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An approach for estimating the magnetization direction of magnetic anomalies
Journal of Applied Geophysics 137 (2017) 1–7
Contents lists available at ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
An approach for estimating the magnetization direction of magnetic anomalies Jinpeng Li ⁎, Yingtang Zhang, Gang Yin, Hongbo Fan, Zhining Li Seventh Department, Mechanical Engineering College, Shijiazhuang 050003, People's Republic of China
a r t i c l e
i n f o
Article history: Received 16 May 2016 Received in revised form 5 December 2016 Accepted 9 December 2016 Available online 11 December 2016 Keywords: Magnetization direction Normalized source strength Reduced-to-the-pole Remanent magnetization Cross-correlation
a b s t r a c t An approach for estimating the magnetization direction of magnetic anomalies in the presence of remanent magnetization through correlation between normalized source strength (NSS) and reduced-to-the-pole (RTP) is proposed. The observation region was divided into several calculation areas and the RTP field was transformed using different assumed values of the magnetization directions. Following this, the cross-correlation between NSS and RTP field was calculated, and it was found that the correct magnetization direction was that corresponding to the maximum cross-correlation value. The approach was tested on both simulated and real magnetic data. The results showed that the approach was effective in a variety of situations and considerably reduced the effect of remanent magnetization. Thus, the method using NSS and RTP is more effective compared to other methods such as using the total magnitude anomaly and RTP. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Magnetic surveys are widely used in different fields, including mining applications, oil and gas exploration, and mapping bedrock topography, among others. In magnetic prospecting, knowledge of the correct magnetization direction of magnetic anomalies is important for calculation of the RTP field, as well as forward calculation and inversion (Li et al., 2010). The total magnetization in a source body is the vector sum of the induced and remanent magnetization. In some cases, the induced magnetization aligns with the direction of earth's ambient field, without remanent magnetization and self-demagnetization effects. However, in most cases, the preexisting remanent magnetization is strong enough to affect the true magnetization direction (Liu et al., 2013, 2015) leading to the erroneous interpretation of magnetic data. Therefore, in recent years, there has been an increasing focus on reducing, or even removing the effects of remanent magnetization in the estimation of magnetization direction. The problem of determining the total magnetization direction has long been of interest in the field of magnetic interpretation. Roest and Pilkington (1993) estimated the magnetization direction by comparing the amplitude of the analytic and horizontal gradient of pseudogravity. Medeiros and Silva (1995) estimated the total magnetization direction using the source moments up to second order derived from the multi⁎ Corresponding author at: The Seventh Department of Mechanical Engineering College, No.97, Hepingxilu Road, Shi Jia-zhuang, He Bei province, PR China. E-mail address: [email protected] (J. Li).
http://dx.doi.org/10.1016/j.jappgeo.2016.12.009 0926-9851/© 2016 Elsevier B.V. All rights reserved.
pole expansion of the magnetic potential. Phillips (2005) proposed direct and indirect algorithms that implemented Helbig's (1963) integrals for estimating the magnetization direction from the first order magnetic moments. Dannemiller and Li (2006) proposed a method to estimate the magnetization direction of 3-D sources based upon the correlation between the vertical and the total gradients of the reduced-to-thepole field. A similar method was proposed by Gerovska et al. (2009), based on the correlation between the reduced-to-the-pole field and the total magnitude anomaly. Shi et al. (2014) presented the crosscorrelation of magnetic dipole sources for determination of magnetization direction from the total magnetic field anomaly. Oliveira et al. (2015) developed a fast total-field anomaly inversion to estimate the magnetization direction of multiple sources with approximately spherical shapes and known centers. However, the aforementioned techniques all suffer from certain drawbacks which limit their accuracy. These limitations include the results being greatly influenced by noise effects, as well as the total magnitude anomaly being insensitive to the magnetization direction while also influencing the precision of the solution. In the present study, a correlation coefficient analysis was carried out to estimate the magnetization direction of magnetic anomalies through correlation between NSS data and RTP in the presence of remanent magnetization. We chose to use NSS data as it is less sensitive to the magnetization direction compared to other transforms of the magnetic data and relates well to the location of the magnetic source. Hence, it has a stronger capacity to reduce the remanence effect. Additionally, the NSS shows a strong relationship to horizontal projections
2
J. Li et al. / Journal of Applied Geophysics 137 (2017) 1–7
where (M, N) is the grid size, ΔTrtp is the RTP field, Tnss is the NSS data, ΔT rtp is mean value of the RTP fields and T nss is mean value of the NSS data. Since the magnetization direction is unknown from the calculations performed using the aforementioned equations, we assumed that the inclination I changed from − 90∘ to 90∘ and the declination D changed from −180∘ to 180∘, both of which with intervals of 1∘. We calculated the RTP field for a series of assumed values of the magnetization direction. Then, using Eq. (4), the correlation coefficient, C, between the NSS and RTP field was calculated. Just as for the NSS, the RTP has a strong relationship to the horizontal projection of the source when the used magnetization and real magnetization directions are aligned. Therefore, the value of C reflects the degree of cross-correlation between the NSS and the RTP and thus is related to the probability of finding the correct magnetization direction of the source. A lower value means lower accuracy of the estimated magnetization direction; therefore, the maximum value of the correlation coefficient indicates the correct magnetization direction.
