Implementation of the singular points method for gravity data by fast Fourier transforms Wenna Zhou, Jiyan Li, and Xiaojuan Du
T
he singular points method was originally implemented using a Fourier series associated w ith the singular point, and it only can be used to interpret the profile data. In this article, the original algorithm is modified using fast Fourier transforms (FF Ts) to eliminate the deficiencies in the Fourier series approach. Meanwhile, we implement the three-dimensional (3-D) computation of the singular points method for interpretation of gravity grid data using FFTs. The same as the standard algorithm, the new algorithm also can be used to estimate the depths of causative sources by means of downward continuation in conjunction with determining the point at which the field becomes singular. It is tested on synthetic models, with satisfactory results. Then we applied it to a real gravity data profile from the Qian-An area in Northeast China, and the 3-D computation was applied to the grid
Digital Object Identifier 10.1109/MPOT.2013.2258074 Date of publication: 20 July 2015
Image courtesy of NASA
data from Dian-Nan in Yunnan province, which was digitalized from a previously published paper (Liu, 2007). Compared with the drilling data and other previous research results, this algorithm demonstrates feasibility and validity of the semi-automatic interpretation. Gravity data are routinely used by underway exploration programmers to evaluate and explore geological complexities hosting hydrocarbon and mineral resources. Various
methods have been introduced to process and interpret gravity data. Semi-automatic interpretation techniques were favored because they are fast and simple, with no need for much prior infor mation. They a re designed to efficiently provide the basic parameters of geologic bodies, such as the central buried depth and horizontal position. The normalized full gradient (NFG) method was one frequently used method. It was introduced by Berezkin (Elysseieva and Pasteka 2009) and subsequently used to locate reservoirs from gravity data. The NFG method does not require many geometric input parameters or assumptions regarding geological properties. The depths of the causative sources can be obtained by downward continuation combined with low-pass filtering of the normalized full gradient values of the gravity data. The theory of the NFG method has been further developed and modified (Zeng et al. 2002), and tremendous progress has recently been made in the implementation of the NFG technique (Fedi and Florio
IEEE Potentials Jul y/August 2015 n 0278-6648/15©2015IEEE
39
g (mGal)
0.15
We compute these functions in the frequency domain as
g
0.1
0.05 0
hg (x, z) = F -1 [hg f (x, 0) $ z (u, z)],(4) 0
50
100
150
200
250 x (m)
300
350
400
450
where
500
(a) 0
5 4 3 2 1
y (m)
20 40 60 80
0
50
100
150
200
250 x (m)
300
350
400
450
500
(b)
2011, Pamukcu and Akcig 2011, Aydin 2007, Aghajani et al. 2011). However, this method requires the use of two high-pass filters: the second derivative filter and downward continuation. As we know, the second derivative filter and downward continuation is unstable and sensitive to noise. Therefore, the increase in instability and noise sensitivity owing to these two high-pass filters may lead to faulty results. The singula r points method (gravity normalized vector analysis method) was introduced by Cianciara and Marcak (B. Cianciara and H. Marcak 1979) as an improvement to the NFG method. Similar to the NFG method, the singular points method can be outlined in three steps. The first step is approximating the gravity anomaly with a Fourier series. The second step is calculating the Hilbert transformation of the Fourier series, instead of the second derivative used in the NFG method. Therefore, the noise sensitivity would be reduced. As in NFG, the next step is estimating the depths of the causative sources by means of downward continuation in conjunction with determining the point(s) where the field becomes singular. It is well known that the Fourier series, which is limited to periodic functions, is a special case of the FFTs.
n
J u l y / A u g u s t 2 0 1 5
The FFTs, on the other hand, are a more universal method. The FFTs are therefore used to implement the singular points method in this article. The 3-D computation of the method is established in the frequency domain. Many synthetic model tests and real field data are used to demonstrate the applied effect.
