce the noise for these methods, we cut the high-wavenumber information, and use a mean-norm criterion to retain a percentage of lowend wavenumbers (presumably nonnoisy) of the magnetic anomaly:

Mean‐normk1 ¼ mean½f k1 − f k1 −Δk ;

(17)

  k1 ; k0

(18)

where k0 denotes the Nyquist wavenumber. When the mean-norm is plotted against k%, the knee of the mean-norm is expected to yield the optimum value of k%.

Figure 8. Reduced-to-pole magnetic-anomaly field from a magnetite ore body in China. Black filled-in diamonds: drillholes with no magnetite; pink filled-in diamonds: drillholes with magnetite. The blue lenticular symbols show approximate areas of geologically inferred magnetite ore occurrences from the drillholes. The line spacing is 50–100 m and the observations along a line are at 10-m interval for the ground magnetic survey.

Figure 7. Edge detection on anomalies in Figure 6a by NSTD with a window size of 15 by 15 grid points (a), tilt angle (b), SECGGT (c), and NAV-Edge with a window size of 15 by 15 grid points, and an anisotropy scale parameters of two (d). The white lines show the synthetic sources’ edges.

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Based on Figure 9, we keep 5% of low-end wavenumbers before applying the C-HGA, tilt angle, THDT, theta map, and SECGGT to suppress noise, whereas NSTD and NAV-Edge operations are performed on Figure 8 directly. The THDT is too noisy because it involves computation of a second derivative of the RTP field, and hence its result is not shown. The C-HGA and NSTD methods divide the anomaly shown in Figure 10 into two parts (marked A and B), and they locate the edges better than tilt angle and theta map.

Figure 9. Mean-norm wavenumber selection for denoising. Wavenumber selection is based on the convergence of the mean-norm closer to zero.

Figure 10. Edge detection by the C-HGA (a), tilt angle (b), theta map (c), and NSTD with the window size of 31 by 31 grid points (d). All of the results of C-HGA, tilt angle, and theta map are constructed from Figure 8 by keeping 5% of low-end wavenumbers to minimize the effect of noise. (a) The thick curvy and discontinuous black lines are created by the open circles indicating the edge locations detected by C-HGA method. Other thin black lines denote the edges interpreted by the respective methods.

The tilt angle and theta map results shown in Figure 10b and 10c, which are effective in balancing the amplitudes of anomaly A and B, but do not resolve their edges well (e.g., the zero contour of tilt angle, and the peak values of theta map drillholes with and without magnetite). Figure 11 shows the edge detection by SECGGT and NAV-Edge, whereas window size for NAV-Edge is defined based on the L2 -norm misfit function shown in Figure 12. The SECGGT result (Figure 11a) divides the anomaly into four parts (A, B, D, and E), and yields a sharper image than the results of Figure 10. Unfortunately many drillholes with magnetite are not located inside the zero contour of SECGGT. Combining this observation with the results of the model studies in Figures 3f and 7c, it appears that the SECGGT method’s results do not identify some of the edges well. The NAV-Edge method (Figure 11b) does not only distinguish well the northwest sources (A, C, and D) from the southwestern sources (B and E), it also splits the northwestern sources into subparts A, C, and D (source C is probably an extension of source A). These enhancements are based on the subtle pinching of contours and the variation in the intensity of highs of the NAV function. We stress that the other methods do not highlight these subtle features. In addition, NAV-Edge appears to see the distinction between features B and E. Notice also that the enhancements produced by the NAV-Edge filter (A–E features in Figure 11b) are not spurious enhancements, they are confirmed by the information provided by boreholes. The NAV-Edge method also separates magnetite-bearing drillholes (pink-filled-in diamonds) from nonmagnetite-bearing ones