of the sources, which makes it a more accurate choice for estimating the magnetization direction. This paper is structured as follows: in Section 2, a new approach for estimating the magnetization direction is described; in Section 3, the results for simulated and real magnetization data are presented. Finally, the conclusions are given in Section 4. 2. Method 2.1. Normalized source strength data In a Cartesian coordinate system, with the x-axis pointing to the geographical east, the y-axis to the north, and the z-axis vertically downwards, the theoretical magnetic gradient tensor data in the observation plane (x, y, z) can be expressed as follows: 2
Bxx G ¼ 4 Byx Bzx
Bxy Byy Bzy
3 Bxz Byz 5; Bzz
ð1Þ
where G is the magnetic gradient tensor matrix and Bαβ(α,β =x, y, z) are the magnetic gradient tensor components. If the eigenvalues of the matrix G are arranged in descending order as λ1 ≥ λ2 ≥ λ3. Then the theoretical normalized source strength data may be expressed as (Zhou and Meng, 2015; Guo et al., 2014; Beiki et al., 2012; Pilkington and Beiki, 2013; Wilson, 1985) uðx; y; zÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −λ2 2 −λ1 λ3 ;
3. Data experiments 3.1. Test on the synthetic magnetic data 3.1.1. Isolated model
ð2Þ
where u(x,y, z) represents the NSS data at an arbitrary station (x,y,z) on the observational surface. 2.2. Correlation coefficient analysis for choosing optimum magnetization direction The RTP turns the measured total magnetic field to anomaly at the north magnetic pole, where the magnetization direction points directly downwards; hence, there is a good correlation between the RTP anomaly and the source horizontal projection. In the presence of remanence, the transformation of RTP requires both the geomagnetic field direction (I0,D0) and the magnetization direction (I, D). The RTP field may be calculated in the frequency domain using the transfer function (Blakely, 1995) H¼
u2 þ v2 ; ½iðu cos I 0 cos D0 þ v cos I 0 sin D0 Þ þ r sin I0 ½iðu cos I cos D þ v cos I sin DÞ þ r sin I
ð3Þ where u is the angular frequency in the x-direction, v is the angular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency in the y-direction, and r ¼ u2 þ v2 . The correlation coefficient C between the NSS data and the RTP fields is defined as M X N X ΔT rtp ði; jÞ−ΔT rtp T nss ði; jÞ−T nss i¼1 j¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; C¼v uM 2 N uX X 2 t ΔT rtp ði; jÞ−ΔT rtp T nss ði; jÞ−T nss i¼1
ð4Þ
j¼1
3.1.1.1. The magnetization. The geometric parameters and actual magnetization directions of these models are in Table 1. The direction of the geomagnetic field used for the sources was I0 = 60∘ and D0 = − 20∘. The observed geometry was a 22 × 22 regular grid with spacing 0.1 m at an altitude of zero. The total magnetic field anomalies for the three model sources were forwardly calculated and shown in Fig. 1(a), (e) and (i). The total magnitude anomalies of the three sources roughly corresponded to their real locations, as shown in Fig. 1(b), (f) and (j). The centers of the sphere, horizontal cylinder and rectangular prism were shifted to the south-east, south, and south-east of the real center, respectively. The results show that the total magnitude anomaly was insensitive to the magnetization direction, while also influenced by remanent magnetization; thus the results are not very convincing. The NSS appeared to correspond fairly well to the real positions of the three sources, as shown in Fig. 1(c), (g) and (k). The centers of NSS data were closer to the real centers of the three sources than the centers of the total magnitude anomaly. This shows that the NSS provides more reliable information about the source geometry when the magnetic source contains remanent magnetization with a different direction to that of the inducing field. The results clearly show that while both the NSS data and the total magnitude anomaly are insensitive to the remanent magnetization, the NSS seems to perform better than the total magnitude anomaly. Thus, in theory, using the correlation between NSS and RTP anomaly to estimate the magnetization direction should provide better results than using the correlation between the total magnitude anomaly and RTP anomaly. Next, we estimated the magnetization directions of the three sources using these two different methods, in order to prove that the NSS is more useful than the total magnitude anomaly. Fig. 1(d), (h) and (l) show the cross-correlation coefficient maps of the three sources using NSS and RTP, and it can be seen that the estimations of
Table 1 The geometric parameters of three sources. I and D, respectively, represent the actual inclination and declination of each of the sources. Source
Center coordinates/m
Length in x-direction/m
Length in y-direction/m
Length in z-direction/m
Magnetization direction (I, D)
Sphere Horizontal Cylinder Rectangular
(1,1,0.3) (1,1,0.3) (1,1,0.3)
0.2 1 0.4
0.2 0.3 0.4
0.2 0.3 0.2
(20∘, −30∘) (15∘, 30∘) (50∘, −40∘)
J. Li et al. / Journal of Applied Geophysics 137 (2017) 1–7
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Fig. 1. The three model sources without interference: (a), (e) and (i) the total magnetic field anomaly, nT; (b), (f) and (j) the total magnitude anomaly, nT; (c), (g) and (k) the normalized source strength, nT/m; (d), (h) and (l) cross-correlation coefficient maps by using NSS and RTP of the three sources. Model values of D and I, marked with ‘∗’ and parameters estimated of D and I for each source, marked with ‘∘’.
