El-mallaha and Zeit basins having depth ranging from. 2 km to 5 km, that indicates the presence of deep magnetic sources. The model region covers 30 km.
International Symposium on Earth Science and Technology 2015
Three-Dimensional Forward Modelling of Geomagnetic Data Using Hexahedral Element with an Application to Zeit Basin Area, Gulf of Suez, Egypt Hassan MOHAMED1, Hideki MIZUNAGA1, Hakim SAIBI1 and Ali ABDELAZIZ2 1
Department of Earth Resources Engineering, Graduate School of Engineering, Kyushu University, Fukuoka 819-0395, Japan 2 Airborne Geophysics Dept., Exploration Division, Nuclear Materials authority, P.O. Box 530, Maadi, Cairo, Egypt
ABSTRACT This study explicitly develops the elegant Gauss-Legendre Quadrature formulation for 3-D forward modelling of geomagnetic data of the hexahedral prism. We have used these results for verification to ensure that three dimensional forward modelling using GLQ integration method had been correctly implemented. Our algorithm assumes that there is no remanent magnetization and that the magnetic data are produced by induced magnetization only. The rate of change of the potential in the third dimension as well as the total magnetic effect, which is related to the magnetic field strength, has been evaluated by looking at the summed effects of the point dipoles that fill out the volume. We tested the algorithm on a synthetic example and compare our result against that from the code developed at the University of British Columbia – Geophysical inversion facility (UBC-GIF) to verify the accuracy of our code, tests showed positive results. We also forward total field magnetic anomaly data of 3-D graben structure as geologic feature. We tested the algorithm on aeromagnetic field data from Zeit Basin area, Southern of Gulf of Suez, Egypt. INTRODUCTION In the 1960s, computer simulations became available for calculating magnetic anomalies. Talwani and Heirtzler (1964) calculated magnetic anomalies across two-dimensional bodies of polygonal cross sections. Bhattacharyya (1964) presented magnetic anomalies over three-dimensional bodies using right rectangular prisms. Another approach to model 3-D bodies using polygonal faces found by Bott (1963), or by stacked, thin, horizontal sheets of polygonal shape as proved by Talwani (1965). The magnetic field of the earth can be approximated roughly by that of a dipole located at the center of the earth and inclined at an angle of 11o from the axis of rotation (Griffiths and King, 1981). Magnetic data can be acquired on the ground by an operator using magnetometer, in the air by aircraft or helicopter. Magnetic measurements are easier and cheaper to make than most other geophysical measurements (Telford et al., 1990). In forward modelling process, initial model for the source body based on geologic and geophysical intuition is constructed. The model's anomaly is calculated and compared with the observed anomaly, and model parameters are adjusted in order to improve the fit between the two anomalies (Blakely, 1995). The three-step process of body adjustment, anomaly calculation and anomaly comparison, is repeated until calculated and observed anomalies are deemed sufficiently alike. In this study to calculate the magnetic effects of the trilinear (hexahedral) element we developed five points Gauss-Legendre formulation. We evaluated the ability of hexahedral element to calculate magnetic effect and its components for complex shape and terrain. The accuracy of calculation becomes higher if the susceptibility distribution, that is more complicated than the cuboid, is set for three
dimensional susceptibility structures. This method is evaluated using synthetic examples and field data measured over Zeit basin region, Gulf of Suez in Egypt. NUMERICAL IMPLEMENTATION Forward problem for magnetic data requires two fundamental relations: (1) One of Maxwell’s equations. (2) A relation for the magnetic field due to a dipole expressed in terms of a scalar potential (Blakely, 1995). In this section development of magnetic effects expressions for arbitrary volume and generalize these effects for specific geometric parameters of the 8-nodes trilinear hexahedral prism is described. The scalar magnetic potential of the 3-D magnetized body is given by: !
!
V r =− ! 𝐦 ∙ ∇! dx ! dy ! dz ! , (1) !" ! where m is the dipole moment, r is the distance connected between the dipole center and the observation point and 𝜇0 is the magnetic permeability of air (Fig.1).
Fig. 1. Magnetic dipole due to equal but opposite monopole separated by infinitesimal distance l. the dipole is located at the source point (x', y', z') at distance r from the observation point (x, y, z).
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The vector field (B) at distance r is derived by integrating equation (1) in each spatial dimension of the 3-D source. 𝛍!
𝐁
!,!,! = − !"# ∇V r dv = !"# ∇ !" 𝐦 ∙ ! ∇! dv. (2) !
Taking the finite volume integrations of equation (1), the magnetic field components of the 3-D source given by: B x= c ! m! (
3ΔxΔy !!
)+m!
B y= c ! m! (
m! (
3ΔzΔy !!
3ΔzΔx !! !ΔxΔy !!
)+m!
B z= c !
m! )+m!
+
!!
m!
