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# Hybrid finite difference–finite element method to incorporate topography and bathymetry for two-dimensional magnetotelluric modeling

- Weerachai Sarakorn
^{1}Email authorView ORCID ID profile and - Chatchai Vachiratienchai
^{2}

**Received:**31 January 2018**Accepted:**14 June 2018**Published:**26 June 2018

## Abstract

## Keywords

- Hybrid method
- Finite difference method
- Finite element method
- 2-D magnetotelluric modeling

## Mathematics Subject Classification

- 65
*N*30 - 65
*N*50 - 74
*S*05

## Introduction

Topography or bathymetry zones may be encountered during magnetotelluric surveys. These zones can affect the apparent resistivity and phase obtained from two-dimensional magnetotelluric (MT) surveys (Schwalenberg and Edwards 2004; Wannamaker et al. 1986). On land, a ridge causes a low apparent resistivity zone beneath it, whereas its foothill causes a high resistivity zone. These effects are observed in the transverse electric (*E*-polarization) and transverse magnetic (*H*-polarization) modes. However, the effects of a ridge on the apparent resistivity are greater for *E*-polarization than for *H*-polarization. The effects of the ridge on the phase are similar to the effects on the apparent resistivity for both *E*- and *H*-polarizations. Furthermore, the effects of a ridge also depend on the period of the EM field to be considered. The effects of a ridge are greater in a short period and reduced for longer periods. The effect of a valley on land is opposite that of a ridge. Beneath the sea, the effects of ridges and valleys on the responses calculated for both the *E*- and *H*-polarizations are the converse of those calculated on land. These effects can result in misleading interpretations but can be corrected with some techniques (Baba and Chave 2005; Matsuno et al. 2007; Nam et al. 2008; Singer 1992). These techniques are approximations and yield numerical errors. Therefore, the appropriate incorporation of topography and bathymetry with less consumption of memory storage and computation time during modeling is necessary to avoid these artifacts and to enable efficient forward modeling for the inversion process.

Finite difference (FD) and finite element (FE) methods are mostly developed and applied as the main process of forward calculation for 2-D and 3-D inversions (Egbert and Kelbert 2012; Franke et al. 2007; Key and Weiss 2006; Kordy et al. 2016a, b; Lee et al. 2009; Mackie et al. 1993; Nam et al. 2007; Ren et al. 2013; Sarakorn 2017; Sharma and Kaikkonen 1998; Siripunvaraporn et al. 2002; Usui 2015; Usui et al. 2017, 2018; Wannamaker et al. 1986, 1987). The FD method is accurate for simple modeling and, because of the limitation of the structured rectangular mesh, consumes less memory storage and computation time. The FE method is accurate for real-world complex modeling, especially topography and bathymetry, because of the greater flexibility of mesh schemes (Franke et al. 2007; Grayver and Kolev 2015; Grayver 2015; Key and Weiss 2006; Kordy et al. 2016a, b; Lee et al. 2009; Nam et al. 2007; Ren et al. 2013; Sarakorn 2017; Sharma and Kaikkonen 1998; Usui 2015; Usui et al. 2017, 2018; Wannamaker et al. 1986, 1987). However, the disadvantages of FE are greater consumption of memory storage and longer computation time. Thus, combining these two methods to exploit their advantages for solving magnetotelluric modeling is of interest and a challenge.

In this paper, we present a hybrid finite difference–finite element method (or hybrid FD–FE method, abbreviated HB) to incorporate topography and bathymetry for 2-D magnetotelluric modeling. The rectangular blocks around the topography (or bathymetry) zone are transformed to quadrilateral elements to perform FE approximation, whereas the FD approximation is performed outside. The system of equations for the hybrid FD–FE method is a combination of the FD and FE systems of equations. FE is mostly used only around the topography zone. Otherwise, the FD method is applied, and the hybrid FD–FE system of equations is, therefore, closer to the FD system of equations. For the same mesh, the computational time of hybrid FD–FE is, thus, closer to that of FD, whereas the accuracy of the hybrid FD–FE method is the same as that of FE.

Within the scope of this work, we begin by reviewing the 2-D magnetotelluric governing equation. Then, the processes of the common FD and FE methods are described, and a theoretical comparison of these methods is discussed. Next, the compatibility conditions between the FD and FE methods are confirmed to construct the hybrid FD–FE scheme. Then, the main concept of the hybrid FD–FE method for 2-D magnetotelluric forward modeling is introduced and explained in detail, and the validity of the hybrid method is confirmed with numerical experiments. Here, the simple automatic mesh algorithm used for these experiments is introduced. Next, the accuracy in terms of relative errors and efficiency in terms of computation time and consumption of memory resources of the hybrid FD–FE method on both nontopographic or nonbathymetric models and topographic or bathymetric models are presented and discussed in comparison with those of the common FD and FE methods. Finally, some conclusions and important remarks are given.

