- Email: [email protected]
Advantages of horizontal directional Theta method to detect the edges of full tensor gravity gradient data
Journal of Applied Geophysics 130 (2016) 53–61
Contents lists available at ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
Advantages of horizontal directional Theta method to detect the edges of full tensor gravity gradient data Yuan Yuan a,b,⁎, Jin-Yao Gao a,b, Ling-Na Chen c a b c
Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China Key Laboratory of Submarine Geoscience, State Oceanic Administration, Hangzhou 310012, China College of Geo-exploration Science and Technology, Jilin University, Changchun 130026, China
a r t i c l e
i n f o
Article history: Received 31 August 2015 Received in revised form 19 April 2016 Accepted 19 April 2016 Available online 22 April 2016 Keywords: Full tensor gravity gradient data Horizontal directional Theta method Edge detection
a b s t r a c t Full tensor gravity gradient data contain nine signal components. They include higher frequency signals than traditional gravity data, which can extract the small-scale features of the sources. Edge detection has played an important role in the interpretation of potential-field data. There are many methods that have been proposed to detect and enhance the edges of geological bodies based on horizontal and vertical derivatives of potentialfield data. In order to make full use of all the measured gradient components, we need to develop a new edge detector to process the full tensor gravity gradient data. We first define the directional Theta and use the horizontal directional Theta to define a new edge detector. This method was tested on synthetic and real full tensor gravity gradient data to validate its feasibility. Compared the results with other balanced detectors, the new detector can effectively delineate the edges and does not produce any additional false edges. © 2016 Elsevier B.V. All rights reserved.
1. Introduction With the developing of the full tensor gravity gradient measuring techniques, more and more gravity gradient tensor data have been widely used in geophysical exploration for its large amount on information and containing higher frequency signals than traditional gravity data. The full tensor gravity gradient technique can simultaneously measure six gradient components. Each component has its own geophysical meaning. The high frequency gravity gradient tensor data can be used to delineate the small scale anomalies and investigate the geological structure details. Edge detection is a required task in interpreting the potential field data, and has been widely used in exploration technology for discovery of mineral resources, energy resources and regional tectonics. The main geological edges are fault lines and the boundaries of geological or rock bodies of different densities, magnetic natures, etc. Many traditional methods are employed to outline the edges. Most of them are based on the horizontal derivatives and vertical derivative of potential field data, such as total horizontal derivative, analytic signal, and so on (Evjen, 1936; Cordell, 1979; Cordell and Grauch, 1985; Roest et al., 1992). However, all of them cannot equal the amplitude of the edges of shallow and deep geological bodies simultaneously. In order to display the edges of large and small anomalies simultaneously, some balanced filters have been proposed (Miller and Singh, 1994; ⁎ Corresponding author at: Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China. E-mail address: [email protected] (Y. Yuan).
http://dx.doi.org/10.1016/j.jappgeo.2016.04.009 0926-9851/© 2016 Elsevier B.V. All rights reserved.
Verduzco and Fairhead, 2004; Wijns et al., 2005; Cooper and Cowan, 2008; Ma and Li, 2012). However, in order to make full use of the full tensor gravity gradient data, new edge detection method is required to interpret these data, especially for small geological structures. The maximum eigenvalue is a widely used tool to interpret the edges. Oruc et al. (2013) and Zhou et al. (2013) used the maximum eigenvalue of the curvature of the gravity gradient tensor data to delineate the edges. Wang et al. (2015) make some changes, using the principal component analysis of curvature of the gravity gradient tensor data, to identify the edges. Sertcelik and Kafadar (2012) and Yuan et al. (2014) used the eigenvalue of structure tensor of gravity gradient data to outline the edges. Zuo and Hu (2015) used the eigenvalue of the gravity gradient tensor matrix to extract the edges. Beyond these, some directional methods have been used to interpret the full tensor gravity gradient data. Cooper and Cowan (2006) and Oruc and Keskinsezer (2008) have defined the directional tilt angles to delineate edges, but only the vertical direction tilt angle, namely the tilt angle described by Miller and Singh (1994), can get the good results. Mikhailov et al. (2007)and Beiki (2010) proposed the directional analytic signal to extract the edges. However, this method cannot equal to the edge amplitude size of different anomalies. Yuan and Yu (2015) have made an improvement on the directional analytic signal. They defined second order directional analytic signal and proposed a normalization method, which can effectively enhance the large and small amplitude anomalies. Marson and Klingele (1993) have pointed out that the resolution of total horizontal derivative is higher than analytic signal. Therefore, Yuan et al. (2015) defined the directional total horizontal derivative and enhanced directional total horizontal derivatives
54
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 1. Plan view and 3D view of the synthetic model.
and use them to define new edge detectors. In order to remove the additional false edges, they introduce a constant parameter in the denominator of the normalization method. However, this removing false edge
method is subjective. In this paper, we propose the directional Theta method, and use the horizontal direction Theta to define a new edge detector to delineate the edges of full tensor gravity gradient data.
Fig. 2. Synthetic gravity gradient tensor data of model 1. (a) Gxx; (b) Gyy; (c) Gzz; (d) Gzx; (e) Gzy; (f) Gyx.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
55
Fig. 3. Edge results of horizontal directional analytic signal, total horizontal derivatives and Theta method. (a) ASx of the data in Fig. 2; (b) ASy of the data in Fig. 2; (c) THDx of the data in Fig. 2; (d) THDy of the data in Fig. 2; (e) ThetaX of the data in Fig. 2; (f) ThetaY of the data in Fig. 2.
