TY - JOUR
AU1 - Karimi,, Kurosh
AU2 - Shirzaditabar,, Farzad
AB - Abstract The analytic signal of magnitude of the magnetic field’s components and its first derivatives have been employed for locating magnetic structures, which can be considered as point-dipoles or line of dipoles. Although similar methods have been used for locating such magnetic anomalies, they cannot estimate the positions of anomalies in noisy states with an acceptable accuracy. The methods are also inexact in determining the depth of deep anomalies. In noisy cases and in places other than poles, the maximum points of the magnitude of the magnetic vector components and Az are not located exactly above 3D bodies. Consequently, the horizontal location estimates of bodies are accompanied by errors. Here, the previous methods are altered and generalized to locate deeper models in the presence of noise even at lower magnetic latitudes. In addition, a statistical technique is presented for working in noisy areas and a new method, which is resistant to noise by using a ‘depths mean’ method, is made. Reduction to the pole transformation is also used to find the most possible actual horizontal body location. Deep models are also well estimated. The method is tested on real magnetic data over an urban gas pipeline in the vicinity of Kermanshah province, Iran. The estimated location of the pipeline is accurate in accordance with the result of the half-width method. analytic signal, magnetic gradient tensor, averaging method, depths mean method 1. Introduction Many methods have been developed to process magnetic data for locating magnetic sources. The analytic signal was introduced to calculate the dip and depth of 2D magnetic sources (Nabighian 1972). The analytic signal of the total magnetic field anomaly is also used to estimate dip, depth, and strike of dikes (Bastani and Pedersen 2001). It has been shown that the maximum value of the analytic signal magnitude from the total field anomaly for a dipolar source is not always located directly at the top of the source (Agarwal and Shaw 1996, Huang et al1997, Salem et al2002). An inversion algorithm based on the absolute value of the total field anomaly for estimating the shape and depth of 2D magnetic sources is also suggested (Stavrev 2001). The use of derivatives of the analytic signal is also shown to be a useful tool for location estimations (Salem and Ravat 2003). After magnetic gradient tensor (MGT) measurements, Pedersen and Rasmussen (1990) showed that the use of MGT data is practical for geophysical prospecting. They discussed in some detail the practical problems encountered in the collection and processing of the MGT. There has been significant progress in developing instrumentation to measure the gradient tensor of the Earth’s magnetic field (Schmidt and Clark 2000, Gamey et al2004, Schmidt and Bracken 2004, Doll et al2006). In a real magnetic survey, in an area with complex geology, long wavelength anomalies due to deeper sources can overcome short wavelength anomalies due to small and superficial sources. Consequently, the separation and exploration of small magnetic objects without additional processing of the measured data is difficult. As we deal with derivatives of magnetic field by MGT measurements, the local variations are amplified and make small, shallow, and weakly magnetic objects detectable. Oruc (2010) analyzed the analytic signals of magnitude of the magnetic field components and their derivatives to determine the locations of simple sources, such as point-dipole and line of dipoles. In Oruc’s paper, the effect of noise is not considered. Oruc emphasized shallow sources rather than deeper ones, and also, in the case of 3D bodies, his approach was accompanied by a remarkable error in horizontal location approximations, especially for deeper bodies, and in middle and low magnetic latitudes. Schneider et al (2013) employed a software tool for the fast and direct inversion of full-tensor data in a superconducting quantum interference device system, which yields high-resolution recording of all components of the Earth’s magnetic field gradient tensor in large land areas. They sought to localize buried magnetic objects, especially dipole-like ones and inhomogeneities. Gang et al (2014) proposed a localization method that uses only MGT data. To measure the quantities required in the localization equations, they designed an MGT system in which finite differences were used to approximate the first- and second-order spatial gradients of the magnetic field components. Guo et al (2015a, 2015b) analyzed the influence of all factors on magnetic anomalies. They established a forward model and performed a series of calculations between model parameters of a pipeline, the geomagnetic field, the measuring trace, and total magnetic anomaly, and examined the influence of model parameters on the shape, amplitude, and magnetic width. Guo et al (2015a, 2015b) took the advantage of the vertical component of the magnetic anomaly and its analytic signal to detect oil field underground pipelines. Based on peaks of the curves, they could find the azimuth and horizontal location of the pipeline. They also analyzed the effect of magnetization, thickness, susceptibility, and depth of the pipeline on horizontal location error. Gang et al (2016) studied the interference with magnetic heading estimation caused by a local near-surface magnetic anomaly and suggested an error compensation method based on the discrete cosine transform. In this paper, as a whole, Oruc’s method is altered for working in noisy areas, deep bodies, and at any latitudes. 