Three-dimensional forward modeling for ... - Springer Link

J. Cent. South Univ. Technol. (2009) 16: 0136−0142 DOI: 10.1007/s11771−009−0023−5

Three-dimensional forward modeling for magnetotelluric sounding by finite element method TONG Xiao-zhong(童孝忠), LIU Jian-xin(柳建新), XIE Wei(谢维), XU Lin-hua(徐凌华), GUO Rong-wen(郭荣文), CHENG Yun-tao(程云涛) (School of Info-physics and Geomatics Engineering, Central South University, Changsha 410083, China) Abstract: A finite element algorithm combined with divergence condition was presented for computing three-dimensional(3D) magnetotelluric forward modeling. The finite element equation of three-dimensional magnetotelluric forward modeling was derived from Maxwell’s equations using general variation principle. The divergence condition was added forcedly to the electric field boundary value problem, which made the solution correct. The system of equation of the finite element algorithm was a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method. In order to prove correctness of the three-dimensional magnetotelluric forward algorithm, the computed results and analytic results of one-dimensional geo-electrical model were compared. In addition, the three-dimensional magnetotelluric forward algorithm is given a further evaluation by computing COMMEMI model. The forward modeling results show that the algorithm is very efficient, and it has a lot of advantages, such as the high precision, the canonical process of solving problem, meeting the internal boundary condition automatically and adapting to all kinds of distribution of multi-substances. Key words: magnetotelluric sounding; three-dimensional forward modeling; finite element method; general variation principle; divergence condition

1 Introduction Calculation of magnetotelluric responses for there-dimensional model of the earth must rely on numerical methods. These numerical solutions are obtained by approximating the relevant differential or integral equations and solving a matrix equation. For magnetotelluric forward modeling, three methods have been used for this approximation, namely, integral equation(IE) method, finite difference(FD) method and finite element(FE) method. IE method for three-dimensional forward problem in magnetotelluric sounding was first developed based on the Maxwell’s equations by HOHMANN[1]. Then, WANNAMAKE[2], XIONG[3], BAO et al[4], and XU and LI[5] further studied IE method for magnetotelluric forward modeling. IE method is usually used to modeling the simple three-dimensional models. FD method used to obtain discrete solutions of Maxwell’s equations include the staggered-grid and balance methods. The staggered-grid scheme is effective for solving the coupled first order Maxwell’s equations[6−9]. The balance method integrates the

original second order differential equations in each cell of the FD grid[10−11], and automatically preserves the current balance in the domain. FE method is still not widely used in three-dimensional magnetotelluric forward modeling even though it is supposed to be more suited to complex geometries than other methods. MOGI[12], ZYSERMAN and SANTOS[13], HUANG and DAI[14], MITSUHATA and UCHIDA[15]) designed different forms of finite element method for three-dimensional magnetotelluric forward modeling and had made a preliminary application. In this work, we chose eight-node hexahedron finite element algorithm combined with divergence conditions for numerical modeling the three-dimensional magnetotelluric response.

2 Basic theory in 3D magnetotelluric sounding Assuming the time harmonic dependence e−iwt and ignoring the displacement current in magnetotelluric (10−3−103 Hz) cases, the first Maxwell’s equations in frequency domain can be written as

Foundation item: Project(60672042) supported by the National Natural Science Foundation of China Received date: 2008−02−27; Accepted date: 2008−04−30 Corresponding author: TONG Xiao-zhong, PhD; Tel: +86−731−8830648; E-mail: [email protected]

J. Cent. South Univ. Technol. (2009) 16: 0136−0142

137

∇ × E = −iωµH

(1)

∇ × H = σE

(2)

where E and H are the electric and magnetic fields, respectively; σ is the spatially variable electrical conductivity; ω is the angular frequency; and µ is the magnetic permeability in free space. It can be acquired from Eqns.(1) and (2) that: ∇ × (∇ × E ) − iωµσE = 0

(3)

x = x0 +

b a c ξ , y = y0 + η , z = z0 + γ 2 2 2

a b c dξ , dy = dη , dz = dγ 2 2 2 abc dxdydz = dξdηdγ 8

(7)

dx =

(8)

where x0, y0, and z0 are the mid-point of slave cell; a, b, and c are length of three sides of slave cell.