2. Directional Theta method
Mikhailov et al. (2007) has defined the directional analytic signal: ASx, ASy and ASz, the expressions are:
Gravity gradient tensors are the space derivatives of the components of the gravity vector g = (gx, gy, gz) in the three orthogonal directions x, y and z. The full tensor gravity gradient can be expressed as 2
∂g x 6 ∂x 6 6 ∂g 6 T ¼6 y 6 ∂x 6 4 ∂g z ∂x
∂g x ∂y ∂g y ∂y ∂g z ∂y
ASy ¼ Gyx 2 þ Gyy 2 þ Gyz 2 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ASz ¼ Gzx 2 þ Gzy 2 þ Gzz 2
3
∂g x ∂z 7 7 2G xx ∂g y 7 7 4 7 ¼ Gyx ∂z 7 Gzx 7 ∂g z 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 2 2 > > xx þ Gxy þ Gxz > ASx ¼ qG < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Gxy Gyy Gzy
3 Gxz Gyz 5: Gzz
ð1Þ
∂z
Based on the gravitational potential satisfy the Laplace equation, and the matrix is symmetric, there are only five independent components Gxx, Gxz, Gyx, Gyy and Gyz. However, the full tensor gravity gradiometer measures the six gradient tensor components Gxx, Gxz, Gyx, Gyy, Gyz and Gzz.
ð2Þ
And Beiki (2010) used it to detect the edges of gravity gradient tensor data. Yuan and Geng (2014) and Yuan et al. (2015) have defined the directional total horizontal derivatives, THDx, THDy and THDz, to interpret the gravity gradient tensor data, the expressions are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > > THDx ¼ Gxy þ ðGxz Þ2 > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 þ G THDy ¼ G yx yz > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 : THDz ¼ ðGzx Þ2 þ Gzy
ð3Þ
56
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 4. Edge results of gravity gradient tensor data shown in Fig. 2. (a) edge of TDX; (b) edge of TDR_THDR; (c) edge of Theta map; (d) edge of ED.
which show a higher resolution than directional analytic signals. Based on these definitions, we propose the directional Theta methods, which can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gxx 2 þ Gyx 2 ThetaX ¼ −qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gxx 2 þ Gxy 2 þ Gxz 2
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gyx 2 þ Gyy 2 ThetaY ¼ −qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gyx 2 þ Gyy 2 þ Gyz 2
ð5Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gzx 2 þ Gzy 2 ThetaZ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gzx 2 þ Gzy 2 þ Gzz 2
ð6Þ
Here, we can see that the horizontal direction ThetaX, ThetaY compared with the vertical direction ThetaZ have negative sign, which is used to enhance the visual effect. The maximum value of ThetaX delineates the N–S edges; the maximum value of ThetaY indicates the W–E edges. The vertical direction ThetaZ is same with the traditional Theta map method (Wijns et al., 2005). The horizontal directional Theta ThetaX and ThetaY include all independent components. Therefore, we can combine ThetaX and ThetaY to define a new edge detector: ED ¼ ThetaX þ ThetaY:
ð7Þ
The maximum values of ED automatically recognize the edges of the source. In order to validate the feasibility of the method proposed above, we choose three another TDX (Cooper and Cowan, 2006), TDR_THDR (Verduzco and Fairhead, 2004) and Theta map (Wijns et al., 2005) to compare results.
3. Synthetic model experiments In this section, we construct a gravity gradient anomaly model, which include four prisms at a depth to top of 1 km (label 1), 1 km (label 2), 2 km (label 3) and 2 km (label 4). Fig. 1 displays the plan view and 3D view of the model. The residual densities of all prisms are 0.2 g/cm3. The synthetic gravity gradient tensor data is calculated on a regular grid with a spacing of 0.2 km, shown in Fig. 2. In order to indicate the advantage of directional Theta method, we compare them with directional analytic signals and directional total horizontal derivatives. The compared results are shown in Fig. 3. We can see that all horizontal directional analytic signal, total horizontal derivative and Theta method can outline the N–S and W–E edges of geological sources. However, the resolution of directional analytic signal and total horizontal derivatives is lower, especially the analytic signal. Also, these two methods cannot equal to the size of the amplitude of the edges. The directional Theta method cannot only extract the N–S and W–E edges precisely and clearly, also can balance the edge amplitude size. Based on this advantage, we can apply the constructed new edge detector ED to delineate the edges of geological sources. Fig. 4 displays the edge results of TDX, TDR_THDR, Theta map and new detector ED. We can see that all these four methods can identify the edges clearly and precisely. But the new detector ED has a higher resolution than the others. By comparing the results of Fig. 4c and d, we can obtain that horizontal directional Theta method has an improvement on the traditional Theta map method. From Figs. 3e, f and 4d, we can find that this new method may bring some individual maximum dots in the identified results. However, each individual dot cannot affect the final lineament structure interpretation. In order to validate the new method, we built a more complex model which contains both positive and negative anomalies. Here, the new model is same with the model above, but the contrasted densities of prisms 1 and 3 are 0.2 g/cm3 and the contrasted densities of prisms 2
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
and 4 are − 0.2 g/cm3. The synthetic gravity gradient anomalies are shown in Fig. 5. The identified edge results are shown in Fig. 6. By comparing the results, we can get that the traditional methods can well delineate the edges of different amplitude anomalies. But, they bring some additional false edge information when the model becomes complex, especially when the model contains both positive and negative anomalies simultaneously. These additional edges will lead to a false interpretation on the geological structure. However, our new method ED can avoid this disadvantage effectively. This method is fully automatic without any personal effect on removing the additional false edge information compared with some other methods, which need to bring a constant number in the normalization factor (Cooper, 2013; Li et al., 2014; Yuan et al., 2015). Therefore, this method is more objective. To further test the stability of the new edge detector, we add 5% Gaussian noise to the gravity gradient tensor data shown in Fig. 5. The edge results of TDX, TDR_THDR, Theta map and ED are shown in Fig. 7. We can see that TDR_THDR cannot delineate the edges of the sources due to the influence of noise. The methods TDX, Theta map and ED can all extract the edges clearly. But, the effect of noise to TDX and Theta map is bigger than the effect to ED.