2. Mathematical tools In this paper, we deal with analytic signals of magnetic vector components (MVC) and MGT. The magnetic anomaly produced by a magnetic source can be written as (modified from Blakely 1996): ΔF=F.f=(Fxfx+Fyfy+Fzfz)1 where f is the unit vector in the direction of the Earth’s magnetic field, whereas fx, fy, and fz are its direction cosines. Fx, Fy, and Fz are the MVC of the magnetic source along thex,y, and z directions (⁠ F=-∇U, where U is magnetic potential). Since ∣ΔF∣ is much lower than the magnitude of the ambient field, equation (1) is valid (Blakely 1996). As we do not know anything about the magnetization of the magnetic sources in real surveys, and for simplicity, we assume that the magnetization of the magnetic source is completely induced by the Earth’s magnetic field. Accordingly, magnetization vector m, and f have the same direction. The magnitude of magnetic vector components (MMVC) is determined by calculating the square root of the summation of the MVC squares: MMVC=Fx2+Fy2+Fz2.2 MGT is the second-order derivatives of the magnetic potential or directional derivatives of magnetic anomaly components. Its amplitude is called ‘Analytic signal of magnetic gradient tensor’ and is given as follows: (Oruc 2010): ∣Ax∣=Fxx2+Fxy2+Fxz2∣Ay∣=Fyx2+Fyy2+Fyz2∣Az∣=Fzx2+Fzy2+Fzz2.3 3. Expressions for line of dipoles The magnetic potential of a line of magnetic dipoles along y axis, where its length is much longer than the distance between the sensor and the body, at depth z0 with dipole moment per unit length m0 in the direction of the Earth’s field, can be written as (modified from Blakely 1996): U=2m0r⃗.f^r24 where r⃗=(x-x0)x^+(z-z0)z^, and (x0, z0) are the coordinates for the line of dipoles on xz plane. If data are collected on the surface of the Earth, then z = 0 in the above relation. Because we deal with a 2D body in this case, we assume that the traverse direction is perpendicular to the strike (i.e. along x axis). Replacing the first and second derivatives of relation (4) in equations (2) and (3) and doing some manipulations, the depth, z0, will be extracted as (Oruc 2010): z0=2MMVCAzx=x0.5 Relation (5) is independent of magnetization. It is obvious that the calculated depth is accurate in x = x0 and it holds with less accuracy for the closest neighboring points of x0. These adjacent points are so worthwhile and we use them to make the solutions more reliable when noise enters the system. 4. Expressions for point-dipole The scalar magnetic potential of a point-dipole with a dipole moment m0 at a depth of z0 under the plane of observation (z is vertically downward) is given by (modified from Blakely 1996): U=m0r⃗.f^r36 where r⃗=(x-x0)x^+(y-y0)y^+(z-z0)z^. As in the previous section, replacing the first and second derivatives of relation (6) in equations (2) and (3), the depth, z0, which is independent of magnetization, will be calculated as (Oruc 2010): z0=3MMVCAz(x,y)=(x0,y0).7 5. Synthetic models Now, it is time to study two synthetic models to see the applicability of our method and its accuracy to estimate the location of models in the absence and presence of Gaussian random noise. These models are isolated line of dipoles and isolated point-dipoles. Such models are representatives of several geological structures, like spherical bodies or pipelines. Then, we go further and examine two close parallel lines of dipoles ‘A’ and ‘B’, and finally, two neighboring point-dipoles ‘A’ and ‘B’ to see the effect of interfering sources on solutions (see table 1). Table 1. Specifications of different models along with the information of the direction of ambient field. Model . Length and/or width of investigation area . Real location(m) . I . D . Line of dipoles 1 200 m [100, 0, 10] 45° 0° Line of dipoles 2 200 m [100, 0, 20] Line of dipoles 3 200 m [100, 0, 50] Point-dipole 1 200 × 200 m [100, 100, 10] 40° 40° Point-dipole 2 200 × 200 m [100, 100, 20] Point-dipole 3 200 × 200 m [100, 100, 50] Line of dipoles A 200 m [85, 0, 15] 60° 0° Line of dipoles B [115, 0, 12] Point-dipole A 200 × 200 m [85, 90, 15] 60° 0° Point-dipole B [110, 110, 12] Model . Length and/or width of investigation area . Real location(m) . I . D . Line of dipoles 1 200 m [100, 0, 10] 45° 0° Line of dipoles 2 200 m [100, 0, 20] Line of dipoles 3 200 m [100, 0, 