Three-Dimensional Marine Magnetotelluric Parallel Forward Modeling in Conductive and Magnetic Anisotropic Medium Using Finite-Element Method Based on Secondary Field

Abstract

The marine magnetotelluric (MMT) method is a significant tool extensively utilized in offshore studies, including the understanding of the Earth’s tectonics and hydrocarbon exploration. Conductive anisotropy and non-zero magnetic susceptibility are common phenomena observed in the Earth’s subsurface, and MMT forward modeling is the basis of practical inversion. However, numerical modeling that incorporates both conductive anisotropy and magnetic susceptibility has received limited attention. Moreover, both accuracy and efficiency are crucial in developing a 3D MMT modeling algorithm. Therefore, we developed a multi-level parallel MMT forward modeling algorithm that is capable of simultaneously modeling conductive and magnetic arbitrary anisotropic models using the vector finite element method based on the secondary field formula. The algorithm’s accuracy was validated through comparisons with previously published results for an arbitrary anisotropic model. The results show that the maximum relative error is below 2%, and the speedup reaches an impressive value of 552.41 when running with 2048 cores. Furthermore, the MMT responses of conductive anisotropy and magnetic susceptibility were comprehensively analyzed by several typical models. Our findings highlight the importance of considering magnetic susceptibility in magnetite-rich regions, particularly as the MMT responses may exhibit opposite responses for anomalies with lower resistivity and higher magnetic susceptibility compared with the surrounding rocks.

1. Introduction

The marine magnetotelluric (MMT) method is a crucial tool in electrical exploration, capable of inferring the subsurface electrical structure of the Earth by measuring naturally varying electromagnetic (EM) signals at the seafloor or ocean surface [1,2]. This technique has been widely employed in offshore research, including studies of the Earth’s tectonics hydrocarbon exploration, and so forth [3,4,5].

Conductivity and magnetic permeability are both essential physical properties of the Earth. Previous studies have shown that the susceptibility of certain geological materials significantly exceeds that of free space [6]. Common minerals exhibit a wide range of magnetic susceptibility variations, with magnetite minerals displaying exceptionally high values (~19.2 SI) [7]. For instance, Figure 1 [8] illustrates that the magnetic susceptibility of magnetite rock ranges from 0.6 SI to 1.0 SI in Huayang, Shaanxi Province, China. Although geological targets often possess both conductivity and permeability characteristics [9], the magnetic permeability in practical and numerical studies is usually assumed to be the free-space permeability ${\mu }_0$ ($4\pi \times {10}^{-7} \mathrm{H}/\mathrm{m}$). Many studies have shown that the influence of magnetic permeability in magnetite-rich regions cannot be ignored. Zhang and Oldenburg [10] demonstrated that assuming a susceptibility value of 0 for strong magnetized examples may result in an incorrect electrical model. Thomas [11] revealed that susceptibilities greater than 0.005 can significantly affect the EM responses in layered Earth models. Sasaki et al. [12] pointed out that ignoring susceptibility may lead to misinterpretation of models with high susceptibility values. Zhang et al. [13] successfully inverted two-dimensional data while simultaneously considering both conductivity and susceptibility, highlighting the significant impact of local magnetic anomalies on MT responses.

In recent years, there has been increased attention on magnetotelluric (MT) studies investigating the influence of conductive anisotropy [14,15,16], particularly in the development of three-dimensional (3D) forward modeling algorithms. Weidelt et al. [17] presented a staggered-grid finite difference algorithm for MT modeling in conductive anisotropic media; however, this algorithm is not convenient for dealing with irregular anomalies. Xu [18] proposed a node-based finite element (FE) method for MT modeling, but its accuracy is insufficient as it fails to satisfy the required condition of discontinuous normal electrical fields at electrical interfaces. To overcome this disadvantage of the node-based FE method, Xiao et al. [19] developed an edge-based FE method with rectangular meshes; however, this technique also faces challenges in dealing with irregular anomalies and terrain. Subsequently, Liu et al. [20] introduced a 3D adaptive FE method employing tetrahedron elements, enabling improved simulation of topography and irregular anisotropic anomalies. Additionally, Xiao et al. [21,22] contributed A-$\phi$ and T-$\Omega$ algorithms based on EM potentials. Zhou et al. [23] further advanced the field by presenting a divergence scheme rooted in the edge-based FE method, while Bai et al. [24] developed a mixed algorithm that combines nodal and edge-based approaches. However, there has been limited focus on incorporating conductive anisotropy and non-zero magnetic susceptibility. Xiao et al. [25] proposed a 3D MT modeling algorithm based on the edge-based FE method, utilizing the total field formula to obtain the EM field. Yu et al. [26] provided another 3D MT modeling scheme based on the A-$\phi$ formula, where the EM field is obtained by differentiating potential. Notably, neither of these algorithms have been parallelized, which limits their computational efficiency and scalability.