57
4. Applied to real data In order to test the application in real case, we apply the new edge detector to real measured full tensor gravity gradient data by Bell Geospace with Air-FTG gradiometer over St. George's Bay, Newfoundland and Labrador. Flight lines were spaced 500 m apart with a northeast–southwest orientation whereas tie-lines were spaced at 5000 m intervals with a northwest–southeast orientation (Pilkigton and Shamsipour, 2014). The total kilometers of flight lines are 5992 km whereas the total kilometers of tie-lines are 610 km. The survey was flown at a nominal altitude of 100 m on a gentle drape surface. The 90 m SRTM elevation model was used for computing the drape surface. The data was leveled and denoised using the FFT method (Sanchez et al., 2005). Based on correlation tests, a density of 2.2 g/cm3 was chosen for terrain correction. The corrected full tensor gravity gradient data was shown in Fig. 8. The gradiometer survey covers most of the Carboniferous Bay St. George Subbasin, which is bounded to the south by the onshore northeast-trending Long Range Fault and constitutes part of the larger Maritimes Basin (Schillerif and Williams, 1979). Carboniferous and
Fig. 5. Synthetic gravity gradient tensor data of model 2. (a) Gxx; (b) Gyy; (c) Gzz; (d) Gzx; (e) Gzy; (f) Gyx.
58
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 6. Same with Fig. 4, but for the gravity gradient tensor data of synthetic model 2.
Fig. 7. Same with Fig. 4, but for the gravity gradient tensor data of model 2 corrupted by 5% Gaussian noise.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
59
Fig. 8. Full tensor gravity gradient data in St. George's Bay. (a) Gxx; (b) Gxz; (c) Gyy; (d) Gyz; (e) Gyx; (f) Gzz.
Upper Devonian sedimentary rocks that constitute the subbasin fill are exposed onshore to the north, east, and south, overlying Precambrian basement. These fill rocks consist of marine and nonmarine sequences of shale, siltstone, sandstone, and limestone, with salt bodies. The edge results of the full tensor gravity gradient data over St. George's Bay are shown in Fig. 9. The black lines in figure are the known edge mapped by Miller et al. (1990). We can find that the traditional method TDR_THDR cannot extract the edge information clearly and precisely, Fig. 9b. Methods TDX, Theta and ED all can recognize the edges. However, there are some individual maximum values areas outlined by red ellipse in Fig. 9c. We can consider that these are not the real edge structure. Also, the new method ED can extract a new edge clearly than the other method outlined by yellow ellipse. Therefore, we can consider that our new detector ED has a higher resolution, the outlined edges are clear and precise.
5. Conclusions Our main contribution by this study was to define the directional Theta method of full tensor gravity gradient data. Comparing with horizontal directional analytic signal and directional total horizontal derivatives, we can find that horizontal directional Theta has a higher resolution and can identify the edge of geological bodies clearly and precisely, which can effectively equal to the edge amplitude size of different amplitude gravity gradient anomalies. Therefore, we use them to define a new edge detector ED to delineate the edges of full tensor gravity gradient data. In order to validate the feasibility of the new method, we demonstrate it on three synthetic models. Compared the result of the new detector with three traditional edge identification methods. It has demonstrated that the new proposed method is an effective tool for edge detection, and can avoid bringing false edges when the geological
60
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 9. Edge results of St. George's Bay. (a) TDX; (b) TDR_THDR; (c) Theta; (d) ED.