50] Point-dipole 1 200 × 200 m [100, 100, 10] 40° 40° Point-dipole 2 200 × 200 m [100, 100, 20] Point-dipole 3 200 × 200 m [100, 100, 50] Line of dipoles A 200 m [85, 0, 15] 60° 0° Line of dipoles B [115, 0, 12] Point-dipole A 200 × 200 m [85, 90, 15] 60° 0° Point-dipole B [110, 110, 12] Open in new tab Table 1. Specifications of different models along with the information of the direction of ambient field. Model . Length and/or width of investigation area . Real location(m) . I . D . Line of dipoles 1 200 m [100, 0, 10] 45° 0° Line of dipoles 2 200 m [100, 0, 20] Line of dipoles 3 200 m [100, 0, 50] Point-dipole 1 200 × 200 m [100, 100, 10] 40° 40° Point-dipole 2 200 × 200 m [100, 100, 20] Point-dipole 3 200 × 200 m [100, 100, 50] Line of dipoles A 200 m [85, 0, 15] 60° 0° Line of dipoles B [115, 0, 12] Point-dipole A 200 × 200 m [85, 90, 15] 60° 0° Point-dipole B [110, 110, 12] Model . Length and/or width of investigation area . Real location(m) . I . D . Line of dipoles 1 200 m [100, 0, 10] 45° 0° Line of dipoles 2 200 m [100, 0, 20] Line of dipoles 3 200 m [100, 0, 50] Point-dipole 1 200 × 200 m [100, 100, 10] 40° 40° Point-dipole 2 200 × 200 m [100, 100, 20] Point-dipole 3 200 × 200 m [100, 100, 50] Line of dipoles A 200 m [85, 0, 15] 60° 0° Line of dipoles B [115, 0, 12] Point-dipole A 200 × 200 m [85, 90, 15] 60° 0° Point-dipole B [110, 110, 12] Open in new tab 5.1. Locating a line of dipoles It is assumed that the inclination and declination of Earth’s magnetic field are 45° and 0°, respectively, and the traverse direction is N–S. The magnetization is completely induced by the ambient field. A line of dipoles is an example of 2D bodies in which the length of the line is much longer than the distance between it and the magnetic sensor. An interval of 1 m is used for calculating the data. Besides, different depths are used for the model to realize the accuracy of the method. Accordingly, three separate lines of dipoles are examined. The coordinates of the center of lines of dipoles ‘1’, ‘2’, and ‘3’ are (100, 0, 10 m), (100, 0, 20 m), and (100, 0, 50 m), respectively, and the length of the profile is 200 m. The strike directions of all of the line of dipoles are in the E–W direction and the traverse direction is fitted on the horizontal component of the total magnetic field. The derivatives of magnetic field anomaly are all calculated in the Fourier domain using the fast Fourier transform by using MATLAB software. To avoid the Gibbs phenomenon in detecting deeper bodies, the data should be tapered with a linear equation (Blakely 1996). The number of tapered data is three times greater than the number of data at half maximum of reduction to the pole (RTP) field’s full-width and it is stated as a percentage of the number of main data. The percentage of tapered data for the line of dipoles ‘1’ is zero, in which the magnetic field begins from a ground value and eventually ends at it. For lines of dipoles ‘2’ and ‘3’, the percentages of tapered data are 30% and 81%, respectively. Random Gaussian noise for each source is considered respectively 0% and 15% of the main data (see table 2). In the case of line of dipoles, the maximum of MMVC and Az lie right above the body and the amounts of I and D cannot affect them. When noise exists in the data, the maximum of MMVC and Az curves might undergo displacement from the top of the bodies. If this displacement occurs, increasing the error is inevitable and we are not able to locate exactly the actual horizontal locations of the bodies. Moreover, these curves face perturbations and as a result, the numerator and denominator of equation (5) will vary. Consequently, the estimated depths are subject to variation and they lose their accuracy. To compensate perturbation effects, we use a ‘moving five points averaging’ method and reinstruct the total magnetic field curves (Oruc 2010). However, the disruptive effect of noise still remains and it is just partially reduced. Since we do not know the real horizontal location of the body (because noise probably shifts the maximum point of MMVC and Az curves), and also because the noise changes the magnitude of curves at their maximum points, we must not restrict ourselves to a single point for approximation of the depth in relation (5). Another step to decrease the probable error due to horizontal shifting of curves is to take an average over the calculated depths from the maximum point and its four surrounding points, that is, two points on either side (5-depths mean). Note that this average is taken over five depths and differs from the ‘moving five points averaging’ method, which is employed to reduce the disturbing effect of noise over the data. To determine normalized RMS error (NRMSE) for line of dipoles, in all solutions we have used the following equation: NRMSE=1z0(x0-est-x0)2+(z0-est-z0)2n×100%8 where (x0–est, z0–est) are the estimated locations for line of dipoles and (x0, z0) are their real coordinates The number