- Email: [email protected]
The antisymmetric factor method for magnetic reduction to the pole at low latitudes
Journal of Applied Geophysics 92 (2013) 103–109
Contents lists available at SciVerse ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
The antisymmetric factor method for magnetic reduction to the pole at low latitudes Lianghui Guo a,⁎, Lei Shi b, Xiaohong Meng a a b
Key Laboratory of Geo-detection (China University of Geosciences, Beijing), Ministry of Education, Beijing 100083, China Institute of Geophysics, China Earthquake Administration, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 3 November 2012 Accepted 19 February 2013 Available online 14 March 2013 Keywords: Total magnetic intensity anomaly Reduction to the pole Low latitude Antisymmetric factor The South China Sea
a b s t r a c t We analyze the characteristics of the wavenumber-domain factor for magnetic reduction to the pole (RTP) at low latitudes, and then propose a new wavenumber-domain method for RTP at low latitudes, herein called the antisymmetric factor method, based on modification of the RTP factor. The method applies the antisymmetric factor in a given scope of directions centered along the magnetic declination to suppress amplification of the RTP factor, stabilizing the RTP. Meanwhile it utilizes the routine RTP factor in other directions to preserve the effective RTP features. The test on the synthetic data demonstrates that the method is robust and effective. Finally, we use the new method, as well as a variable magnetic inclinations algorithm, to perform RTP on the real data of total magnetic intensity anomalies in the South China Sea, and obtain the reliable RTP anomalies. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The reduction-to-the-pole (RTP) of magnetic anomalies is an important task in the interpretation that transforms total magnetic intensity (TMI) anomalies in oblique magnetization into that as in a vertical magnetization. Thus, the complexity of the TMI anomalies caused by oblique magnetization can be eliminated. However, the RTP factor is one of the amplifying transformation factors. It is related directly to the magnetic inclination. At low latitudes (an absolute inclination less than 20°), the smaller the absolute value of the magnetic inclination is, the stronger the amplification effect of the RTP factor will be. Such amplification upon the noise in the TMI anomalies (which always exists in the real world) causes linear artifacts along the direction of magnetic declination. Therefore, the RTP procedure at low latitudes is much more troublesome than at high latitudes (an absolute inclination larger than 20°). To overcome the difficulty of RTP at low latitudes, several special RTP methods or techniques were proposed, such as the equivalent source inversion method (Silva, 1986), the Werner filtering (Hansen and Pawlowski, 1989), the energy balance technique (Keating and Zerbo, 1996), the inversion-based method (Li and Oldenburg, 2001), the pseudo inclination method (Macleod et al., 1993), the azimuthal filtering (Phillips, 1997) and the suppressing factor method (Yao et al., 2003). The equivalent source inversion method solves the RTP instability by inversing the magnetic data to derive equivalent sources and then doing forward modeling to produce the RTP data. However, the inversion requires expensive computation, making the method unsuitable ⁎ Corresponding author. Tel.: +86 1082322648. E-mail address: [email protected] (L. Guo). 0926-9851/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jappgeo.2013.02.018
for large-scale magnetic data. The Wiener filtering method utilizes a regularized RTP operator based on Wiener filter (Clarke, 1969; Wiener, 1949), which is a denoising filter. But the method tend to overly smooth the RTP result and to lose signal at short wavelengths (Li, 2008). The energy balance technique suppresses noise effects by iterative blanking so that the average energy is similar in all directions while retaining some signal at short wavelengths. However, the technique is subjective in choosing the related noise assumption and control parameters. The inverse-based method constructs the RTP anomalies model by inversing the observed magnetic data in the wavenumber domain with explicit regularization and an imposition of power spectral decay. But the application of the method is limited due to the complicated inversion and its expensive computation. All of the pseudo inclination method, the azimuthal filtering and the suppressing filter method are based on modifying the RTP factor in the wavenumber domain to suppress the amplification effect along and near the direction of magnetic declination. The pseudo inclination method replaces the actual magnetic inclination with a larger pseudo inclination in the RTP calculations. The azimuthal filtering tapers the RTP factor by a special sine function, while the suppressing filter method applies a special cosine function. However, all the three methods apt to lose parts of the effective RTP anomalies, respectively because the pseudo inclination method also suppresses the amplification effect in other directions outside the declination, the suppressing filter method overly suppress the amplification effect along the declination (Shi et al., 2012) and the azimuthal filtering has the similar limitation. In this paper, we first analyze the characteristics of the RTP factors of the routine RTP method and the pseudo inclination method, as well as their drawbacks in applications in RTP at low latitudes. Then we optimize the RTP factor and propose an antisymmetric factor method
104
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
for RTP at low latitudes. The method applies the antisymmetric factor in a certain scope of directions centered along the magnetic declination, while it utilizes the routine RTP factor in other directions. Finally, we test the method both on the synthetic magnetic data and on the real TMI anomalies data in the South China Sea. The routine RTP method and the pseudo inclination method are also used to test the data for comparisons. 2. Method 2.1. The RTP methods based on modifying the RTP factor In magnetic exploration, we usually observe the TMI anomaly. The corresponding observation direction coincides with the geomagnetic field. Suppose that the remanent magnetization can be neglected and the magnetization direction is consistent with the geomagnetic field. Then the RTP factor in a wavenumber domain can be written in a polar coordinate system as (Gunn, 1975; Macleod et al., 1993; Spector and Grant, 1970) H ðr; θÞ ¼ HðθÞ ¼
1 ; ½ sinðIÞ þ i cosðIÞ cosðD−θÞ2
ð1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where, r ¼ u2 þ v2 , θ = tg−1(v/u), u and v are respectively the wavenumbers in x and y directions, I and D areprespectively the inclinaffiffiffiffiffiffiffiffi tion and declination of magnetization, and i ¼ −1. The RTP factor H(θ) in Eq. (1) is one of the amplifying transformation factors, which is a mono-function of angle θ, while directly related to magnetic inclination I and declination D. At low latitudes, the absolute value of I is relatively small. When θ approaches D ± 90°, the amplitude of H(θ) grows rapidly to large values. In extreme conditions, I = 0 and θ = D ± 90°, there is H(θ) → − ∞. Such amplification effect of H(θ) at low latitudes makes calculation of RTP very unstable, yielding notable stripes (linear artifacts) along the magnetic declination D in the RTP results. Therefore, it is necessary to modify the RTP factor to suppress the amplification effect along the magnetic declination, so that calculation will be stable and the stripes will be reduced or even vanish. Three such methods include the pseudo inclination method (Macleod et al., 1993), the azimuthal filtering (Phillips, 1997) and the suppressing factor method (Yao et al., 2003). Here, we select the pseudo inclination method as an example to analyze the characteristics of the RTP factor. The RTP factor of the pseudo inclination (PI) method in a polar coordinate system is written as (Macleod et al., 1993)
Fig. 1. The features of the RTP factor of the PI method at the magnetic equator. The thin solid line, dotted line, dashed line and thick solid line correspond to the pseudo inclination of 0°, 30°, 60° and 90°, respectively.