Theory of the singular points method using FFTs The two-dimensional computation for profile data The singular points method was introduced in 1979 (B. Cianciara and H. Marcak 1979), and can be summed up by the following equation: 1 GH (x, z) = G (x, z) / M
g f (x, 0) = F [g (x, 0)](5)
hg f (x, 0) = F [hg (x, 0)] .(6)
In (5) and (6), F -1 [] denotes the inverse Fourier transform and F [] is evaluated using FFTs. It is generally known that the dow nwa rd cont inuat ion factor, z (u, z), can be expressed as
G (x, z) =
+ 6hg (x, z)@2
rkz r 2 k 2 z 2 Q (z) = exp c M - 2 2 m .(8) b M
So in frequency domain (new algorithm), it can be calculated using
z (u, z) = exp (2r u z) /
exp ((2r u z/b) 2),
(9)
where exp ((2r u z/b) 2 denotes a smoothing factor. It can be seen that the new algorithm is only related to b, and independent on the harmonic number. So during the calculation, the new algorithm only needs the value of the b. As Wu et al. (1996) have pointed out, b can be calculated by
b = - 0.0684a 2
+ 1.0639a + 0.3158,
(10)
where a denotes the number of layers of the downward continuation. . (2)
The denominator 1/M / i = 1 G (x, z) is the arithmetic mean over each downward continuation level of G (x, z) . In (2), g (x, z) is the gravity anomaly of the lower hemi-space obtained through downward continuation, and hg (x, z) is the Hilbert transform of g (x, z) .
IEEE Potentials
i=1
6g (x, z)@2
z (u, z) = exp (2r u z) .(7)
In the singular points method (standard algorithm), the optimized downward continuation factor is expressed as (B. Cianciara and H. Marcak 1979)
M
/ G (x, z),(1)
where
fig1 (a) A gravity anomaly from the single horizontal cylinder. (b) The interpretation result for the single horizontal cylinder model without noise, where the dotted line is the true position of the horizontal cylinder.
40
g (x, z) = F -1 [g f (x, 0) $ z (u, z)](3)
M
The 3-D computation for grid data It is easy to extend the algorithm from two-dimensional (2-D) to 3-D in the frequent domain. In 3-D computation, the equation of singular points method can be rewritten as
GH (x, y, z) = G (x, y, z) / M N 1 / G (x, y, z), M$N / i=1 j=1 (11)
M
N
and 1/M $ N / / G (x, y, z) is the i=1 j=1 same meaning as the 2-D computation—it is the arithmetic mean over each downward continuation level of G (x, y, z) . M and N are the length of survey station and line in x- and ydirections, respectively. These functions in the frequency domain can be written as g (x, y, z) = F [g (x, y, 0) $ z (u, v, z)] -1
g
0
50
100
150
200
h y g (x, y, z) = F -1 [h y g f (x, y, 0) $ z (u, v, z)], (15)
where
g f (x, y, 0) = F [g (x, y, 0)](16)
h x g f (x, y, 0) = F [h x g (x, y, 0)](17)
h y g f (x, y, 0) = F [h y g (x, y, 0)] .(18)
As above, F -1 [] denotes the inverse Fourier transform and F [] is evaluated using FFTs. The 2-D Hilbert transform operator is defined in the frequency domain as (Cooper 2009) H (u, v) = - i sign (u, v) (19) u sign (u, v) = et u2 + v2 x v + et , 2 2 y (20) u +v
and u, v are the Fourier wave numbers; et x and et y are the unit vectors in the x- and y- directions, respectively. The downward continuation factor can be defined as
0 20 40 60 80
Application to synthetic models Model 1: Single horizontal cylinder model without noise Our first model contains a single horizontal cylinder with the following parameters: the depth of the center point is 30 m, the radius is 10 m, the
400
450
500
2.5 2 1.5 1 0.5 0
50
100
150
200
250 x (m)
data spacing is 5 m, and the residual density is 1 g/cm3. Figure 1(a) shows the gravity data of the horizontal cylinder. Figure 1(b) shows the interpretation result of the algorithm using FFTs, where the dotted line is the true position of the horizontal cylinder. From Fig. 1, we find that the maximum value position accurately corresponds to the central point position of the horizontal cylinder. The interpretation result is reasonable. The new algorithm of the singular points method is able to locate the single horizontal cylinder accurately. The FFTs can
300
350
400
450
500
therefore be used to implement the singular points method.
Model 2: Model with two horizontal cylinders without noise To capture the mutual influence of various adjacent buried geologic bodies, we construct another model containing two horizontal cylinders. The parameters are that the buried depths of the center points are 30 m and 40 m and the radius is 10 m and 15 m, respectively. The data spacing is 1 m, and the residual density is 1 g/cm 3. Figure 2(a) shows the
0.3
g
0.2 0.1 0 -0.1
0
50
100
150
200
2
250 x (m)
300
350
400
450
500
(a)
y (m)
The b can be calculated by (10) as above.