To address the aforementioned problems, we developed a robust and scalable multi-level parallelized 3D MMT modeling scheme that is capable of handling conductive and magnetic anisotropic anomalies. Based on the second field formula, the edge-based FE method is adopted to discretize the electric governing equation. To ensure accuracy, the algorithm enables the directly computation of secondary fields by solving equations with the SuperLU_dist direct solver [27]. To enhance efficiency, a multi-level parallel algorithm has been developed. This paper is organized as follows: Firstly, we provided a concise overview of the methodology for MMT modeling in conductive and magnetic anisotropic media. Subsequently, we presented crucial components necessary for achieving accurate and efficient implementation based on the secondary field approach. Following this, we conducted tests on different typical models, and we analyzed the impact of conductive anisotropy and magnetic susceptibility.

2. Problem Formulation

2.1. Differential Equations

By adopting the time dependence of $e^{-i\omega t}$, the Maxwell equations can be written as follows:

$$\nabla \times E=i\omega \widehat{\mu }H,$$

(1)

$$\nabla \times H=\left(\tilde{\sigma }-i\omega \epsilon \right)E,$$

(2)

$$\nabla \cdot H=0,$$

(3)

$$\nabla \cdot E={\rho }_Q.$$

(4)

where $E={(E_x,E_y,E_z)}^T$ and $H={(H_x,H_y,H_z)}^T$ are the electrical and magnetic fields, respectively; $\widehat{\mu }$ and $\epsilon$ represent the magnetic permeability and the dielectric constant of the medium, respectively; $\omega$ is angular frequency; $\tilde{\sigma }$ is electrical conductivity of the medium; and ${\rho }_Q$ is the accumulated free charge. Both $\tilde{\sigma }$ and $\widehat{\mu }$ are tensors in conductive and magnetic anisotropic media, and the resistivity $\tilde{\rho }$ utilized in this context is the reciprocal of the conductivity, namely $\tilde{\rho }={\tilde{\sigma }}^{-1}$. See [28,29] for more details of those parameters.

Displacement currents can be neglected at the frequencies used in the MMT method, namely $\tilde{\sigma }-i\omega \epsilon \approx \tilde{\sigma }$. Two definitions are available to define $\tilde{\sigma }$ and $\widehat{\mu }$ [23,24]. The former, as shown in Figure 2, is adopted here.

$$\tilde{\sigma }=\left(\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{yx}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{zx}& {\sigma }_{zy}& {\sigma }_{zz}\end{array}\right),$$

(5)

$$\widehat{\mu }=\left(\begin{array}{ccc}{\mu }_{xx}& {\mu }_{xy}& {\mu }_{xz}\\ {\mu }_{yx}& {\mu }_{yy}& {\mu }_{yz}\\ {\mu }_{zx}& {\mu }_{zy}& {\mu }_{zz}\end{array}\right)={\mu }_0\left(\begin{array}{ccc}1+{\chi }_{xx}& {\chi }_{xy}& {\chi }_{xz}\\ {\chi }_{yx}& 1+{\chi }_{yy}& {\chi }_{yz}\\ {\chi }_{zx}& {\chi }_{zy}& 1+{\chi }_{zz}\end{array}\right).$$

(6)

where $\chi$ is the magnetic susceptibility. By solving Equation (1) for $E$ and substituting the resulting value into Equation (2), it can be obtained that

$$\nabla \times \left({\displaystyle \frac{\nabla \times E}{{\widehat{\mu }}_r}}\right)-i\omega {\mu }_0\widehat{\sigma }E=0.$$

(7)

The total electric field $E$ can be separated into the background field $E_P$ and the second field $E_S$.

$$E=E_P+E_S.$$

(8)

Similarly, the curl equation for the primary electric field can be obtained.

$$\nabla \times \left({\displaystyle \frac{\nabla \times E_p}{{\widehat{\mu }}^P}}\right)-i\omega {\widehat{\sigma }}^P{E}_p=0.$$

(9)

where ${\tilde{\sigma }}^P$ and ${\widehat{\mu }}^p$ are the conductivity and magnetic permeability of the background, which is usually modeled as a homogeneous half-space or layered model. If the background model has a magnetic susceptibility of 0, then the magnetic permeability for the primary field model is the same as that in vacuum. Under these circumstances, Equation (9) can be modified as follows:

$$\nabla \times \nabla \times E_p-i\omega {\mu }_0{\widehat{\sigma }}^P{E}_p=0.$$

(10)

Subtracting Equation (10) from Equation (7) and considering Equation (8), then,

$$\nabla \times \left({\displaystyle \frac{\nabla \times E_S}{{\widehat{\mu }}_r}}\right)+\nabla \times \left({\displaystyle \frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}}\right)-i\omega {\mu }_0\tilde{\sigma }{E}_S-i\omega {\mu }_0{\tilde{\sigma }}^a{E}_P=0.$$