inclination (I′) is, the stronger such suppression becomes. Extremely, when I′ = 90°,there is HPI(θ) = − 1, and the suppression reaches the maximum. However, the RTP factor of the PI method suppresses the amplification effects not only along the magnetic declination but also in other directions. The suppression of other directions unexpectedly weakens the RTP characteristics and thus reduces RTP precision. To solve this problem, Li (2008) proposed an improved algorithm for the PI method, which uses the RTP factor of the PI method (HPI(θ)) to suppress amplification within a wedge-shaped segment centered along the magnetic declination, and uses the routine RTP factor (H(θ)) in other directions to preserve amplification. Nevertheless, such an algorithm brings about the problem of discontinuity at the conjunction between the two different factors.
2
H PI ðθÞ ¼
½ sinðIÞ−i cosðIÞ cosðD−θÞ ; sin I′ þ cos2 I′ cos2 ðD−θÞ ⋅ sin2 ðIÞ þ cos2 ðIÞ cos2 ðD−θÞ 2
ð2Þ where I′ is the pseudo inclination defined by the user, which is larger than the real one I. If |I′| b |I|, there is I′ = I. In practice, the absolute value of I′ is often set between 20° and 30° (Macleod et al., 1993), and it needs to be larger for stronger noise in the observed anomalies data (Li, 2008). In the extreme case of the magnetic equator, where I = 0°, the substitution into Eq. (2) yields H PI ðθÞ ¼
−1 : sin2 I′ þ cos2 I ′ cos2 ðD−θÞ
ð3Þ
Then, substituting the pseudo inclinations of I′ = 0°, 30°, 60° and 90° into Eq. (3), respectively, we obtain the corresponding RTP factors (thin solid line, dotted line, dashed line and thick solid line in Fig. 1, respectively). When I′ = 0°, there is HPI(θ) = H(θ), i.e., the RTP factor is equivalent to that of the routine RTP method without suppression of amplification effects. When I′ > 0°, the RTP factors suppress amplification in every direction to stabilize RTP. The larger the pseudo
Fig. 2. The features of the RTP factor of the AF method at the magnetic equator. The black, red, blue and green lines correspond to the threshold angles of 90°, 75°, 60° and 45°, respectively. The purple curve is the RTP factor of the PI method with a pseudo inclination of 30°.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
105
Fig. 3. The theoretical TMI anomaly of the synthetic model at the magnetic equator (a), and that at the magnetic pole (b), and the noisy TMI anomaly at the magnetic equator (c). The contour interval of the anomaly is 5 nT.
Fig. 4. The RTP results of the noisy TMI anomalies by different methods: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°; (c) the routine RTP method. The contour interval of the anomaly is 5 nT.
Fig. 5. The RTP results of the noisy TMI anomalies along the profiles of X = 23 m and X = 33 m by different methods. The black, green, blue and red lines correspond to the theoretical RTP anomalies without noise, the routine RTP method, the PI method and the AF method, respectively.
2.2. The antisymmetric factor method Based on the RTP factor characteristics stated above, we propose a new RTP method at low latitudes, herein called the antisymmetric factor (AF) method, based on modification of the RTP factor. We set
a threshold angle θ0(45° ≤ θ0 ≤ 90°) for suppression. We design a special RTP factor to suppress the amplification of RTP factor when |D − θ| > θ0, while adopt the routine RTP factor to preserve the amplification when |D − θ| ≤ θ0. Meanwhile, to make the transition from the routine RTP factor to the special RTP factor continuous and natural at
106
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 6. The RTP results of the denoised TMI anomalies by different methods: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°; (c) the routine RTP method. The contour interval of the anomaly is 5 nT.
the threshold angle, we take the antisymmetric mapping of the routine RTP factor as the special factor, with respect to the central axis of the threshold angle. Here we call the special factor the antisymmetric factor. The RTP factor of the AF method is defined in a polar coordinate system as 8 When ðD−θ0 Þ≤θ≤ðD þ θ0 Þ; H AF ðθÞ ¼ H ðθÞ > > < When D−90B ≤θbðD−θ0 Þ; HAF ðθÞ ¼ 2HðD þ θ0 Þ−H½2ðD−θ0 Þ−θ > > : When ðD þ θ Þbθ≤D þ 90B; H ðθÞ ¼ 2H ðD þ θ Þ−H ½2ðD þ θ Þ−θ; 0 AF 0 0
As θ0 becomes smaller, the suppression area becomes wider, and the AF method performs wider and stronger suppression. In general, the threshold angle needs to be smaller for stronger noise in the observed anomalies data. In contrast to the PI method, the AF method suppresses amplification in the suppression area (θ0 ∼ 90°) but not in the non-suppression area (0 ~ θ0). Consequently, it stabilizes the RTP and, at the same time, avoids any loss of RTP characteristics in the non-suppression area, resulting in enhanced RTP precision. It is noted that in curve features, the RTP factor of the AF method using θ0 = 45° is close to that of the PI method using I′ = 30°.
