An efficient spectral element method for two-dimensional magnetotelluric modeling

Abstract

We introduce a new efficient spectral element approach to solve the two-dimensional magnetotelluric forward problem based on Gauss–Lobatto–Legendre polynomials. It combines the high accuracy of the spectral technique and the perfect flexibility of the finite element approach, which can significantly improve the calculation accuracy. This method mainly includes two steps: 1) transforming the boundary value problem in the partial differential form into the variational problem in the integral form and 2) solving large symmetric sparse systems based on the combination of incomplete LU factorization and the double conjugate gradient stability algorithm through the spectral element with quadrilateral meshes. We imply the spectral element method on a resistivity half-space model to obtain a simple analytical solution and find that the magnetic field solutions simulated by the spectral element approach matched closely to the exact solutions. The experiment result shows that the spectral element solution has high accuracy with coarse meshes. We further compare the numerical results of the spectral element, finite difference, and finite element approaches on the COMMEMI 2D-1 and smooth models, respectively. The numerical results of the spectral element procedure are highly consistent with the other two techniques. All these comparison results suggest that the spectral element technique can not only give high accuracy for modeling results but also provide more detailed information. In particular, a few nodes are required in this method relative to the finite difference and finite element methods, which can decrease the relative errors. We then deduce that the spectral element method might be an alternative approach to simulate the magnetotelluric responses in two- or three-dimensional structures.

1 Introduction

As a special geo-electromagnetic method, magnetotelluric sounding can identify the resistivity or conductivity distributions in a geological medium based on harmonic and variable electromagnetic fields (Chave and Jones, 2012). Magnetotelluric sounding is based on naturally occurring electromagnetic fields, which can provide a comprehensive and continuous spectrum of the geo-electromagnetic field. This electrical resistivity, measured by comparing the electric field’s horizontal component to the magnetic field on the surface, can detect a depth of several tens of kilometers associated with the acquisition frequency. With the rapid advancement in magnetotelluric modeling and inversion, it has become one of the essential tools for recognizing deep geological structures (Unsworth, 2010; Avdeeva et al., 2012; Azeez et al., 2017; Nagarjuna et al., 2021) and geophysical investigations, such as geothermal exploration (Barcelona et al., 2013; Patro, 2017; Tarek et al., 2023), mineral deposit exploration (Benjamin et al., 2018; Jiang et al., 2022), and gas exploration (Zhang et al., 2014).

There are some numerical approaches for solving two-dimensional magnetotelluric forward problems, such as finite difference and finite element, and they are applied to two-dimensional magnetotelluric inversion (deGroot-Hedlin and Constable, 1990; Rodi and Mackie, 2001; Siripunvaraporn and Egbert, 2007; Lee et al., 2009; Kelbert et al., 2014; Guo et al., 2020; Liao et al., 2022). The finite difference numerical approach can solve partial differential equations for approximating first-order or second-order derivatives with a difference scheme (Pek and Verner, 1997; deGroot-Hedlin, 2006; Rao and Babu, 2006; Kumar et al., 2011). They also investigated the efficiency of computing two-dimensional magnetotelluric responses. This method also has high accuracy on the electric field and magnetic field components (Guo et al., 2018; Kalscheuer et al., 2018; Sarakorn and Vachiratienchai, 2018). Unfortunately, it is not easy to compute the resulting fields’ accurate apparent resistivities and phases. As another important numerical approach, finite element can be applied to solve the two-dimensional magnetotelluric forward problem (Wannamaker et al., 1986; Key and Weiss, 2006; Franke et al., 2007; Lee et al., 2009; Sarakorn, 2017; Yao et al., 2021). It involves the hypothetical functional form of the model and the field in a small area of the specified geometry. The finite element method can introduce complex information from the real world to construct the initial model, including surface topography, and can also improve the flexibility of mesh discretization. However, it requires fine meshing to obtain high accuracy, which results in high computational costs. Some other numerical methods are also used to simulate two-dimensional magnetotelluric forward modeling, such as the boundary element (Xu and Zhou, 1997), the finite-volume (Du et al., 2016; Wang et al., 2019), the mesh-free (Wittke and Tezkan, 2014; Wittke and Tezkan, 2021), the domain decomposition (Bihlo et al., 2017), the numerical manifold (Liang et al., 2021), and the pseudo-spectral methods (Tong et al., 2020). These numerical methods provide an essential practical basis for two-dimensional magnetotelluric forward modeling.