magnetization direction for the different models were very close to the real values, which shows that this method is suitable for many different types of source models. Table 2 shows the estimated results for the different methods. In the case of the sphere, the estimations for inclination and declination obtained using NSS and RTP differed by 2∘. For the horizontal cylinder, the estimation for inclination was almost equal to the true value and the estimation error for declination was 5∘. For the rectangular prism, the estimates for declination and inclination were almost equal to the true values. However, the estimations of the method using total magnitude anomaly and RTP were far from the true values, and in particular the maximum error for inclination was 20∘. The absolute errors of the method using NSS and RTP were in the range of 1 − 2∘ for inclination and 2 − 5∘ for declination, whereas the absolute errors using
Table 2 Obtain the estimated magnetization direction by using different methods. The absolute error E is the estimation minus the true value. I and D, respectively, represent the estimated inclination and declination of each of the sources. Cmax is the maximum of the crosscorrelation coefficient C between the RTP, obtained for tentative values of magnetization direction and NSS. Source
Sphere Horizontal cylinder Rectangular prism
Parameters estimated for each source using NSS and RTP
Parameters estimated for each source using total magnitude anomaly and RTP
I∘(E)
D∘(E)
Cmax
I∘(E)
D∘(E)
Cmax
18(−2) 14(−1) 51(1)
-28(2) 35(5) -38(2)
0.9613 0.9616 0.9231
23(3) 13(−2) 71(20)
−27(3) 39(9) -36(4)
0.8731 0.9012 0.9466
4
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Fig. 2. The four magnetized sources.
the total magnitude anomaly and RTP were in the range of 2 − 20∘ for inclination and 3 − 9∘ for declination. This comparison provides compelling evidence that using NSS and RTP provides more stable results than using total magnitude anomaly and RTP. Thus, the former method can effectively reduce the effect of remanent magnetization in estimating the magnetization direction.
3.1.2. Multiple models To demonstrate the effectiveness of the method applied for multiple models, we used a combination of four bodies: a sphere, a horizontal cylinder and two rectangular prisms, as shown in Fig. 2. The inclination and declination of the geomagnetic fields of these models were, 60∘ and −20∘, respectively. The geometric parameters for the four sources are given in Table 3. The model and estimated declination, D, and inclination, I, of each of the four sources both without and with noise added are shown in Table 4. The observed geometry is a 22 × 22 regular grid with spacing 0.1 m at altitude 0 m. The total magnetic field anomaly
Table 3 The geometric parameters of four sources from Fig. 2. Source
Center point coordinates/m
Length in x-direction/m
Length in y-direction/m
Length in z-direction/m
1 2 3 4
(0.8.0.6.0.3) (0.4.1.6.0.3) (1.7.0.6.0.3) (1.5.1.6.0.3)
0.2 0.2 0.1 0.8
0.6 0.2 0.1 0.4
0.2 0.2 0.1 0.4
and the normalized source strength of the four model sources are shown in Fig. 3 with the noise added to the theoretical data. The standard deviation of the added noise was 5% of the maximum amplitude of the anomaly. Fig. 4 presents the cross-correlation coefficient map of four sources, in order that the distance between the true and the estimated values (marked with ‘∗’ and ‘∘’, respectively, for D and I) can be seen more clearly. The observed geometry was divided into four areas, so that only the area encompassing the target anomaly was used in the estimation for each individual source (see Fig. 3(b)). Fig. 3(a) shows that the negative and positive anomaly of the sources showed some mutual interference due to being positioned adjacent to each other. The absolute errors obtained using the method of NSS and RTP without noise were in the range of 1 − 4∘ for inclination and in the range of 4 − 16∘ for declination, hence the estimations of the inclinations were more accurate. In addition, the estimated absolute errors of the multiple models were bigger compared to those of the isolated models, as shown in Tables 2 and 4. As Table 4 shows, the estimations of the inclination and declination for the sources with noise were close to those without noise. Therefore, these results indicate that the method used for estimating the magnetization direction can effectively reduce the effects of remanent magnetization, and also shows a good resistance to noise.
3.2. Test on the real magnetic data Next, we tested the proposed method using real magnetic data collected from an area located in Shijiazhuang, China. The study area covered 2.1 × 1.9 m2 and the observed geometry was a 22 × 20 regular
Table 4 The declination, D, and inclination, I, are the model and estimated magnetization directions of the four sources without noise and with noise. Cmax is the maximum of the cross-correlation coefficient C between the RTP, obtained for tentative values of magnetization direction and NSS. Source
Model parameters I
1 2 3 4
∘
−50 45 60 15
Parameters estimated for each source without noise ∘
∘
∘
Parameters estimated for each source with noise
D
I (E)
D (E)
Cmax
I∘(E)
D∘(E)
Cmax
135 −20 −30 −5
−47(3) 44(−1) 56(−4) 19(4)
151(16) −10(−10) −20(10) 9(4)
0.9594 0.9576 0.9759 0.9175
−46(4) 44(−1) 56(−4) 19(4)
153(18) −10(10) −21(9) 9(4)
0.9581 0.9540 0.9722 0.9169
J. Li et al. / Journal of Applied Geophysics 137 (2017) 1–7
5
Fig. 3. (a) The noisy total magnetic field anomaly with added random Gaussian noise, nT, white line represents the actual profile of the four models; (b) The normalized source strength, nT/m, red line represents the window sides of the calculation areas.
grid with a spacing of 0.1 m. The inclination and declination of the geomagnetic fields of the observed area were 57∘ and −16∘, respectively. We carried out two experiments, one using an isolated ferromagnetic horizontal cylinder and the other on a combination of two ferromagnetic rectangular prisms. The observational data from the first experiment are
shown in Fig. 5(a) and (b). The ferromagnetic horizontal cylinder, which is denoted as ‘1’ in the figure, had a radius of 0.05 m and length of 0.46 m, with its center located at Easting = 0.9 m and Northing = 1 m. The observational data from the second experiment are shown in Fig. 6(a) and (b). The sizes of the two ferromagnetic rectangular prisms,
Fig. 4. Cross-correlation coefficient maps of the four sources. Model values of D and I, marked with ‘∗’ and parameters estimated of D and I for each source with noise, marked with‘∘’.