bodies contain both positive and negative anomalies simultaneously, while tradition methods TDX, TDR_THDR and Theta may produce false edges. Finally, we apply it on real measured full tensor gravity gradient data over St. George's Bay, which get a well edge results. Acknowledgments We gratefully acknowledge the Natural Resources Canada for providing the full tensor gravity gradient data measured by Air-FTG in St. George's Bay, Canada. This work was financially supported by the basic scientific research business special fund project of Second Institute of Oceanography, State Oceanic Administration (14275-10); Comprehensive Environmental Investigation and Assessment of North and South Poles (CHINARE2015-01-03; CHINARE2015-04-01); National Marine Public Welfare Projects (2013418035). References Beiki, M., 2010. Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics 75 (6), I59–I74. Cooper, G.R.J., 2013. Reply to a discussion about the “hyperbolic tilt angle method” by Zhou et al. Comput. Geosci. 53, 496–497. Cooper, G.R.J., Cowan, D.R., 2006. Enhancing potential field data using filters based on the local phase. Comput. Geosci. 32, 1585–1591. Cooper, G.R.J., Cowan, D.R., 2008. Edge enhancement of potential-field data using normalized statistics. Geophysics 73, H1–H4. Cordell, L., 1979. Gravimetric expression of graben faulting in Santa Fe Country and the Espanola Basin, New Mexico. 30th Field Conf., 1979. New Mexico Geo. Soc. Guidebook, pp. 59–64. Cordell, L., Grauch, V.J.S., 1985. Mapping basement magnetization zones from aeromagnetic data in the San Juan Basin, New Mexico. In: Hinze, W.J. (Ed.), The Utility of Regional Gravity and Magnetic Maps. Society of Exploration Geophysics, pp. 181–197.
Evjen, H.M., 1936. The place of the vertical gradient in gravitational interpretations. Geophysics 1, 127–136. Li, L., Huang, D., Han, L., Ma, G., 2014. Optimised edge detection filters in the interpretation of potential field data. Explor. Geophys. 45 (3), 171–176. Ma, G.Q., Li, L.L., 2012. Edge detection in potential fields with the normalized total horizontal derivatives. Comput. Geosci. 41, 83–87. Marson, I., Klingele, E.E., 1993. Advantages of using the vertical gradient of gravity for 3-D interpretation. Geophysics 58 (11), 1588–1595. Mikhailov, V., Pajot, G., Diament, M., Price, A., 2007. Tensor deconvolution: a method to locate equivalent sources from full tensor gravity data. Geophysics 72 (5), 161–169. Miller, H.G., Singh, V., 1994. Potential field tilt—a new concept for location of potential field sources. J. Appl. Geophys. 32, 213–217. Miller, H.G., Kilfoil, G.J., Peavy, S.T., 1990. An integrated geophysical interpretation of the Carboniferous Bay St. George Subbasin, Western Newfoundland. Bull. Can. Petrol. Geol. 38, 320–331. Oruc, B., Keskinsezer, A., 2008. Structural setting of the northeastern Biga Peninsula (Turkey) from tilt derivatives of gravity gradient tensors and magnitude of horizontal gravity components. Pure Appl. Geophys. 165, 1913–1927. Oruc, B., Sertcelik, I., Kafadar, O., Selim, H., 2013. Structural interpretation of the Erzurum Basin, eastern Turkey, using curvature gravity gradient tensor and gravity inversion of basement relief. J. Appl. Geophys. 88, 105–113. Pilkigton, M., Shamsipour, P., 2014. Noise reduction procedures for gravity-gradiometer data. Geophysics 79 (5), G69–G78. Roest, W.R., Verhoef, J., Pilkington, M., 1992. Magnetic interpretation using the 3-D analytic signal. Geophysics 57, 116–125. Sanchez, V., Sinex, D., Li, Y., Nabighian, M., 2005. Processing and inversion of magnetic gradient tensor data for UXO applications. 18th EEGS Symposium on the Applications of Geophysics to Engineering and Environmental Problems, Extended Abstracts, pp. 1193–1202. Schillerif, S., Williams, H., 1979. Geology of the Stephenville map area, Newfoundland. Current Research, Part A, Geological Survey of Canada, Paper 79-1A, pp. 327–332. Sertcelik, I., Kafadar, O., 2012. Application of edge detection to potential field data using eigenvalue analysis of structure tensor. J. Appl. Geophys. 84, 127–136. Verduzco, B., Fairhead, J.D., 2004. New insights into magnetic derivatives for structural mapping. Lead. Edge 23 (2), 116–119. Wang, J., Meng, X.H., Li, F., 2015. Improved curvature gravity gradient tensor with principal component analysis and its application in edge detection of gravity data. J. Appl. Geophys. 118, 106–114.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61 Wijns, C., Perez, C., Kowalczyk, P., 2005. Theta map: edge detection in magnetic data. Geophysics 70, L39–L43. Yuan, Y., Geng, M., 2014. Directional total horizontal derivatives of gravity gradient tensor and their application to delineate the edges. 76th EAGE Conference & Exhibition 2014. Yuan, Y., Yu, Q., 2015. Edge detection in potential-field gradient tensor data by use of improved horizontal analytic signal methods. Pure Appl. Geophys. 172 (2), 461–472. Yuan, Y., Huang, D., Yu, Q., Lu, P., 2014. Edge detection of potential field data with improved structure tensor methods. J. Appl. Geophys. 108, 35–42.
61
Yuan, Y., Huang, D., Yu, Q., 2015. Using enhanced directional total horizontal derivatives to detect the edges of potential-field full tensor data. Chin. J. Geophys. 58 (7), 2556–2565 (in Chinese). Zhou, W.N., Du, X.J., Li, J.Y., 2013. The limitation of curvature gravity gradient tensor for edge detection and a method for overcoming it. J. Appl. Geophys. 98, 237–242. Zuo, B.X., Hu, X.Y., 2015. Edge detection of gravity field using eigenvalue analysis of gravity gradient tensor. J. Appl. Geophys. 114, 263–270.