(11)

where $I$ is a $3\times 3$ unit matrix, and ${\tilde{\sigma }}^a$ represents the distribution of anomalous electrical resistivity, defined as follows:

$${\tilde{\sigma }}^a=\tilde{\sigma }-{\tilde{\sigma }}^p.$$

(12)

2.2. Finite Element Analysis

The Galerkin weighted residuals method is adopted here to obtain the variational equation [30]. Multiplying both sides of Equation (11) by $\delta E_S$ and integrating over the entire domain, Equation (13) can be obtained:

$${\displaystyle \underset{v}{\int }\left[\nabla \times \left(\frac{\nabla \times E_S}{{\widehat{\mu }}_r}\right)\cdot \delta E_s+\nabla \times \left(\frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}\right)\cdot \delta E_s-i\omega {\mu }_0\tilde{\sigma }{E}_S\cdot \delta E_s-i\omega {\mu }_0{\tilde{\sigma }}^a{E}_P\cdot \delta E_s\right]}dv=0.$$

(13)

In this equation, $\delta E_S$ represents the variation of $E_S$, namely $\delta E_S=\delta E_{Sx}+\delta E_{Sy}+\delta E_{Sz}$.

The vector identity equation and the divergence theorem equation are expressed as follows:

$$\nabla \cdot \left(A\times B\right)=\left(\nabla \times A\right)\cdot B-A\cdot \left(\nabla \times B\right),$$

(14)

$${\displaystyle {\int }_v\nabla \cdot A}dv={\displaystyle {\oint }_{\Gamma }A\cdot nd\Gamma }.$$

(15)

where $n$ is the outward normal vector of the boundary. According to Equations (14) and (15), the first term of Equation (13) can be transformed into

$${\displaystyle {\int }_v\nabla \times \left(\frac{\nabla \times E_S}{{\widehat{\mu }}_r}\right)\cdot \delta E_S}dv={\displaystyle {\int }_v\nabla \times \delta E_S\cdot \frac{\nabla \times E_S}{{\widehat{\mu }}_r}}dv+{\displaystyle {\int }_{\Gamma }\left(\frac{\nabla \times E_S}{{\widehat{\mu }}_r}\right)}\times \delta E_S\cdot d\Gamma .$$

(16)

where $v$ represents the entire study region, $\Gamma$ represents the boundary surface of the entire region.

The second term of Equation (13) transforms into

$${\displaystyle {\int }_v\nabla \times \left(\frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}\right)\cdot \delta E_S}dv={\displaystyle {\int }_v\nabla \times \delta E_S\cdot \frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}}dv+{\displaystyle {\int }_{\Gamma }\left(\frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}\right)}\times \delta E_S\cdot d\Gamma .$$

(17)

The first type of boundary condition is adopted, then the field value on the boundary is zero. Consequently, Equation (13) turns into

$${\displaystyle \underset{V}{\int }\left[\nabla \times \delta E_S\cdot \frac{\nabla \times E_S}{{\widehat{\mu }}_r}+\nabla \times \delta E_S\cdot \frac{\nabla \times E_P}{{\widehat{\mu }}_r/(I-{\widehat{\mu }}_r)}-i\omega {\mu }_0\tilde{\sigma }{E}_S\cdot \delta E_S-i\omega {\mu }_0{\tilde{\sigma }}^a{E}_P\cdot \delta E_S\right]}dv=0.$$

(18)

The region is discretized into hexahedral elements, as shown in Figure 3. The edge lengths in the x-, y-, and z- directions are denoted by a, b, and c, respectively, and the center coordinates are $\left(x_0,y_0,z_0\right)$.

We adopted a Whitney vector as the basis [30]. When the tangential field is assigned to the edges, the field components of the background field and the secondary field in the three directions can be expressed as follows, respectively:

$$E_P^e={\displaystyle \sum _{i=1}^4\left(N_{xi}{E}_{Pxi}\hat{\mathrm{x}}+N_{yi}{E}_{Pyi}\hat{\mathrm{y}}+N_{zi}{E}_{Pzi}\widehat{z}\right)},$$

(19)

$$E_S^e={\displaystyle \sum _{i=1}^4\left(N_{xi}{E}_{Sxi}\hat{\mathrm{x}}+N_{yi}{E}_{Syi}\hat{\mathrm{y}}+N_{zi}{E}_{Szi}\widehat{z}\right)}.$$

(20)

where $N$ is the interpolation basis function [31]. Consequently, Equation (14) turns into

$${\displaystyle \sum _V{\displaystyle \underset{e}{\int }\nabla }\times \delta E_S\cdot \frac{\nabla \times E_S}{{\widehat{\mu }}_r}dv}+{\displaystyle \sum _V{\displaystyle \underset{e}{\int }}\nabla \times \delta E_S\cdot \frac{\nabla \times E_P}{{\widehat{\mu }}_r/(1-{\widehat{\mu }}_r)}dv}-{\displaystyle \sum _V{\displaystyle \underset{e}{\int }i\omega {\mu }_0\tilde{\sigma }\delta E_S\cdot E_S}dv}-{\displaystyle \sum _V{\displaystyle \underset{e}{\int }i\omega {\mu }_0{\tilde{\sigma }}^a\delta E_S\cdot E_P}dv}=0.$$