Fig. 5. The real magnetic field anomalies of first experiment: (a) the total magnetic field anomaly of the horizontal cylinder, nT; (b) the normalized source strength of the horizontal cylinder, nT/m. White line represents the actual profile of the horizontal cylinder and red line represents the window sides of the calculation area.
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Fig. 6. The real magnetic field anomalies of second experiment: (a) the total magnetic field anomaly of the rectangular composition models, nT; (b) the normalized source strength of the rectangular composition models, nT/m. White line represents the actual profile of the two ferromagnetic rectangular prisms and red line represents the window sides of the calculation areas.
denoted as ‘2’ and ‘3’ in Fig. 6, were as follows: prism 2 was of length 0.4 m and width 0.2 m with its center located at Easting = 0.65 m and Northing = 1.1 m; prism 3 was of length 0.4 m and width 0.2 m with its center located at Easting = 1.5 m and Northing = 0.45 m. We used the NSS map to isolate the target anomalies whose magnetization directions were to be estimated. The white line represents the actual profile of the three target anomalies and the red line represents the window sides of the calculation areas. By ignoring the influence of remanent magnetization and using the geomagnetic field direction as the magnetization direction, the RTP fields of the target anomalies were obtained, as shown in Fig. 7. It can be seen from the figure that the RTP results are not consistent with the real locations of the magnetic anomalies. Given that the assumed magnetization is aligned with the inducing field, this result shows that the preexisting remanent magnetization was strong enough to lead to a clear deviation from the true magnetization direction. Therefore, it is evident that NSS
data would need to be utilized to reduce the effects of remanent magnetization. The estimated magnetization direction of each target anomaly was calculated through correlation between the NSS data and RTP. For the isolated horizontal cylinder, the estimated magnetization direction was obtained as (I, D) = (16∘, 13∘), and the corresponding (I, D) values obtained for anomalies 2 and 3 were (12∘, −5∘) and (3∘, −13∘), respectively. We calculated the RTP fields of the three windows for the estimated values of the magnetization direction as shown in Fig. 8, where the white lines represent the actual profiles of the three target anomalies. It is apparent that the positions of the white lines are quite consistent with the main positive extreme value distributions of the RTP fields of the target anomalies. Thus, the analysis performed using the real magnetic data indicates that the NSS data is insensitive to remanent magnetization and the method using NSS and RTP to estimate the magnetization direction gives good results.
Fig. 7. The RTP anomaly of the target anomalies in the window sides of the three calculation areas, nT, when the geomagnetic field direction is used as the magnetization direction: A is the RTP field of isolated ferromagnetic horizontal cylinder which named 1; B is the RTP field of the ferromagnetic rectangular which named 2; C is the RTP field of the ferromagnetic rectangular which named 3. White line represents the actual profile of the target anomalies.
J. Li et al. / Journal of Applied Geophysics 137 (2017) 1–7
7
Fig. 8. The RTP anomaly of the target anomalies in the window sides of the three calculation areas, nT, when the estimated magnetization direction is used as the magnetization direction: A′ is the RTP field of isolated ferromagnetic horizontal cylinder which named 1; B′ is the RTP field of the ferromagnetic rectangular which named 2; C′ is the RTP field of the ferromagnetic rectangular which named 3. White line represents the actual profile of the target anomalies.
4. Conclusions We proposed a method of estimating the magnetization direction of magnetization anomalies through correlation between NSS data and RTP. In this approach, the RTP field was transformed using different assumed values of the magnetization directions. The cross-correlation between NSS and RTP field was then calculated, with the magnetization direction corresponding to the maximum of the cross-correlation. Tests using both simulated and real magnetic data showed that this method is suitable for many different types of source models and also has a good noise resistance. In addition, compared to that of the total magnitude anomaly, the NSS center is closer to the real center and the approach using the NSS and RTP is less sensitive to the remanent magnetization than the method which uses the total magnitude anomaly and RTP. Thus, we conclude that when remanence exists, our proposed method produces more reliable results compared to other methods. Acknowledgement This work was supported by the National Natural Science Foundation of China (No.51305454). References Beiki, M., Clark, D.A., Austin, J.R., Foss, C.A., 2012. Estimating source location using normalized magnetic source strength calculated from magnetic gradient tensor data. Geophysics 77, J23–J37. Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications. Cambidge University Press. Dannemiller, N., Li, Y.G., 2006. A new method for determination of magnetization direction. Geophysics 71, L69–L73.