of error elements in each case is n, which in this case is 2. The worth of our ‘5-depths mean’ method is more evident in noisy, deeper, and lower latitude cases. By using this technique, the estimated depths of models greatly improve and the effect of noise decreases remarkably. To show the efficiency of our method, the calculated depths from merely the maximum point of MMVC (Oruc’s method) as well as the calculated depths via the ‘5-depths mean’ method (i.e. our method) are reported in table 2. One can see that in the noisy case, the estimated locations from the single maximum point is severely disturbed, whereas with the ‘5-depths mean’ method it is clear that the estimated locations are still acceptable and the horizontal errors in estimations, which were not considered by Oruc (2010), contribute to NRMSEs (see table 2). The characteristics of line of dipoles ‘3’ with the actual location of (100, 0, 50 m) is indicated in table 1. In the noisy state, with the first type of solution (Oruc method) it is clear that line of dipoles ‘3’ is (99, 0, 37.25 m) with the NRMSE of 18.08%, while this estimated location from the second method, i.e. our method, is (99, 0, 47.83 m) with the NRMSE of 4.78%. To see the solutions for the lines of dipoles ‘1’ and ‘2’, refer to table 2. Figures 1(a) and (b) show the MMVC and Az curves for line of dipoles ‘2’ in the absence of noise, while figures 1(c) and (d) illustrate them in the presence of 15% noise. The reason that estimated locations in noiseless cases are not accurate is that the MMVC and Az curves are obtained from Fourier transform of the data, not direct calculations from potential field relation. Table 2. Estimated locations in conjunction with NRMSEs for all three separate lines of dipoles and a binary system of line of dipoles. . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Percentage of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Line of dipoles 1 0 Max point [100, 0, 9.92] [99, 0, 8.72] 0.56 11.48 5-depths mean [100, 0, 10.12] [99, 0, 10.54] 0.85 8.04 Line of dipoles 2 30 Max point [100, 0, 19.64] [99, 0, 5.77] 1.27 50.43 5-depths mean [100, 0, 19.69] [99, 0, 19.89] 1.10 3.56 Line of dipoles 3 81 Max point [100, 0, 47.47] [99, 0, 37.25] 3.57 18.08 5-depths mean [100, 0, 47.49] [99, 0, 47.83] 3.55 4.78 Line of dipoles A 18 5-depths mean [82, 0, 15.10] [80, 0, 14.98] 14.15 23.59 Line of dipoles B [116, 0, 11.76] [115, 0, 11.54] 6.06 2.71 . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Percentage of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Line of dipoles 1 0 Max point [100, 0, 9.92] [99, 0, 8.72] 0.56 11.48 5-depths mean [100, 0, 10.12] [99, 0, 10.54] 0.85 8.04 Line of dipoles 2 30 Max point [100, 0, 19.64] [99, 0, 5.77] 1.27 50.43 5-depths mean [100, 0, 19.69] [99, 0, 19.89] 1.10 3.56 Line of dipoles 3 81 Max point [100, 0, 47.47] [99, 0, 37.25] 3.57 18.08 5-depths mean [100, 0, 47.49] [99, 0, 47.83] 3.55 4.78 Line of dipoles A 18 5-depths mean [82, 0, 15.10] [80, 0, 14.98] 14.15 23.59 Line of dipoles B [116, 0, 11.76] [115, 0, 11.54] 6.06 2.71 Open in new tab Table 2. Estimated locations in conjunction with NRMSEs for all three separate lines of dipoles and a binary system of line of dipoles. . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Percentage of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Line of dipoles 1 0 Max point [100, 0, 9.92] [99, 0, 8.72] 0.56 11.48 5-depths mean [100, 0, 10.12] [99, 0, 10.54] 0.85 8.04 Line of dipoles 2 30 Max point [100, 0, 19.64] [99, 0, 5.77] 1.27 50.43 5-depths mean [100, 0, 19.69] [99, 0, 19.89] 1.10 3.56 Line of dipoles 3 81 Max point [100, 0, 47.47] [99, 0, 37.25] 3.57 18.08 5-depths mean [100, 0, 47.49] [99, 0, 47.83] 3.55 4.78 Line of dipoles A 18 5-depths mean [82, 0, 15.10] [80, 0, 14.98] 14.15 23.59 Line of dipoles B [116, 0, 11.76] [115, 0, 11.54] 6.06 2.71 . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Percentage of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Line of dipoles 1 0 Max point [100, 0, 9.92] [99, 0, 8.72] 0.56 11.48 5-depths mean [100, 0, 10.12] [99, 0, 10.54] 0.85 8.04 Line of dipoles 2 30 Max point [100, 0, 19.64] [99, 0, 5.77] 1.27 50.43 5-depths mean [100, 0, 19.69] [99, 0, 19.89] 1.10 3.56 Line of dipoles 3 81 Max point [100, 0, 47.47] [99, 0, 37.25] 3.57 18.08 5-depths mean [100, 0, 47.49] [99, 0, 47.83] 3.55 4.78 Line of dipoles A 18 5-depths mean [82, 0, 15.10] [80, 0, 14.98] 14.15 23.59 Line of dipoles B [116, 0, 11.76] [115, 0, 11.54] 6.06 2.71 Open in new tab Figure 1. Open in new tabDownload slide Curves for line of dipoles ‘2’. (a) Az curve in noiseless state; (b) MMVC in noiseless state; (c) Az curve in noisy state (15%); (d) MMVC curve in noisy state (15%). Maximum of the MMVC curves are represented by red squares along with their relevant coordinates. Figure 1. Open in new tabDownload slide Curves for line of dipoles ‘2’. (a) Az curve in noiseless state; (b) MMVC in noiseless state; (c) Az curve in noisy state (15%); (d) MMVC curve in noisy state (15%). Maximum of the MMVC curves are represented by red squares along with their relevant coordinates. 