## Two-dimensional magnetotelluric modeling

### Governing equations

*t*is time and \(\omega\) is an angular frequency (\(\omega =2\pi /T\),

*T*is a period). Assuming a strike direction that parallels the

*x*- direction, i.e., \(\rho =\rho (y,z)\), the governing equation, in general form, is given by

*a*and

*b*and the variable \(\phi\) are notations that depend on the two polarizations:

*x*-direction, respectively, and \(\mu _{0}\) is the magnetic permeability in free space (

*Vs*/

*Am*). The bounded region

*D*is defined by \(D=D_{1}\cup D_{2}\cup \varGamma \cup \varGamma ^{\mathrm{int}}\), where \(D_{1}\) is the air subregion, \(D_{2}\) is the earth subregion, \(\varGamma\) is the outer boundary and \(\varGamma ^{\mathrm{int}}\) is the air–earth interface, where the electrical conductivity \(\rho\) is discontinuous. An example of domain

*D*with terrain is shown in Fig. 1 (top). The governing equation (1) is subjected to the Dirichlet boundary conditions

## Hybrid finite difference–finite element method

### Finite difference method

*i*,

*j*) is given by

*i*,

*j*) and the top node \((i,j-1)\), left node \((i-1,j)\), right node \((i+1,j)\) and bottom node \((i,j+1)\), respectively. The remaining coefficient \(A_{C}^{ij}\) is a self-coefficient at node (

*i*,

*j*). Note that these coefficients are expressed in terms of block sizes and block resistivity. Finally, applying the Dirichlet boundary conditions obtained by solving the 1-D problem, rearranging and grouping all coefficients in (5) together, the obtained system of equations is given by

**A**is sparse, symmetric and contains five bands with complex number coefficients but is not Hermitian. \(\tilde{\varvec{\Upphi }}\) is the unknown interior field vector, and

**s**is a source vector. Equation (6) can be solved by either a direct or iterative solver. Here, we use the direct solver for a sparse matrix in MATLAB to obtain the solution to (6). Finally, the MT responses including impedances, apparent resistivity and phases at each site for each EM period are calculated by

### Quadrilateral element-based finite element method

*i*,

*j*) and its neighbor, we obtain the discrete equation

*i*,

*j*) and left-top node \((i-1,j-1)\), center-top \((i,j-1)\), right-top node \((i+1,j-1)\), left-center node \((i-1,j)\), right-center node \((i+1,j)\), left-bottom node \((i-1,j+1)\), center-bottom node \((i,j+1)\) and right-bottom node \((i+1,j+1)\), respectively, and \(B_{C}^{ij}\) is a self-coefficient. As with the FD approach, these coefficients are all expressed in terms of element size and element resistivity. For the last step, applying Dirichlet boundary conditions and assembling coefficients and unknowns, the obtained system of equations is given by

**B**is large, sparse and symmetric and contains nine bands with complex number but is not Hermitian. \(\tilde{\varvec{\Upphi }}\) is the unknown vector, and

**s**is the source vector corresponding to the boundary conditions. As with FD, the direct solver in MATLAB is also used to solve (10). Then, the impedances, apparent resistivity and phases are finally computed at each site.

### Theoretical comparison of the FD and FE methods

As shown in Figs. 1 and 2, both the FD and quadrilateral-based FE methods define electric (magnetic) fields at each node with the same coordinates. Then, the numbers of unknowns \(\tilde{{\phi }}\) generated by the FE and FD methods are, therefore, identical when their meshes are the same. Furthermore, the coefficient matrices **A** and **B** have the same dimension. However, the matrix **A** has only five bands, whereas the matrix **B** has nine, and the number of nonzero coefficients of **A** is, therefore, less than that of **B**. Thus, the FD method requires less memory resources than the FE method. In addition, the calculation time required by the direct solver in MATLAB to solve (6) is expected to be less than that for (10).