Gerovska, D., Arauzo-Bravo, M.J., Stavrev, P., 2009. Estimating the magnetization direction of sources from Southeast Bulgaria through correlation between reduced-to-the-pole and total magnitude anomalies. Geophys. Prospect. 57, 491–505. Guo, L.H., Meng, X.H., Zhang, G.L., 2014. Three-dimensional correlation imaging for total amplitude magnetic anomaly and normalized source strength in the presence of strong remanent magnetization. J. Appl. Geophys. 111, 121–128. Helbig, K., 1963. Some integrals of magnetic anomalies and their relation to the parameters of the disturbing body. Z. Geophys. 29, 83–97. Li, Y., Shearer, S.E., Haney, M.M., Dannemiller, N., 2010. Comprehensive approaches to 3D inversion of magnetic data affected by remanent magnetization. Geophysics 75, L1–L11. Liu, S., Feng, J., Gao, W.L., Qiu, L.Q., Liu, T.Y., Hu, X.Y., 2013. 2D inversion for borehole magnetic data in the presence of significant remanence and demagnetization. Chin. J. Geophys. 56, 4297–4309 (in Chinese). Liu, S., Hu, X.Y., Xi, Y.F., Liu, T., Xu, S., 2015. 2D sequential inversion of total magnitude and total magnetic anomaly data affected by remanent magnetization. Geophysics 80, K1–K12. Medeiros, W.E., Silva, J.B.C., 1995. Simultaneous estimation of total magnetization direction and 3-D spatial orientation. Geophysics 60, 1365–1377. Oliveira Jr., V.C., Sales, D.P., Barbosa, V.C.F., Uieda, L., 2015. Estimation of the total magnetization direction of approximately spherical bodies. Nonlinear Process. Geophys. 22, 215–232. Phillips, J.D., 2005. Can we estimate total magnetization directions from aeromagnetic data using Helbig's integrals. Earth Planets Space 57, 681–689. Pilkington, M., Beiki, M., 2013. Mitigating remanent magnetization effects in magnetic data using the normalized source strength. Geophysics 78, J25–J32. Roest, W.R., Pilkington, M., 1993. Identifying remanent magnetization effects in magnetic data. Geophysics 58, 653–659. Shi, L., Meng, X.H., Guo, L.H., Chen, S., Xu, W.M., 2014. A simple algorithm for estimating the magnetization direction of magnetic bodies under the influence of remanent magnetization. Prog. Geophys. 29, 1748–1751 (In Chinese). Wilson, H.S., 1985. Analysis of the magnetic gradient tensor. Defence research establishment Pacific, technical. Memorandum 8, 5–13. Zhou, J., Meng, X.H., 2015. Three-dimensional cross-gradient joint inversion of gravity and normalized magnetic source strength data in the presence of remanent magnetization. J. Appl. Geophys. 119, 51–60.
Contents lists available at ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
An approach for estimating the magnetization direction of magnetic anomalies Jinpeng Li ⁎, Yingtang Zhang, Gang Yin, Hongbo Fan, Zhining Li Seventh Department, Mechanical Engineering College, Shijiazhuang 050003, People's Republic of China
a r t i c l e
i n f o
Article history: Received 16 May 2016 Received in revised form 5 December 2016 Accepted 9 December 2016 Available online 11 December 2016 Keywords: Magnetization direction Normalized source strength Reduced-to-the-pole Remanent magnetization Cross-correlation
a b s t r a c t An approach for estimating the magnetization direction of magnetic anomalies in the presence of remanent magnetization through correlation between normalized source strength (NSS) and reduced-to-the-pole (RTP) is proposed. The observation region was divided into several calculation areas and the RTP field was transformed using different assumed values of the magnetization directions. Following this, the cross-correlation between NSS and RTP field was calculated, and it was found that the correct magnetization direction was that corresponding to the maximum cross-correlation value. The approach was tested on both simulated and real magnetic data. The results showed that the approach was effective in a variety of situations and considerably reduced the effect of remanent magnetization. Thus, the method using NSS and RTP is more effective compared to other methods such as using the total magnitude anomaly and RTP. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Magnetic surveys are widely used in different fields, including mining applications, oil and gas exploration, and mapping bedrock topography, among others. In magnetic prospecting, knowledge of the correct magnetization direction of magnetic anomalies is important for calculation of the RTP field, as well as forward calculation and inversion (Li et al., 2010). The total magnetization in a source body is the vector sum of the induced and remanent magnetization. In some cases, the induced magnetization aligns with the direction of earth's ambient field, without remanent magnetization and self-demagnetization effects. However, in most cases, the preexisting remanent magnetization is strong enough to affect the true magnetization direction (Liu et al., 2013, 2015) leading to the erroneous interpretation of magnetic data. Therefore, in recent years, there has been an increasing focus on reducing, or even removing the effects of remanent magnetization in the estimation of magnetization direction. The problem of determining the total magnetization direction has long been of interest in the field of magnetic interpretation. Roest and Pilkington (1993) estimated the magnetization direction by comparing the amplitude of the analytic and horizontal gradient of pseudogravity. Medeiros and Silva (1995) estimated the total magnetization direction using the source moments up to second order derived from the multi⁎ Corresponding author at: The Seventh Department of Mechanical Engineering College, No.97, Hepingxilu Road, Shi Jia-zhuang, He Bei province, PR China. E-mail address: [email protected] (J. Li).
http://dx.doi.org/10.1016/j.jappgeo.2016.12.009 0926-9851/© 2016 Elsevier B.V. All rights reserved.
pole expansion of the magnetic potential. Phillips (2005) proposed direct and indirect algorithms that implemented Helbig's (1963) integrals for estimating the magnetization direction from the first order magnetic moments. Dannemiller and Li (2006) proposed a method to estimate the magnetization direction of 3-D sources based upon the correlation between the vertical and the total gradients of the reduced-to-thepole field. A similar method was proposed by Gerovska et al. (2009), based on the correlation between the reduced-to-the-pole field and the total magnitude anomaly. Shi et al. (2014) presented the crosscorrelation of magnetic dipole sources for determination of magnetization direction from the total magnetic field anomaly. Oliveira et al. (2015) developed a fast total-field anomaly inversion to estimate the magnetization direction of multiple sources with approximately spherical shapes and known centers. However, the aforementioned techniques all suffer from certain drawbacks which limit their accuracy. These limitations include the results being greatly influenced by noise effects, as well as the total magnitude anomaly being insensitive to the magnetization direction while also influencing the precision of the solution. In the present study, a correlation coefficient analysis was carried out to estimate the magnetization direction of magnetic anomalies through correlation between NSS data and RTP in the presence of remanent magnetization. We chose to use NSS data as it is less sensitive to the magnetization direction compared to other transforms of the magnetic data and relates well to the location of the magnetic source. Hence, it has a stronger capacity to reduce the remanence effect. Additionally, the NSS shows a strong relationship to horizontal projections
2
J. Li et al. / Journal of Applied Geophysics 137 (2017) 1–7
where (M, N) is the grid size, ΔTrtp is the RTP field, Tnss is the NSS data, ΔT rtp is mean value of the RTP fields and T nss is mean value of the NSS data. Since the magnetization direction is unknown from the calculations performed using the aforementioned equations, we assumed that the inclination I changed from − 90∘ to 90∘ and the declination D changed from −180∘ to 180∘, both of which with intervals of 1∘. We calculated the RTP field for a series of assumed values of the magnetization direction. Then, using Eq. (4), the correlation coefficient, C, between the NSS and RTP field was calculated. Just as for the NSS, the RTP has a strong relationship to the horizontal projection of the source when the used magnetization and real magnetization directions are aligned. Therefore, the value of C reflects the degree of cross-correlation between the NSS and the RTP and thus is related to the probability of finding the correct magnetization direction of the source. A lower value means lower accuracy of the estimated magnetization direction; therefore, the maximum value of the correlation coefficient indicates the correct magnetization direction.