ð4Þ where H(D + θ0) represents the value of the routine RTP factor at the threshold angle. When θ = D ± θ0 (i.e., at the threshold angle), the antisymmetric factor HAF(θ) = 2H(D + θ0) − H[2(D ± θ0) − (D ± θ0)] = 2H(D + θ0) − H(D ± θ0) = H(D + θ0), equal to the routine RTP factor. Again, taking the case of the magnetic equator as an example, where I = 0°, we calculate the antisymmetric factors using the θ0 values of 45°, 60°, 75° and 90°, respectively, and plot their curves in Fig. 2. As shown in Fig. 2, in the suppression area (θ0 ∼ 90°), the RTP factor of the AF method is antisymmetric with that of the routine RTP method with respect to the central axis of the threshold angle, and it transits naturally and continuously from one side to the other side of this angle. When θ0 = 90°, there is HAF(θ) = H(θ), i.e., the RTP factor is equivalent to that of the routine RTP method without suppression of amplification effects.
3. Data experiments 3.1. Test on the synthetic data The synthetic model is composed of a single prism in a nonsusceptible background. Assume a 64 × 64 observational grid with spacing 1 m at altitude 0 m. Using this grid model, we calculate the TMI anomaly of horizontal magnetization at the magnetic equator and that of vertical magnetization at the magnetic pole, respectively shown in Fig. 3(a) and (b). Fig. 3(a) clearly shows that the negative values dominate the TMI anomaly of horizontal magnetization coupled with small-scale positive values, while Fig. 3(b) shows the central high above the prism and four negative sidelobes. Then, we add Gaussian random noise, with zero
Fig. 7. The RTP results of the denoised TMI anomalies along the profiles of X = 23 m and X = 33 m by different methods. The black, green, blue and red lines correspond to the theoretical RTP anomalies without noise, the routine RTP method, the PI method and the AF method, respectively.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 8. The real TMI anomalies in the southwestern sub-basin of the South China Sea and its surroundings.
107
the anomaly is stretched in the direction perpendicular to the declination, due to the unexpected suppression in other directions outside the declination. Fig. 4(c) shows that the serious strips along the declination completely obscure the real anomaly, due to no suppression of amplification by the routine RTP method. Fig. 5 shows that the RTP result of the AF method is closer to the true values than the other two methods. It also shows that short-wavelength noise has an obvious influence on both the AF method and the PI method. Thus, in practice, application of either is better after noise removal or reduction. Therefore, we denoise the noisy TMI anomaly by the low-pass filtering with a cut-off wavelength of 10 m, and then conduct the RTP test again by using the three RTP method, respectively. In this case, a pseudo inclination of 30° is taken for the PI method, and the threshold angle of 60° for the AF method. Fig. 6 displays the RTP results of the three methods, and Fig. 7 demonstrates their RTP results along the profiles of X = 0 m and Y = 0 m. From the two figures we see that all the three RTP results get better after noise reduction. The result of the AF method is much closer to the true values than the other two. The strips in the result of the routine RTP method are greatly alleviated but still serious. 3.2. Test on the real data
mean and a standard deviation of 1nT, to the TMI anomaly of horizontal magnetization, shown in Fig. 3(c). We then conduct RTP tests on the noisy TMI anomaly of horizontal magnetization by using the AF method, the PI method and the routine RTP method, respectively. In a wavenumber domain, the random noise concentrates largely in short-wavenumber components. As an amplifying function, the RTP factor must enlarge noise. Thus, typically short-wavenumber noise should be eliminated or reduced first before the RTP. Nevertheless, here, we perform RTP directly to the noisy anomaly, in order to test the RTP effects by different methods as well as to evaluate the influence of short-wavenumber noise on each method. In this case, a pseudo inclination of 30° is taken for the PI method, and the threshold angle of 60° for the AF method. Fig. 4 displays the RTP results of the three methods, and Fig. 5 demonstrates their RTP results along the profiles of X = 0 m and Y = 0 m. Fig. 4(a) clearly shows the central high above the prism and two negative sidelobes with little striping along the magnetic declination, indicating effective suppression of amplification by the AF method. The amplitude of the RTP result agrees well with the true value, except that gradients in the east–west direction are weaker than their true values. Fig. 4(b) shows a similar result to Fig. 4(a), implying effective suppression of amplification by the PI method. However, gradients in the east–west direction are much weaker than their true values and
The real TMI anomalies data come from the magnetic survey in the southwestern sub-basin of the South China Sea (SCS) and its surroundings. We assemble the data from the database of East Asian geomagnetic anomalies compiled by the Geological Survey of Japan and Coordinating Committee for Coastal and Offshore Geoscience Programmes in East and Southeast Asia (CCOP) (1996). This database collects the ship-borne and air-borne magnetic survey results in the SCS over the years, with remarkable coverage and accuracy. Fig. 8 shows the assembled TMI anomalies in the studied area with a data grid of 0.02° × 0.02°. We also calculate the magnetic inclination and declination of the geomagnetic field in this area from the IGRF model, respectively shown in Fig. 9(a) and (b). They demonstrate that the studied area locates at low magnetic latitudes, and the magnetic inclination changes much from south to north in a range of 9.37°–18.44°, while the magnetic declination varies gently in a range of −1.35° to −0.48°. Thus, an appropriate RTP method with a variable magnetic inclinations algorithm should be used in this area in order to obtain a good RTP result. Here, we suppose to neglect the influence of remanent magnetization in this area and use the AF method to test the data. Our algorithm for RTP with variable magnetic inclinations is described in brief as the following. First, the studied area is divided into several equal-interval latitudinal zones, each of which is regarded as a small sub-area and
Fig. 9. The inclination (a) and declination (b) of the geomagnetic field in the studied area. The units of the inclination and declination are degree.