(21)

For a given element, the first term in Equation (21) turns into

$$\begin{array}{l}{\displaystyle {\int }_e\nabla \times \delta E_S\cdot \left(\tilde{v}\nabla \times E_S\right)dv}\\ ={\displaystyle {\int }_e\left\{{\left[\begin{array}{c}\delta E_{Sx}^e\\ \delta E_{Sy}^e\\ \delta E_{Sz}^e\end{array}\right]}^T\left[\begin{array}{ccc}0& \frac{\partial N_x^e}{\partial z}& -\frac{\partial N_x^e}{\partial y}\\ -\frac{\partial N_y^e}{\partial z}& 0& \frac{\partial N_y^e}{\partial x}\\ \frac{\partial N_z^e}{\partial y}& -\frac{\partial N_z^e}{\partial x}& 0\end{array}\right]\left(\begin{array}{ccc}{v}_{xx}& v_{xy}& v_{xz}\\ v_{yx}& v_{yy}& v_{yz}\\ v_{zx}& v_{zy}& v_{zz}\end{array}\right)\left[\begin{array}{ccc}0& -{\left(\frac{\partial N_y^e}{\partial z}\right)}^T& {\left(\frac{\partial N_z^e}{\partial y}\right)}^T\\ {\left(\frac{\partial N_x^e}{\partial z}\right)}^T& 0& -{\left(\frac{\partial N_z^e}{\partial x}\right)}^T\\ -{\left(\frac{\partial N_x^e}{\partial y}\right)}^T& {\left(\frac{\partial N_y^e}{\partial x}\right)}^T& 0\end{array}\right]\left[\begin{array}{c}{E}_{Sx}^e\\ E_{Sy}^e\\ E_{Sz}^e\end{array}\right]\right\}dv}\\ ={(\delta E^e)}^T{K}_1^e{E}^e.\end{array}$$

(22)

where $\tilde{v}=\left(\begin{array}{ccc}{v}_{xx}& v_{xy}& v_{xz}\\ v_{yx}& v_{yy}& v_{yz}\\ v_{zx}& v_{zy}& v_{zz}\end{array}\right)={{\widehat{\mu }}_r}^{-1}$, and the second term in Equation (21) becomes

$$\begin{array}{l}{\displaystyle {\int }_e\nabla \times \delta E_S\cdot \left((I-{\widehat{\mu }}_r)\tilde{v}\nabla \times E_S\right)dv}\\ ={\displaystyle {\int }_e\left\{{\left[\begin{array}{c}\delta E_{Sx}^e\\ \delta E_{Sy}^e\\ \delta E_{Sz}^e\end{array}\right]}^T\left[\begin{array}{ccc}0& \frac{\partial N_x^e}{\partial z}& -\frac{\partial N_x^e}{\partial y}\\ -\frac{\partial N_y^e}{\partial z}& 0& \frac{\partial N_y^e}{\partial x}\\ \frac{\partial N_z^e}{\partial y}& -\frac{\partial N_z^e}{\partial x}& 0\end{array}\right]\left(\begin{array}{ccc}1-{\mu }_{xx}& {\mu }_{xy}& {\mu }_{xz}\\ {\mu }_{yx}& 1-{\mu }_{yy}& {\mu }_{yz}\\ {\mu }_{zx}& {\mu }_{zy}& 1-{\mu }_{zz}\end{array}\right)\left(\begin{array}{ccc}{v}_{xx}& v_{xy}& v_{xz}\\ v_{yx}& v_{yy}& v_{yz}\\ v_{zx}& v_{zy}& v_{zz}\end{array}\right)\left[\begin{array}{ccc}0& -{\left(\frac{\partial N_y^e}{\partial z}\right)}^T& {\left(\frac{\partial N_z^e}{\partial y}\right)}^T\\ {\left(\frac{\partial N_x^e}{\partial z}\right)}^T& 0& -{\left(\frac{\partial N_z^e}{\partial x}\right)}^T\\ -{\left(\frac{\partial N_x^e}{\partial y}\right)}^T& {\left(\frac{\partial N_y^e}{\partial x}\right)}^T& 0\end{array}\right]\left[\begin{array}{c}{E}_{Px}^e\\ E_{Py}^e\\ E_{Pz}^e\end{array}\right]\right\}dv}\\ ={(\delta E^e)}^T{K}_2^e{E}^e.\end{array}$$

(23)