Compared to other numerical approaches, the finite element method requires fine grids to obtain higher calculation accuracy. This will bring challenges, especially when computational resources are limited. Moreover, in practical geophysical applications, when the discrete meshes need be refined to a geo-electrical model, the convergence rate will decrease gradually, while the number of meshes and the computational cost can increase largely (Key and Weiss, 2006). The spectral method, as a novel approach, can provide the numerical approximation of partial differential equations (Zou and Cheng, 2018). In this numerical approach, the field in the computational domain can be approximated by polynomials or Fourier expansions. Since high-order orthogonal basis functions are applied in the spectral method, it has exponential convergence. In addition, the spectral interpolation points are densely distributed at the boundary, which can avoid the Runge phenomenon in the traditional high-order interpolation (Tong et al., 2020). The method that combines the finite element and spectral method is called the spectral element method. In the past 20 years, geophysicists have dedicated these numerical methods to developing efficiency and accuracy. Some recent developments found that the spectral element approach can be seen as a high-order finite element method and its high-accuracy is derived from the properties of the spectral method (Patera, 1984). It can combine the high-accuracy of the spectral method and the flexibility of the finite element technique. Compared with the classical finite element method, the Runge phenomenon of isometric interpolation can be avoided using Gaussian orthogonal basis functions and Gaussian points (Xu et al., 2022). There are two types of spectral element methods, one based on Legendre polynomials and another based on Chebyshev polynomials. It is widely used in applications for wave propagation (Komatitsch and Tromp, 1999; Seriani and Oliveria, 2008; Luo et al., 2013; Trinh et al., 2019; Lyu et al., 2020), forward gravity modeling (Ghariti et al., 2018; Martin et al., 2017), and for geo-electromagnetic forward modeling problems (Zhou et al., 2016; Huang et al., 2019; Yin et al., 2019; Zhu et al., 2020; Huang et al., 2021; Weiss et al., 2023). However, it is rarely used in two-dimensional magnetotelluric forward modeling.

This paper proposes an efficient and accurate spectral element approach to compute the two-dimensional magnetotelluric responses of the boundary problem without measuring Earth’s curvature. To benchmark the accuracy, we compare the numerical results of the spectral element forward algorithm with the analytical solutions and numerical results of the finite difference and finite element schemes. Although our approach can be applied to any two-dimensional geo-electromagnetic forward modeling, in this study, we demonstrate its implementation mainly in numerical experiments.

2 Boundary value problem

2.1 Electromagnetic equations

We define the z-axis at the depth and the x-axis along the geologic strike, as shown in Figure 1. Using a time-harmonic factor , Maxwell’s equations can then be written as (Wannamaker et al., 1987; Yao et al., 2021)where E means the electric field, H represents the magnetic field, denotes the angular frequency, is the magnetic permeability, and is the conductivity.

FIGURE 1

For a two-dimensional structure, due to and , we can expand the curl operators in Eqs 1, 2as follows:

and

The electromagnetic equations are more complex than homogeneous media for two-dimensional modeling, where resistivity changes occur in the y-axis and z-axis. According to Eqs 4–6, the TE mode can yield a second-order Helmholtz equation:

Meanwhile, for the TM mode, yields another second-order Helmholtz equation:

Then the electric field or the magnetic field in the Helmholtz-type equation can be rewritten aswhere u, , and represent different meanings relying on the different polarized modes. In the TE mode,

and in the TM mode,

2.2 Boundary conditions

We restrict the computational region for Eq. 11 to a two-dimensional bounded domain , as shown in Figure 1. Therefore, the boundary conditions in the computational domain can be expressed aswhere .

3 Spectral element formulation

3.1 Discretization of a variational problem

The magnetotelluric fields can be simulated by the Helmholtz equation of Eq. 12 under the boundary conditions of Eq. 14. Using the variational principle (Pozrikidis, 2014), the boundary value problem of the partial differential form displayed in Eq. 12 and Eq. 14 can be written as the variational problem of the integral form:

Within spectral element approximation, the magnetotelluric field can be expanded with two-dimensional interpolation basis functions:where is the number of primary procedures for an element and are the essential functions.

The integral of all elements, Eq. 15, can be rewritten as

This will lead to a discrete linear equation as follows:where u represents the values of the unknown magnetic field or electric field.

3.2 Spectral basis functions

Its spectral accuracy characterizes the spectral element, i.e., the numerical error depends on the order of the basis functions (Lee and Liu, 2005). We choose Gauss–Lobatto–Legendre (GLL) element discretization for the magnetotelluric forward problem. The Nth-order GLL basis functions in a one-dimensional reference element can be expressed byfor , where is the Nth-order Legendre polynomial and is its derivative. On a reference element with , the two-dimensional basis function can be written as

For example, if order p = 4, there are 25 basis functions to the interpolation nodes. Figure 2 shows two-dimensional basis functions of the 4th order in part, and four of the nodal basis functions corresponding to , , , and are represented.

FIGURE 2

3.3 Spectral element equation

In the spectral element method, a physical sub-element needs to be mapped into a reference parent element and the element coefficient matrix can be achieved in the reference element. Figure 3 shows a mapping example of a two-dimensional spectral element -coordinate and the normalized reference coordinate.