350
fig2 (a) A gravity anomaly from the two horizontal cylinders. (b) The interpretation result for the model with two horizontal cylinders without noise, where the dotted line is the true position of the horizontal cylinders.
exp ((2r u + v z/b) ) (21) 2
300
(b)
z (u, v, z) = exp (2r u 2 + v 2 z) / 2
250 x (m) (a)
(13)
h x g (x, y, z) = F -1 [h x g f (x, y, 0) $ z (u, v, z)] (14)
0.25 0.2 0.15 0.1 0.05 0
f
g (mGal)
+ 6h x g (x, y, z)@2 (12) + 6h y g (x, y, z)@2
G (x, y, z) =
y (m)
6g (x, y, z)@2
g (mGal)
where
0 20 40 60 80
2.5 2 1.5 1 0.5 0
50
100
150
200
250 x (m)
300
350
400
450
500
(b) fig3 (a) A gravity anomaly from Fig. 2, when uniformly distributed random noise is added, with amplitude equal to 5% of the data amplitude. (b) The interpretation result for the model with two horizontal cylinders with random noise, where the dotted lines are the true position of the cylinders.
IEEE Potentials Jul y/August 2015
n
41
Field data processing
g (mGal)
0.2
In this section, we apply the new algorithm to a real gravity data profile from the Qian-An area in Northeast China. The 3-D computation of the method is applied to the grid data from Dian-Nan in Yunnan province (Liu 2007).
g
0.1 0 -0.1 -0.2
0
200
400
600
800
1,000 1,200 1,400 1,600 1,800 x (m)
Profile data processing
(a) 0
4 3.5 3 2.5 2 1.5 1 0.5
y (m)
100 200 300 400 500 600
0
200
400
600
800
1,000 1,200 1,400 1,600 1,800 x (m) (b)
fig4 (a) A residual gravity anomaly of the real gravity profile. (b) The interpretation result for the real gravity profile.
synthetic data set consisting of the gravity anomalies from these two cylinders. Figure 2(b) shows the result obtained with the new algorithm using FFTs, and the dotted lines are the true positions of causative sources. The same as single horizontal cylinder model of Fig. 1, the maximum value corresponds to the horizontal cylinder central point position. So both cylinders are successfully located, but the shallow one is located more accurately, and the deep one tends to offset upward to the surface. The downward continuation is unstable when it passes through an anomaly target, and the location of the deeper geologic body is therefore less accurate. The maximum value occurs at a depth more shallow than the true position of the deep geologic source.
equal to 5% of the data amplitude has been added. Figure 3(b) shows the result of the new algorithm, and the true positions of the horizontal cylinders are given by dotted lines. The new algorithm can locate the depth accurately, but it is susceptible to noise. It can be seen that there is much noise interference. However, it is reassuring that the errors caused by the interference are concentrated in the shallow layer (near the surface). A deep geologic source is not surrounded by interference and false anomaly, and a more advantageous interpretation can therefore be obtained. Nevertheless, because of the instability of downward continuation, the noise should first be filtered out in the field data processing.
15
Noise interference is inevitable when processing real data, especially for downward continuation, which requires high-precision data and is extremely vulnerable to noise. It is therefore very important to quantify the effects of noise. Figure 3(a) shows the synthetic gravity data from Fig. 2 when uniformly distributed random noise with amplitude
10
42
n
J u l y / A u g u s t 2 0 1 5
y (km)
Two horizontal cylinders modeled with random noise
5
5
10 x (km)
mGal 5 4 3 2 1 0 -1 -2 -3 15
fig5 A gravity grid data from Dian-Nan in the Yunnan province of China, where the dotted line is an interpretation profile.
IEEE Potentials
Qian-An lies in the northeast of China, and the research area is rich in iron ore. The profile is 1.86 km in length and contains 95 stations. The purpose of the research is to estimate the depth and extent of the iron ore. The residual gravity data are shown in Fig. 4(a) and displays a major positive anomaly at the distance from 800 m to 1,000 m. The interpretation result is shown in Fig. 4(b), and the primary anomaly range is 50–350 m as seen in this figure. According to the drilling and ore mining data, the ore body depth is around 150–400 m. The interpretation results are approximately consistent with the drilling data. The errors maybe produced from the noise interference or the shallow geologic bodies.
Grid data processing The grid data is digitized from a residual Bouguer anomaly map (Tian-you 2007). The intervals of the grid data is 500 m (and the Bouguer anomaly map shown in Fig. 5), and the dotted line is the position of an interpretation profile. Figure 6 includes the drilling data interpretation result of the profile shown in Fig. 5. The range of the salt mine is given by the shadow part of Fig. 6. The residual density of the salt mine is 0.4 g/cm3. Figure 7 is a 3-D slice map of the imaging result from the new approach, which is calculated by (11) for 3-D. The position of the salt mine is clearly displayed. The maximum value corresponds to the central depth of the mine, and the result accurately coincides with Fig. 6. The results in this section demonstrate the efficacy of the new algorithm using FFTs in practical applications. It can be effectively applied to realistic geological data.