The third term turns into

$$\begin{array}{l}i\omega {\mu }_0\tilde{\sigma }{E}_S\cdot \delta E_S=i\omega {\mu }_0\delta E_S\cdot \tilde{\sigma }{E}_S\\ =i\omega {\mu }_0{\displaystyle \sum _{i=1}^4{\displaystyle \sum _{j=1}^4\left\{\left[\begin{array}{ccc}\delta E_{Sxj}& \delta E_{Syj}& \delta E_{Szj}\end{array}\right]\left[\begin{array}{ccc}{N}_{xi}& 0& 0\\ 0& N_{yi}& 0\\ 0& 0& N_{zi}\end{array}\right]\cdot \left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{yx}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{zx}& {\sigma }_{zy}& {\sigma }_{zz}\end{array}\right]\left[\begin{array}{ccc}{N}_{xj}& 0& 0\\ 0& N_{yj}& 0\\ 0& 0& N_{zj}\end{array}\right]\left[\begin{array}{c}{E}_{Sxi}\\ E_{Syi}\\ E_{Szi}\end{array}\right]\right\}}}\\ ={\left(\delta E_S^e\right)}^T{K}_3^e{E}_S^e.\end{array}$$

(24)

The fourth term turns into

$$\begin{array}{l}i\omega {\mu }_0\delta E_S\cdot {\tilde{\sigma }}^a{E}_P\\ =i\omega {\mu }_0{\displaystyle \sum _{i=1}^4{\displaystyle \sum _{j=1}^4\left\{\left[\begin{array}{ccc}\delta E_{Sxj}& \delta E_{Syj}& \delta E_{Szj}\end{array}\right]\left[\begin{array}{ccc}{N}_{xi}& 0& 0\\ 0& N_{yi}& 0\\ 0& 0& N_{zi}\end{array}\right]\cdot \left[\begin{array}{ccc}{\sigma }_{xx}^a& {\sigma }_{xy}^a& {\sigma }_{xz}^a\\ {\sigma }_{yx}^a& {\sigma }_{yy}^a& {\sigma }_{yz}^a\\ {\sigma }_{zx}^a& {\sigma }_{zy}^a& {\sigma }_{zz}^a\end{array}\right]\left[\begin{array}{ccc}{N}_{xj}& 0& 0\\ 0& N_{yj}& 0\\ 0& 0& N_{zj}\end{array}\right]\left[\begin{array}{c}{E}_{Pxi}\\ E_{Pyi}\\ E_{Pzi}\end{array}\right]\right\}}}\\ ={\left(\delta E_S^e\right)}^T{K}_4^e{E}_P^e.\end{array}$$

(25)

Extending it to all elements, then Equation (26) can be obtained for the entire computational region.

$$\begin{array}{l}{\displaystyle \sum _V{\displaystyle \underset{e}{\int }\nabla }\times \delta E_S\cdot \frac{\nabla \times E_S}{{\widehat{\mu }}_r}dv}+{\displaystyle \sum _V{\displaystyle \underset{e}{\int }}\nabla \times \delta E_S\cdot \frac{\nabla \times E_P}{{\widehat{\mu }}_r/(1-{\widehat{\mu }}_r)}dv}-{\displaystyle \sum _V{\displaystyle \underset{e}{\int }i\omega {\mu }_0\tilde{\sigma }\delta E_S\cdot E_S}dv}-{\displaystyle \sum _V{\displaystyle \underset{e}{\int }i\omega {\mu }_0{\tilde{\sigma }}^a\delta E_S\cdot E_P}dv}\\ ={\displaystyle \sum _1^{n_e}\left[\delta {E_S}^T(K_1^e-K_3^e)E_S+\delta {E_S}^T(K_2^e-K_4^e)E_P\right]}.\end{array}$$

(26)

Considering that $\delta E_S$ does not always equal zero, the equation system is as follows:

$$(K_1-K_3)E_S=(-K_2+K_4)E_P.$$

(27)

The values of $K_1^e$, $K_2^e$, $K_3^e$, and $K_4^e$ in an element are shown in Appendix A. The SuperLU_DIST [27], which is a parallel direct solver, is utilized to solve Equation (27), and the technique for computing $E_P$ is consistent with the work in [14].

Two orthogonal generating sources, namely Source A in the y-direction and Source B in the x-direction, located on the top surface of this study’s domain, are used in this paper.

$$E_{\mathrm{SA}}=\left(E_{\mathrm{SAx}},E_{\mathrm{SAy}},E_{\mathrm{SAz}}\right)=\left(0,1,0\right),E_{\mathrm{SB}}=\left(E_{\mathrm{SBx}},E_{\mathrm{SBy}},E_{\mathrm{SBz}}\right)=\left(1,0,0\right).$$

(28)

3. Multi-Level Parallel Algorithm

There are several available parallel schemes in the area of MT or controlled-source EM modeling [31,32]. Our MMT parallel algorithm is implemented in the C programming language using MPICH 3.2.1, with GCC 9.3.0 as the compiler and SuperLU_dist 8.0.0 as the direct solver for solving equations. As shown in Figure 4, the overall workflow of the parallel algorithm consists of six main steps: input, pre-processing, equation assembly, equation solving, post-processing, and output. There are N processes in MPI_COMMON_WORLD. The computational domain (for all frequencies) is first divided into several groups in parallel. Then, the frequencies in each group are executed one by one, while the computations for each frequency are carried out in parallel.