The model of three-dimensional magnetotelluric sounding is shown in Fig.1. Assuming that the polarization direction of initial electric field corresponds with x axis, boundary conditions can be written as follows[16]. (1) In ABCD face, there exist Ex=1, Ey=0, Ez=0

(4)

(2) In four vertical faces, there exist E×H⊥Γ (Γ is model border)

(5)

(3) In EFGH face, there exist Ex=ce−kz, Ey=0, Ez=0

(6)

where c is the invariable, k = − iωµσ , and σ is invariable conductivity below EFGH face.

Fig.1 Model of 3D magnetotelluric sounding

3 Finite element method analysis 3.1 Finite element method The generalization of a quadrilateral threedimensional is a hexahedron, also known in the finite element literature as brick. A hexahedron is topologically equivalent to a cube. It has eight corns, twelve edges or sides, and six faces. In this paper, we chose an eight-node hexahedron finite element mesh (shown in Fig.2). The coordinate relationship between master cell and slave cell can be described as Eqn.(7), and the differential relationship between master cell and slave cell can be described as Eqn.(8).

Fig.2 Eight-node hexahedron element: (a) Master cell; (b) Slave cell

The shape functions are

1 1 N 1e = (1 − ξ )(1 − η )(1 − γ ), N 2e = (1 − ξ )(1 − η )(1 + γ ) 8 8 1 1 e e N 3 = (1 − ξ )(1 + η )(1 + γ ), N 4 = (1 − ξ )(1 + η )(1 − γ ) 8 8 1 1 e e N 5 = (1 + ξ )(1 − η )(1 − γ ), N 6 = (1 + ξ )(1 − η )(1 + γ ) 8 8 1 1 e e N 7 = (1 + ξ )(1 + η )(1 + γ ) , N 8 = (1 + ξ )(1 + η )(1 − γ ) 8 8 (9) These eight formulas can be summarized in a single


138

expression: 1 N ie = (1 + ξ iξ )(1 + η iη )(1 + γ i γ ) 8

e e ∂N ie ∂N j e ∂N ie ∂N j e ⎤ E yj − E zj ⎥ dxdydz = 0 ∂y ∂x ∂z ∂x ⎥⎦

(10)

ξi, ηi and γi denote the coordinates of the ith

where node.

3.2 General variation method The function can be generated by the general variation principle as follows:

F[ E ] =

1 [(∇ × E ) ⋅ (∇ × E ) − k 2 E ⋅ E ]dV 2 V∫

In the same way, we can obtain 8 ⎡ ∂N e ∂N ej ∂Fe i E xje + = ∑ ∫ ⎢− ∂E yi j =1 e ⎢ ∂x ∂y ⎣

⎞ ⎛ ∂N e ∂N ej ∂N e ∂N ej i ⎜ i − k 2 N ie N ej ⎟ E yje − + ⎟ ⎜ ∂x ∂x ∂z ∂z ⎠ ⎝

(11)

e ∂N ie ∂N j ⎤ e ⎥ E zj dxdydz = 0 ∂z ∂y ⎥ ⎦

It can be acquired from Eqn.(11) that: F [ E ] = ∑ Fe [ E e ] =

1 ∑ ∫ 2 [ (∇ × E e ) ⋅ (∇ × E e ) − k 2 E e ⋅ E e ]dV (12) V

⎧ e 1 ⎪⎛⎜ ∂E ze ∂E y ⎞⎟ F [ E ] = ∑ Fe [ E ] = ∑ ∫ ⎨ − + 2 ⎪⎜ ∂y ∂z ⎟ e ⎝ ⎠ ⎩

e ⎞ ⎤ ∂N ie ∂N j − k 2 N ie N ej ⎟ E zje ⎥ dxdydz = 0 ⎟ ⎥ ∂y ∂y ⎠ ⎦

2

⎛ ∂E xe ∂E ze ⎜ − ⎜ ∂z ∂x ⎝

k

2⎡

(E e ) 2 ⎢ x ⎣

+

2

e ⎛ ∂E e ⎞ ⎟ + ⎜ y − ∂E x ⎟ ⎜ ∂x ∂y ⎠ ⎝

2 ( E ye )