108
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 10. The RTP results of the real TMI anomalies by different methods with the variable magnetic inclinations algorithm: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°.
given its distinct average inclination and declination. Then, the RTP is performed on the observed data of each sub-area in turn using their own average inclination and declination. Lastly, the RTP results of all sub-areas are merged by using a linear weighted function. In this case, the threshold angle of 60° is taken for the AF method and a latitude interval of 0.5° for the variable inclinations algorithm. We also use the PI method with a pseudo inclination of 30° to test the data for comparisons. Considering the contamination of short-wavelength noise in the real data, we denoise the data before the RTP. Fig. 10 shows the RTP results of the two RTP methods, and Fig. 11 displays their RTP results along the profiles of 114°E and 115°E.
From the two figures, we see that both two RTP results seem highly the same in comparison, however, the amplitude of the RTP anomalies by the AF method is slightly stronger than that of the PI method, implying that the former retains more effective RTP characteristics than the latter. After the RTP, the negative values in the original TMI anomalies are mostly transformed into positive values, while the original positive values mostly change into negative. In the east, middle and south of the studied area, alternating positive and negative linear magnetic anomalies are clearly present, trending in a northeast direction. They are interpreted to be caused by the basaltic oceanic crust, which is formed jointly by the Cenozoic seafloor spreading and the repeated reversals
Fig. 11. The RTP results of the real TMI anomalies along the profiles of 114°E and 115°E by different methods. The black, red and blue lines correspond to the observed magnetic total field anomalies, the AF method and the PI method, respectively.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
of the Earth's magnetic field during varied ages. The magnetic anomalies in the northwestern area are analyzed to be primarily caused by igneous or magmatic rocks there. 4. Conclusions We have presented the newly antisymmetric factor method for magnetic reduction to the pole at low latitudes. In this method, the antisymmetric factor is used in a given scope of directions centered along the magnetic declination to suppress amplification of the RTP factor, making RTP stable. Meanwhile the routine RTP factor is employed in other directions, so that the effective RTP features and the accuracy are preserved as much as possible. Similar to other RTP methods at low latitudes in the wavenumber domain, which are based on modification of the RTP factor, the antisymmetric factor method has the advantages of few control parameters, fast and stable RTP, and applicable to large-scale magnetic data. The tests on the synthetic data and the real data from the South China Sea have proved the effectiveness of this new method. Acknowledgments We thank two anonymous reviewers for their helpful comments and valuable suggestions. We are grateful for the financial support of the National Natural Science Foundation of China (40904033), the Fundamental Research Funds for the Central Universities and the SinoProbe projects (201011039, 201011049-03).
109
References Clarke, G.K.C., 1969. Optimum second-derivative and downward continuation filters. Geophysics 34, 424–437. Geological Survey of Japan and Coordinating Committee for Coastal and Offshore Geoscience Programmes in East and Southeast Asia (CCOP), 1996. Magnetic anomaly map of East Asia 1:4,000,000 CD-ROM Version. Digital Geoscience Map 2(P-1). Gunn, P.J., 1975. Linear transformations of gravity and magnetic fields. Geophysical Prospecting 23, 300–312. Hansen, R.O., Pawlowski, R.S., 1989. Reduction to the pole at low latitudes by Wiener filtering. Geophysics 54, 1607–1613. Keating, P., Zerbo, L., 1996. An improved technique for reduction to the pole at low latitudes. Geophysics 61, 131–137. Li, X., 2008. Magnetic reduction-to-the-pole at low latitudes: observations and considerations. The Leading Edge 27, 990–1002. Li, Y., Oldenburg, D.W., 2001. Stable reduction to the pole at the magnetic equator. Geophysics 66, 571–578. Macleod, I.N., Jones, K., Dai, T.F., 1993. 3-D analytic signal in the interpretation of total magnetic field data at low magnetic latitudes. Exploration Geophysics 24, 679–688. Phillips, J.D., 1997. Potential-Field Geophysical Software for the PC, Version 2.2. USGS Open-File Report 97–725. Shi, L., Guo, L.H., Meng, X.H., et al., 2012. The modified pseudo inclination method for magnetic reduction to the pole at low latitudes. Chinese Journal of Geophysics 55, 1775–1783 (in Chinese). Silva, J.B.C., 1986. Reduction to the pole as an inverse problem and its application to low-latitude anomalies. Geophysics 51, 369–382. Spector, A., Grant, F.S., 1970. Statistical models for interpreting aeromagnetic data. Geophysics 35, 293–302. Wiener, N., 1949. Extrapolation, interpolation, and smoothing of stationary time series. John Wiley & Sons Inc., USA. Yao, C.L., Guan, Z.N., Gao, D.Z., et al., 2003. Reduction to the pole of magnetic anomalies at low latitude with suppression filter. Chinese Journal of Geophysics 46, 690–696 (in Chinese).