To illustrate, consider a scenario with six frequencies, 12 processes, and three groups. Initially, the total of 12 processes is divided into three groups, each group consisting of two frequencies and four processes. Each group has all the necessary information for computation, thereby reducing MPI traffic to some extent. Consequently, computations are performed in parallel both between groups and within each frequency; however, frequencies within each group are executed sequentially. Notably, all components within each frequency, including pre-processing, element analysis, matrix assembly, boundary conditions handling, equation solving, and post-processing are carried out in parallel.

4. Numerical Results

4.1. Accuracy Validation

To validate the algorithm’s correctness, a 3D conductive and magnetic arbitrary anisotropic model is designed, as shown in Figure 5. A 3D anomaly is embedded in an isotropic half-space with a resistivity of 100 $\Omega \cdot \mathrm{m}$. The dimensions of the anomaly are 600 m × 600 m × 600 m, and its top depth is 220 m. The principal resistivities are 50 $\Omega \cdot \mathrm{m}$, 100 $\Omega \cdot \mathrm{m}$, and 200 $\Omega \cdot \mathrm{m}$, respectively. The conductive Euler angles are 20°, 30°, and 10°, respectively. Its principal susceptibilities are 1, 0.5, and 0.8, and magnetic Euler angles are 45°, 20°, and 30°, respectively. The frequency computed is 10 Hz.

The comparison between the results of this study and those of Xiao et al. [25] is shown in Figure 6 and Figure 7, respectively. In Figure 6, the upper row corresponds to Xiao’s results, which include the apparent resistivities and phases. The middle row corresponds to the results of this paper, and the bottom row shows the comparison errors. The first and second columns correspond to the apparent resistivity of the xy- and yx-modes, respectively, while the third and fourth columns correspond to the phases of the xy- and yx-modes, respectively.

The electric fields $E_x$ along y = 0 m at the Earth’s surface generated by Source B are further validated. In Figure 7, the electric fields $E_x$ from this paper and those from Xiao et al. (2020) [25] are shown in the upper subplot, while the relative errors are displayed in the bottom subplot.

4.2. Scalability Test

The scalability of the parallel algorithm is tested by computing a model with over one million unknowns (16 frequencies are equally sampled in logarithmic space between 0.001 Hz and 100 Hz). The grid dimensions are 76, 76, and 71 in the x-, y-, and z-directions, respectively.

A series of strong scalability tests were conducted on the Tianhe supercomputer with FT processors (National Supercomputer Center in Tianjin, Tianjin, China). Each computation node contains 64 cores running at a frequency of 2.0 GHz frequency, with 128 GB memory, while also exhibiting lower power consumption. The speedup and parallel efficiency are shown in Figure 8.

4.3. Typical Models

To eliminate the influence of magnetic susceptibility in marine environments, several representative marine models have been designed.

4.3.1. Cases with Single Anomaly

As shown in Figure 9, there is a layered marine medium with a 3D rectangular block embedded in the underlying half-space. The resistivities of the air, seawater, first layer (layer1), and half-space are 10¹² $\Omega \cdot \mathrm{m}$, 0.4 $\Omega \cdot \mathrm{m}$, 10 $\Omega \cdot \mathrm{m}$ and 100 $\Omega \cdot \mathrm{m}$, respectively. The thickness of the seawater layer is 300 m, and the thickness of the first layer is 100 m. The dimensions of the 3D anomaly are 1000 m × 1000 m × 100 m in the x-, y-, and z-directions, respectively. Its top depth is 200 m. A magnetic case and a conductive anisotropic and magnetic case are studied, respectively.

For the magnetic case, the 3D anomaly’s resistivity is set to 100 $\Omega \cdot \mathrm{m}$, which is the same as the half-space, while its susceptibility is set to be −0.5, 0, 0.5, 1, 1.5, and 2.0, respectively. The apparent resistivities and phases of 10 Hz are shown in Figure 10 and Figure 11, respectively. The first to second rows correspond to the xy- and yx-modes, respectively, while the first to fifth columns correspond to susceptibilities of −0.5, 0, 0.5, 1, 1.5, and 2, respectively.