5.2. Locating a point-dipole A point-dipole is an example of a 3D body. Since for 3D bodies the inclination of Earth’s field can shift the maximum point of MMVC and Az maps, particularly in the middle and low magnetic latitudes, the horizontal positions of maximum of MMVC and Az maps are shifted from the top of the source. This displacement differs from the horizontal shifting due to the noise in the system. This horizontal displacement error in location estimation increases by increasing the depth of the body and decreasing the inclination of Earth’s field. Therefore, specifically in this case, Oruc’s method is challenged and the estimations possess excessive errors. In the middle and low magnetic latitudes, the maximum points of the MMVC and Az maps are shifted and we cannot estimate the true horizontal location of the body, and consequently, (x0, y0) are not the true ones. In addition to horizontal location, the calculated depth undergoes a high error. Now, if the dipole source is deeper or at a lower magnetic latitude or both, particularly when the data are noisy, the problem becomes more and more complex. The first step to find the best estimation for horizontal location is to reduce the total magnetic field to the pole (RTP) (see the appendix). As a result, the maximum of the RTP contour map lies right above the body and the horizontal position of the source is determined. Nevertheless, noise still might shift the maximum point of the RTP field. This shift is much smaller than the displacement due to being at the middle or low magnetic latitudes. As already mentioned, it is unavoidable. By applying the RTP technique, the horizontal location of the body is very well specified, and then the use of our method increases its accuracy. Now, it is possible to apply it in a wide range of magnetic latitudes, even though the body is rather deep. For studying our synthetic models, we assume that both the inclination and declination of the ambient field are 40°. The magnetization is completely induced by the ambient field. The cell size of 1 × 1 m is used for computing the data. The dimensions of the grid map are 200 × 200 m. As in the preceding section, we examine three separate models (three point-dipoles) in different depths: point-dipole ‘1’ in (100, 100, 10 m), point-dipole ‘2’ in (100, 100, 20 m), and point-dipole ‘3’ in (100, 100, 50 m). The number of tapered data in x andy directions are evenly 0% for point-dipole ‘1’, 30% for point-dipole ‘2’, and 90% for point-dipole ‘3’ (it is three times greater than the number of data at half maximum of the RTP field’s full-width and is stated in terms of percentage). We study the estimated source locations in the absence and presence of 15% random Gaussian noise. To reduce the effect of noise, we use the ‘moving 25 point averaging’ method (maximum point of MMVC and its 24 close surrounding points). As can be seen in table 3, there are two types of calculations. One which is obtained with the aid of an RTP contour map and the use of the ‘25-depths mean’ method (an average over 25 depths), which is shown in table 3 with ‘RTP & 25-depths mean’ (this average is taken over 25 depths, i.e. the depth calculated from MMVC and Azof the estimated horizontal position and its 24 surrounding points. Note that it is different from the ‘moving 25 point averaging’ method, which is used to damp the effect of noise on the data). The other type is represented by the expression ‘MMVC & max point method’, which means that MMVC contours are used for locating the horizontal locations and we just use the maximum point of MMVC (Oruc’s method) for finding depth. The estimated locations for dipoles ‘1’ and ‘2’ in the presence of 15% noise by the ‘MMVC & max point’ method are (99, 99, 12.43 m) and (97, 97, 20.74 m), with the NRMSE of 16.23% and 12.43%, respectively, while the calculated depths from the ‘RTP & 25-depths mean’ method for dipoles ‘1’ and ‘2’ are (100, 100, 11.10 m) and (100, 100, 20.16 m) with the NRMSE of 6.35% and 0.46%, respectively. The difference between these two types of solutions is obvious. The reason that the second solution is much better than the first one is twofold: (1) in place of using the maximum of the MMVC contour, we used the maximum of the RTP field’s contour, which is much closer to the model, and (2) we applied, first, the ‘moving 25 points averaging’ method over the data to compensate the effect of noise and second, the averaging of 25 depths instead of just one depth (25-depths mean). As in the previous section, the reason for the high total errors is that there are horizontal errors, not vertical ones (equation (9)). Figures 2(a) and (b) show the MMVC and RTP grid maps for point-dipole ‘3’ in the noisy state (15%). In both figures, the white points are representatives of estimated horizontal locations and the