0 400
K1m
K1m
K2me
r = 2.64
r = 2.60 r = 2.64
r = 2.20
K 1p
800 1,200
Depth (km)
fig6 The drilling data interpretation results (r denotes the density).
0 0.4 0.8 1.2 1.6 0
8 6 4 2
5
y ) (m
10 15
0
5
10
15
x (m)
fig7 The interpretation result for the gravity grid data using 3-D computation.
Conclusion A new implementation of the singular points method is presented in this article. The FFTs algorithm is used to replace the Fourier series used in the standard method. Because FFTs are a fast algorithm, the speed of computation is accelerated, and it can easily implement 3-D computation for grid data in the frequency domain. We have described the equations of 3-D computation. It makes the singular points method more suitable for gravity data processing. After that, the 2-D computation of the new algorithm is tested on synthetic data and on real gravity profile data from the Qian-An area in Northeast China. The 3-D computation of the algorithm is applied to the gravity grid data from Dian-Nan. Many accurate results have been obtained. It has been demonstrated that the Fourier series can be replaced by FFTs in the implementation of the singular points method. The new algorithm is more suitable for data processing because it can be used to interpret profile data as well as grid data directly. As a supplement to the singular points method, this method has the potential for widespread application. It is
an effective semi-automatic interpretation method for gravity data.
Acknowledgment This work was supported by the National Science and Technology Support Program (2009BAB43B01).
Read more about it ••I. S. Elysseieva and R. Pasteka, “Direct interpretation of 2D potential fields for deep structures by means of the quasi-singular points method,” Geophys. Prospecting, vol. 57, no. 4, pp. 683–705, July 2009. ••H. Zeng, X. Meng, C. Yao, X. Li, H. Lou, Z. Guang, and Z. Li, “Detection of reservoirs from normalized full gradient of gravity anomalies and its application to Shengli oil field, East China,” Geophysics, vol. 67, pp. 1138–1147, July–Aug. 2002. ••M. Fedi and G. Florio, “Normalized downward continuation of potential fields within the quasi-harmonic region,” Geophys. Prospecting, vol. 59, pp. 1087–1100, Nov. 2011. ••O. A. Pamukcu and Z. Akcig, “Isostasy of the Eastern Anatolia (Turkey) and discontinuities of its crust,” Pure Appl. Geophys., vol. 168, pp. 901–917, May 2011.
••A. Aydin, “Interpretation of gravity anomalies with the normalized full gradient method and an example,” Pure Appl. Geophys., vol. 164, pp. 2329–2344, Dec. 2007. ••H. Aghajani, A. Moradzadeh, and H. Zeng, “Detection of highpotential oil and gas fields using normalized full gradient of gravity anomalies: A case study in the Tabas Basin, Eastern Iran,” Pure Appl. Geophys., vol. 168, pp. 1851– 1863, Oct. 2011. ••B. Cianciara and H. Marcak, “Geophysical anomaly interpretation of potential fields by means of singular points method and filtering,” Geophys. Prospecting, vol. 27, no. 1, pp. 251–260, 1979. ••Y. G. Wu, R. J. Shan, F. X. Zhou, X. B. Lu, C. L. Tian, and F. J. Du, “A study of appending of total gradient of gravity and its phase,” Global Geol. (in Chinese), vol. 15, no. 3, pp. 84–90, 1996. ••G. R. J. Cooper, “Balancing images of potential-field data,” Geophysics, vol. 74, pp. L17–L20, May– June 2009. • T. Y. Liu, New Data Processing Methods for Potential Field Exploration. Beijing, China: Science Press, 2007.
About the authors Wenna Zhou (zhouwn09@mails. jlu.edu.cn) is a Ph.D. student at the College of Geo-Exploration Science and Technology, Jilin University, China. His research interests are the potential field data processing and interpretation. Jiyan Li (
[email protected]) is a lecturer at the College of Petrochemical Technology, Lanzhou University of Technology. She earned her Ph.D. degree at the College of Earth Science, Jilin University, China. She is currently working on petroleum geologic features research. Xiaojuan Du (
[email protected]) is a professor of the College of Geoexploration Science and Technology, Jilin University, China, where she teaches courses on the application of gravity and magnetic potential field data processing.
IEEE Potentials Jul y/August 2015
n
43