FIGURE 3

The derivatives and the volume in the -coordinate system in Eq. 17 can be transformed to the -coordinate system as follows:where J is the Jacobian matrix.

The first-term integral in Eq. 17 iswhere

The second-term integral in Eq. 17 iswhere .

The third-term integral in Eq. 17 iswhere can be obtained with a one-dimensional line integral.

Considering the Dirichlet boundary condition at , we can derive the linear equations with the spectral element approach to build the two-dimensional magnetotelluric forward modeling:where is the coefficient matrix containing conductivity parameters , and is the number of all elements. The right-side vector p is interrelated to the Dirichlet boundary conditions. The coefficient matrix K is a sparse, positive-defined, symmetric matrix. Figure 4 shows the sparse elements’ distribution of the coefficient matrix for a 3 × 3 grid with the 4th polynomial order (just for illustration purposes). We use the biconjugate gradient-stabilized algorithm (van der Vorst, 1992; Chen et al., 2002), preconditioned with incomplete LU factorization (Pan et al., 2022), for our forward problem.

FIGURE 4

After obtaining , the corresponding magnetic field component can be solved by Eq. 4 for the TE mode. Since we acquired , the corresponding electric component can be solved by Eq. 7 for the TM mode. Then, we can obtain the two-dimensional impedance tensor from

The impedance can be used to calculate apparent resistivities

and impedance phases

4 Accuracy of the method

For all the spectral element numerical approaches, the numerical solution of the boundary value problem depends on two parameters: (1) the size of each spectral element and (2) the interpolating polynomial order. To verify our spectral element method numerically, we consider the Dirichlet boundary for a Helmholtz equationwith the exact solution

The physical domain is discretized using a uniform mesh with 5 × 5 elements. Figure 5 shows the spectral element solutions obtained using different interpolating polynomial orders. The maximum absolute errors of spectral element solutions for 2, 3, and 4th polynomial orders are 1.06E-4, 2.89E-5, and 7.84E-6, respectively. The numerical results computed by our spectral element approach are generally consistent with the series solutions.

FIGURE 5

5 Model studies and discussion

5.1 Homogeneous half-space

We developed a half-space resistivity model to test the high-accuracy benchmark of our spectral element scheme. The half-space resistivity is designed as 10 and the computational domain is set as 20 km × 4 km. During numerical simulation, each spectral element includes 4 GLL points in the y-direction and z-direction, as shown in Figure 6. In addition, we assume that only one measuring point is located on the ground, marked by a green triangle. The numerical magnetic field H_x is simulated at f = 10 Hz.

FIGURE 6

We set the number of elements in the horizontal direction to 10 (i.e., N_y = 10), while the number of elements in the depth direction is designed to 5 and 10, respectively (i.e., N_z = 10 and 5). Figure 7 shows the numerical solution of the magnetic field H_x for the homogeneous half-space frequency of f = 10 Hz. They also offer that the real part and imaginary part of the magnetic field H_x calculated by the spectral element method agree with the analytical solution. This phenomenon also shows the correlation between the number of discrete elements and computational accuracy. Furthermore, it indicates that the number of discrete components does not affect the computational accuracy under high-polynomial order conditions. These results suggest that the spectral element approach can improve the accuracy for the two-dimensional magnetotelluric forward modeling.

FIGURE 7

To further verify the applicability of our spectral element approach, we increase the number of elements in horizontal and depth directions to 20. We then calculated the magnetotelluric response, including apparent resistivity and phase, at f = 0.1, 1.1, 10, 10, and 100 Hz frequencies in the TM mode and TE mode. The computing time of our code is about 1.6 s for each frequency. The apparent resistivity for each frequency is identical to the true resistivity 10 , and the phase was equal to 45°. The RMS errors of the magnetotelluric responses are given in Table 1. It is clear that the RMS error is proportional to the frequency, with the minimum error at f = 0.01 Hz and the maximum error at f = 100 Hz. In general, the RMS error value falls within the frequency band that we set, which is also the acceptable error range. The results show that our spectral element approach has high computational accuracy.

5.2 COMMEMI 2D-1 model

We conducted a numerical experiment to compare with the finite difference method. This numerical experiment coincides with the COMMEMI 2D-1 example (Zhdanov et al., 1997), which can test the accuracy and reliability of the spectral element forward algorithm. The COMMEMI 2D-1 model is shown in Figure 8. A symmetrical, rectangular, low-resistivity body is inserted in a homogeneous conductive half-space. The rectangular anomaly body has a width of 1,000 m, a length of 2,000 m, and a burial depth of 250 m from the ground surface. The resistivity of the anomaly is set as , and the half-space resistivity is designed as . The frequency we applied in the next two experiments was f = 0.1 Hz.

