# Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric

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## Abstract

**:**

## 1. Introduction

## 2. IE Method Foundation for MT Modeling

## 3. Improved Treatments

#### 3.1. Analytical Method for Computation of Bessel Function

#### 3.2. Rapid Implementation of Coefficient Matrix-Vector Multiplication

**A**and sort them in a 3-D array

**x**is

**a**. Using the discrete convolution theorem

## 4. Model Test

#### 4.1. COMMEMI 3D-1A Model

#### 4.2. Dublin Test Model 1

#### 4.3. COMMEMI 3D-2A Model

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Analytical Formula of Bessel Function Integral

## References

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**Figure 1.**Illustration of modeling domain discretization: (

**a**) discretizing anomalous domain and part of background; (

**b**) discretizing anomalous domain only. Dots denote observation sites.

**Figure 4.**The apparent resistivity and phase of COMMEMI 3D-1A model for 0.1 Hz. The circles with error bars are the mean apparent resistivity of COMMEMI with their standard deviations [50]; the black solid lines are the solutions of Farquharson and Miensopust [5]; the symbols ‘*’ denote the values of IE method in this study. Stations locate across the center of conductive rectangular prism.

**Figure 5.**The relative residual norm during iterations for solving electric fields of COMMEMI 3D-1A model. The number 1 and 2 in the legend indicate the electric field of the incident wave is in the x- and y-direction, respectively.

**Figure 6.**Comparison of 100 random ${\bm{\gamma}}_{1}$ and ${\bm{\gamma}}_{2}$ calculated by digital filtering and analytical formulas, respectively. (

**a**) The real part of ${\bm{\gamma}}_{1}$; (

**b**) The imaginary part of ${\bm{\gamma}}_{1}$; (

**c**) The real part of ${\bm{\gamma}}_{2}$; (

**d**) The imaginary part of ${\bm{\gamma}}_{2}$.

**Figure 8.**Comparison of apparent resistivity and phase components at the station located at coordinate (0, 0, 0) m of DTM1 with 21 periods ranging from 10 s to 10,000 s (4 periods per decade). Our results are compared with those provided by other authors (see [51] for details).

**Figure 9.**The relative residual norm during iterations for solving electric fields of the Dublin Test Model 1.

**Figure 10.**The COMMEMI 3D-2A model [50]: (

**a**) plan view, (

**b**) section view. The filled nabla on the plan view denotes the profile where apparent resistivity values are calculated.

**Figure 11.**The apparent resistivities of model 3D-2A for 0.1, 0.01, and 0.001 Hz. The solid lines are the solutions of Farquharson and Miensopust [5], and circles indicate the solutions of this study. The first to third rows are for 0.1 Hz, 0.01 Hz and 0.001 Hz, respectively.

Extend in x (km) | Extend in y (km) | Extend in z (km) | $\mathbf{Resistivity}\text{}\left(\mathbf{\Omega}\mathbf{m}\right)$ | |
---|---|---|---|---|

Block 1 | −20 to 20 | −2.5 to 2.5 | 5 to 20 | 10 |

Block 2 | −15 to 0 | −2.5 to 22.5 | 20 to 25 | 1 |

Block 3 | 0 to 15 | −22.5 to 2.5 | 20 to 50 | 10,000 |

Computation Cost | Number of Cells in x-, y-, z-Direction for Anomalous Bodies | ||||
---|---|---|---|---|---|

Iterations | Time (s)/Period | Peak RAM(GB) | Body 1 | Body 2 | Body 3 |

30 | 25 | 0.047 | $40\times 5\times 15$ | $15\times 25\times 5$ | $15\times 25\times 30$ |

20 | 108 | 0.2 | $64\times 8\times 24$ | $24\times 40\times 8$ | $24\times 40\times 48$ |

20 | 234 | 0.4 | $80\times 10\times 30$ | $30\times 50\times 10$ | $30\times 50\times 60$ |

20 | 2061 | 3.19 | $160\times 20\times 60$ | $60\times 100\times 20$ | $60\times 100\times 120$ |

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**MDPI and ACS Style**

Luo, T.; Chen, L.; Hu, X.
Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric. *Minerals* **2022**, *12*, 504.
https://doi.org/10.3390/min12050504

**AMA Style**

Luo T, Chen L, Hu X.
Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric. *Minerals*. 2022; 12(5):504.
https://doi.org/10.3390/min12050504

**Chicago/Turabian Style**

Luo, Tianya, Longwei Chen, and Xiangyun Hu.
2022. "Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric" *Minerals* 12, no. 5: 504.
https://doi.org/10.3390/min12050504