For the conductive anisotropic and magnetic case, the axial resistivities are 30 $\Omega \cdot \mathrm{m}$, 50 $\Omega \cdot \mathrm{m}$, and 60 $\Omega \cdot \mathrm{m}$ in the x-, y-, and z-directions, respectively. Its susceptibility values are set to be 0, 0.5, 1, 1.5, and 2.0, respectively. The apparent resistivities and phases at 10 Hz are shown in Figure 12 and Figure 13, respectively. The first to second rows correspond to the xy- and yx-modes, respectively, while the first to fifth columns correspond to susceptibilities of 0, 0.5, 1, 1.5, and 2, respectively.

4.3.2. Case with Synthetic Anomalies

The synthetic model is shown in Figure 14. There is another 3D anomaly (anomaly2) located beneath the first 3D anomaly (anomaly1). The dimensions of anomaly2 are 1000 m × 1000 m × 160 m in the x-, y-, and z-directions, respectively. In this case, anomaly1 is an axial conductive anisotropic block with three principal resistivities of 30 $\Omega \cdot \mathrm{m}$, 50 $\Omega \cdot \mathrm{m}$, and 60 $\Omega \cdot \mathrm{m}$, respectively. While anomaly2 is a magnetic block with a susceptibility of 0, 1, and 2, respectively. The frequency computed here is also 10 Hz.

The apparent resistivities and phases are presented in Figure 15 and Figure 16, respectively. The first to second rows correspond to the xy-, and yx-modes, respectively, while the first to fifth columns correspond to susceptibilities of 0, 0.5, 1, 1.5, and 2, respectively.

5. Analysis and Discussion

5.1. Accuracy Validation

The conductive and magnetic arbitrary anisotropic model is utilized to comprehensively verify the accuracy of this algorithm. As shown in Figure 6, the relative errors of the xy-mode apparent resistivities are below 1.2%, and those of the yx-mode apparent resistivities are less than 0.15%. Moreover, the errors of the xy-mode phases do not exceed 0.3254°, and for the yx-mode phases, they remain below 0.03°. As shown in Figure 7, the relative errors of the electric fields $E_x$ of Source B along the y = 0 m at the Earth’s surface are less than 0.05%. Given that the relative errors of the apparent resistivities are significantly lower than 5%, the errors of the phases are considerably less than 1°, and the relative errors of electric fields are far below 0.05%, the results of this algorithm exhibit a high level of agreement with the findings of Xiao et al. [25].

5.2. Scalability Test

In this section, the degree of freedom (Dof), the run-time, and the number of nonzero elements (NoN) are discussed in detail.

With the grid dimensions of 76, 76, and 71 in the x-, y-, and z-directions, respectively, the Dof of one frequency is 1,263,647, which is more than 1,000,000. For the 16 frequencies tested in this Section, the Dof is 20,218,352, which exceeds 20,000,000. The serial computation time is 116.92 h, while the parallel computation time using 2048 processes only needs 12.70 min. As shown in Figure 7, the speedup shows a remarkable improvement with the increase in processes. For the parallelized algorithm running with 2048 processes, the speedup over the serial version reaches an impressive value of 552.41 and is accompanied by a parallel efficiency of 26.97%. Due to the independence of calculations between different frequencies, this naturally leads to a task-level parallelism among them. Notably, the speedup will further increase with an increasing number of unknowns or processes, and the computation time will further decrease with a higher CPU frequency. The scalability test reveals that the multi-level parallel algorithm exhibits exceptional scalability.

The memory required to run this code is closely related to two aspects: (1) the storage strategy of matrices, which is further closely connected to the NoN of the coefficient matrices in Equation (27); and (2) the choice of the direct or iterative solver to solving Equation (27), which is also closely related to the NoN. Therefore, we only analyze the NoN here. Since it is the hexahedral cell and the edge-based FE method are adopted in this context, the NoN of the left or right coefficient matrices in Equation (27) can be obtained by

$$\begin{array}{l}NoN=33\times NL-26\times (NX\times (NY-1)+NY\times (NX-1)+\\ \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}NX\times (NZ-1)+NZ\times (NX-1)+NY\times (NZ-1)+\\ \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}NZ\times (NY-1)))-48\times (NX+NY+NZ).\end{array}$$

(29)

where $NL$ represents the edges of the entire computational domain, and $NX$, $NY$, $NZ$ denote the number of cells in the x-, y-, and z-directions, respectively.

For each frequency, two orthogonal sources are adopted. Therefore the $NoN$ of the tested model with 16 frequencies is 64 times Formula (29), amounting to 2,613,227,456. Furthermore, given that the matrix coefficients are complex values with double precision, it takes 38.94 GB to store the non-zero coefficients alone. The data are shown in

Table 1. Statistics including Dofs, NoNs, runtime, and memory required.

Modes	Dofs	NoNs	Runtime (m)	Memory (GB)
Serial	/	/	7015.20	1.22
Parallel	20,218,352	2,613,227,456	12.70	38.94

.

5.3. Typical Models

For the convenience of expression, in this section we hereby define $\chi$ as the magnetic susceptibility of anomalies, ${\rho }^P$ as the resistivity of the surrounding rock where the anomaly is embedded, and ${\rho }_x$ and ${\rho }_y$ as the principal resistivities in the x- and y-directions, respectively.