Contents lists available at ScienceDirect
Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
Advantages of horizontal directional Theta method to detect the edges of full tensor gravity gradient data Yuan Yuan a,b,⁎, Jin-Yao Gao a,b, Ling-Na Chen c a b c
Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China Key Laboratory of Submarine Geoscience, State Oceanic Administration, Hangzhou 310012, China College of Geo-exploration Science and Technology, Jilin University, Changchun 130026, China
a r t i c l e
i n f o
Article history: Received 31 August 2015 Received in revised form 19 April 2016 Accepted 19 April 2016 Available online 22 April 2016 Keywords: Full tensor gravity gradient data Horizontal directional Theta method Edge detection
a b s t r a c t Full tensor gravity gradient data contain nine signal components. They include higher frequency signals than traditional gravity data, which can extract the small-scale features of the sources. Edge detection has played an important role in the interpretation of potential-field data. There are many methods that have been proposed to detect and enhance the edges of geological bodies based on horizontal and vertical derivatives of potentialfield data. In order to make full use of all the measured gradient components, we need to develop a new edge detector to process the full tensor gravity gradient data. We first define the directional Theta and use the horizontal directional Theta to define a new edge detector. This method was tested on synthetic and real full tensor gravity gradient data to validate its feasibility. Compared the results with other balanced detectors, the new detector can effectively delineate the edges and does not produce any additional false edges. © 2016 Elsevier B.V. All rights reserved.
1. Introduction With the developing of the full tensor gravity gradient measuring techniques, more and more gravity gradient tensor data have been widely used in geophysical exploration for its large amount on information and containing higher frequency signals than traditional gravity data. The full tensor gravity gradient technique can simultaneously measure six gradient components. Each component has its own geophysical meaning. The high frequency gravity gradient tensor data can be used to delineate the small scale anomalies and investigate the geological structure details. Edge detection is a required task in interpreting the potential field data, and has been widely used in exploration technology for discovery of mineral resources, energy resources and regional tectonics. The main geological edges are fault lines and the boundaries of geological or rock bodies of different densities, magnetic natures, etc. Many traditional methods are employed to outline the edges. Most of them are based on the horizontal derivatives and vertical derivative of potential field data, such as total horizontal derivative, analytic signal, and so on (Evjen, 1936; Cordell, 1979; Cordell and Grauch, 1985; Roest et al., 1992). However, all of them cannot equal the amplitude of the edges of shallow and deep geological bodies simultaneously. In order to display the edges of large and small anomalies simultaneously, some balanced filters have been proposed (Miller and Singh, 1994; ⁎ Corresponding author at: Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China. E-mail address: [email protected] (Y. Yuan).
http://dx.doi.org/10.1016/j.jappgeo.2016.04.009 0926-9851/© 2016 Elsevier B.V. All rights reserved.
Verduzco and Fairhead, 2004; Wijns et al., 2005; Cooper and Cowan, 2008; Ma and Li, 2012). However, in order to make full use of the full tensor gravity gradient data, new edge detection method is required to interpret these data, especially for small geological structures. The maximum eigenvalue is a widely used tool to interpret the edges. Oruc et al. (2013) and Zhou et al. (2013) used the maximum eigenvalue of the curvature of the gravity gradient tensor data to delineate the edges. Wang et al. (2015) make some changes, using the principal component analysis of curvature of the gravity gradient tensor data, to identify the edges. Sertcelik and Kafadar (2012) and Yuan et al. (2014) used the eigenvalue of structure tensor of gravity gradient data to outline the edges. Zuo and Hu (2015) used the eigenvalue of the gravity gradient tensor matrix to extract the edges. Beyond these, some directional methods have been used to interpret the full tensor gravity gradient data. Cooper and Cowan (2006) and Oruc and Keskinsezer (2008) have defined the directional tilt angles to delineate edges, but only the vertical direction tilt angle, namely the tilt angle described by Miller and Singh (1994), can get the good results. Mikhailov et al. (2007)and Beiki (2010) proposed the directional analytic signal to extract the edges. However, this method cannot equal to the edge amplitude size of different anomalies. Yuan and Yu (2015) have made an improvement on the directional analytic signal. They defined second order directional analytic signal and proposed a normalization method, which can effectively enhance the large and small amplitude anomalies. Marson and Klingele (1993) have pointed out that the resolution of total horizontal derivative is higher than analytic signal. Therefore, Yuan et al. (2015) defined the directional total horizontal derivative and enhanced directional total horizontal derivatives
54
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 1. Plan view and 3D view of the synthetic model.
and use them to define new edge detectors. In order to remove the additional false edges, they introduce a constant parameter in the denominator of the normalization method. However, this removing false edge
method is subjective. In this paper, we propose the directional Theta method, and use the horizontal direction Theta to define a new edge detector to delineate the edges of full tensor gravity gradient data.
Fig. 2. Synthetic gravity gradient tensor data of model 1. (a) Gxx; (b) Gyy; (c) Gzz; (d) Gzx; (e) Gzy; (f) Gyx.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
55
Fig. 3. Edge results of horizontal directional analytic signal, total horizontal derivatives and Theta method. (a) ASx of the data in Fig. 2; (b) ASy of the data in Fig. 2; (c) THDx of the data in Fig. 2; (d) THDy of the data in Fig. 2; (e) ThetaX of the data in Fig. 2; (f) ThetaY of the data in Fig. 2.