black ones show the real horizontal locations of point-dipole ‘3’. The figures show that RTP contour is so useful for finding the horizontal location of a body and it gives horizontal location of the source in both x and y directions, while there is 7 m displacement in both x and y directions in the case of using the MMVC contour. Consequently, it is normal that the estimated depth using the maximum of the MMVC contour gives a high NRMSE, which is calculated by the following relation: NRMSE=1z0×(x0-est-x0)2+(y0-est-y0)2+(z0-est-z0)2n×100%9 where (x0-est, y0-est, z0-est) are the estimated locations for a point-dipole and (x0, y0, z0) are its real coordinates. The number of error elements is n, which in this case is 3. Table 3. Estimated locations in conjunction with NRMSEs for all three separate point-dipoles and a binary system of point-dipoles. . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Number of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Point-dipole 1 0 RTP & 25-depths mean [100, 100, 10.32] [100, 100, 11.10] 1.85 6.35 MMVC & max point [99, 99, 10.23] [99, 99, 12.43] 8.27 16.23 Point-dipole 2 30 RTP & 25-depths mean [100, 100, 19.98] [100, 100, 20.16] 0.18 0.46 MMVC & max point [97, 98, 20.22] [97, 97, 20.74] 10.43 12.43 Point-dipole 3 90 RTP & 25-depths mean [100,100,49.61] [100, 100, 47.68] 0.45 2.68 MMVC & max point [93, 94, 49.92] [93, 93, 36.25] 10.64 19.56 Point-dipole A 24 RTP & 25-depths mean [85, 90, 15.37] [85, 90, 15.60] 1.42 2.31 Point-dipole B [110, 110, 12.25] [110, 110, 12.31] 1.20 1.49 . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Number of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Point-dipole 1 0 RTP & 25-depths mean [100, 100, 10.32] [100, 100, 11.10] 1.85 6.35 MMVC & max point [99, 99, 10.23] [99, 99, 12.43] 8.27 16.23 Point-dipole 2 30 RTP & 25-depths mean [100, 100, 19.98] [100, 100, 20.16] 0.18 0.46 MMVC & max point [97, 98, 20.22] [97, 97, 20.74] 10.43 12.43 Point-dipole 3 90 RTP & 25-depths mean [100,100,49.61] [100, 100, 47.68] 0.45 2.68 MMVC & max point [93, 94, 49.92] [93, 93, 36.25] 10.64 19.56 Point-dipole A 24 RTP & 25-depths mean [85, 90, 15.37] [85, 90, 15.60] 1.42 2.31 Point-dipole B [110, 110, 12.25] [110, 110, 12.31] 1.20 1.49 Open in new tab Table 3. Estimated locations in conjunction with NRMSEs for all three separate point-dipoles and a binary system of point-dipoles. . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Number of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Point-dipole 1 0 RTP & 25-depths mean [100, 100, 10.32] [100, 100, 11.10] 1.85 6.35 MMVC & max point [99, 99, 10.23] [99, 99, 12.43] 8.27 16.23 Point-dipole 2 30 RTP & 25-depths mean [100, 100, 19.98] [100, 100, 20.16] 0.18 0.46 MMVC & max point [97, 98, 20.22] [97, 97, 20.74] 10.43 12.43 Point-dipole 3 90 RTP & 25-depths mean [100,100,49.61] [100, 100, 47.68] 0.45 2.68 MMVC & max point [93, 94, 49.92] [93, 93, 36.25] 10.64 19.56 Point-dipole A 24 RTP & 25-depths mean [85, 90, 15.37] [85, 90, 15.60] 1.42 2.31 Point-dipole B [110, 110, 12.25] [110, 110, 12.31] 1.20 1.49 . . . Estimated location (m) . NRMSE in estimated location (%) . Model . Number of tapered data (%) . Method . No noise . With 15% noise . No noise . With 15% noise . Point-dipole 1 0 RTP & 25-depths mean [100, 100, 10.32] [100, 100, 11.10] 1.85 6.35 MMVC & max point [99, 99, 10.23] [99, 99, 12.43] 8.27 16.23 Point-dipole 2 30 RTP & 25-depths mean [100, 100, 19.98] [100, 100, 20.16] 0.18 0.46 MMVC & max point [97, 98, 20.22] [97, 97, 20.74] 10.43 12.43 Point-dipole 3 90 RTP & 25-depths mean [100,100,49.61] [100, 100, 47.68] 0.45 2.68 MMVC & max point [93, 94, 49.92] [93, 93, 36.25] 10.64 19.56 Point-dipole A 24 RTP & 25-depths mean [85, 90, 15.37] [85, 90, 15.60] 1.42 2.31 Point-dipole B [110, 110, 12.25] [110, 110, 12.31] 1.20 1.49 Open in new tab Figure 2. Open in new tabDownload slide Grid map for point-dipole ‘3’. (a) MMVC in noisy state (15%); (b) RTP in noisy state (15%). Black spots are the real positions of models, while the white spots are the estimated horizontal locations that are also the maximum points of the grid maps. Figure 2. Open in new tabDownload slide Grid map for point-dipole ‘3’. (a) MMVC in noisy state (15%); (b) RTP in noisy state (15%). Black spots are the real positions of models, while the white spots are the estimated horizontal locations that are also the maximum points of the grid maps. 5.3. Interfering sources In this section, at first, we analyze two parallel lines of dipoles, which are in the neighborhood of E–W strikes. Central location of line of dipoles ‘A’ is in (85, 0, 15 m) and the center of line of dipoles ‘B’ is in (115, 0, 12 m) (see table 1 and figure 3(a)). The inclination and declination of Earth’s field in this situation are 60° and 0°, respectively. In this case, the data are tapered with respect to the deeper body, ‘A’, which is 18%. By using our method, the estimated location for line of dipoles ‘B’ in the absence of noise is (116, 