2

K1e = (kij ) =

⎫

+

⎦⎪ ⎭

(13)

The equivalent equation is set to solve the stagnation point of the first variation as follows: ∂F ( E x ) ∂F =∑ e =0 ∂E xi ∂E xi

∂F ( E y )

(15)

∂F ∂F ( E z ) =∑ e =0 ∂E xz ∂E zi

(16)

∂E yi

where i=1, 2, ···, Np, and Np is the total node. For node i of any element, we have

⎧⎪⎛ ∂E e ∂E e ∂Fe z = ∫ ⎨⎜ x − ∂E xi e ⎪⎜⎝ ∂z ∂x ⎩ ⎛ ∂E ye ⎜ ⎜ ∂x ⎝

k 2 E xe

−

∂E xe ∂y

⎞⎡ ∂ ⎟⎢ ⎟ ⎢ ∂E xi ⎠⎣

⎞⎡ ∂ ⎟⎢ ⎟ ⎢ ∂E xi ⎠⎣

⎛ ∂E xe ⎜ ⎜ ∂z ⎝

⎛ ∂E xe ⎜− ⎜ ∂y ⎝

⎞⎤ ⎟⎥ − ⎟⎥ ⎠⎦

∂N ej ∂y

+

⎤ ⎥ ⎥ ⎥ e ∂N e e ∂N e e ∂N e ∂N i ∂N i ∂N i j j j ⎥dxdydz − + − ⎥ ∂x ∂x ∂z ∂z ∂z ∂y ⎥ e ∂N e ⎥ e ∂N e e ∂N e ∂N i ∂N i ∂N i j j j − + ⎥ ∂y ∂z ∂x ∂x ∂y ∂y ⎦⎥

e ∂N ie ∂N j ∂z ∂z

−

e ∂N ie ∂N j ∂x ∂y

−

e ∂N ie ∂N j ∂x ∂z

−

e ∂N ie ∂N j ∂y ∂x

−

e ∂N ie ∂N j ∂z ∂x

⎤ ⎡ N ie N ej 0 0 ⎥ ⎢ ⎥dxdydz k 2 ∫ ⎢ 0 N ie N ej 0 ⎥ ⎢ e ⎢ 0 0 N ie N ej ⎥ ⎦ ⎣

(20)

If the boundary face of finite element falls over the EFGH face, the boundary function can be written as follows: I [ E ] = ∫∫ [(∇ × E ) ⋅ E ]ds = s

∫∫ E x

⎞⎤ ⎟⎥ + ⎟⎥ ⎠⎦

s

∂E x dxdy = ∫∫ k ( E x ) 2 dxdydz ∂z s

(21)

Using the stagnation point of variation, we obtain ∂I e ∂E xie

⎫ ∂ ( E xe )⎬dxdydz = ∂E xi ⎭

⎡⎛ ∂N e ∂N e ∂N e ∂N e ⎞ j j ∑ ∫ ⎢⎢⎜⎜ ∂yi ∂y + ∂zi ∂z − k 2 N ie N ej ⎟⎟ E xje − j =1 e ⎝ ⎠ ⎣ 8

⎡ ∂N e ⎢ i ⎢ ∂y ⎢ ⎢ ∫⎢ e⎢ ⎢ ⎢ ⎣⎢

(14)

∂Fe =0 ∂E yi

=∑

(19)