Contents lists available at SciVerse ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
The antisymmetric factor method for magnetic reduction to the pole at low latitudes Lianghui Guo a,⁎, Lei Shi b, Xiaohong Meng a a b
Key Laboratory of Geo-detection (China University of Geosciences, Beijing), Ministry of Education, Beijing 100083, China Institute of Geophysics, China Earthquake Administration, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 3 November 2012 Accepted 19 February 2013 Available online 14 March 2013 Keywords: Total magnetic intensity anomaly Reduction to the pole Low latitude Antisymmetric factor The South China Sea
a b s t r a c t We analyze the characteristics of the wavenumber-domain factor for magnetic reduction to the pole (RTP) at low latitudes, and then propose a new wavenumber-domain method for RTP at low latitudes, herein called the antisymmetric factor method, based on modification of the RTP factor. The method applies the antisymmetric factor in a given scope of directions centered along the magnetic declination to suppress amplification of the RTP factor, stabilizing the RTP. Meanwhile it utilizes the routine RTP factor in other directions to preserve the effective RTP features. The test on the synthetic data demonstrates that the method is robust and effective. Finally, we use the new method, as well as a variable magnetic inclinations algorithm, to perform RTP on the real data of total magnetic intensity anomalies in the South China Sea, and obtain the reliable RTP anomalies. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The reduction-to-the-pole (RTP) of magnetic anomalies is an important task in the interpretation that transforms total magnetic intensity (TMI) anomalies in oblique magnetization into that as in a vertical magnetization. Thus, the complexity of the TMI anomalies caused by oblique magnetization can be eliminated. However, the RTP factor is one of the amplifying transformation factors. It is related directly to the magnetic inclination. At low latitudes (an absolute inclination less than 20°), the smaller the absolute value of the magnetic inclination is, the stronger the amplification effect of the RTP factor will be. Such amplification upon the noise in the TMI anomalies (which always exists in the real world) causes linear artifacts along the direction of magnetic declination. Therefore, the RTP procedure at low latitudes is much more troublesome than at high latitudes (an absolute inclination larger than 20°). To overcome the difficulty of RTP at low latitudes, several special RTP methods or techniques were proposed, such as the equivalent source inversion method (Silva, 1986), the Werner filtering (Hansen and Pawlowski, 1989), the energy balance technique (Keating and Zerbo, 1996), the inversion-based method (Li and Oldenburg, 2001), the pseudo inclination method (Macleod et al., 1993), the azimuthal filtering (Phillips, 1997) and the suppressing factor method (Yao et al., 2003). The equivalent source inversion method solves the RTP instability by inversing the magnetic data to derive equivalent sources and then doing forward modeling to produce the RTP data. However, the inversion requires expensive computation, making the method unsuitable ⁎ Corresponding author. Tel.: +86 1082322648. E-mail address: [email protected] (L. Guo). 0926-9851/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jappgeo.2013.02.018
for large-scale magnetic data. The Wiener filtering method utilizes a regularized RTP operator based on Wiener filter (Clarke, 1969; Wiener, 1949), which is a denoising filter. But the method tend to overly smooth the RTP result and to lose signal at short wavelengths (Li, 2008). The energy balance technique suppresses noise effects by iterative blanking so that the average energy is similar in all directions while retaining some signal at short wavelengths. However, the technique is subjective in choosing the related noise assumption and control parameters. The inverse-based method constructs the RTP anomalies model by inversing the observed magnetic data in the wavenumber domain with explicit regularization and an imposition of power spectral decay. But the application of the method is limited due to the complicated inversion and its expensive computation. All of the pseudo inclination method, the azimuthal filtering and the suppressing filter method are based on modifying the RTP factor in the wavenumber domain to suppress the amplification effect along and near the direction of magnetic declination. The pseudo inclination method replaces the actual magnetic inclination with a larger pseudo inclination in the RTP calculations. The azimuthal filtering tapers the RTP factor by a special sine function, while the suppressing filter method applies a special cosine function. However, all the three methods apt to lose parts of the effective RTP anomalies, respectively because the pseudo inclination method also suppresses the amplification effect in other directions outside the declination, the suppressing filter method overly suppress the amplification effect along the declination (Shi et al., 2012) and the azimuthal filtering has the similar limitation. In this paper, we first analyze the characteristics of the RTP factors of the routine RTP method and the pseudo inclination method, as well as their drawbacks in applications in RTP at low latitudes. Then we optimize the RTP factor and propose an antisymmetric factor method
104
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
for RTP at low latitudes. The method applies the antisymmetric factor in a certain scope of directions centered along the magnetic declination, while it utilizes the routine RTP factor in other directions. Finally, we test the method both on the synthetic magnetic data and on the real TMI anomalies data in the South China Sea. The routine RTP method and the pseudo inclination method are also used to test the data for comparisons. 2. Method 2.1. The RTP methods based on modifying the RTP factor In magnetic exploration, we usually observe the TMI anomaly. The corresponding observation direction coincides with the geomagnetic field. Suppose that the remanent magnetization can be neglected and the magnetization direction is consistent with the geomagnetic field. Then the RTP factor in a wavenumber domain can be written in a polar coordinate system as (Gunn, 1975; Macleod et al., 1993; Spector and Grant, 1970) H ðr; θÞ ¼ HðθÞ ¼
1 ; ½ sinðIÞ þ i cosðIÞ cosðD−θÞ2
ð1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where, r ¼ u2 þ v2 , θ = tg−1(v/u), u and v are respectively the wavenumbers in x and y directions, I and D areprespectively the inclinaffiffiffiffiffiffiffiffi tion and declination of magnetization, and i ¼ −1. The RTP factor H(θ) in Eq. (1) is one of the amplifying transformation factors, which is a mono-function of angle θ, while directly related to magnetic inclination I and declination D. At low latitudes, the absolute value of I is relatively small. When θ approaches D ± 90°, the amplitude of H(θ) grows rapidly to large values. In extreme conditions, I = 0 and θ = D ± 90°, there is H(θ) → − ∞. Such amplification effect of H(θ) at low latitudes makes calculation of RTP very unstable, yielding notable stripes (linear artifacts) along the magnetic declination D in the RTP results. Therefore, it is necessary to modify the RTP factor to suppress the amplification effect along the magnetic declination, so that calculation will be stable and the stripes will be reduced or even vanish. Three such methods include the pseudo inclination method (Macleod et al., 1993), the azimuthal filtering (Phillips, 1997) and the suppressing factor method (Yao et al., 2003). Here, we select the pseudo inclination method as an example to analyze the characteristics of the RTP factor. The RTP factor of the pseudo inclination (PI) method in a polar coordinate system is written as (Macleod et al., 1993)
Fig. 1. The features of the RTP factor of the PI method at the magnetic equator. The thin solid line, dotted line, dashed line and thick solid line correspond to the pseudo inclination of 0°, 30°, 60° and 90°, respectively.