3Δy ! !!! !!
3Δx ! !!!
m!
3ΔzΔy !! !ΔzΔx !!
3Δz ! !!! !!
dx ! dy ! dz ! , (3-a) + dx ! dy ! dz ! , (3-b) + dx ! dy ! dz ! ,
(3-c)
x=N1 x1 +N2 x2 +N3 x3 +N4 x4 +N5 X 5 +N6 x6 +N7 x7 +N8 x8 , y=N1 y1 +N2 y2 +N3 y3 +N4 y4 +N5 y5 +N6 y6 +N7 y7 +N8 y8 ,
(5)
z=N1 z1 +N2 z2 +N3 z3 +N4 z4 +N5 z5 +N6 z6 +N7 z7 + N8 z8 .
N1 to N8 are called shape functions which are given by: 𝑁! = 𝑁! = 𝑁! =
!!! !!! !!! ! !!! !!! !!! ! !!! !!! !!! ! !!! !!! !!! !
, 𝑁! = , 𝑁! = , N! = , 𝑁! =
!
!
!
B = c!
f(x ξ, η, ζ , y ξ, η, ζ , z ξ, η, ζ ) !! !! !!
(7) where JD is the determinant of the Jacobi matrix. After Gauss-Legendre Quadrature decomposition, equation (7) becomes: J! 𝑑𝜉 𝑑𝜂 𝑑𝜁,
!
!
!
!
!
!
B = c!
where Δx = x-x', Δy = y-y' and Δz = z-z'. Bx, By and Bz are the magnetic vector components respectively. The total magnetic field anomaly of the 3-D source for geomagnetic inclination (𝜃I) and declination (𝜃D) can be calculated using: B! x, y, z = B! (cos θ! cos θ! ) + B! (cosθ! sinθ! ) + B! (sin θ! ). (4) To calculate the magnetic field due to hexahedral prism, let the observation point in the three-dimensional orthogonality coordinate system (x, y, z) is defined as p (x, y, z), and the coordinates of eight vertexes of the region of interest are defined from (x1, y1, z1) to (x8, y8, z8) as shown in Fig. 2. One of the most crucial part in the development of the computer program was the implementation of the appropriate shape functions for the hexahedral prism, 3-D element. Before using equations 3 (a-c) and 4 to calculate the magnetic anomaly measured at point p, transformation done from general element to a standard element using these linear equations:
𝑁! =
The previous equations can be generalized as following:
!!! !!! !!! ! !!! !!! !!!
(6)
! !!! !!! !!! ! !!! !!! !!! !
,
where 𝜉, 𝜂 and 𝜁 are the parameters in local unit element (Fig. 2).
C!"# ∙
f x ξ ! , η ! , ζ ! , y ξ ! , η ! , ζ ! , z ξ ! , η ! , ζ ! J! . (8) Interestingly, to integrate the symmetric integral form in equation 7, a discrete formulation like equation 8 can be defined after GLQ decomposition where ξ! , η! , ζ! are sampling points or Gaussian nodes within the hexahedral element and are weighted by Gauss-Legendre coefficients (Cijk). The left portion of equation 8 is the magnetic effect of the 3-D hexahedral prism. B
A (x4, y4, z4)
(x3, y3, z3)
(1,1,-1)
(-1,1,-1)
(1,-1,-1)
(-1,-1,-1) (x2, y2, z2)
(x1, y1, z1)
(x7, y7, z7)
(x8, y8, z8)
(1,1,1)
(-1,1,1) (-1,-1,1)
(x5, y5, z5)
y
(1,-1,1)
(x6, y6, z6)
𝜂
x z
𝜉 𝜁
Fig.2. Coordinates transformation from general hexahedral element (A) to a local unit element (B).
Synthetic Examples Once the computer code was completed, evaluation is necessary to ensure that 3-D forward modelling using GLQ integration method had been correctly implemented. Simple 3-D magnetized body is used in this model to calculate magnetic anomaly. The model has a width from -5 m to 5 m in each horizontal direction and it extends from a depth 3 m to 9 m, the ambient field has an inclination of 90o and a declination of 0o, 2 m grid interval used to calculate the surface magnetic anomaly. The model is divided into four regions, magnetization assumed to be constant 10 A/m within the magnetic body as shown in Fig. 3. Total magnetic field at each 169 data points collected over the model region with 13 x 13 grid of 2 m interval along x and y directions is calculated (Fig. 3). For more accuracy, we calculated the magnetic
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effects for the hexahedral prism using GLQ integration of 5 points formula listed in Table.1. Fig. 4 shows how the data over this body, the observation range from -12 to 12 m in eastern and northern directions.
Fig. 3. Synthetic 3-D model involves a buried magnetic block.