Therefore, the coefficient matrix at any element can be written

⎞ ⎟ − ⎟ ⎠

⎪ ( E ze ) 2 ⎤ ⎬dxdydz ⎥

(18)

e e 8 ⎡ ∂N e ∂N ej ∂Fe ∂N ie ∂N j e ⎛⎜ ∂N ie ∂N j i E xje − E yj + = ∑ ∫ ⎢− + ⎜ ∂x ∂x ∂E zi j =1 e ⎢ ∂x ∂z ∂y ∂z ⎣ ⎝

It can be obtained from Eqn.(12) that: e

(17)

= ∫∫ k N ie N ej E xje dxdy

(22)

s

Therefore,

K 2e

⎡ K 11 ⎢ =⎢ 0 ⎢0 ⎣

0 0 0

0⎤ ⎥ 0⎥ 0⎥ ⎦

(23)


139

K 3e = (k ij ) =

where K 11 = (k ij ) = ∫∫ k N ie N ej dxdy . s

3.3 Divergence condition In order to guarantee that the solutions conserve electric current especially in the area with very small conductivity in air and at the low frequencies, we enforce the divergence free conditions on the field. It is often necessary to explicitly enforce divergence conditions, that ∇ ⋅ σE = 0

(24)

∇⋅E =0

(25)

in the earth and air, respectively. Using the general variation principle, we obtain

2

⎛ ∂E e ∂E ye ∂E e ⎞ 1 ∑ ⎜ x + ∂y + ∂zz ⎟⎟ dxdydz 2 ∫e ⎜ ∂x ⎝ ⎠

(26)

Using the stagnation point of variation, we obtain ⎛ ∂E e ∂E ye ∂E e ⎞ ⎡ ∂ z ⎟ =⎜ x + + ⎢ ⎜ ∂x ∂y ∂z ⎟ ⎢ ∂E xie ⎝ ⎠⎣

⎛ ∂E xe ⎜ ⎜ ∂x ⎝

⎞⎤ ⎟⎥ = ⎟⎥ ⎠⎦

⎛ ∂N ej ∂N ej e e ⎜ E + ∑ ∫ ⎜ ∂x xj ∂y E yj + j =1 e ⎝ 8

⎞ ∂N e E zje ⎟ i dxdydz = ⎟ ∂x ∂z ⎠

∂N ej

e ⎛ ∂N e ∂N ej ∂N ie ∂N j e e ⎜ i + E ∑ ∫ ⎜ ∂x ∂x xj ∂x ∂y E yj + j =1 e ⎝ 8

e ∂N ie ∂N j e ⎞⎟ E zj dxdydz=0 ⎟ ∂x ∂z ⎠

(27)

In the same way, we can obtain ∂Ge ∂E yie

⎛ ∂N e ∂N ej = ∑∫⎜ i E xe + ⎜ ∂x j =1 e ⎝ ∂y 8

e e ∂N ie ∂N j e ∂N ie ∂N j e ⎞⎟ Ey + E z d xd yd z=0 ⎟ ∂y ∂y ∂y ∂z ⎠

∂Ge ∂E zie

∂x ∂N ej ∂x ∂N ej ∂x

e ∂N ie ∂N j ∂x ∂y e ∂N ie ∂N j ∂y ∂y e ∂N ie ∂N j ∂z ∂y

e ∂N ie ∂N j ⎤ ⎥ ∂x ∂z ⎥ e⎥ ∂N ie ∂N j ⎥ dxdydz ∂y ∂z ⎥ ⎥ e ∂N ie ∂N j ⎥ ⎥ ∂z ∂z ⎥ ⎦

(28)

e 8 ⎛ ∂N e ∂N ej ∂N ie ∂N j e = ∑∫⎜ i E xe + Ey + ⎜ ∂x ∂z ∂y j =1 e ⎝ ∂z e ∂N ie ∂N j e ⎞⎟ E z dxdydz=0 ⎟ ∂z ∂z ⎠ Therefore,

(29)

(30)

Finally, the total stiffness matrix can be written as (31)

So a linear equation can be obtained as follows: KE=0

1 G[ E ] = ∫ (∇ ⋅ E ) 2 dV = 2V

∂E xie

∂N ej

K = ∑ ( K 1e + K 2e + K 3e )

and

∂Ge

⎡ ∂N e ⎢ i ⎢ ∂x ⎢ e ⎢ ∂N i ∫ ⎢ ∂y e⎢ ⎢ ∂N e ⎢ i ⎢⎣ ∂z

(32)

The system of equation of the proposed finite element algorithm is a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method[17−18]. Considering boundary conditions, we can obtain the electric field of each point by solving linear equation.