FIGURE 8

First, we simulated the numerical solutions for the COMMEMI 2D-1 model using the spectral element algorithm and the finite difference method (Tong et al., 2018). In this example, the uniform meshes of the model for the whole calculation area are set to . For spectral element numerical modeling, each spectral element includes two GLL points in the y-direction and z-direction, and has nine points for each element. The numerical apparent resistivities and phases in the TM mode are shown in Figure 9. The results show that the numerical results calculated by the spectral element method match well with those of the finite difference method. Compared to finite difference results, the spectral element method shows a clear and continuous downward smoothness in TM mode calculations. The numerical results for the TE mode are shown in Figure 10. These results also matched well for the numerical magnetotelluric responses in the TE mode. The computing time of the COMMEMI 2D-1 model is about 0.5 s for the finite difference algorithm and about 1.8 s for the spectral element algorithm.

FIGURE 9

FIGURE 10

We also compare the numerical apparent resistivities calculated by the spectral element scheme and the finite difference approach with the averaged numerical solutions of the COMMEMI project (Zhdanov et al., 1997), showing that the numerical apparent resistivities of the spectral element scheme agree well with the averaged numerical solutions of the COMMEMI project compared to those measured by the finite difference approach (Figure 9 top; Figure 10 top). It means that the modeling precision of the spectral element scheme is higher than that of the finite difference method in calculating the magnetotelluric responses using the same mesh size.

In the second example, we only simulate the spectral element solutions with different polynomial orders in the COMMEMI locations. In this experiment, we designed a non-uniform grid of the model over the entire computational domain. To make a more precise comparison with the resistivity values published by the committee experiments, in Table 2, we list the standard deviation from Table B.11 (Zhdanov et al., 1997) along with the numerical resistivity values simulated by the spectral element approach. From Table 2, the values produced by the spectral element method match well with the numerical results published in the COMMEMI experiments. The accuracy of the apparent resistivity simulated by the spectral element might depend on the polynomial order.

5.3 Topographical model

To compute the magnetotelluric responses of the two-dimensional undulating terrain, we applied our spectral element code to a ridge topographical model, as shown in Figure 11, which is the same as that used by other researchers (Wannamaker et al., 1986; Liang et al., 2021). The ridge model has a width of 2,400 m with a height of 100 m, and its resistivity value of half-space is . The calculated magnetotelluric responses obtained by the finite element subroutine (Wannamaker et al., 1986) are the reference for this simulation.

FIGURE 11

In this study, the non-uniform meshes in the TM mode and TE mode are set as 15 × 10 and 20 × 10, respectively (in which 10 km is the air media and its resistivity is equal to 10¹⁵), and each spectral element has four GLL points in the y-direction and z-direction. The measurement profile along the atmospheric grounding interface varies from −2,000 to 2,000 m. The frequency to be tested is only 10 Hz. Figure 12 shows the comparison of the finite element results from Wannamaker et al. (1986) and our spectral element forward code, and the results match well. The maximum relative apparent resistivity error between the two forward schemes is equal to 0.15% in the TM mode and 0.07% in the TE mode, respectively. The maximum relative phase error is equal to 0.04% in the TM mode and 0.05% in the TE mode, respectively. As shown in Figure 12A, the topographic resistivity distortion of the TE mode is smaller than that of the TM mode.

FIGURE 12

5.4 Smooth resistivity model

In this numerical example, a smooth resistivity model is set to 18 km × 4 km, as shown in Figure 13. We calculate the response of a two-dimensional magnetotelluric model with a smooth resistivity distribution. The least-square iterative algorithm calculated the inversion of the resistivity distribution for this model with the MT2DInvMatlab subroutine (Lee et al., 2009) for a fault model tested by Sasaki (1989).

FIGURE 13

We chose nine frequencies to test this model, which are 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, and 50.0 Hz (Lee et al., 2009). The computational domain was set as 200 km × 100 km, and the resistivity value for the extended region was designed as 50 . We chose four GLL points for each element in the y-direction and z-direction, and 25 points per element. Figures 14, 15 show the pseudo-sections of the apparent resistivity and phase for this smooth model in the TM mode and the TE mode. By comparing the spectral element results and the finite element results, we found that the accuracy of the two ways is almost the same, and the results agree well.