of the sources, which makes it a more accurate choice for estimating the magnetization direction. This paper is structured as follows: in Section 2, a new approach for estimating the magnetization direction is described; in Section 3, the results for simulated and real magnetization data are presented. Finally, the conclusions are given in Section 4. 2. Method 2.1. Normalized source strength data In a Cartesian coordinate system, with the x-axis pointing to the geographical east, the y-axis to the north, and the z-axis vertically downwards, the theoretical magnetic gradient tensor data in the observation plane (x, y, z) can be expressed as follows: 2
Bxx G ¼ 4 Byx Bzx
Bxy Byy Bzy
3 Bxz Byz 5; Bzz
ð1Þ
where G is the magnetic gradient tensor matrix and Bαβ(α,β =x, y, z) are the magnetic gradient tensor components. If the eigenvalues of the matrix G are arranged in descending order as λ1 ≥ λ2 ≥ λ3. Then the theoretical normalized source strength data may be expressed as (Zhou and Meng, 2015; Guo et al., 2014; Beiki et al., 2012; Pilkington and Beiki, 2013; Wilson, 1985) uðx; y; zÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −λ2 2 −λ1 λ3 ;
3. Data experiments 3.1. Test on the synthetic magnetic data 3.1.1. Isolated model
ð2Þ
where u(x,y, z) represents the NSS data at an arbitrary station (x,y,z) on the observational surface. 2.2. Correlation coefficient analysis for choosing optimum magnetization direction The RTP turns the measured total magnetic field to anomaly at the north magnetic pole, where the magnetization direction points directly downwards; hence, there is a good correlation between the RTP anomaly and the source horizontal projection. In the presence of remanence, the transformation of RTP requires both the geomagnetic field direction (I0,D0) and the magnetization direction (I, D). The RTP field may be calculated in the frequency domain using the transfer function (Blakely, 1995) H¼
u2 þ v2 ; ½iðu cos I 0 cos D0 þ v cos I 0 sin D0 Þ þ r sin I0 ½iðu cos I cos D þ v cos I sin DÞ þ r sin I
ð3Þ where u is the angular frequency in the x-direction, v is the angular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency in the y-direction, and r ¼ u2 þ v2 . The correlation coefficient C between the NSS data and the RTP fields is defined as M X N X ΔT rtp ði; jÞ−ΔT rtp T nss ði; jÞ−T nss i¼1 j¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; C¼v uM 2 N uX X 2 t ΔT rtp ði; jÞ−ΔT rtp T nss ði; jÞ−T nss i¼1
ð4Þ
j¼1
3.1.1.1. The magnetization. The geometric parameters and actual magnetization directions of these models are in Table 1. The direction of the geomagnetic field used for the sources was I0 = 60∘ and D0 = − 20∘. The observed geometry was a 22 × 22 regular grid with spacing 0.1 m at an altitude of zero. The total magnetic field anomalies for the three model sources were forwardly calculated and shown in Fig. 1(a), (e) and (i). The total magnitude anomalies of the three sources roughly corresponded to their real locations, as shown in Fig. 1(b), (f) and (j). The centers of the sphere, horizontal cylinder and rectangular prism were shifted to the south-east, south, and south-east of the real center, respectively. The results show that the total magnitude anomaly was insensitive to the magnetization direction, while also influenced by remanent magnetization; thus the results are not very convincing. The NSS appeared to correspond fairly well to the real positions of the three sources, as shown in Fig. 1(c), (g) and (k). The centers of NSS data were closer to the real centers of the three sources than the centers of the total magnitude anomaly. This shows that the NSS provides more reliable information about the source geometry when the magnetic source contains remanent magnetization with a different direction to that of the inducing field. The results clearly show that while both the NSS data and the total magnitude anomaly are insensitive to the remanent magnetization, the NSS seems to perform better than the total magnitude anomaly. Thus, in theory, using the correlation between NSS and RTP anomaly to estimate the magnetization direction should provide better results than using the correlation between the total magnitude anomaly and RTP anomaly. Next, we estimated the magnetization directions of the three sources using these two different methods, in order to prove that the NSS is more useful than the total magnitude anomaly. Fig. 1(d), (h) and (l) show the cross-correlation coefficient maps of the three sources using NSS and RTP, and it can be seen that the estimations of
Table 1 The geometric parameters of three sources. I and D, respectively, represent the actual inclination and declination of each of the sources. Source
Center coordinates/m
Length in x-direction/m
Length in y-direction/m
Length in z-direction/m
Magnetization direction (I, D)
Sphere Horizontal Cylinder Rectangular
(1,1,0.3) (1,1,0.3) (1,1,0.3)
0.2 1 0.4
0.2 0.3 0.4
0.2 0.3 0.2
(20∘, −30∘) (15∘, 30∘) (50∘, −40∘)
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Fig. 1. The three model sources without interference: (a), (e) and (i) the total magnetic field anomaly, nT; (b), (f) and (j) the total magnitude anomaly, nT; (c), (g) and (k) the normalized source strength, nT/m; (d), (h) and (l) cross-correlation coefficient maps by using NSS and RTP of the three sources. Model values of D and I, marked with ‘∗’ and parameters estimated of D and I for each source, marked with ‘∘’.