4 Numerical simulation 4.1 One-dimensional model The two-layer model is simulated for a layer 600 Ω·m homogeneous half-space containing a layer 100 Ω·m between 0 and 1 800 m in depth. The comparison of magnetotelluric forward result between one-dimensional synthetic data and three-dimensional modeling is shown in Fig.3. It can be seen that the apparent resistivity and impedance phase curve of forward modeling are consistent with the synthetic data[19]. The maximum relative error of the apparent resistivity is 6.12% the and the maximum relative error of impedance phase is 1.56%, which shows that the three-dimensional forward algorithm is more efficient. 4.2 COMMEMI 3D−1 model The COMMEMI 3D−1 model(Fig.4) was tested by many methods and adopted in the COMMEMI project [20]. The model has a relatively high-conductivity compared with a 1 km × 2 km × 2 km conductive block(0.5 Ω·m) embedding a homogeneous background earth (100 Ω·m). In forward simulation, using 24×30×22 elements (including five air layers) for the subdivision, the extension of the air layer is 90.75 km. The grid units along x direction of the interval(unit: km) are as follows: 60.75 20.25 6.75 2.25 0.75 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.75 2.25 6.75 20.25. The grid units along y direction of the interval(unit:

140

km) are as follows: 60.75 20.25 6.75 2.25 0.75 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.75 2.25 6.75 20.25. The grid units along z direction of the interval(unit: km) are as follows: 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.75 2.25 6.75 20.25. Extension of the air unit interval (unit: km) are as follows:


0.75 2.25 6.75 20.25 60.75 We computed the three-dimensional magnetotelluric responses of the COMMEMI 3D−1 model respectively at 0.1 and 10 Hz. Figs.5 and 6 show the contour maps of the apparent resistivity and impedance phase by three-dimensional FE method, which basically reflect the geo-electrical parameters of the model. By the comparison of three-dimensional forward results between the algorithm and Ref.[20], there is excellent

Fig.3 Comparison of magnetotelluric forward results between 1D and 3D modeling: (a) Apparent resistivity; (b) Impedance phase

Fig.4 COMMEMI 3D−1 model: (a) Section view; (b) Plan view; (c) Cut-away volume view


141

Fig.5 COMMEMI 3D−1 model forward results at 0.1 Hz: (a) Apparent resistivity of xy-mode; (b) Impedance phase of xy-mode; (c) Apparent resistivity of yx-mode; (d) Impedance phase of yx-mode

Fig.6 COMMEMI 3D−1 model forward results at 10 Hz: (a) Apparent resistivity of xy-mode; (b) Impedance phase of xy-mode; (c) Apparent resistivity of yx-mode; (d) Impedance phase of yx-mode


142

overall agreement between COMMEMI project results and apparent resistivity responses finite element method obtained by the proposed, with large discrepancies in impedance phase near. Therefore, the algorithm for three-dimensional magnetotelluric forward modeling is efficient.

[6]

[7] [8]

5 Conclusions [9]

(1) An algorithm of finite element method combined with divergence condition for 3D magnetotelluric forward modeling is presented. The algorithm can eliminate the spurious solutions and the divergence corrections can guarantee the divergence-free conditions in low frequency magnetotelluric field. (2) The system of equations of the finite element algorithm is a large sparse, banded, symmetric, ill-conditioned, and non-Hermitian complex matrix equation. (3) The validity of this algorithm is confirmed by comparing modeling results with other synthetic results or numerical results. This efficient algorithm will help to study the distribution laws of 3D magnetotelluric responses and to setup basis for research of three-dimensional inversion.