0, 11.76 m) with the NRMSE of 6.06% (note that the dominant portion of this total error belongs to horizontal error, not vertical). In the presence of noise, it will be (115, 0, 11.54 m) with the NRMSE of 2.71%. The interfering effect has caused 1 m shifting in the horizontal location for line of dipoles ‘B’ in the noiseless state. The good result for the noisy state is obvious. This means that although noise exists in the system, our ‘5-depths mean’ method annuls the disturbing effect of both interference and noise. To see the calculated locations for line of dipoles ‘A’, refer to table 2. Figures 4(a) and (b) show the Az and MMVC curves for this binary system shown in figure 3(a) in the presence of 15% random noise. The reason for maximum point displacement of MMVC curve over body ‘A’ is the effect of interference and noise together. In the last step in the examination of the synthetic models, we consider a binary system of point-dipoles close together with coordinates of (85, 90, 15 m) and (110, 110, 12 m) in a grid map for point-dipoles ‘A’ and ‘B’, respectively (table 1 and figure 3(b)). It should be noted that these sources can have a mutual affect on each other. The closer the sources, the more the interfering effects. The data are tapered with regard to the contour of the deeper model, ‘A’, which is 24%. As can be seen in table 3, neither the noise nor interfering effects of the sources can influence the maximums of the contour maps. It is clear that noise and interfering effects of the sources, together partially deviate the estimated depths from the true ones, e.g. the estimated location for point-dipole ‘B’, without noise, is (110, 110, 12.25 m) with the NRMSE of 1.20%, while this location is estimated in the presence of 15% noise as (110, 110, 12.31 m). NRMSE in this case is 1.49%. Figure 5 indicates the RTP field for this binary system. As before, the black spots are representative of true horizontal locations and white ones show the estimated horizontal locations. Under this condition, one can see that these black and white spots coincide. Despite the fact that there is noise in the data, it is commendable that the solutions are useful and their accuracy is acceptable. Figure 3. Open in new tabDownload slide (a) Binary system of lines of dipoles (‘A’ and ‘B’); (b) binary system of point-dipoles (‘A’ and ‘B’). The centers of the models are indicated by white squares alongside their relevant coordinates. Figure 3. Open in new tabDownload slide (a) Binary system of lines of dipoles (‘A’ and ‘B’); (b) binary system of point-dipoles (‘A’ and ‘B’). The centers of the models are indicated by white squares alongside their relevant coordinates. Figure 4. Open in new tabDownload slide Curves for the binary system of lines of dipoles ‘A’ and ‘B’ in a noisy state (15%). (a) Az; (b) MMVC. Maximum of the MMVC curves are represented with black squares alongside their relevant coordinates. Figure 4. Open in new tabDownload slide Curves for the binary system of lines of dipoles ‘A’ and ‘B’ in a noisy state (15%). (a) Az; (b) MMVC. Maximum of the MMVC curves are represented with black squares alongside their relevant coordinates. Figure 5. Open in new tabDownload slide RTP grid map for the binary system of point-dipoles (‘A’ and ‘B’) in noisy state (15%). Black spots are the real positions of models, while the white spots are the estimated horizontal locations, which are the maximum points of grid maps too. One can see that both black and white spots coincide. Figure 5. Open in new tabDownload slide RTP grid map for the binary system of point-dipoles (‘A’ and ‘B’) in noisy state (15%). Black spots are the real positions of models, while the white spots are the estimated horizontal locations, which are the maximum points of grid maps too. One can see that both black and white spots coincide. 6. Field example In this section, we study the validity of our method over a too long buried urban gas pipe around Kermanshah, Iran. The inclination and declination of the total magnetic field are 52° 30́ and 4° 33́, respectively. Inasmuch as we have a long profile and shallow buried body, tapering is not necessary. Since the strike direction of this pipe is roughly N–S, we collected the data in an E–W direction. The height of the magnetometer from the surface is 1.85 m. The profile length is 30 m and the distance between the data points is 0.5 m. Figure 6 shows the measured total magnetic field ‘F’, computed horizontal field ‘Fx’, vertical field ‘Fz’, second derivatives ‘Fxx’ and ‘Fxz’, ‘MMVC’ and ‘Az’ curves. The maximum point of the MMVC curve occurs at 16 m from the beginning of the profile. This means that the gas pipe is laid at x = 16 m and digging evidence in the area confirms the correctness of this distance. The estimated depth by our method is 2.87 m (note that this depth is the distance from the magnetometer sensor to the pipe). On the other hand, the calculated depth from full-width of the RTP field at