Edge detection using NAV-Edge (black-filled-in diamonds). Note that there is a noticeable separation between areas A and D (near line 2, Figure 11b). Even though the gap between the features has drillholes with magnetite, at the drillhole number 1 located on line 2, we have a deeper and thinner magnetic ore body (see Table 1). Table 1 also shows that drillhole 5 on line 4 (pink diamond in Figure 11b) containing magnetite; however,

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this drillhole is not located inside the edge of the source as indicated by the zero contour of the NAV function. This occurs because the magnetite ore body is thinner and deeper in this location. Furthermore, we note that the NAV-Edge result separates features B and E, whereas other methods shown in Figure 10 do not. As shown in Figure 11b, drillholes with and without magnetite are located on line 5 (between areas B and E). In this case, the magnetite-bearing drillhole appears to lie on the high of the NAV function, whereas the nonmagnetite-bearing drillhole is in the location where a pinch out of the zero contour occurs. If the method can truly separate features over such small distances, the resolution that NAV-Edge is able to deliver is quite superb. We think that the superiority of this method over the others is that the noise is not enhanced as much as opposed to methods involving field derivatives.

CONCLUSIONS NAV-Edge is an effective processing tool for the edge detection of potential-field sources. It has characteristics similar to those of the tilt angle (i.e., the zero value corresponds to the edge, Figure 11. Edge detection by the SECGGT (a), and NAV-Edge method with an and the positive and negative values correspond anisotropy scale parameter of 5 (b). The optimal value of the window size of NAV-Edge to source areas and nonsource areas, respecis 31 by 31 data points (based on Figure 12). We note that the SECGGT and NAV-Edge tively). Because NAV-Edge does not directly use zero values (continuous black lines showing zero contours) to locate the edges. use any derivatives, it is more stable than the traditional methods based on derivatives (e.g., THDT). Because the zero crossings of NAV function show the same location as the maxima of the horizontal derivative of the potential field, NAV-Edge works for the same conditions as do the existing methods, i.e., it works for gravity data or the pole reduced magnetic data. Two parameters need to be selected by the user to calculate NAV-Edge: window size and cof (anisotropy coefficient). We suggest using the L2 -norm misfit function to deduce the optimal window size, but the anisotropy scale is most easily chosen in an empirical manner (by observing the change in the NAV images with respect the value of cof). The comparisons of NAV-Edge with the CHGA, tilt angle, THDT, theta map, NSTD, and SECGGT discussed in this paper indicate that NAV-Edge yields superior results and produces better interpretive images than the other methods.

Figure 12. The L2 -norm misfit function from the NAV-Edge method with respect to window size used in selecting the optimum window size based on the location of the knee of the curve.

ACKNOWLEDGMENTS H. Zhang thanks the University of Kentucky for making his possible research visit. We thank J. D. Phillips for providing his code of

Table 1. The depth and total thickness of magnetite ore in the drillholes in the study area. The drillholes located on lines 2 and 4 show that the magnetite ore there is deeper and thinner, but variable depending on the location of the drillhole. Line numbers and drillhole numbers are shown in Figures 8, 10, and 11. Line number Drillhole number Depth of the magnetite ore Total thickness of the magnetite ore

Line 1 1 349 m 87 m

2 210 m 62 m

Line 2 3 224 m 35 m

1 445 m 4m

2 236 m 16 m

Line 3 1 242 m 30 m

3 313 m 56 m

Line 4 4 314 m 35 m

5 435 m 1m

Line 5 2 243 m 58 m

Zhang et al.

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the curvature method. H. Zhang expresses his sincere appreciation for the detailed suggestions and valuable comments provided by M. Lee. Many thanks go to R. Lane, R. Smith, S. Goussev, and other anonymous reviewers for their insightful comments that improved the manuscript. We thank associate editor Barbosa for her valuable comments and insights. This work was in part financially supported by the Fundação de Amparao à Pesquisa do Estado de São Paulo (2012/00593-9), China Postdoctoral Science Foundation (2011M501257), Geological exploration project (12120113101800), Hubei Provincial Natural Science Foundation, China (2011CDA123), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUG120815, CUG120501, CUG120116, and CUG130103). D. Ravat thanks the U.S. National Science Foundation for partial support through award EAR-1246921.