### Concept of the hybrid method

In many previous works, the solutions to (1) obtained by the FD and FE schemes are not very different when the models do not include topography and bathymetry. However, when topography or bathymetry appears in the model, the FE scheme with relaxed elements, i.e., triangular or quadrilateral elements, is more suitable for handling an interface with topography or bathymetry than the rectangular blocks of the FD scheme. This advantage can be seen in Fig. 2. The FE method may be more accurate than FD. Evidence in support of this advantage has been provided by 2-D resistivity modeling (Erdoğan et al. 2008). For these reasons, it is interesting and challenging to introduce a hybrid FD–FE method that can combine the advantages of the FD and quadrilateral element-based FE methods while avoiding their deficiencies. Hybrid FD–FE approaches are not new and have been proposed and applied to solve various science and geoscience problems. Evidence of their applications can be found in elastic wave modeling (Galis et al. 2008; Jianfeng and Tielin 2002), hydrology (Simpson and Clement 2003) and direct current resistivity modeling (Vachiratienchai et al. 2010).

Usually, the dimensions of the systems of equations generated for the same discrete model by the FD and FE methods are identical. The obtained approximated solutions at each node should be similar. There is some evidence that the FE and FD methods provide similar discrete approximate solutions when applied to the same mesh (Zienkiewicz and Cheung 1965), supporting the construction of a hybrid method. If there are greater differences between the two approximations, then a hybrid method will require a transition zone from FE to FD or vice versa to ensure validity during modeling (Galis et al. 2008).

In this work, the construction of the hybrid FD–FE method begins by separating a given model into FD and FE zones. As shown in Fig. 3 (left), some FE zones are set for any topographic and bathymetric zones and optimized. Note that separated multiple FE zones can be allowed during modeling. By contrast, the FD zone is set in a simple or flat zone where topography and bathymetry do not exist. Next, the initial mesh with rectangular blocks is generated and applied in the FD zones. Then, the rectangular blocks located in the FE zones are relaxed to the quadrilateral shapes to handle topography and bathymetry appropriately. Examples of meshes for the FD and FE zones are shown in Fig. 3 (right). Note that the number of quadrilateral elements in the FE zone is equal to the initial number of rectangular blocks.

## Numerical experiments

In this section, numerical experiments are designed to investigate the accuracy and efficiency of the hybrid FD–FE method. Experiments are performed on both nontopographic and topographic models. The two-layered model is used for the nontopographic case. The topographic and bathymetric models including two models, land topographic and bathymetry models are selected as the case studies.

*y*- and

*z*-directions are automatically generated by

*r*is the growth rate and \(p=y\) or

*z*. The upper bound of \(\delta _{i}^{p}\) must be set to prevent a huge mesh size. If \(\delta _{0}^{p}\) is greater than the size of some subregions or the site distance, then it is divided by an appropriate proportion. Generating the sequence in (12) for the

*y*- and

*z*-directions transforms the continuous model into a discrete model with rectangular blocks. Clearly, the parameters in (12) control the roughness or fineness of the mesh inside the discrete model and directly affect the number of nodes and rectangular blocks in the domain. Thus, the selection of these parameters to generate mesh data requires optimization to produce a trade-off between accuracy and efficiency. When the three numerical methods finish the task completely, the obtained numerical results are presented, compared and discussed. Note that the computing tool used for executing FD, FE and hybrid FD–FE codes in all experiments is a MacBook Pro with CPU 2.3 GHz Core i5 and RAM 8 GB.

### Nontopographic model

*r*, the error of each method increases when \(\delta _{0}^{z}\) increases. For fixed \(\delta _{0}^{z}\), the error increases when

*r*increases. Thus, both

*r*and \(\delta _{0}^{z}\) affect the accuracy of the three methods. However, \(\delta _{z}^{0}\) appears to play a significantly more important role than the growth rate. Next, the relative errors of the phases are considered and shown in Fig. 7.

The errors of the phases are less than 0.35% and small compared to those of the apparent resistivity. Similar to the errors of the apparent resistivity, the FE method is more accurate than the FD and hybrid FD–FE methods. The errors obtained by the hybrid FD–FE method are also more similar to those obtained by the FD than the FE method. The mesh parameters *r* and \(\delta _{0}^{z}\) both still affect the accuracy of the three methods.

### Topographic and bathymetric models

In this section, two more complex and realistic 2-D models with topography and bathymetry are designed and used in numerical experiments. The accuracy in terms of numerical errors is estimated, compared and discussed, whereas the efficiency in terms of CPU time and consumption of memory resources is collected, compared and discussed.

#### Topographic model

The hybrid FD–FE approach takes the shortest time to complete the task for 4 periods, whereas FD takes the longest time. Note that the FE with fine mesh is excluded in this comparison. These results indicate that the hybrid method is more powerful than the FE and FD methods when topography exists in the model.

#### Bathymetric model

*H*-polarization. The total CPU times required to finish the task for 4 periods are shown in Fig. 13.

The hybrid FD–FE method still takes the shortest time to complete the task for 4 periods, whereas the FD method still takes the longest time. These results also indicate that the hybrid method is more powerful than the FE and FD methods when bathymetry exists in the model.