As the second synthetic example, we forward totalled field anomaly data produced by a slightly complicated model. The model consists of a 3-D graben buried in a non-susceptible half-space (graben model), the depressed block bordered by parallel normal faults. The bounded blocks extends from a depth at 50 m to 550 m, while the depressed block extends from a depth at 250 m to 550 m. The model region extends from -100 m to 1400 m and from -100 m to 600 m along eastern and northern direction, respectively. The situation is represented in Fig. 6. Constant magnetization intensity of 0.5 A/m is assumed in this model. Under an inducing magnetic field with an inclination of 90o and a declination of 0o, the surface total magnetic anomaly produced by the graben model is shown in Fig. 7. Data consists of 465 calculated points collected over a line spacing 50 m with a grid interval 50 m.
Table.1. Five sample points (Abscissas) and weights Abscissas
Weights
±0.9061798459
0.2369268851
±0.5384693101 0.0000000000
0.4786286705 0.5688888888
The shape of the buried magnetized block is clearly defined and solid black line indicate its true position. The maximum magnetic anomaly produced by the buried block is 240 nT. Magnetic methods can be used for detecting some geologic features (and sometimes their characteristics). Some geological models for common magnetic anomaly sources are represented in Fig. 5.
Fig.6. 3-D perspective view of graben model.
Fig.7. Total magnetic field anomaly produced by the graben model.
Fig. 4. Total magnetic field anomaly produced by the buried block using GLQ.
Fig. 5. Geological models for common magnetic anomaly sources. Adapted from Breiner (1999).
For further verification the results were compared with that from the program developed at the University of British Columbia (UBC) – Geophysical inversion facility UBC-GIF (2005) and Li and Oldenburg (2000). The UBC program uses the numerical finite volume discretization (FVD) solutions to the magnetostatics problem, a set of rectangular cells and a constant value of susceptibility within each cell are assumed in this solution. Same model parameters have been used to compare results. Fig. 8 shows the buried magnetic blocks extending from -5 m to 5 m meters along eastern and northern directions and its thickness is six meters. The contour map of the surface total field anomaly is shown in Fig. 9. Similarly, the shape of the buried magnetized block is clearly defined with maximum
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magnetic anomaly of 240 nT. To compare the results to assess the accuracy of our forward modelling code, we used the same profile at same interval 2 m runs from west to east direction (Fig. 10).
Fig. 8. Synthetic 3-D model involves a buried magnetic block.
Fig. 9. Total field anomaly produced by the buried block using FVD.
A
B
Fig. 10. Magnetic anomaly profile across the center line (A-B) from west to east calculated by GLQ and FVD methods.
Fig. 10 shows great identity between both results calculated from our numerical implementation
method (GLQ) and UBC computer code using FVD method. Field Example The aeromagnetic survey was carried out over Zeit basin, which is located on the western coast of the Gulf of Suez, Egypt (Fig. 11). In oil exploration, geological structure in oil-bearing sedimentary layers is controlled by topographic features (such as ridges or faults) on the basement that can be mapped using aeromagnetic data (Aboud et al., 2005). The study area has a great importance due to its hydrocarbon resources potential, therefore several seismic surveys have been conducted in this area to delineate the subsurface structure and its relation to hydrocarbon prospects. However, it was deduced that the seismic energy was damped by a Pre-Miocene salt formation (Taha et al., 2002). Seismic mapping of horizons below the Pre-Miocene salt is difficult and unreliable, therefore, other geophysical methods are recommended to delineate the subsurface structure (Aboud et al., 2005).
Fig. 11. Location and geologic map of Gebel El-Zeit area, Gulf of Suez, Egypt (after Conoco, 1987).
The aeromagnetic survey was carried out in 1984 by the Aero-Service Division of Western Geophysical Company of America. Data were obtained with resolution of 0.01 nT at a mean terrain clearance 120 m. Data were acquired with 91 m intervals and 1 km spaced lines (AeroService, 1984). The aeromagnetic data were transformed reduced to the pole (RTP) using a magnetic inclination of 40° and a declination of 2.0° (Fig. 12). In practice, a strategy for trial-and-error forward modeling has been used (Fig. 13). Initial estimation for the model parameters then anomaly calculation, and anomaly comparison is repeated as well as changing the coordinates and magnetization until calculated and observed anomalies are match reasonably well.
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with a good general agreement with the original field data, higher values at region A and B reflects that, there is susceptibility contrast and depth variation. The depth to the magnetized sources range from a depth 1.2 km to 3 km at the boundary and approximately from 4.7 km to 5 km at the central portion of the studied area, with magnetization intensity varies from 7 to 9 A/m.
Fig. 12. Reduced-to-pole aeromagnetic data of Zeit Basin area.
Aboud et. al. (2005) found that Zeit basin has NW-SE major trend, and Euler solution shows that, Esh El-mallaha and Zeit basins having depth ranging from 2 km to 5 km, that indicates the presence of deep magnetic sources. The model region covers 30 km and 15 km along eastern and northern direction, respectively.