2. Directional Theta method
Mikhailov et al. (2007) has defined the directional analytic signal: ASx, ASy and ASz, the expressions are:
Gravity gradient tensors are the space derivatives of the components of the gravity vector g = (gx, gy, gz) in the three orthogonal directions x, y and z. The full tensor gravity gradient can be expressed as 2
∂g x 6 ∂x 6 6 ∂g 6 T ¼6 y 6 ∂x 6 4 ∂g z ∂x
∂g x ∂y ∂g y ∂y ∂g z ∂y
ASy ¼ Gyx 2 þ Gyy 2 þ Gyz 2 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ASz ¼ Gzx 2 þ Gzy 2 þ Gzz 2
3
∂g x ∂z 7 7 2G xx ∂g y 7 7 4 7 ¼ Gyx ∂z 7 Gzx 7 ∂g z 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 2 2 > > xx þ Gxy þ Gxz > ASx ¼ qG < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Gxy Gyy Gzy
3 Gxz Gyz 5: Gzz
ð1Þ
∂z
Based on the gravitational potential satisfy the Laplace equation, and the matrix is symmetric, there are only five independent components Gxx, Gxz, Gyx, Gyy and Gyz. However, the full tensor gravity gradiometer measures the six gradient tensor components Gxx, Gxz, Gyx, Gyy, Gyz and Gzz.
ð2Þ
And Beiki (2010) used it to detect the edges of gravity gradient tensor data. Yuan and Geng (2014) and Yuan et al. (2015) have defined the directional total horizontal derivatives, THDx, THDy and THDz, to interpret the gravity gradient tensor data, the expressions are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > > THDx ¼ Gxy þ ðGxz Þ2 > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 þ G THDy ¼ G yx yz > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 : THDz ¼ ðGzx Þ2 þ Gzy
ð3Þ
56
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 4. Edge results of gravity gradient tensor data shown in Fig. 2. (a) edge of TDX; (b) edge of TDR_THDR; (c) edge of Theta map; (d) edge of ED.
which show a higher resolution than directional analytic signals. Based on these definitions, we propose the directional Theta methods, which can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gxx 2 þ Gyx 2 ThetaX ¼ −qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gxx 2 þ Gxy 2 þ Gxz 2
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gyx 2 þ Gyy 2 ThetaY ¼ −qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gyx 2 þ Gyy 2 þ Gyz 2
ð5Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gzx 2 þ Gzy 2 ThetaZ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gzx 2 þ Gzy 2 þ Gzz 2
ð6Þ
Here, we can see that the horizontal direction ThetaX, ThetaY compared with the vertical direction ThetaZ have negative sign, which is used to enhance the visual effect. The maximum value of ThetaX delineates the N–S edges; the maximum value of ThetaY indicates the W–E edges. The vertical direction ThetaZ is same with the traditional Theta map method (Wijns et al., 2005). The horizontal directional Theta ThetaX and ThetaY include all independent components. Therefore, we can combine ThetaX and ThetaY to define a new edge detector: ED ¼ ThetaX þ ThetaY:
ð7Þ
The maximum values of ED automatically recognize the edges of the source. In order to validate the feasibility of the method proposed above, we choose three another TDX (Cooper and Cowan, 2006), TDR_THDR (Verduzco and Fairhead, 2004) and Theta map (Wijns et al., 2005) to compare results.
3. Synthetic model experiments In this section, we construct a gravity gradient anomaly model, which include four prisms at a depth to top of 1 km (label 1), 1 km (label 2), 2 km (label 3) and 2 km (label 4). Fig. 1 displays the plan view and 3D view of the model. The residual densities of all prisms are 0.2 g/cm3. The synthetic gravity gradient tensor data is calculated on a regular grid with a spacing of 0.2 km, shown in Fig. 2. In order to indicate the advantage of directional Theta method, we compare them with directional analytic signals and directional total horizontal derivatives. The compared results are shown in Fig. 3. We can see that all horizontal directional analytic signal, total horizontal derivative and Theta method can outline the N–S and W–E edges of geological sources. However, the resolution of directional analytic signal and total horizontal derivatives is lower, especially the analytic signal. Also, these two methods cannot equal to the size of the amplitude of the edges. The directional Theta method cannot only extract the N–S and W–E edges precisely and clearly, also can balance the edge amplitude size. Based on this advantage, we can apply the constructed new edge detector ED to delineate the edges of geological sources. Fig. 4 displays the edge results of TDX, TDR_THDR, Theta map and new detector ED. We can see that all these four methods can identify the edges clearly and precisely. But the new detector ED has a higher resolution than the others. By comparing the results of Fig. 4c and d, we can obtain that horizontal directional Theta method has an improvement on the traditional Theta map method. From Figs. 3e, f and 4d, we can find that this new method may bring some individual maximum dots in the identified results. However, each individual dot cannot affect the final lineament structure interpretation. In order to validate the new method, we built a more complex model which contains both positive and negative anomalies. Here, the new model is same with the model above, but the contrasted densities of prisms 1 and 3 are 0.2 g/cm3 and the contrasted densities of prisms 2
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
and 4 are − 0.2 g/cm3. The synthetic gravity gradient anomalies are shown in Fig. 5. The identified edge results are shown in Fig. 6. By comparing the results, we can get that the traditional methods can well delineate the edges of different amplitude anomalies. But, they bring some additional false edge information when the model becomes complex, especially when the model contains both positive and negative anomalies simultaneously. These additional edges will lead to a false interpretation on the geological structure. However, our new method ED can avoid this disadvantage effectively. This method is fully automatic without any personal effect on removing the additional false edge information compared with some other methods, which need to bring a constant number in the normalization factor (Cooper, 2013; Li et al., 2014; Yuan et al., 2015). Therefore, this method is more objective. To further test the stability of the new edge detector, we add 5% Gaussian noise to the gravity gradient tensor data shown in Fig. 5. The edge results of TDX, TDR_THDR, Theta map and ED are shown in Fig. 7. We can see that TDR_THDR cannot delineate the edges of the sources due to the influence of noise. The methods TDX, Theta map and ED can all extract the edges clearly. But, the effect of noise to TDX and Theta map is bigger than the effect to ED.