inclination (I′) is, the stronger such suppression becomes. Extremely, when I′ = 90°,there is HPI(θ) = − 1, and the suppression reaches the maximum. However, the RTP factor of the PI method suppresses the amplification effects not only along the magnetic declination but also in other directions. The suppression of other directions unexpectedly weakens the RTP characteristics and thus reduces RTP precision. To solve this problem, Li (2008) proposed an improved algorithm for the PI method, which uses the RTP factor of the PI method (HPI(θ)) to suppress amplification within a wedge-shaped segment centered along the magnetic declination, and uses the routine RTP factor (H(θ)) in other directions to preserve amplification. Nevertheless, such an algorithm brings about the problem of discontinuity at the conjunction between the two different factors.
2
H PI ðθÞ ¼
½ sinðIÞ−i cosðIÞ cosðD−θÞ ; sin I′ þ cos2 I′ cos2 ðD−θÞ ⋅ sin2 ðIÞ þ cos2 ðIÞ cos2 ðD−θÞ 2
ð2Þ where I′ is the pseudo inclination defined by the user, which is larger than the real one I. If |I′| b |I|, there is I′ = I. In practice, the absolute value of I′ is often set between 20° and 30° (Macleod et al., 1993), and it needs to be larger for stronger noise in the observed anomalies data (Li, 2008). In the extreme case of the magnetic equator, where I = 0°, the substitution into Eq. (2) yields H PI ðθÞ ¼
−1 : sin2 I′ þ cos2 I ′ cos2 ðD−θÞ
ð3Þ
Then, substituting the pseudo inclinations of I′ = 0°, 30°, 60° and 90° into Eq. (3), respectively, we obtain the corresponding RTP factors (thin solid line, dotted line, dashed line and thick solid line in Fig. 1, respectively). When I′ = 0°, there is HPI(θ) = H(θ), i.e., the RTP factor is equivalent to that of the routine RTP method without suppression of amplification effects. When I′ > 0°, the RTP factors suppress amplification in every direction to stabilize RTP. The larger the pseudo
Fig. 2. The features of the RTP factor of the AF method at the magnetic equator. The black, red, blue and green lines correspond to the threshold angles of 90°, 75°, 60° and 45°, respectively. The purple curve is the RTP factor of the PI method with a pseudo inclination of 30°.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
105
Fig. 3. The theoretical TMI anomaly of the synthetic model at the magnetic equator (a), and that at the magnetic pole (b), and the noisy TMI anomaly at the magnetic equator (c). The contour interval of the anomaly is 5 nT.
Fig. 4. The RTP results of the noisy TMI anomalies by different methods: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°; (c) the routine RTP method. The contour interval of the anomaly is 5 nT.
Fig. 5. The RTP results of the noisy TMI anomalies along the profiles of X = 23 m and X = 33 m by different methods. The black, green, blue and red lines correspond to the theoretical RTP anomalies without noise, the routine RTP method, the PI method and the AF method, respectively.
2.2. The antisymmetric factor method Based on the RTP factor characteristics stated above, we propose a new RTP method at low latitudes, herein called the antisymmetric factor (AF) method, based on modification of the RTP factor. We set
a threshold angle θ0(45° ≤ θ0 ≤ 90°) for suppression. We design a special RTP factor to suppress the amplification of RTP factor when |D − θ| > θ0, while adopt the routine RTP factor to preserve the amplification when |D − θ| ≤ θ0. Meanwhile, to make the transition from the routine RTP factor to the special RTP factor continuous and natural at
106
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 6. The RTP results of the denoised TMI anomalies by different methods: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°; (c) the routine RTP method. The contour interval of the anomaly is 5 nT.
the threshold angle, we take the antisymmetric mapping of the routine RTP factor as the special factor, with respect to the central axis of the threshold angle. Here we call the special factor the antisymmetric factor. The RTP factor of the AF method is defined in a polar coordinate system as 8 When ðD−θ0 Þ≤θ≤ðD þ θ0 Þ; H AF ðθÞ ¼ H ðθÞ > > < When D−90B ≤θbðD−θ0 Þ; HAF ðθÞ ¼ 2HðD þ θ0 Þ−H½2ðD−θ0 Þ−θ > > : When ðD þ θ Þbθ≤D þ 90B; H ðθÞ ¼ 2H ðD þ θ Þ−H ½2ðD þ θ Þ−θ; 0 AF 0 0
As θ0 becomes smaller, the suppression area becomes wider, and the AF method performs wider and stronger suppression. In general, the threshold angle needs to be smaller for stronger noise in the observed anomalies data. In contrast to the PI method, the AF method suppresses amplification in the suppression area (θ0 ∼ 90°) but not in the non-suppression area (0 ~ θ0). Consequently, it stabilizes the RTP and, at the same time, avoids any loss of RTP characteristics in the non-suppression area, resulting in enhanced RTP precision. It is noted that in curve features, the RTP factor of the AF method using θ0 = 45° is close to that of the PI method using I′ = 30°.