References [1] [2]

[3]

[4]

[5]

HOHMANN G W. Three-dimensional induced polarization and electromagnetic modeling [J]. Geophysics, 1975, 40(2): 309−324. WANNMAKER P E. Advances in three-dimensional magnetotelluric modeling using integral equations [J]. Geophysics, 1991, 56(11): 1716−1728. XIONG Z. EM modeling of three-dimensional structures by the method of system iteration using integral equations [J]. Geophysics, 1992, 57(12): 1556−1561. BAO Guan-shu, ZHANG Bi-xing, JING Rong-zhong, MAO Xian-jin. Numerical modeling for 3-D EM responses using integral equation solution [J]. Journal of Central South University of Technology: Natural Science, 1999, 30(5): 472−474. (in Chinese) XU Kai-jun, LI Tong-lin. Three-dimensional magnetotelluric forward modeling using integral equation [J]. Northwestern

[10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

Seismological Journal, 2006, 28(2): 104−107. (in Chinese) MACKIE R L, MADDEN T R, WANNAMAKER P E. Three-dimensional magnetotelluric modeling using difference equations—Theory and comparisons to integral equation solutions [J]. Geophysics, 1993, 58(2): 215−226. XIONG Z. Domain decomposition for 3D electromagnetic modeling [J]. Earth Planets Space, 1999, 51(4): 1013−1018. FOMENKO E Y, MOGI T. A new computation method for a staggered grid of 3D EM field conservative modeling [J]. Earth Planets Space, 2002, 54(7): 499−509. TAN H D, YU Q F, WEI W B, BOOKER J. Magnetotelluric three-dimensional modeling using the staggered-grid finite difference method [J]. Chinese Journal of Geophysics, 2003, 46(5): 705−711. (in Chinese) ZHDANOV M S, SPICHAK V V. Mathematical modeling of electromagnetic fields in three-dimensional inhomogeneous media [M]. Moscow: Nauka Publishing House, 1992. MEHANEE S A. Multidimensional finite difference electromagnetic modeling and inversion based on the balance method [D]. Salt lake City: University of Utah, 2003. MOGI T. Three-dimensional modeling of magnetotelluric data using finite-element method [J]. Journal of Applied Geophysics, 1996, 35(2): 185−189. ZYSERMAN F I. SANTOS J E. Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modeling [J]. Journal of Applied Geophysics, 2000, 44(4): 337−351. HUANG Lin-ping, DAI Shi-kun. Finite element calculation method of 3D electromagnetic field under complex condition [J]. Earth Science—Journal of China University of Geosciences, 2002, 27(6): 775−779. (in Chinese) MITSUHATA Y, UCHIDA T. 3D magnetotelluric modeling using the T-Ω finite-element method [J]. Geophysics, 2004, 69(7): 108−119. XU Shi-zhe. Finite element method for geophysics [M]. Beijing: Science Press, 1994. (in Chinese) SMITH J T. Conservative modeling of 3D electromagnetic fields; Part Ⅱ: Biconjugate gradient solution and an accelerator [J]. Geophysics, 1996, 61(10): 1319−1324. HABER E, ASCHER U M, ARULIAH, OLDERBURG D W. Fast simulation of 3D electromagnetic problems using potentials [J]. Journal of Computational Physics, 2000, 163(2): 150−171. CHEN Le-shou, LIU Ren, WANG Tian-sheng. The magnetotelluric sounding data processing and interpretation methods [M]. Beijing: Petroleum Industry Press, 1989. (in Chinese) ZHDANOV M S, VARENTOV I M. Methods for modeling electromagnetic fields: Results from COMMEMI—The international project on the comparison of modeling methods for electromagnetic induction [J]. Journal of Applied Geophysics, 1997, 37(3): 133−271. (Edited by YANG You-ping)