Three-dimensional magnetotelluric modeling using the finite element model reduction algorithm

doi:10.1016/j.cageo.2021.104750

Computers & Geosciences

Volume 151, June 2021, 104750

https://doi.org/10.1016/j.cageo.2021.104750 Get rights and content

Highlights

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Model reduction algorithm was combined with the vector finite element method to solve the large sparse complex system.
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Finite element model reduction approach was applied to three-dimensional magnetotelluric modeling to save considerable time.
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We adopted model reduction algorithm based on repeated poles.

Abstract

This study presents a fast and efficient vector finite element method, based on the model reduction algorithm, to simulate the 3-D magnetotelluric response. Firstly, we derive a finite element governing equation based on electric field vector from Maxewell's equations Then we express the electric field by a function of frequency parameter to get a transfer function that can be approximated by the Krylov subspace model reduction algorithm. The transfer function after model reduction has far lower dimensions than the original one, and it can be directly solved by the Pardiso solver. Therefore, the computational efficiency is significantly improved. For validation, we compare the proposed algorithm with the 3-D direct finite element method (3DFEM) by a three-layer model, and the results show that the computation time of the former is less than 1/10 of the latter for the same accuracy. We also compare the electromagnetic response obtained by the proposed method with that obtained by two classical models, COMMEMI 3D-2A Model and Dublin Test Model 1. These results agree well, indicating that the proposed method is feasible and efficient for 3-D magnetotelluric forward modeling.

Introduction

The magnetotelluric method is often used in detecting the electrical characteristics of underground targets, due to its wide band range (10⁻⁴-10⁴) and large detection depth. It also has been widely used in the exploration of oil and gas, geothermal resources, mineral resources and coal mine, and has been proved to be useful in investigating deep rock structures, including the mantle and crust (Unsworth et al., 2005; Becken et al., 2011; Guo et al., 2018).

Recent researches on the magnetotelluric method focus on the three-dimensional (3D) forward numerical simulation and inversion. Forward modeling is very important in investigating the magnetotelluric response of the earth structure and is the basis of inversion. The methods for forward modeling include the integral equation method (Avdeev et al., 2002; Farquharson et al., 2006; Zhdanov et al., 2006), the finite difference method (Haber et al., 2000; Weiss and Newman, 2003; Newman and Alumbaugh, 2002; Streich, 2009; Li et al., 2012) and the finite element method (Mitsuhata and Uchida 2004; Key and Weiss, 2006; Nam et al., 2007; Farquharson and Miensopust, 2011; Cai et al., 2014, Cai et al., 2015, Xiao et al., 2018; Zhao et al., 2020). The finite element method is flexible in mesh generation, so it is often applied in the 3D forward numerical simulation and inversion (Key and Weiss, 2006; Franke et al., 2007; Liu et al., 2008; Ren et al., 2013, 2014b; Li et al., 2016a, Li et al., 2016b; Yin et al., 2017; Cao et al., 2018).

The 3D magnetotelluric forward modeling using the finite element method has huge calculation loads, because of the multiple frequencies and large amount of unknowns. One way to improve the computation efficiency is to perform parallel calculations on a high performance computing platform, in which the calculations of different frequencies are done by different processors simultaneously (Key, 2011; Puzyrev et al., 2013; Qin et al., 2017). Another way for improving the calculation efficiency is to optimize the algorithms. For example, the aggregation multigrid algorithm (AGMG) can be used as the pre-processing operator of the Krylov subspace iteration method to accelerate error convergence and reduce iteration (Chen et al., 2018); the domain decomposition algorithm (Xiong, 1999; Zyserman and Santos, 2000; Puzyrev et al., 2013; Ren et al., 2014a) can decompose the calculation region into sub-regions, for which the parallel algorithm is adopted to improve the solution speed; combining finite element with infinite element can greatly reduce the infinite mesh boundary, thereby reducing the degree of freedom and improving the efficiency (Zhang et al., 2017). The model order reduction method (Börner et al., 2008, Börner et al., 2015; Kordy et al., 2017; Zhou et al., 2018) uses the transfer function to convert large matrix to small matrix. Model order reduction is independent from frequency, so one reduction can get multiple frequency electromagnetic responses. Thus, this method is particularly suitable for the fast forward modeling of broad band magnetotelluric method.

Two approaches are often used for the 3D magnetotelluric finite element forward modeling. The first one derives the governing equations directly from the electric field or the magnetic field (Mackie et al., 1994; Newman and Alumbaugh, 1995; Smith, 1996; Sasaki, 1999; Zanoubi et al., 1999; Tong et al., 2009; Farquharson and Miensopust, 2011; Cai et al., 2017; Nunes and Régis, 2020) and forms a system of equations (Haber et al., 2000; Aruliah et al., 2001). Due to discontinuity of the electric field normal direction, divergence correction is often used to alleviate possible spurious solutions for the nodal finite elements that are based on electric field (Mackie et al., 1994; Sasaki, 2001; Zhang et al., 2009). The other approach uses vector and scalar potentials to get the electric field or magnetic field (Badea et al., 2001; Mitsuhata and Uchida, 2004; Stalnaker et al., 2006). Since vector potential and scalar potential are intermediate variables, the components of electric field and magnetic field can only be obtained through numerical differentiation, and have low accuracy.