magnetization direction for the different models were very close to the real values, which shows that this method is suitable for many different types of source models. Table 2 shows the estimated results for the different methods. In the case of the sphere, the estimations for inclination and declination obtained using NSS and RTP differed by 2∘. For the horizontal cylinder, the estimation for inclination was almost equal to the true value and the estimation error for declination was 5∘. For the rectangular prism, the estimates for declination and inclination were almost equal to the true values. However, the estimations of the method using total magnitude anomaly and RTP were far from the true values, and in particular the maximum error for inclination was 20∘. The absolute errors of the method using NSS and RTP were in the range of 1 − 2∘ for inclination and 2 − 5∘ for declination, whereas the absolute errors using
Table 2 Obtain the estimated magnetization direction by using different methods. The absolute error E is the estimation minus the true value. I and D, respectively, represent the estimated inclination and declination of each of the sources. Cmax is the maximum of the crosscorrelation coefficient C between the RTP, obtained for tentative values of magnetization direction and NSS. Source
Sphere Horizontal cylinder Rectangular prism
Parameters estimated for each source using NSS and RTP
Parameters estimated for each source using total magnitude anomaly and RTP
I∘(E)
D∘(E)
Cmax
I∘(E)
D∘(E)
Cmax
18(−2) 14(−1) 51(1)
-28(2) 35(5) -38(2)
0.9613 0.9616 0.9231
23(3) 13(−2) 71(20)
−27(3) 39(9) -36(4)
0.8731 0.9012 0.9466
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Fig. 2. The four magnetized sources.
the total magnitude anomaly and RTP were in the range of 2 − 20∘ for inclination and 3 − 9∘ for declination. This comparison provides compelling evidence that using NSS and RTP provides more stable results than using total magnitude anomaly and RTP. Thus, the former method can effectively reduce the effect of remanent magnetization in estimating the magnetization direction.
3.1.2. Multiple models To demonstrate the effectiveness of the method applied for multiple models, we used a combination of four bodies: a sphere, a horizontal cylinder and two rectangular prisms, as shown in Fig. 2. The inclination and declination of the geomagnetic fields of these models were, 60∘ and −20∘, respectively. The geometric parameters for the four sources are given in Table 3. The model and estimated declination, D, and inclination, I, of each of the four sources both without and with noise added are shown in Table 4. The observed geometry is a 22 × 22 regular grid with spacing 0.1 m at altitude 0 m. The total magnetic field anomaly
Table 3 The geometric parameters of four sources from Fig. 2. Source
Center point coordinates/m
Length in x-direction/m
Length in y-direction/m
Length in z-direction/m
1 2 3 4
(0.8.0.6.0.3) (0.4.1.6.0.3) (1.7.0.6.0.3) (1.5.1.6.0.3)
0.2 0.2 0.1 0.8
0.6 0.2 0.1 0.4
0.2 0.2 0.1 0.4
and the normalized source strength of the four model sources are shown in Fig. 3 with the noise added to the theoretical data. The standard deviation of the added noise was 5% of the maximum amplitude of the anomaly. Fig. 4 presents the cross-correlation coefficient map of four sources, in order that the distance between the true and the estimated values (marked with ‘∗’ and ‘∘’, respectively, for D and I) can be seen more clearly. The observed geometry was divided into four areas, so that only the area encompassing the target anomaly was used in the estimation for each individual source (see Fig. 3(b)). Fig. 3(a) shows that the negative and positive anomaly of the sources showed some mutual interference due to being positioned adjacent to each other. The absolute errors obtained using the method of NSS and RTP without noise were in the range of 1 − 4∘ for inclination and in the range of 4 − 16∘ for declination, hence the estimations of the inclinations were more accurate. In addition, the estimated absolute errors of the multiple models were bigger compared to those of the isolated models, as shown in Tables 2 and 4. As Table 4 shows, the estimations of the inclination and declination for the sources with noise were close to those without noise. Therefore, these results indicate that the method used for estimating the magnetization direction can effectively reduce the effects of remanent magnetization, and also shows a good resistance to noise.