5.3.1. Cases with Single Anomaly

Cases with a single anomaly include the magnetic case and the conductive anisotropic and magnetic case.

Magnetic case. As shown in Figure 10 and Figure 11, when $\chi$ is −0.5 the apparent resistivities and phases are somewhat similar to those of a low-resistive anomaly. When $\chi$ is 0, the apparent resistivities and phases remain unchanged due to the magnetic case turning into the layered background model. However, as susceptibility increases, the apparent resistivities become higher and the phases become lower. This indicates that a positive susceptibility in a magnetic anomaly will result in higher apparent resistivities and lower phases, somewhat resembling a high-resistive anomaly, while a negative susceptibility in a magnetic anomaly will result in lower apparent resistivities and higher phases, resembling a lower-resistive anomaly to some extent. These findings are consistent with previous research by Xiao et al. [25], who explained this phenomenon using the analytical solutions of an axial conductive and magnetic anisotropic half-space.

Conductive anisotropic and magnetic case. The results presented in Figure 12 and Figure 13 demonstrate a transition from lower to higher apparent resistivities, accompanied by significant variations in the phases. Specifically, the xy-mode apparent resistivities exhibit noticeably lower values when $\chi$ is 0, 0.5, 1, or 1.5, while they tend to manifest higher values when $\chi$ is 2. Through parameter analysis, it is observed that the value $(1+\chi ){\rho }_x$ in the x-direction increases when $\chi$ varies from 0 to 2, and the maximum value 90 $\Omega \cdot \mathrm{m}$ approaches ${\rho }^P$ (100 $\Omega \cdot \mathrm{m}$). On the other hand, the yx-mode’s apparent resistivities display obviously lower values when $\chi$ is set at either 0 or 0.5 but show clearly higher values when $\chi$ is set at 1.0, 1.5, or 2.0. Parameter analysis reveals that the value of $(1+\chi ){\rho }_y$ in the y-direction is no less than ${\rho }^P$ when $\chi \ge 1$. This phenomenon is also consistent with the conclusions in [25].

5.3.2. Case with Synthetic Anomalies

The results, as shown in Figure 15 and Figure 16, demonstrate a transition from lower to higher apparent resistivities with significant variations in phases. When the magnetic susceptibility of anomaly2 is 0, indicating that it is identical to the surrounding rocks, the apparent resistivities and phases are solely influenced by anomaly1, resulting in lower apparent resistivities. The xy-mode apparent resistivities manifest as obvious lower apparent resistivities when $\chi$ is set to 0, 0.5, 1, or 1.5, while they exhibit almost higher values when $\chi$ equals 2. On the other hand, the yx-mode apparent resistivities manifest as noticeably lower apparent resistivities for $\chi$ values of 0 and 0.5, while they manifest obvious higher values for $\chi$ values of 1, 1.5, or 2.

In this case, anomaly1 is a conductive axial anisotropic yet non-magnetic block, while anomaly2 beneath it is a magnetic but non-conductive block. Additionally, the size of anomaly2 is larger than that of anomaly1. Despite the presence of these two distinct anomalies, the explanation provided in Section 5.3.1 remains applicable for elucidating the behavior of MMT’s responses. However, it is crucial to note that MMT’s responses are not solely dependent on conductivity and magnetic susceptibility; they are also closely linked to the size and location of these anomalies.

6. Conclusions

In this study, we developed an accurate and efficient 3D MMT modeling algorithm based on the secondary field formula, utilizing the edge-based finite element method and integrating multi-level parallel technology to investigate MMT responses in complex media with conductive and magnetic anisotropy. The accuracy of the algorithm was validated through comparisons with previously published results for a conductive and magnetic arbitrary anisotropic model, while its efficiency was confirmed via scalability tests. The scalability test indicates that the computational efficiency can be significantly improved with the implementation of parallel technology. Therefore, the algorithm is beneficial for practical inverse problems that rely on a fast-forward modeling scheme.

More importantly, it can be concluded that the susceptibility has a considerable influence on MMT’s responses in practical MMT works conducted in magnetite-rich regions, as demonstrated through several typical modeling tests in this study:

(1)

For a single conductive and magnetic anomaly, such as an anomaly with lower resistivity and higher susceptibility than the surrounding rocks, the apparent resistivities and phases can even be reversed due to the influence of susceptibility.

(2)

For synthetic anomalies, such as the combination of a conductive anomaly and a magnetic anomaly, the responses of the conductive anomaly can be affected or even covered by the influence of the magnetic anomaly.

(3)

The marine geological environment of practical MMT works is extremely intricate. It needs more attention for the influences of susceptibility, especially in magnetite-rich regions though non-zero susceptibility and conductive anisotropy complicate MMT’s responses.