## Conclusions

In this paper, a hybrid FD–FE method is developed. The common FD method and the FE method with quadrilateral elements are used to construct the hybrid method. The FD method is applied in areas where topography and bathymetry do not exist. By contrast, the FE method is applied around topography and bathymetry zones. The characteristics of the resulting system of equations of 2-D magnetotelluric modeling are a combination of those of the FD and FE methods. The hybrid FD–FE method is validated and compared to the common FD and FE methods. The accuracy and efficiency of the hybrid method in terms of relative errors and memory storage and in terms of calculation time, respectively, are presented for various models, including nontopographic model with two layers, topographic models and bathymetric models. For nontopographic models, the efficiency and accuracy of the hybrid FD–FE method are close to those of the FD method. For topographic and bathymetric models, the hybrid FD–FE method can handle topography and bathymetry appropriately as well as the FE method. Furthermore, the hybrid FD–FE method can provide accuracy equivalent to the FE method, but the calculation is dramatically reduced compared to that of the FE scheme. By contrast, the FD method provides poor accuracy and requires more calculation time even though a mesh with a huge number of nodes and blocks is used.

Based on these results, the hybrid method may represent a new, better option for the forward calculation routine for 2-D magnetotelluric inversion to interpret real field magnetotelluric data in the future. Note that the hybrid FD–FE method that has been presented uses a structured mesh. Some extra mesh refinement can appear in the projective subregion where topography or bathymetry exists. This extra mesh refinement will not appear when an unstructured quadrilateral mesh is applied in the FE zone instead. The unstructured quadrilateral mesh has a local refinement mesh as well as an unstructured triangular mesh but provides better accuracy without adaptive routines (Sarakorn 2017). The size of the elements around the boundary of the FE zone is optimized nearly equally. The connected rectangular blocks in the FD zone then have a conformal shape and a good edge ratio. Thus, the hybrid method is enhanced by this feature. However, some procedures, such as mesh generation and node and element numbering, need to be revisited and modified. Furthermore, the concept of the hybrid FD–FE method can be extended and applied to solve 3-D MT modeling in the future.

## Declarations

### Authors' contributions

WS developed the theoretical concepts, performed the numerical simulations and contributed to the final version of the manuscript. CV supervised the project. Both authors read and approved the final manuscript.

### Acknowlegements

The authors also would like to thank Assoc. Prof. Dr. Weerachai Siripunvaraporn, Department of Physics, Faculty of Science, Mahidol University, for many useful suggestions for improving this research.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

All the data and models in this manuscript are available.

### Funding

This work has been supported by 1. The DPST Research grant (Grant Number 002/2557). 2. The Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2560 (RAAPF). 3. The National Research Council of Thailand and Khon Kaen University, Thailand (Grant Number 6100148).