In this paper, we develop a new 3D vector finite element model reduction algorithm based on electric field formula for the 3D multi-frequency magnetotelluric modeling. The algorithm is validated by a three-layer model that has an analytical solution. We also compare the numerical result of the proposed method with that obtained by the COMMEMI 3D-2 model and Dublin Test Model 1 (DTM1).

Section snippets

Electric field equations

In quasi-static approximation, the conduction current is dominant. With $σ ≫ ω ε$ , the effect of the displacement current can be neglected. $ω$ = $2 π f$ is the angular frequency, with ƒ denoting the frequency that depends on the transmitting source. $ε$ is the dielectric constant and $σ$ is the electrical conductivity. Assuming that the initial field is a plane wave field and the time factor is $e^{i ω t}$ , the equations of the electric field ( $E$ ) and magnetic field ( $H$ ) in the frequency domain are $\nabla \times E = - i ω μ H$ $\nabla \times H = σ E$ where $μ$

The model reduction method

From equation (18b), a transfer function $g^{ω} (A)$ can be derived as follows (Jiang, 2010; Börner et al., 2008): $\begin{array}{l} E (ω) = {(C + i ω M)}^{- 1} b \\ = {(M^{- 1} C + i ω I) -}^{1} M^{- 1} b \\ = {(A + i ω I)}^{- 1} X \\ = g^{ω} (A) X \end{array}$ where $g^{ω} (A) = {(A + i ω I)}^{- 1}, A = M^{- 1} C, X = M^{- 1} b$ , and $I \in {ℝ^{n}}^{\times n}$ denotes the identity matrix.

The transfer function $g^{ω} (A)$ is a large scale matrix, which is difficult to solve directly.

In this paper, we propose a model reduction method based on Krylov subspace to approximate the transfer function by an $m$ order rational function $r_{m} (A)$ and we get $E \approx r_{m} (A) \cdot X$ where $r_{m} ($

Numerical experiments

In order to verify the feasibility of the model reduction algorithm, we established a three-layer formation model. The XY-mode and YX-mode responses calculated by the model reduction program were compared with the results of the directly solved 3D forward program and the 1D forward program in terms of relative errors and calculation efficiency. Then we calculated the electromagnetic responses of the COMMEMI 3D-2 model and the Dublin Test Model 1 (DTM1), and compared them with the proposed ones

Conclusions

This study presents a finite element model reduction algorithm for the 3D magnetotelluric forward modeling. Different from traditional frequency domain modeling approaches, the proposed modeling method is independent of frequencies, so it is particularly suitable for the broadband magnetotellurics modeling. We tested the efficiency and feasibility of the proposed algorithm using a three layered model and a COMMEMI 3D-2 model. We also compared this method with the 3DFEM in terms of running-time

Computer code availability

The code named “3D_MT_FORWARD_BY_MODEL_REDUCTION” was developed by Jiren Liu and Jifeng Zhang. The contact address is Chang'an University in China. The telephone number is 86–15129062903 and e-mail is [email protected]. The program has been available since Feb. 2020. The program can be run on any computer installed with windows7 or above. It requires two software: the visual studio 2013 and Intel visual FORTRAN 2013 version or above. The program language is Fortran90 and the size of program is

CRediT authorship contribution statement

Jifeng Zhang: conceived the idea and revised the manuscript. Jiren Liu: Writing – original draft. Bing Feng: interpreted the results. Yi'an Zheng: Formal analysis. Jianbo Guan: Formal analysis. Zhongqiang Liu: Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We are grateful for the financial support provided by the National Key Research and Development Program of China, (2017YFC0602202), the National Natural Science Foundation of China, (41830101), Shaanxi Natural Science Foundation (2021JM-159) and Emei-hanyang highway engineering research project (LH-HT-45).

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View full text

Three-dimensional magnetotelluric modeling using the finite element model reduction algorithm

Highlights

Abstract

Introduction

Section snippets

Electric field equations

The model reduction method

Numerical experiments

Conclusions

Computer code availability

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

Comput. Geosci.

Comput. Geosci.

J. Appl. Geophys.

Comput. Geosci.

J. Comput. Phys.

J. Comput. Phys.

J. Appl. Geophys.

J. Appl. Geophys.

J. Appl. Geophy.

J. Appl. Geophys.

A method for the forward modelling of 3Delectromagnetic quasi-static problems

Math. Model Methods Appl. Sci.

Three-dimensional induction logging problems, part I: an integral equation solution and model comparisons

Geophysics

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Geophysics

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Nature

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Surv. Geophys.

Fast 3D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection

Geophys. J. Int.

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Three-dimensional marine controlled-source electromagnetic modeling in anisotropic medium using finite element method

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J. Jilin Univ. (Sci. Ed.)

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The Finite Element Method in Electromagnetics

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Adaptive finite-element modeling using unstructured grids: the 2D magnetotelluric example

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Chin. J. Geophys.