FIGURE 14

FIGURE 15

6 Conclusion

The spectral element method combined with the GLL point interpolating scheme has been developed for the first time to solve the two-dimensional magnetotelluric forward problem. We presented the spectral element formulas and implemented this algorithm. Compared with the finite difference scheme and the finite element technique, our spectral element approach requires fewer elements and produces accurate results. In the first investigation, we apply the spectral element strategy on a simple half-space geo-electric model to test its high accuracy. We presented the comparison results of the finite difference algorithm and the spectral element algorithm for the COMMEMI 2D-1 model. The accuracy of our spectral element method is better than that of the finite difference approach. We compare the numerical results from Wannamaker et al. (1986) and our spectral element scheme for a ridge topographical model, and they agree well. These results demonstrate the effectiveness and flexibility of the spectral element forward algorithm. We also applied the spectral element method to a model with a smooth resistivity structure and compared the simulation results with those of the finite element code (Lee et al., 2009). This shows that the calculation results of the spectral element algorithm are as smooth and accurate as those of the finite element method. These measurements and comparative results suggest that the spectral element method can provide another effective scheme for the two-dimensional magnetotelluric forward problem.

References

• 1

  Avdeeva A. Avdeev D. Jegen M. (2012). Detecting a salt dome overhang with magnetotellurics: 3D inversion methodology and synthetic model studies. Geophysics77 (4), 251–263. 10.1190/geo2011-0167.1

  • CrossRef
  • Google Scholar

• 2

  Azeez K. K. Patro P. K. Harinarayana T. Sarma S. V. (2017). Magnetotelluric imaging across the tectonic structures in the eastern segment of the Central Indian Tectonic Zone: Preserved imprints of polyphase tectonics and evidence for suture status of the Tan Shear. Precambrian Res.298, 325–340. 10.1016/j.precamres.2017.06.018

  • CrossRef
  • Google Scholar

• 3

  Barcelona H. Favetto A. Peri V. Pomposiello C. Ungarelli C. (2013). The potential of audio magnetotellurics in the study of geothermal fields: A case study from the northern segment of the La candelaria range, northwestern Argentina. J. Appl. Geophys.88 (1), 83–93. 10.1016/j.jappgeo.2012.10.004

  • CrossRef
  • Google Scholar

• 4

  Benjamin M. L. Unsworth J. H. Jeremy P. R. Jean M. L. Legault J. M. (2018). 3D joint inversion of magnetotelluric and airborne tipper data: A case study from the morrison porphyry Cu–Au–Mo deposit, British columbia, Canada. Geophys. Prospect.66 (2), 397–421. 10.1111/1365-2478.12554

  • CrossRef
  • Google Scholar

• 5

  Bihlo A. Farquharson C. G. Haynes R. D. Loredo-Osti J. C. (2017). Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem. Comput. Geosci.21 (1), 117–129. 10.1007/s10596-016-9598-8

  • CrossRef
  • Google Scholar

• 6

  Chave A. D. Jones A. G. (2012). The magnetotelluric method: Theory and practice. Cambridge: Cambridge University Press.

  • Google Scholar

• 7

  Chen J. Haber E. Oldenburg D. W. (2002). Three-dimensional numerical modeling and inversion of magnetometric resistivity data. Geophys. J. Int.149 (3), 679–697. 10.1046/j.1365-246X.2002.01688.x

  • CrossRef
  • Google Scholar

• 8

  deGroot-Hedlin C. (2006). Finite-difference modeling of magnetotelluric fields: Error estimates for uniform and nonuniform grids. Geophysics71 (3), 97–106. 10.1190/1.2195991

  • CrossRef
  • Google Scholar

• 9

  deGroot-Hedlin N. C. Constable S. (1990). Occam's inversion generates smooth, two-dimensional models from magnetotelluric data. Geophysics55 (12), 1613–1624. 10.1190/1.1442813

  • CrossRef
  • Google Scholar

• 10

  Du H. K. Ren Z. Y. Tang J. T. (2016). A finite-volume approach for 2D magnetotellurics modeling with arbitrary topographies. Studia Geophys. Geod.60 (2), 332–347. 10.1007/s11200-014-1041-9

  • CrossRef
  • Google Scholar

• 11

  Franke A. Börner R. U. Spitzer K. (2007). Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography. Geophys. J. Int.171 (1), 71–86. 10.1111/j.1365-246X.2007.03481.x

  • CrossRef
  • Google Scholar

• 12

  Gharti H. N. Tromp J. Zampini S. (2018). Spectral-infinite-element simulations of gravity anomalies. Geophys. J. Int.215 (2), 1098–1117. 10.1093/gji/ggy324

  • CrossRef
  • Google Scholar

• 13

  Guo R. Li M. Yang F. Xu S. Abubakar A. (2020). Application of supervised descent method for 2D magnetotelluric data inversion. Geophysics85 (4), 53–65. 10.1190/geo2019-0409.1

  • CrossRef
  • Google Scholar

• 14

  Guo Z. Q. Egbert G. D. Wei W. B. (2018). Modular implementation of magnetotelluric 2D forward modeling with general anisotropy. Comput. Geosciences118, 27–38. 10.1016/j.cageo.2018.05.004