ð4Þ where H(D + θ0) represents the value of the routine RTP factor at the threshold angle. When θ = D ± θ0 (i.e., at the threshold angle), the antisymmetric factor HAF(θ) = 2H(D + θ0) − H[2(D ± θ0) − (D ± θ0)] = 2H(D + θ0) − H(D ± θ0) = H(D + θ0), equal to the routine RTP factor. Again, taking the case of the magnetic equator as an example, where I = 0°, we calculate the antisymmetric factors using the θ0 values of 45°, 60°, 75° and 90°, respectively, and plot their curves in Fig. 2. As shown in Fig. 2, in the suppression area (θ0 ∼ 90°), the RTP factor of the AF method is antisymmetric with that of the routine RTP method with respect to the central axis of the threshold angle, and it transits naturally and continuously from one side to the other side of this angle. When θ0 = 90°, there is HAF(θ) = H(θ), i.e., the RTP factor is equivalent to that of the routine RTP method without suppression of amplification effects.
3. Data experiments 3.1. Test on the synthetic data The synthetic model is composed of a single prism in a nonsusceptible background. Assume a 64 × 64 observational grid with spacing 1 m at altitude 0 m. Using this grid model, we calculate the TMI anomaly of horizontal magnetization at the magnetic equator and that of vertical magnetization at the magnetic pole, respectively shown in Fig. 3(a) and (b). Fig. 3(a) clearly shows that the negative values dominate the TMI anomaly of horizontal magnetization coupled with small-scale positive values, while Fig. 3(b) shows the central high above the prism and four negative sidelobes. Then, we add Gaussian random noise, with zero
Fig. 7. The RTP results of the denoised TMI anomalies along the profiles of X = 23 m and X = 33 m by different methods. The black, green, blue and red lines correspond to the theoretical RTP anomalies without noise, the routine RTP method, the PI method and the AF method, respectively.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 8. The real TMI anomalies in the southwestern sub-basin of the South China Sea and its surroundings.
107
the anomaly is stretched in the direction perpendicular to the declination, due to the unexpected suppression in other directions outside the declination. Fig. 4(c) shows that the serious strips along the declination completely obscure the real anomaly, due to no suppression of amplification by the routine RTP method. Fig. 5 shows that the RTP result of the AF method is closer to the true values than the other two methods. It also shows that short-wavelength noise has an obvious influence on both the AF method and the PI method. Thus, in practice, application of either is better after noise removal or reduction. Therefore, we denoise the noisy TMI anomaly by the low-pass filtering with a cut-off wavelength of 10 m, and then conduct the RTP test again by using the three RTP method, respectively. In this case, a pseudo inclination of 30° is taken for the PI method, and the threshold angle of 60° for the AF method. Fig. 6 displays the RTP results of the three methods, and Fig. 7 demonstrates their RTP results along the profiles of X = 0 m and Y = 0 m. From the two figures we see that all the three RTP results get better after noise reduction. The result of the AF method is much closer to the true values than the other two. The strips in the result of the routine RTP method are greatly alleviated but still serious. 3.2. Test on the real data
mean and a standard deviation of 1nT, to the TMI anomaly of horizontal magnetization, shown in Fig. 3(c). We then conduct RTP tests on the noisy TMI anomaly of horizontal magnetization by using the AF method, the PI method and the routine RTP method, respectively. In a wavenumber domain, the random noise concentrates largely in short-wavenumber components. As an amplifying function, the RTP factor must enlarge noise. Thus, typically short-wavenumber noise should be eliminated or reduced first before the RTP. Nevertheless, here, we perform RTP directly to the noisy anomaly, in order to test the RTP effects by different methods as well as to evaluate the influence of short-wavenumber noise on each method. In this case, a pseudo inclination of 30° is taken for the PI method, and the threshold angle of 60° for the AF method. Fig. 4 displays the RTP results of the three methods, and Fig. 5 demonstrates their RTP results along the profiles of X = 0 m and Y = 0 m. Fig. 4(a) clearly shows the central high above the prism and two negative sidelobes with little striping along the magnetic declination, indicating effective suppression of amplification by the AF method. The amplitude of the RTP result agrees well with the true value, except that gradients in the east–west direction are weaker than their true values. Fig. 4(b) shows a similar result to Fig. 4(a), implying effective suppression of amplification by the PI method. However, gradients in the east–west direction are much weaker than their true values and
The real TMI anomalies data come from the magnetic survey in the southwestern sub-basin of the South China Sea (SCS) and its surroundings. We assemble the data from the database of East Asian geomagnetic anomalies compiled by the Geological Survey of Japan and Coordinating Committee for Coastal and Offshore Geoscience Programmes in East and Southeast Asia (CCOP) (1996). This database collects the ship-borne and air-borne magnetic survey results in the SCS over the years, with remarkable coverage and accuracy. Fig. 8 shows the assembled TMI anomalies in the studied area with a data grid of 0.02° × 0.02°. We also calculate the magnetic inclination and declination of the geomagnetic field in this area from the IGRF model, respectively shown in Fig. 9(a) and (b). They demonstrate that the studied area locates at low magnetic latitudes, and the magnetic inclination changes much from south to north in a range of 9.37°–18.44°, while the magnetic declination varies gently in a range of −1.35° to −0.48°. Thus, an appropriate RTP method with a variable magnetic inclinations algorithm should be used in this area in order to obtain a good RTP result. Here, we suppose to neglect the influence of remanent magnetization in this area and use the AF method to test the data. Our algorithm for RTP with variable magnetic inclinations is described in brief as the following. First, the studied area is divided into several equal-interval latitudinal zones, each of which is regarded as a small sub-area and
Fig. 9. The inclination (a) and declination (b) of the geomagnetic field in the studied area. The units of the inclination and declination are degree.