57
4. Applied to real data In order to test the application in real case, we apply the new edge detector to real measured full tensor gravity gradient data by Bell Geospace with Air-FTG gradiometer over St. George's Bay, Newfoundland and Labrador. Flight lines were spaced 500 m apart with a northeast–southwest orientation whereas tie-lines were spaced at 5000 m intervals with a northwest–southeast orientation (Pilkigton and Shamsipour, 2014). The total kilometers of flight lines are 5992 km whereas the total kilometers of tie-lines are 610 km. The survey was flown at a nominal altitude of 100 m on a gentle drape surface. The 90 m SRTM elevation model was used for computing the drape surface. The data was leveled and denoised using the FFT method (Sanchez et al., 2005). Based on correlation tests, a density of 2.2 g/cm3 was chosen for terrain correction. The corrected full tensor gravity gradient data was shown in Fig. 8. The gradiometer survey covers most of the Carboniferous Bay St. George Subbasin, which is bounded to the south by the onshore northeast-trending Long Range Fault and constitutes part of the larger Maritimes Basin (Schillerif and Williams, 1979). Carboniferous and
Fig. 5. Synthetic gravity gradient tensor data of model 2. (a) Gxx; (b) Gyy; (c) Gzz; (d) Gzx; (e) Gzy; (f) Gyx.
58
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 6. Same with Fig. 4, but for the gravity gradient tensor data of synthetic model 2.
Fig. 7. Same with Fig. 4, but for the gravity gradient tensor data of model 2 corrupted by 5% Gaussian noise.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
59
Fig. 8. Full tensor gravity gradient data in St. George's Bay. (a) Gxx; (b) Gxz; (c) Gyy; (d) Gyz; (e) Gyx; (f) Gzz.
Upper Devonian sedimentary rocks that constitute the subbasin fill are exposed onshore to the north, east, and south, overlying Precambrian basement. These fill rocks consist of marine and nonmarine sequences of shale, siltstone, sandstone, and limestone, with salt bodies. The edge results of the full tensor gravity gradient data over St. George's Bay are shown in Fig. 9. The black lines in figure are the known edge mapped by Miller et al. (1990). We can find that the traditional method TDR_THDR cannot extract the edge information clearly and precisely, Fig. 9b. Methods TDX, Theta and ED all can recognize the edges. However, there are some individual maximum values areas outlined by red ellipse in Fig. 9c. We can consider that these are not the real edge structure. Also, the new method ED can extract a new edge clearly than the other method outlined by yellow ellipse. Therefore, we can consider that our new detector ED has a higher resolution, the outlined edges are clear and precise.
5. Conclusions Our main contribution by this study was to define the directional Theta method of full tensor gravity gradient data. Comparing with horizontal directional analytic signal and directional total horizontal derivatives, we can find that horizontal directional Theta has a higher resolution and can identify the edge of geological bodies clearly and precisely, which can effectively equal to the edge amplitude size of different amplitude gravity gradient anomalies. Therefore, we use them to define a new edge detector ED to delineate the edges of full tensor gravity gradient data. In order to validate the feasibility of the new method, we demonstrate it on three synthetic models. Compared the result of the new detector with three traditional edge identification methods. It has demonstrated that the new proposed method is an effective tool for edge detection, and can avoid bringing false edges when the geological
60
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61
Fig. 9. Edge results of St. George's Bay. (a) TDX; (b) TDR_THDR; (c) Theta; (d) ED.