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## Authors’ Affiliations

## References

- Baba K, Chave AD (2005) Correction of seafloor magnetotelluric data for topographic effects during inversion. J Geophys Res B Solid Earth 110(12):1–16Google Scholar
- Egbert GD, Kelbert A (2012) Computational recipes for electromagnetic inverse problems. Geophys J Int 189(1):251–267View ArticleGoogle Scholar
- Erdoğan E, Demirci I, Candansayar ME (2008) Incorporating topography into 2D resistivity modeling using finite-element and finite-difference approaches. Geophysics 73(3):F135–F142View ArticleGoogle Scholar
- 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):71–86View ArticleGoogle Scholar
- Galis M, Moczo P, Kristek J (2008) A 3-D hybrid finite-difference–finite-element viscoelastic modelling of seismic wave motion. Geophys J Int 175(1):153–184View ArticleGoogle Scholar
- Grayver AV (2015) Parallel three-dimensional magnetotelluric inversion using adaptive finite-element method. Part I: theory and synthetic study. Geophys J Int 202(1):584–603View ArticleGoogle Scholar
- Grayver AV, Kolev TV (2015) Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method. Geophysics 80(6):277–291View ArticleGoogle Scholar
- Jianfeng Z, Tielin L (2002) Elastic wave modelling in 3D heterogeneous media: 3D grid method. Geophys J Int 150(3):780–799View ArticleGoogle Scholar
- Key K, Weiss C (2006) Adaptive finite-element modeling using unstructured grids: the 2D magnetotelluric example. Geophysics 71(6):G291–G299View ArticleGoogle Scholar
- Kordy M, Wannamaker P, Maris V, Cherkaev E, Hill G (2016a) 3-D magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on SMP computers—part I: forward problem and parameter jacobians. Geophys J Int 204(1):74–93View ArticleGoogle Scholar
- Kordy M, Wannamaker P, Maris V, Cherkaev E, Hill G (2016b) 3-dimensional magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on symmetric multiprocessor computers—part II: direct data-space inverse solution. Geophys J Int 204(1):94–110View ArticleGoogle Scholar
- Lee S, Kim H, Song Y, Lee C-K (2009) MT2DInvMatlab—a program in MATLAB and FORTRAN for two-dimensional magnetotelluric inversion. Comput Geosci 35(8):1722–1734View ArticleGoogle Scholar
- Mackie RL, Madden TR, Wannamaker PE (1993) Three-dimensional magnetotelluric modeling using difference equations—theory and comparisons to integral equation solutions. Geophysics 58(2):215–226View ArticleGoogle Scholar
- Matsuno T, Seama N, Baba K (2007) A study on correction equations for the effect of seafloor topography on ocean bottom magnetotelluric data. Earth Planets Space 59(8):981–986. https://doi.org/10.1186/BF03352037 View ArticleGoogle Scholar
- Nam MJ, Kim HJ, Song Y, Lee TJ, Son J-S, Suh JH (2007) 3D magnetotelluric modelling including surface topography. Geophys Prospect 55(2):277–287View ArticleGoogle Scholar
- Nam MJ, Kim HJ, Song Y, Lee TJ, Suh JH (2008) Three-dimensional topography corrections of magnetotelluric data. Geophys J Int 174(2):464–474View ArticleGoogle Scholar
- Ren Z, Kalscheuer T, Greenhalgh S, Maurer H (2013) A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling. Geophys J Int 194(2):700–718View ArticleGoogle Scholar
- Sarakorn W (2017) 2-D magnetotelluric modeling using finite element method incorporating unstructured quadrilateral elements. J Appl Geophys 139:16–24View ArticleGoogle Scholar
- Schwalenberg K, Edwards R (2004) The effect of seafloor topography on magnetotelluric fields: an analytical formulation confirmed with numerical results. Geophys J Int 159(2):607–621View ArticleGoogle Scholar
- Sharma SP, Kaikkonen P (1998) An automated finite element mesh generation and element coding in 2-D electromagnetic inversion. Geophysica 34(3):93–114Google Scholar
- Simpson M, Clement T (2003) Comparison of finite difference and finite element solutions to the variably saturated flow equation. J Hydrol 270(1):49–64View ArticleGoogle Scholar
- Singer BS (1992) Correction for distortions of magnetotelluric fields: limits of validity of the static approach. Surv Geophys 13(4):309–340View ArticleGoogle Scholar
- Siripunvaraporn W, Egbert G, Lenbury Y (2002) Numerical accuracy of magnetotelluric modeling: a comparison of finite difference approximations. Earth Planets Space 54(6):721–725. https://doi.org/10.1186/BF03351724 View ArticleGoogle Scholar
- Usui Y (2015) 3-D inversion of magnetotelluric data using unstructured tetrahedral elements: applicability to data affected by topography. Geophys J Int 202(2):828–849View ArticleGoogle Scholar
- Usui Y, Ogawa Y, Aizawa K, Kanda W, Hashimoto T, Koyama T, Yamaya Y, Kagiyama T (2017) Three-dimensional resistivity structure of Asama Volcano revealed by data-space magnetotelluric inversion using unstructured tetrahedral elements. Geophys J Int 208(3):1359–1372View ArticleGoogle Scholar
- Usui Y, Kasaya T, Ogawa Y, Iwamoto H (2018) Marine magnetotelluric inversion with an unstructured tetrahedral mesh. Geophys J Int 214(2):952–974View ArticleGoogle Scholar
- Vachiratienchai C, Boonchaisuk S, Siripunvaraporn W (2010) A hybrid finite difference–finite element method to incorporate topography for 2D direct current (DC) resistivity modeling. Phys Earth Planet Inter 183(3):426–434View ArticleGoogle Scholar
- Wannamaker PE, Stodt JA, Rijo L (1986) Two-dimensional topographic responses in magnetotellurics modeled using finite elements. Geophysics 51(11):2131–2144View ArticleGoogle Scholar
- Wannamaker PE, Stodt JA, Rijo L (1987) A stable finite element solution for two-dimensional magnetotelluric modelling. Geophys J R Astron Soc 88(1):277–296View ArticleGoogle Scholar
- Zhdanov MS, Varentsov IM, Weaver JT, Golubev NG, Krylov VA (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):133–271View ArticleGoogle Scholar
- Zienkiewicz O, Cheung Y (1965) Finite element in the solution of field problems. Engineering 220:507–510Google Scholar