  • CrossRef
  • Google Scholar

• 15

  Huang X. Farquharson C. G. Yin C. Yan L. Cao X. Zhang B. (2021). A 3D forward-modeling approach for airborne electromagnetic data using a modified spectral-element method. Geophysics86 (5), 343–354. 10.1190/geo2020-0004.1

  • CrossRef
  • Google Scholar

• 16

  Huang X. Yin C. Farquharson C. G. Cao X. Zhang B. Huang W. et al (2019). Spectral-element method with arbitrary hexahedron meshes for time-domain 3D airborne electromagnetic forward modeling. Geophysics84 (1), 37–46. 10.1190/geo2018-0231.1

  • CrossRef
  • Google Scholar

• 17

  Jiang W. Duan J. Doublier M. Clark A. Schofield A. Brodie R. et al (2022). Application of multiscale magnetotelluric data to mineral exploration: An example from the east tennant region, northern Australia. Geophys. J. Int.229 (2), 1628–1645. 10.1093/gji/ggac029

  • CrossRef
  • Google Scholar

• 18

  Kalscheuer T. Juhojuntti N. Vaittinen K. (2018). Two-dimensional magnetotelluric modelling of ore deposits: Improvements in model constraints by inclusion of borehole measurements. Surv. Geophys.39 (3), 467–507. 10.1007/s10712-017-9454-y

  • CrossRef
  • Google Scholar

• 19

  Kelbert A. Meqbel N. Egbert G. D. Tandon K. (2014). ModEM: A modular system for inversion of electromagnetic geophysical data. Comput. Geosciences66, 40–53. 10.1016/j.cageo.2014.01.010

  • CrossRef
  • Google Scholar

• 20

  Key K. Weiss C. (2006). Adaptive finite-element modeling using unstructured grids: The 2D magnetotelluric example. Geophysics71 (6), 291–299. 10.1190/1.2348091

  • CrossRef
  • Google Scholar

• 21

  Komatitsch D. Tromp J. (1999). Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int.139 (3), 806–822. 10.1046/j.1365-246x.1999.00967.x

  • CrossRef
  • Google Scholar

• 22

  Kumar K. Gupta P. K. Niwas S. (2011). Efficient two-dimensional magnetotellurics modelling using implicitly restarted Lanczos method. J. Earth Syst. Sci.120 (4), 595–604. 10.1007/s12040-011-0093-2

  • CrossRef
  • Google Scholar

• 23

  Lee J. H. Liu Q. H. (2005). An efficient 3-D spectral-element method for Schrodinger equation in nanodevice simulation. IEEE Trans. Computer-Aided Des. Integr. Circuits Syst.24 (2), 1848–1858. 10.1109/TCAD.2005.852675

  • CrossRef
  • Google Scholar

• 24

  Lee S. K. Kim H. J. Song Y. Lee C. K. (2009). MT2Dinvmatlab—A program in MATLAB and FORTRAN for two-dimensional magnetotelluric inversion. Comput. Geosciences35 (8), 1722–1734. 10.1016/j.cageo.2008.10.010

  • CrossRef
  • Google Scholar

• 25

  Liang J. W. Tong D. F. Tan F. Jiao Y. Y. Yan C. W. (2021). Two-Dimensional magnetotelluric modelling based on the numerical manifold method. Eng. Analysis Bound. Elem.124 (1), 87–97. 10.1016/j.enganabound.2020.12.011

  • CrossRef
  • Google Scholar

• 26

  Liao X. Shi Z. Zhang Z. Yan Q. Liu P. (2022). 2D inversion of magnetotelluric data using deep learning technology. Acta Geophys.70, 1047–1060. 10.1007/s11600-022-00773-z

  • CrossRef
  • Google Scholar

• 27

  Luo Y. Tromp J. Denel B. Calandra H. (2013). 3D coupled acoustic-elastic migration with topography and bathymetry based on spectral-element and adjoint methods. Geophysics78 (4), 193–202. 10.1190/geo2012-0462.1

  • CrossRef
  • Google Scholar

• 28

  Lyu C. Capdeville Y. Zhao L. (2020). Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation. Geophysics85 (1), 33–43. 10.1190/geo2019-0087.1

  • CrossRef
  • Google Scholar

• 29

  Martin R. Chevrot S. Komatitsch D. Seoane L. Spangenberg H. Wang Y. et al (2017). A high-order 3D spectral-element method for the forward modelling and inversion of gravimetric data - application to the Western Pyrenees. Geophys. J. Int.209 (1), ggx010–424. 10.1093/gji/ggx010