APPENDIX A HOW THE NAV-EDGE SIGN CHANGE DETECTS EDGES

i¼3 X ¯ ½ðfðiÞ − fÞQðiÞ i¼1

¼

hP

i

i¼3 i¼1 ðfðiÞQðiÞÞ

− f¯

1 ¼ p1 t1 þ p0 t0 þ p2 t2 − ðp1 þ p0 þ p2 Þ 3    P2 1 1 ¼ 2 p0 − 2 k¼1 pk 3 − t1   ¼ ððp0 − p1 Þ − ðp2 − p0 ÞÞ 13 − t1 :

p5 f ¼ 4 p4 p8

p1 p0 p3

3 p6 p2 5: p7

t1 t0 t1

3 t2 t1 5 . t2

(A-3)

P Once again, we set Q ¼ 1 to simplify the illustration. Then, we obtain t0 ¼ 1 − 4t1 − 4t2 and t0 > ð1∕9Þ. The numerator in equation 11 is



i;j¼3 X

¯ ½ðfði; jÞ − fÞQði; jÞ

i;j¼1

¼



 Pi;j¼3

i;j¼1 ½fði; jÞQði; jÞ

− f¯

P P ¼ t1 4k¼1 pk þ t2 8k¼5 pk þ t0 p0   P P − 19 p0 þ 19 4k¼1 pk þ 19 8k¼5 pk     X 1 4 t0 − 19 p0 þ 4t1 − 49 p 4 k¼1 k   8 4 1X þ 4t2 − p: 9 4 k¼5 k

(A-4)

P If P we assume that the average of 14 4k¼1 pk is equal to the average 1 8 of 4 k¼5 pk , equation A-4 can be rewritten as

    4 1 8 1X T ¼ t0 − p0 þ 4t1 þ 4t2 − p 9 9 4 k¼1 k    P4 1 1 ¼ p0 − 4 k¼1 pk t0 − 9 :

(A-5)

Recall that θ ¼ 0, so

p4 ¼ p0 . p2 ¼ p0 (A-1)

Because p0 − p1 and p2 − p0 denote the finite differences, equation A-1 shows that the sign of the NAV value is controlled by the derivative of f. When p0 − p1 ¼ p2 − p0 , the window center is located at the maximum of the derivative of f, which is associated with the source edge in the method. Second, we use a window size of M ¼ 3 by M ¼ 3 to describe to examine equation 11 (the 2D map form). Setting

2

t2 Q ¼ 4 t1 t2

¼

The sign of the NAV function is controlled by the numerator. For illustration, first, we use a window length of M ¼ 3 points to examine equation 12 (the profile case). Setting up f ¼ fp1 ; p0 ; p2 g, where p1 , p0 , and p2 are the potential-field values, corresponding to the P Gaussian coefficients Q ¼ ft1 ; t0 ; t2 g, we additionally set Q ¼ 1 to simplify the illustration. Considering the symmetry of Gaussian function in equation 13, we have t1 ¼ t2 and t0 > t1 , so t1 < ð1∕3Þ. The numerator of the NAV function in equation 12 is



2

(A-6)

Combining equations A-5 and A-6, we obtain

    1X 1 1 T ¼ p0 − p t − 2 k¼1;3 k 2 0 9   1 ¼ ððp0 − p1 Þ − ðp3 − p0 ÞÞ t0 − 9 :

(A-7)

Similar to equation A-1, equation A-7 shows that the sign of the NAV value is controlled by the derivative of f. When p0 − p1 ¼ p3 − p0 , the window center is located at the maximum of the derivative of f, which is taken to be the source edge in the method.

(A-2)

Setting θ ¼ 0 and cof ¼ 1, the Gaussian coefficients can be written as

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