half maximum, H (Telford et al1990), is 2.94 m (figure 7). This value equals the depth of the pipe. By comparison of these two depths, one can see that our method estimates the location of the pipe very well. Figure 6. Open in new tabDownload slide (a) Measured total magnetic field, F, in an E–W direction over a buried gas pipe around Kermanshah, Iran; (b) calculated horizontal magnetic field from total magnetic field, i.e. ‘Fx’; (c) calculated vertical magnetic field from total magnetic field, i.e. ‘Fz’; (d) first derivatives ‘Fxz’ and ‘Fxx’; (e) computed ‘MMVC’ curve from magnetic components. Note that the maximum point of the curve occurs at x = 16 m; (f) computed Az curve using first derivatives. Figure 6. Open in new tabDownload slide (a) Measured total magnetic field, F, in an E–W direction over a buried gas pipe around Kermanshah, Iran; (b) calculated horizontal magnetic field from total magnetic field, i.e. ‘Fx’; (c) calculated vertical magnetic field from total magnetic field, i.e. ‘Fz’; (d) first derivatives ‘Fxz’ and ‘Fxx’; (e) computed ‘MMVC’ curve from magnetic components. Note that the maximum point of the curve occurs at x = 16 m; (f) computed Az curve using first derivatives. Figure 7. Open in new tabDownload slide Computed RTP field. The full-width at half of maximum, H, equals the depth of the pipe. This value is 2.94 m. Maximum point and full-width at half of maximum of the RTP field are marked by a red square and red line, respectively. Figure 7. Open in new tabDownload slide Computed RTP field. The full-width at half of maximum, H, equals the depth of the pipe. This value is 2.94 m. Maximum point and full-width at half of maximum of the RTP field are marked by a red square and red line, respectively. 7. Conclusion Due to the nature of Earth’s magnetic field, the locations of MMVC and Az, in 3D cases, are not directly over the anomalous body. The existence of noise on the data may increase this displacement. Employing a ‘moving 5 or 25 point averaging’ method can decrease the turbulent influence of noise. In 2D cases, we solved this problem by using an average of estimated depths from the maximum point of the MMVC curve and its four neighboring points (5-depths mean). This method really works and lessens the high error at depth, which was just estimated by using the maximum point of MMVC. In 3D cases, when we are involved with grid maps, true horizontal location of the maximum of the MMVC undergoes horizontal shifting and it does not lie over the top of the body, especially at lower latitudes and for deeper models. Therefore, using the location of the maximum of the MMVC in this state yields a depth of the body with considerable error. To solve this problem, we used the RTP map for finding the horizontal location and used the average of 25 estimated depths for adjacent points around the estimated horizontal location (25-depths mean). Testing the method on synthetic data showed better results than the previous work. Applying the method on real data yielded a result that was close to the result from a known half-width method. Appendix RTP transformation Positive gravity anomalies tend to be located over mass concentrations, but the same is not necessarily true for magnetic anomalies when the magnetization and ambient field are both not directed vertically. That is to say, the maximum of the MMVC curves are displaced over their relevant masses, unless the m and f are both vertical, i.e. in the poles. Consequently, we need a tool to give us the true horizontal location of a magnetic mass. The RTP of magnetic field can serve this purpose. The relation between Fourier transforms of the RTP field and the field in any other place on the Earth is (modified from Blakely 1996): (FRTP)=(ψRTP)(F)A where (FRTP) and ℱ(F) are the Fourier transformation of the RTP magnetic field and anomalous magnetic field, respectively. (ψRTP) is the RTP operator (in Fourier domain) and equals to (modified from Blakely 1996): (ψRTP)=1(θm)(θf) in which θm=mz+imxkx+myky∣k∣ and θf=fz+ifxkx+fyky∣k∣.θm is the phase factor of magnetization, m, and θf is the phase factor of induced field, f. The wave number k is expressed as k = (kx, ky) where ∣k∣=kx2+ky2. In order to attain the RTP field, FRTP, we must take inverse Fourier transform from relation (A). By doing this, we have: FRTP=-1[(ψRTP)(F)].B References Agarwal B N P , Shaw R K . , 1996 Comment on ‘An analytic signal approach to the interpretation of total field magnetic anomalies’ by Shuang Qin , Geophys. 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TI - Using the ratio of the magnetic field to the analytic signal of the magnetic gradient tensor in determining the position of simple shaped magnetic anomalies
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2140/aa68bb
DA - 2017-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/using-the-ratio-of-the-magnetic-field-to-the-analytic-signal-of-the-Okpt4mQe0v
SP - 769
VL - 14
IS - 4
DP - DeepDyve
ER -