  • CrossRef
  • Google Scholar

• 30

  Nagarjuna D. Rao C. K. Pavankumar G. Kumar A. Manglik A. (2021). Magnetotelluric evidence for an Archaean – proterozoic lithospheric assemblage within the Cambay rift basin, Western India, and its role in channeling of plume-derived fluids within the basin. Tectonophysics818, 229064. 10.1016/j.tecto.2021.229064

  • CrossRef
  • Google Scholar

• 31

  Pan K. Wang J. Hu S. Ren Z. Cui T. Guo R. et al (2022). An efficient cascadic multigrid solver for 3-D magnetotelluric forward modelling problems using potentials. Geophys. J. Int.230 (3), 1834–1851. 10.1093/gji/ggac152

  • CrossRef
  • Google Scholar

• 32

  Patera A. T. (1984). A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys.54 (3), 468–488. 10.1016/0021-9991(84)90128-1

  • CrossRef
  • Google Scholar

• 33

  Patro P. K. (2017). Magnetotelluric studies for hydrocarbon and geothermal resources: Examples from the asian region. Surv. Geophys.38, 1005–1041. 10.1007/s10712-017-9439-x

  • CrossRef
  • Google Scholar

• 34

  Pek J. Verner T. (1997). Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media. Geophys. J. Int.128 (3), 505–521. 10.1111/j.1365-246X.1997.tb05314.x

  • CrossRef
  • Google Scholar

• 35

  Pozrikidis C. (2014). Introduction to finite and spectral element methods using MATLAB. New York: Chapman and Hall Press.

  • Google Scholar

• 36

  Rao K. P. Babu G. A. (2006). EMOD2D—A program in C++ for finite difference modeling of magnetotelluric TM mode responses over 2D Earth. Comput. Geosciences32 (9), 1499–1511. 10.1016/j.cageo.2006.02.017

  • CrossRef
  • Google Scholar

• 37

  Rodi W. Mackie R. L. (2001). Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion. Geophysics66 (1), 174–187. 10.1190/1.1444893

  • CrossRef
  • Google Scholar

• 38

  Sarakorn W. (2017). 2-D magnetotelluric modeling using finite element method incorporating unstructured quadrilateral elements. J. Appl. Geophys.139, 16–24. 10.1016/j.jappgeo.2017.02.005

  • CrossRef
  • Google Scholar

• 39

  Sarakorn W. Vachiratienchai C. (2018). Hybrid finite difference–finite element method to incorporate topography and bathymetry for two-dimensional magnetotelluric modeling. Earth, Planets Space70 (1), 103. 10.1186/s40623-018-0876-7

  • CrossRef
  • Google Scholar

• 40

  Sasaki Y. (1989). Two-dimensional joint inversion of magnetotelluric and dipole-dipole resistivity data. Geophysics54 (2), 254–262. 10.1190/1.1442649

  • CrossRef
  • Google Scholar

• 41

  Seriani G. Oliveira S. P. (2008). Dispersion analysis of spectral element methods for elastic wave propagation. Wave Motion45 (8), 729–744. 10.1016/j.wavemoti.2007.11.007

  • CrossRef
  • Google Scholar

• 42

  Siripunvaraporn W. Egbert G. D. (2007). Data space conjugate gradient inversion for 2-D magnetotelluric data. Geophys. J. Int.170 (3), 986–994. 10.1111/j.1365-246X.2007.03478.x

  • CrossRef
  • Google Scholar

• 43

  Tarek A. H. Mohamed A. Z. Gad E. Q. Hossam M. Samah E. Yasuhiro F. (2023). Deep heat source detection using the magnetotelluric method and geothermal assessment of the Farafra Oasis, Western Desert, Egypt. Geothermics109, 102648. 10.1016/j.geothermics.2023.102648

  • CrossRef
  • Google Scholar

• 44

  Tong X. Guo Y. Xie W. (2018). Finite difference algorithm on non-uniform meshes for modeling 2D magnetotelluric responses. Algorithms11 (2), 203–217. 10.3390/a11120203

  • CrossRef
  • Google Scholar

• 45

  Tong X. Sun Y. Guo R. (2020). A Chebyshev pseudo-spectral approach for simulating magnetotelluric TM-mode responses on 2D structures. J. Appl. Geophys.179, 104085. 10.1016/j.jappgeo.2020.104085

  • CrossRef
  • Google Scholar

• 46

  Trinh R. T. Brossier R. Metiver L. Tavard L. Virieus J. (2019). Efficient time-domain 3D elastic and viscoelastic full-waveform inversion using a spectral-element method on flexible Cartesian-based mesh. Geophysics84 (1), 61–83. 10.1190/geo2018-0059.1

  • CrossRef
  • Google Scholar

• 47

  Unsworth M. (2010). Magnetotelluric studies of active continent-continent collisions. Surv. Geophys.31 (2), 137–161. 10.1007/s10712-009-9086-y