108
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
Fig. 10. The RTP results of the real TMI anomalies by different methods with the variable magnetic inclinations algorithm: (a) the AF method with the threshold angle of 60°; (b) the PI method with the pseudo inclination of 30°.
given its distinct average inclination and declination. Then, the RTP is performed on the observed data of each sub-area in turn using their own average inclination and declination. Lastly, the RTP results of all sub-areas are merged by using a linear weighted function. In this case, the threshold angle of 60° is taken for the AF method and a latitude interval of 0.5° for the variable inclinations algorithm. We also use the PI method with a pseudo inclination of 30° to test the data for comparisons. Considering the contamination of short-wavelength noise in the real data, we denoise the data before the RTP. Fig. 10 shows the RTP results of the two RTP methods, and Fig. 11 displays their RTP results along the profiles of 114°E and 115°E.
From the two figures, we see that both two RTP results seem highly the same in comparison, however, the amplitude of the RTP anomalies by the AF method is slightly stronger than that of the PI method, implying that the former retains more effective RTP characteristics than the latter. After the RTP, the negative values in the original TMI anomalies are mostly transformed into positive values, while the original positive values mostly change into negative. In the east, middle and south of the studied area, alternating positive and negative linear magnetic anomalies are clearly present, trending in a northeast direction. They are interpreted to be caused by the basaltic oceanic crust, which is formed jointly by the Cenozoic seafloor spreading and the repeated reversals
Fig. 11. The RTP results of the real TMI anomalies along the profiles of 114°E and 115°E by different methods. The black, red and blue lines correspond to the observed magnetic total field anomalies, the AF method and the PI method, respectively.
L. Guo et al. / Journal of Applied Geophysics 92 (2013) 103–109
of the Earth's magnetic field during varied ages. The magnetic anomalies in the northwestern area are analyzed to be primarily caused by igneous or magmatic rocks there. 4. Conclusions We have presented the newly antisymmetric factor method for magnetic reduction to the pole at low latitudes. In this method, the antisymmetric factor is used in a given scope of directions centered along the magnetic declination to suppress amplification of the RTP factor, making RTP stable. Meanwhile the routine RTP factor is employed in other directions, so that the effective RTP features and the accuracy are preserved as much as possible. Similar to other RTP methods at low latitudes in the wavenumber domain, which are based on modification of the RTP factor, the antisymmetric factor method has the advantages of few control parameters, fast and stable RTP, and applicable to large-scale magnetic data. The tests on the synthetic data and the real data from the South China Sea have proved the effectiveness of this new method. Acknowledgments We thank two anonymous reviewers for their helpful comments and valuable suggestions. We are grateful for the financial support of the National Natural Science Foundation of China (40904033), the Fundamental Research Funds for the Central Universities and the SinoProbe projects (201011039, 201011049-03).
109
References Clarke, G.K.C., 1969. Optimum second-derivative and downward continuation filters. Geophysics 34, 424–437. Geological Survey of Japan and Coordinating Committee for Coastal and Offshore Geoscience Programmes in East and Southeast Asia (CCOP), 1996. Magnetic anomaly map of East Asia 1:4,000,000 CD-ROM Version. Digital Geoscience Map 2(P-1). Gunn, P.J., 1975. Linear transformations of gravity and magnetic fields. Geophysical Prospecting 23, 300–312. Hansen, R.O., Pawlowski, R.S., 1989. Reduction to the pole at low latitudes by Wiener filtering. Geophysics 54, 1607–1613. Keating, P., Zerbo, L., 1996. An improved technique for reduction to the pole at low latitudes. Geophysics 61, 131–137. Li, X., 2008. Magnetic reduction-to-the-pole at low latitudes: observations and considerations. The Leading Edge 27, 990–1002. Li, Y., Oldenburg, D.W., 2001. Stable reduction to the pole at the magnetic equator. Geophysics 66, 571–578. Macleod, I.N., Jones, K., Dai, T.F., 1993. 3-D analytic signal in the interpretation of total magnetic field data at low magnetic latitudes. Exploration Geophysics 24, 679–688. Phillips, J.D., 1997. Potential-Field Geophysical Software for the PC, Version 2.2. USGS Open-File Report 97–725. Shi, L., Guo, L.H., Meng, X.H., et al., 2012. The modified pseudo inclination method for magnetic reduction to the pole at low latitudes. Chinese Journal of Geophysics 55, 1775–1783 (in Chinese). Silva, J.B.C., 1986. Reduction to the pole as an inverse problem and its application to low-latitude anomalies. Geophysics 51, 369–382. Spector, A., Grant, F.S., 1970. Statistical models for interpreting aeromagnetic data. Geophysics 35, 293–302. Wiener, N., 1949. Extrapolation, interpolation, and smoothing of stationary time series. John Wiley & Sons Inc., USA. Yao, C.L., Guan, Z.N., Gao, D.Z., et al., 2003. Reduction to the pole of magnetic anomalies at low latitude with suppression filter. Chinese Journal of Geophysics 46, 690–696 (in Chinese).