3.2. Test on the real magnetic data Next, we tested the proposed method using real magnetic data collected from an area located in Shijiazhuang, China. The study area covered 2.1 × 1.9 m2 and the observed geometry was a 22 × 20 regular
Table 4 The declination, D, and inclination, I, are the model and estimated magnetization directions of the four sources without noise and with noise. Cmax is the maximum of the cross-correlation coefficient C between the RTP, obtained for tentative values of magnetization direction and NSS. Source
Model parameters I
1 2 3 4
∘
−50 45 60 15
Parameters estimated for each source without noise ∘
∘
∘
Parameters estimated for each source with noise
D
I (E)
D (E)
Cmax
I∘(E)
D∘(E)
Cmax
135 −20 −30 −5
−47(3) 44(−1) 56(−4) 19(4)
151(16) −10(−10) −20(10) 9(4)
0.9594 0.9576 0.9759 0.9175
−46(4) 44(−1) 56(−4) 19(4)
153(18) −10(10) −21(9) 9(4)
0.9581 0.9540 0.9722 0.9169
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Fig. 3. (a) The noisy total magnetic field anomaly with added random Gaussian noise, nT, white line represents the actual profile of the four models; (b) The normalized source strength, nT/m, red line represents the window sides of the calculation areas.
grid with a spacing of 0.1 m. The inclination and declination of the geomagnetic fields of the observed area were 57∘ and −16∘, respectively. We carried out two experiments, one using an isolated ferromagnetic horizontal cylinder and the other on a combination of two ferromagnetic rectangular prisms. The observational data from the first experiment are
shown in Fig. 5(a) and (b). The ferromagnetic horizontal cylinder, which is denoted as ‘1’ in the figure, had a radius of 0.05 m and length of 0.46 m, with its center located at Easting = 0.9 m and Northing = 1 m. The observational data from the second experiment are shown in Fig. 6(a) and (b). The sizes of the two ferromagnetic rectangular prisms,
Fig. 4. Cross-correlation coefficient maps of the four sources. Model values of D and I, marked with ‘∗’ and parameters estimated of D and I for each source with noise, marked with‘∘’.
Fig. 5. The real magnetic field anomalies of first experiment: (a) the total magnetic field anomaly of the horizontal cylinder, nT; (b) the normalized source strength of the horizontal cylinder, nT/m. White line represents the actual profile of the horizontal cylinder and red line represents the window sides of the calculation area.
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Fig. 6. The real magnetic field anomalies of second experiment: (a) the total magnetic field anomaly of the rectangular composition models, nT; (b) the normalized source strength of the rectangular composition models, nT/m. White line represents the actual profile of the two ferromagnetic rectangular prisms and red line represents the window sides of the calculation areas.
denoted as ‘2’ and ‘3’ in Fig. 6, were as follows: prism 2 was of length 0.4 m and width 0.2 m with its center located at Easting = 0.65 m and Northing = 1.1 m; prism 3 was of length 0.4 m and width 0.2 m with its center located at Easting = 1.5 m and Northing = 0.45 m. We used the NSS map to isolate the target anomalies whose magnetization directions were to be estimated. The white line represents the actual profile of the three target anomalies and the red line represents the window sides of the calculation areas. By ignoring the influence of remanent magnetization and using the geomagnetic field direction as the magnetization direction, the RTP fields of the target anomalies were obtained, as shown in Fig. 7. It can be seen from the figure that the RTP results are not consistent with the real locations of the magnetic anomalies. Given that the assumed magnetization is aligned with the inducing field, this result shows that the preexisting remanent magnetization was strong enough to lead to a clear deviation from the true magnetization direction. Therefore, it is evident that NSS
data would need to be utilized to reduce the effects of remanent magnetization. The estimated magnetization direction of each target anomaly was calculated through correlation between the NSS data and RTP. For the isolated horizontal cylinder, the estimated magnetization direction was obtained as (I, D) = (16∘, 13∘), and the corresponding (I, D) values obtained for anomalies 2 and 3 were (12∘, −5∘) and (3∘, −13∘), respectively. We calculated the RTP fields of the three windows for the estimated values of the magnetization direction as shown in Fig. 8, where the white lines represent the actual profiles of the three target anomalies. It is apparent that the positions of the white lines are quite consistent with the main positive extreme value distributions of the RTP fields of the target anomalies. Thus, the analysis performed using the real magnetic data indicates that the NSS data is insensitive to remanent magnetization and the method using NSS and RTP to estimate the magnetization direction gives good results.
Fig. 7. The RTP anomaly of the target anomalies in the window sides of the three calculation areas, nT, when the geomagnetic field direction is used as the magnetization direction: A is the RTP field of isolated ferromagnetic horizontal cylinder which named 1; B is the RTP field of the ferromagnetic rectangular which named 2; C is the RTP field of the ferromagnetic rectangular which named 3. White line represents the actual profile of the target anomalies.
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Fig. 8. The RTP anomaly of the target anomalies in the window sides of the three calculation areas, nT, when the estimated magnetization direction is used as the magnetization direction: A′ is the RTP field of isolated ferromagnetic horizontal cylinder which named 1; B′ is the RTP field of the ferromagnetic rectangular which named 2; C′ is the RTP field of the ferromagnetic rectangular which named 3. White line represents the actual profile of the target anomalies.
4. Conclusions We proposed a method of estimating the magnetization direction of magnetization anomalies through correlation between NSS data and RTP. In this approach, the RTP field was transformed using different assumed values of the magnetization directions. The cross-correlation between NSS and RTP field was then calculated, with the magnetization direction corresponding to the maximum of the cross-correlation. Tests using both simulated and real magnetic data showed that this method is suitable for many different types of source models and also has a good noise resistance. In addition, compared to that of the total magnitude anomaly, the NSS center is closer to the real center and the approach using the NSS and RTP is less sensitive to the remanent magnetization than the method which uses the total magnitude anomaly and RTP. Thus, we conclude that when remanence exists, our proposed method produces more reliable results compared to other methods. Acknowledgement This work was supported by the National Natural Science Foundation of China (No.51305454). References Beiki, M., Clark, D.A., Austin, J.R., Foss, C.A., 2012. Estimating source location using normalized magnetic source strength calculated from magnetic gradient tensor data. Geophysics 77, J23–J37. Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications. Cambidge University Press. Dannemiller, N., Li, Y.G., 2006. A new method for determination of magnetization direction. Geophysics 71, L69–L73.
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