Fig. 15. RTP map overlaid by calculated magnetic anomaly.
Start
Guess initial model parameters (p1, p2, p3,..)
Calculate magnetic anomaly using GLQ integration
Compare between calculated and field data
Do they match?
New (p1, p2, p3,..)
Adjust model parameters
No
Yes Stop
Fig.13. Flow chart for forward modelling process used for calculating magnetic anomaly. Redraw from Blakely (1995).
Magnetic field are calculated at 275 data points collected with 3 km spaced lines and interval of 600 m. Fig. 14 shows 3-D model for Zeit basin to be dominated by normal faults and tilted blocks trending NW-SE.
Fig. 14. 3-D perspective view of the magnetic anomaly model constructed to represent aeromagnetic field data.
Fig. 15 shows the recovered total magnetic anomaly
The most regional trend strikes NW-SE. Forward modeling result shows that the basement in the studied area was overlain by sedimentary succession up to 5 km thick likewise found by Aboud et al. (2005) and Jackson et al. (1988). Conclusion Accurate and efficient numerical calculations for the magnetic anomalies using hexahedral element have been achieved by using GLQ integration method. Using a higher number of nodes (5 nodes) to ensure that, the distance to the observation point is greater than the node spacing, allows our algorithm to compute effective and accurate magnetic values on the surface of the prism. A simple synthetic example was used to examine our computer program. The calculated magnetic result shows great correlation with that obtained from UBC program. Another synthetic example on the scale relevant to hydrocarbon exploration (graben model) has been used. Applications of our program to synthetic examples have produced magnetic data representative of the true model. 3-D forward modelling of aeromagnetic field data from Zeit basin area, Gulf of Suez in Egypt, has produced graben system structure taking the direction of NW-SE, which is consistent with the known geology in the studied area. Finally, results of our program demonstrate the ability of the algorithm to calculate the magnetic anomaly for complex shapes using hexahedral prism. ACKNOWLEDGEMENTS Authors would like to thank the department of airborne geophysics of Nuclear Materials Authority of Egypt for providing aeromagnetic data. REFERENCES Aboud, E., Salem A., and Ushijima K., 2005. Subsurface structural mapping of Gebel El-Zeit area,
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Gulf of Suez, Egypt using aeromagnetic data. Aero Service, 1984. Final operational report of airborne magnetic/radiation survey in the Eastern Desert, Egypt, for the Egyptian General Petroleum Corporation, Aero Service Division, Houston, Texas, Six Volumes, Western Geophysical Company of America. Bhattacharyya, B. K., 1964. Magnetic anomalies due to prism-shaped bodies with arbitrary polarization: Geophysics, 29, pp. 517–531. Blakely, 1995. Potential theory in gravity and magnetic applications: Cambridge University Press. Bott, M. H. P., 1963. Two methods applicable to computers for evaluating magnetic anomalies due to finite three dimensional bodies: Geophysical Prospecting, 11, pp. 292–299. Breiner, S., 1999. Applications manual for portable magnetometers, Geometrics 2190 fortune drive San Jose, California 95131, U.S.A.
Conoco, Geologic map of Egypt, 1987. Scale 1: 500 000 Coral Inc. Cairo, Egypt. Griffiths, D. H. and King, R.F., 1981. Applied Geophysics for geologist and Engineers, 2nd edn, pergamon press, New York, 230 pp. Jackson, J. A., N. J. White, Z. Garfunkel, and H. Anderson, 1988. Relations between normal-fault geometry, tilting and vertical motions in the extensional terrains: an example from the southern Gulf of Suez, Journal of Structural Geology, 10(2), pp. 155–170. Li, Y. and Oldenburg, D. W., 2000. Joint inversion of surface and three-component borehole magnetic data, Geophysics, 65, pp. 540-552. Taha, A., Hoda, B., Fadel, A., 2002. Minimizing the Exploration Risk by Using 3DVSP: Cairo International Petroleum Conference and Exhibition, Cairo, Egypt. Talwani, M., 1965. Computation with the help of a digital computer of magnetic anomalies caused by bodies of arbitrary shape: Geophysics, 30, pp. 797– 817. Talwani, M., and J. R. Heirtzler, 1964. Computation of magnetic anomalies caused by two-dimensional structures of arbitrary shape: Stanford University Publications of the Geological Sciences, Computers in the Mineral Industries. Telford, W. M., Geldart, L. P. and Sheriff, R. E., 1990. Applied Geophysics. 2nd edn, Cambridge University press, New York, 770pp. UBC-GIF, 2005. MAG3D inversion 4.0: A program library for forward modelling and inversion of magnetic data over 3D structures: The University of British Columbia (Vancouver, BC, Canada), Department of Earth and Ocean Sciences, Geophysical Inversion Facility.
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