  • CrossRef
  • Google Scholar

• 48

  van der Vorst H. A. (1992). Bi-CGSTAB: A fast and smoothly converging variant of Bi-cg for the solution of nonsymmetric linear systems. SIAM J. Sci. Comput.13 (2), 631–644. 10.1137/0913035

  • CrossRef
  • Google Scholar

• 49

  Wang N. Tang J. Ren Z. Xiao X. Huang X. (2019). Two-dimensional magnetotelluric anisotropic forward modeling using finite-volume method. Chin. J. Geophys.62 (10), 3912–3922. 10.6038/cjg2019M0498

  • CrossRef
  • Google Scholar

• 50

  Wannamaker P. E. Stodt J. A. Rijo L. (1987). A stable finite element solution for two-dimensional magnetotelluric modelling. Geophys. J. Int.88 (1), 277–296. 10.1111/j.1365-246X.1987.tb01380.x

  • CrossRef
  • Google Scholar

• 51

  Wannamaker P. E. Stodt J. A. Rijo L. (1986). Two-dimensional topographic responses in magnetotellurics modeled using finite elements. Geophysics51 (11), 2131–2144. 10.1190/1.1442065

  • CrossRef
  • Google Scholar

• 52

  Weiss M. Kalscheuer T. Ren Z. (2023). Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes. Geophys. J. Int.232 (2), 1427–1454. 10.1093/gji/ggac358

  • CrossRef
  • Google Scholar

• 53

  Wittke J. Tezkan B. (2014). Meshfree magnetotelluric modelling. Geophys. J. Int.198 (2), 1255–1268. 10.1093/gji/ggu207

  • CrossRef
  • Google Scholar

• 54

  Wittke J. Tezkan B. (2021). Two-dimensional meshless modelling and TE-mode inversion of magnetotelluric data. Geophys. J. Int.226 (2), 1250–1261. 10.1093/gji/ggab147

  • CrossRef
  • Google Scholar

• 55

  Xu J. Tang J. Xiao X. (2022). A hybrid spectral element-infinite element approach for 3D controlled-source electromagnetic modeling. J. Appl. Geophys.200, 104619. 10.1016/j.jappgeo.2022.104619

  • CrossRef
  • Google Scholar

• 56

  Xu S. Zhou H. (1997). Modelling the 2D terrain effect on MT by the boundary-element method. Geophys. Prospect.45 (6), 931–943. 10.1046/j.1365-2478.1997.610301.x

  • CrossRef
  • Google Scholar

• 57

  Yao H. Ren Z. Chen H. Tang J. Li Y. Liu X. (2021). Two-dimensional magnetotelluric finite element modeling by a hybrid Helmholtz-curl formulae system. J. Comput. Phys.443, 110533. 10.1016/j.jcp.2021.110533

  • CrossRef
  • Google Scholar

• 58

  Yin C. Liu L. Liu Y. Zhang B. Qiu C. Huang X. (2019). 3D frequency-domain airborne EM forward modelling using spectral element method with Gauss-Lobatto-Chebyshev polynomials. Explor. Geophys.50 (5), 461–471. 10.1080/08123985.2019.1614162

  • CrossRef
  • Google Scholar

• 59

  Zhang K. Wei W. Lu Q. Dong H. Li Y. (2014). Theoretical assessment of 3-D magnetotelluric method for oil and gas exploration: Synthetic examples. J. Appl. Geophys.106, 23–36. 10.1016/j.jappgeo.2014.04.003

  • CrossRef
  • Google Scholar

• 60

  Zhdanov M. S. Varentsov I. M. Weaver J. T. Golubev N. G. Krylov V. A. (1997). Methods for modelling electromagnetic fields results from COMMEMI—The international project on the comparison of modelling methods for electromagnetic induction. J. Appl. Geophys.37 (3-4), 133–271. 10.1016/S0926-9851(97)00013-X

  • CrossRef
  • Google Scholar

• 61

  Zhou Y. Shi L. Liu N. Zhu C. Liu H. Liu Q. (2016). Spectral element method and domain decomposition for low-frequency subsurface EM simulation. IEEE Geoscience Remote Sens. Lett.13 (4), 550–554. 10.1109/LGRS.2016.2524558

  • CrossRef
  • Google Scholar

• 62

  Zhu J. Yin C. Liu Y. Liu Y. Liu L. Yang Z. et al (2020). 3-D dc resistivity modelling based on spectral element method with unstructured tetrahedral grids. Geophys. J. Int.220 (3), 1748–1761. 10.1093/gji/ggz534

  • CrossRef
  • Google Scholar

• 63

  Zou P. Cheng J. (2018). Pseudo-spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media. Geophys. Prospect.66 (1), 47–61. 10.1111/1365-2478.12543

  • CrossRef
  • Google Scholar