bodies contain both positive and negative anomalies simultaneously, while tradition methods TDX, TDR_THDR and Theta may produce false edges. Finally, we apply it on real measured full tensor gravity gradient data over St. George's Bay, which get a well edge results. Acknowledgments We gratefully acknowledge the Natural Resources Canada for providing the full tensor gravity gradient data measured by Air-FTG in St. George's Bay, Canada. This work was financially supported by the basic scientific research business special fund project of Second Institute of Oceanography, State Oceanic Administration (14275-10); Comprehensive Environmental Investigation and Assessment of North and South Poles (CHINARE2015-01-03; CHINARE2015-04-01); National Marine Public Welfare Projects (2013418035). References Beiki, M., 2010. Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics 75 (6), I59–I74. Cooper, G.R.J., 2013. Reply to a discussion about the “hyperbolic tilt angle method” by Zhou et al. Comput. Geosci. 53, 496–497. Cooper, G.R.J., Cowan, D.R., 2006. Enhancing potential field data using filters based on the local phase. Comput. Geosci. 32, 1585–1591. Cooper, G.R.J., Cowan, D.R., 2008. Edge enhancement of potential-field data using normalized statistics. Geophysics 73, H1–H4. Cordell, L., 1979. Gravimetric expression of graben faulting in Santa Fe Country and the Espanola Basin, New Mexico. 30th Field Conf., 1979. New Mexico Geo. Soc. Guidebook, pp. 59–64. Cordell, L., Grauch, V.J.S., 1985. Mapping basement magnetization zones from aeromagnetic data in the San Juan Basin, New Mexico. In: Hinze, W.J. (Ed.), The Utility of Regional Gravity and Magnetic Maps. Society of Exploration Geophysics, pp. 181–197.
Evjen, H.M., 1936. The place of the vertical gradient in gravitational interpretations. Geophysics 1, 127–136. Li, L., Huang, D., Han, L., Ma, G., 2014. Optimised edge detection filters in the interpretation of potential field data. Explor. Geophys. 45 (3), 171–176. Ma, G.Q., Li, L.L., 2012. Edge detection in potential fields with the normalized total horizontal derivatives. Comput. Geosci. 41, 83–87. Marson, I., Klingele, E.E., 1993. Advantages of using the vertical gradient of gravity for 3-D interpretation. Geophysics 58 (11), 1588–1595. Mikhailov, V., Pajot, G., Diament, M., Price, A., 2007. Tensor deconvolution: a method to locate equivalent sources from full tensor gravity data. Geophysics 72 (5), 161–169. Miller, H.G., Singh, V., 1994. Potential field tilt—a new concept for location of potential field sources. J. Appl. Geophys. 32, 213–217. Miller, H.G., Kilfoil, G.J., Peavy, S.T., 1990. An integrated geophysical interpretation of the Carboniferous Bay St. George Subbasin, Western Newfoundland. Bull. Can. Petrol. Geol. 38, 320–331. Oruc, B., Keskinsezer, A., 2008. Structural setting of the northeastern Biga Peninsula (Turkey) from tilt derivatives of gravity gradient tensors and magnitude of horizontal gravity components. Pure Appl. Geophys. 165, 1913–1927. Oruc, B., Sertcelik, I., Kafadar, O., Selim, H., 2013. Structural interpretation of the Erzurum Basin, eastern Turkey, using curvature gravity gradient tensor and gravity inversion of basement relief. J. Appl. Geophys. 88, 105–113. Pilkigton, M., Shamsipour, P., 2014. Noise reduction procedures for gravity-gradiometer data. Geophysics 79 (5), G69–G78. Roest, W.R., Verhoef, J., Pilkington, M., 1992. Magnetic interpretation using the 3-D analytic signal. Geophysics 57, 116–125. Sanchez, V., Sinex, D., Li, Y., Nabighian, M., 2005. Processing and inversion of magnetic gradient tensor data for UXO applications. 18th EEGS Symposium on the Applications of Geophysics to Engineering and Environmental Problems, Extended Abstracts, pp. 1193–1202. Schillerif, S., Williams, H., 1979. Geology of the Stephenville map area, Newfoundland. Current Research, Part A, Geological Survey of Canada, Paper 79-1A, pp. 327–332. Sertcelik, I., Kafadar, O., 2012. Application of edge detection to potential field data using eigenvalue analysis of structure tensor. J. Appl. Geophys. 84, 127–136. Verduzco, B., Fairhead, J.D., 2004. New insights into magnetic derivatives for structural mapping. Lead. Edge 23 (2), 116–119. Wang, J., Meng, X.H., Li, F., 2015. Improved curvature gravity gradient tensor with principal component analysis and its application in edge detection of gravity data. J. Appl. Geophys. 118, 106–114.
Y. Yuan et al. / Journal of Applied Geophysics 130 (2016) 53–61 Wijns, C., Perez, C., Kowalczyk, P., 2005. Theta map: edge detection in magnetic data. Geophysics 70, L39–L43. Yuan, Y., Geng, M., 2014. Directional total horizontal derivatives of gravity gradient tensor and their application to delineate the edges. 76th EAGE Conference & Exhibition 2014. Yuan, Y., Yu, Q., 2015. Edge detection in potential-field gradient tensor data by use of improved horizontal analytic signal methods. Pure Appl. Geophys. 172 (2), 461–472. Yuan, Y., Huang, D., Yu, Q., Lu, P., 2014. Edge detection of potential field data with improved structure tensor methods. J. Appl. Geophys. 108, 35–42.
61
Yuan, Y., Huang, D., Yu, Q., 2015. Using enhanced directional total horizontal derivatives to detect the edges of potential-field full tensor data. Chin. J. Geophys. 58 (7), 2556–2565 (in Chinese). Zhou, W.N., Du, X.J., Li, J.Y., 2013. The limitation of curvature gravity gradient tensor for edge detection and a method for overcoming it. J. Appl. Geophys. 98, 237–242. Zuo, B.X., Hu, X.Y., 2015. Edge detection of gravity field using eigenvalue analysis of gravity gradient tensor. J. Appl. Geophys. 114, 263–270.